<!-- %W sporsolv.xml GAP 4 package CTblLib Thomas Breuer -->
<Chapter Label=
"chap:sporsolv">
<Heading>Solvable Subgroups of Maximal Order in Sporadic Simple Groups</Heading>
<P/>
Date: May 14th, 2012
<P/>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
We determine the orders of solvable subgroups of maximal orders in sporadic
simple groups and their automorphism groups,
using the information in the
&ATLAS; of Finite Groups <Cite Key=
"CCN85"/> and
the &GAP; system <Cite Key=
"GAP"/>,
in particular its Character Table Library <Cite Key=
"CTblLib"/>
and its library of Tables of Marks <Cite Key=
"TomLib"/>.
<P/>
We also determine the conjugacy classes of these solvable subgroups
in the big group, and the maximal overgroups.
<P/>
A first
version of this document, which was based on &GAP; 4.4.10,
had been accessible in the web since August 2006.
The differences to the current
version are as follows.
<P/>
<List>
<Item>
The format of the &GAP; output was adjusted to the changed behaviour
of &GAP; 4.5.
</Item>
<Item>
The (too wide) table of results was split into two tables,
the first one lists the orders and indices of the subgroups,
the second one lists the structure of subgroups and the maximal overgroups.
</Item>
<Item>
The distribution of the solvable subgroups of maximal orders in the
Baby Monster group and the Monster group to conjugacy classes is now
proved.
</Item>
<Item>
The sporadic simple Monster group has exactly one class of maximal
subgroups of the
type PSL<M>(2, 41)</M> (see <Cite Key=
"NW12"/>),
and has no maximal subgroups which have the socle PSL<M>(2, 27)</M>
(see <Cite Key=
"Wil10"/>).
This does not affect the arguments in Section <Ref Subsect=
"sect:M"/>,
but some statements in this section had to be corrected.
</Item>
<!-- % \item -->
<!-- % Several computations can be made more explicit, due to the fact that -->
<!-- % more character tables of almost simple groups and maximal subgroups of -->
<!-- % such groups are available in the &GAP; Character Table Library. -->
<!-- % The computations from the original version have been kept, -->
<!-- % since they show how one can use the &ATLAS; of Finite Groups <Cite Key="CCN85"/> -->
<!-- % to get information about a group. -->
<!-- % The more automatic computations appear in an appendix -->
<!-- % (see Section <Ref Subsect="appendix"/>), -->
<!-- % references to this appendix were added to the text. -->
</List>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=
"sect:result">
<Heading>The Result</Heading>
The tables I and II list information about
solvable subgroups of maximal order in sporadic simple groups and their
automorphism groups.
The first column in each table gives the names of the almost simple groups
<M>G</M>, in alphabetical order.
The remaining columns of Table I
contain the order and the index of a solvable subgroup <M>S</M>
of maximal order in <M>G</M>,
the value <M>\log_{|G|}(|S|)</M>,
and the page number in the &ATLAS; <Cite Key=
"CCN85"/> where
the information about maximal subgroups of <M>G</M> is listed.
The second and third columns of Table II
show a structure description
of <M>S</M> and the structures of the maximal subgroups that contain <M>S</M>;
the value <Q><M>S</M></Q> in the third column means that <M>S</M> is itself maximal in <M>G</M>.
The fourth and fifth columns list the pages in the &ATLAS;
with the information about the maximal subgroups of <M>G</M>
and the section in this note with the proof of the table row,
respectively.
In the fourth column, page numbers in brackets refer to the &ATLAS; pages
with information about the maximal subgroups of nonsolvable quotients
of the maximal subgroups of <M>G</M> listed in the third column.
<P/>
Note that in the case of nonmaximal subgroups <M>S</M>,
we do not claim to describe the <E>module</E> structure of <M>S</M>
in the third column of the table;
we have kept the &ATLAS; description of the normal subgroups of the maximal
overgroups of <M>S</M>.
For example, the subgroup <M>S</M> listed for <M>Co_2</M> is contained in maximal
subgroups of the types <M>2^{1+8}_+:S_6(2)</M> and <M>2^{4+10}(S_4 \times S_3)</M>,
so <M>S</M> has normal subgroups
of the orders <M>2</M>, <M>2^4</M>, <M>2^9</M>, <M>2^{14}</M>, and <M>2^{16}</M>;
more &ATLAS; conformal notations would be
<M>2^{[14]}(S_4 \times S_3)</M> or <M>2^{[16]}(S_3 \times S_3)</M>.
<P/>
As a corollary (see Section <Ref Subsect=
"sect:corollary"/>),
we read off the following.
<P/>
Corollary:
<P/>
Exactly the following almost simple groups <M>G</M> with sporadic simple socle
contain a solvable subgroup <M>S</M> with the property <M>|S|^2 \geq |G|</M>.
<
Display Mode=
"M">
Fi_{23}, J_2, J_2.2, M_{11}, M_{12}, M_{22}.2.
</
Display>
<P/>
The existence of the subgroups <M>S</M> of <M>G</M> with the structure
and the order stated in Table I and II follows from the &ATLAS;:
It is obvious in the cases where <M>S</M> is maximal in <M>G</M>,
and in the other cases, the &ATLAS; information about a nonsolvable
factor group of a maximal subgroup of <M>G</M> suffices.
<P/>
In order to show that the table rows for the group <M>G</M> are correct,
we have to show the following.
<List>
<Item>
<M>G</M> does not contain solvable subgroups
of order larger than <M>|S|</M>.
</Item>
<Item>
<M>G</M> contain exactly the conjugacy classes of solvable subgroups
of order <M>|S|</M> that are listed in the second column
of Table II.
</Item>
<Item>
<M>S</M> is contained exactly in the maximal subgroups listed in the third
column of Table II.
</Item>
</List>
<P/>
<E>Remark:</E>
<List>
<Item>
Each of the groups <M>M_{12}</M> and <M>He</M> contains two classes
of isomorphic solvable subgroups of maximal order.
</Item>
<Item>
Each of the groups <M>Ru</M>, <M>Th</M>, and <M>M</M> contains two classes
of nonisomorphic solvable subgroups of maximal order.
</Item>
<Item>
The solvable subgroups of maximal order in <M>McL.2</M> have the structure
<M>3^{1+4}_+:4S_4</M>,
the subgroups are maximal in the maximal subgroups of the
structures <M>3^{1+4}_+:4S_5</M> and <M>U_4(3).2_3</M> in <M>McL.2</M>.
Note that the &ATLAS; claims another structure for these
maximal subgroups of <M>U_4(3).2_3</M>,
see <Cite Key=
"CCN85" Where=
"p. 52"/>.
</Item>
<Item>
The solvable subgroups of maximal order in <M>Co_3</M> are the normalizers
of Sylow <M>3</M>-subgroups of <M>Co_3</M>.
</Item>
</List>
<P/>
<Table Align=
"|l|r|r|r|r|r|">
<Caption>Table I: Solvable subgroups of maximal order – orders and indices</Caption>
<HorLine/>
<Row>
<Item><M>G</M></Item>
<Item><M>|S|</M></Item>
<Item><M>|G/S|</M></Item>
<Item><M>\log_{|G|}(|S|)</M></Item>
<Item>p.</Item>
</Row>
<HorLine/>
<HorLine/>
<Row>
<Item><M>M_{11}</M></Item>
<Item><M>144</M></Item>
<Item><M>55</M></Item>
<Item><M>0.5536</M></Item>
<Item><M>18</M></Item>
</Row>
<Row>
<Item><M>M_{12}</M></Item>
<Item><M>432</M></Item>
<Item><M>220</M></Item>
<Item><M>0.5294</M></Item>
<Item><M>33</M></Item>
</Row>
<Row>
<Item><M>M_{12}.2</M></Item>
<Item><M>432</M></Item>
<Item><M>440</M></Item>
<Item><M>0.4992</M></Item>
<Item><M>33</M></Item>
</Row>
<Row>
<Item><M>J_1</M></Item>
<Item><M>168</M></Item>
<Item><M>1\,045</M></Item>
<Item><M>0.4243</M></Item>
<Item><M>36</M></Item>
</Row>
<Row>
<Item><M>M_{22}</M></Item>
<Item><M>576</M></Item>
<Item><M>770</M></Item>
<Item><M>0.4888</M></Item>
<Item><M>39</M></Item>
</Row>
<Row>
<Item><M>M_{22}.2</M></Item>
<Item><M>1\,152</M></Item>
<Item><M>770</M></Item>
<Item><M>0.5147</M></Item>
<Item><M>39</M></Item>
</Row>
<Row>
<Item><M>J_2</M></Item>
<Item><M>1\,152</M></Item>
<Item><M>525</M></Item>
<Item><M>0.5295</M></Item>
<Item><M>42</M></Item>
</Row>
<Row>
<Item><M>J_2.2</M></Item>
<Item><M>2\,304</M></Item>
<Item><M>525</M></Item>
<Item><M>0.5527</M></Item>
<Item><M>42</M></Item>
</Row>
<Row>
<Item><M>M_{23}</M></Item>
<Item><M>1\,152</M></Item>
<Item><M>8\,855</M></Item>
<Item><M>0.4368</M></Item>
<Item><M>71</M></Item>
</Row>
<Row>
<Item><M>HS</M></Item>
<Item><M>2\,000</M></Item>
<Item><M>22\,176</M></Item>
<Item><M>0.4316</M></Item>
<Item><M>80</M></Item>
</Row>
<Row>
<Item><M>HS.2</M></Item>
<Item><M>4\,000</M></Item>
<Item><M>22\,176</M></Item>
<Item><M>0.4532</M></Item>
<Item><M>80</M></Item>
</Row>
<Row>
<Item><M>J_3</M></Item>
<Item><M>1\,944</M></Item>
<Item><M>25\,840</M></Item>
<Item><M>0.4270</M></Item>
<Item><M>82</M></Item>
</Row>
<Row>
<Item><M>J_3.2</M></Item>
<Item><M>3\,888</M></Item>
<Item><M>25\,840</M></Item>
<Item><M>0.4486</M></Item>
<Item><M>82</M></Item>
</Row>
<Row>
<Item><M>M_{24}</M></Item>
<Item><M>13\,824</M></Item>
<Item><M>17\,710</M></Item>
<Item><M>0.4935</M></Item>
<Item><M>96</M></Item>
</Row>
<Row>
<Item><M>McL</M></Item>
<Item><M>11\,664</M></Item>
<Item><M>77\,000</M></Item>
<Item><M>0.4542</M></Item>
<Item><M>100</M></Item>
</Row>
<Row>
<Item><M>McL.2</M></Item>
<Item><M>23\,328</M></Item>
<Item><M>77\,000</M></Item>
<Item><M>0.4719</M></Item>
<Item><M>100</M></Item>
</Row>
<Row>
<Item><M>He</M></Item>
<Item><M>13\,824</M></Item>
<Item><M>291\,550</M></Item>
<Item><M>0.4310</M></Item>
<Item><M>104</M></Item>
</Row>
<Row>
<Item><M>He.2</M></Item>
<Item><M>18\,432</M></Item>
<Item><M>437\,325</M></Item>
<Item><M>0.4305</M></Item>
<Item><M>104</M></Item>
</Row>
<Row>
<Item><M>Ru</M></Item>
<Item><M>49\,152</M></Item>
<Item><M>2\,968\,875</M></Item>
<Item><M>0.4202</M></Item>
<Item><M>126</M></Item>
</Row>
<Row>
<Item><M>Suz</M></Item>
<Item><M>139\,968</M></Item>
<Item><M>3\,203\,200</M></Item>
<Item><M>0.4416</M></Item>
<Item><M>131</M></Item>
</Row>
<Row>
<Item><M>Suz.2</M></Item>
<Item><M>279\,936</M></Item>
<Item><M>3\,203\,200</M></Item>
<Item><M>0.4557</M></Item>
<Item><M>131</M></Item>
</Row>
<Row>
<Item><M>O
'N
<Item><M>25\,920</M></Item>
<Item><M>17\,778\,376</M></Item>
<Item><M>0.3784</M></Item>
<Item><M>132</M></Item>
</Row>
<Row>
<Item><M>O
'N.2
<Item><M>51\,840</M></Item>
<Item><M>17\,778\,376</M></Item>
<Item><M>0.3940</M></Item>
<Item><M>132</M></Item>
</Row>
<Row>
<Item><M>Co_3</M></Item>
<Item><M>69\,984</M></Item>
<Item><M>7\,084\,000</M></Item>
<Item><M>0.4142</M></Item>
<Item><M>134</M></Item>
</Row>
<Row>
<Item><M>Co_2</M></Item>
<Item><M>2\,359\,296</M></Item>
<Item><M>17\,931\,375</M></Item>
<Item><M>0.4676</M></Item>
<Item><M>154</M></Item>
</Row>
<Row>
<Item><M>Fi_{22}</M></Item>
<Item><M>5\,038\,848</M></Item>
<Item><M>12\,812\,800</M></Item>
<Item><M>0.4853</M></Item>
<Item><M>163</M></Item>
</Row>
<Row>
<Item><M>Fi_{22}.2</M></Item>
<Item><M>10\,077\,696</M></Item>
<Item><M>12\,812\,800</M></Item>
<Item><M>0.4963</M></Item>
<Item><M>163</M></Item>
</Row>
<Row>
<Item><M>HN</M></Item>
<Item><M>2\,000\,000</M></Item>
<Item><M>136\,515\,456</M></Item>
<Item><M>0.4364</M></Item>
<Item><M>166</M></Item>
</Row>
<Row>
<Item><M>HN.2</M></Item>
<Item><M>4\,000\,000</M></Item>
<Item><M>136\,515\,456</M></Item>
<Item><M>0.4479</M></Item>
<Item><M>166</M></Item>
</Row>
<Row>
<Item><M>Ly</M></Item>
<Item><M>900\,000</M></Item>
<Item><M>57\,516\,865\,560</M></Item>
<Item><M>0.3562</M></Item>
<Item><M>174</M></Item>
</Row>
<Row>
<Item><M>Th</M></Item>
<Item><M>944\,784</M></Item>
<Item><M>96\,049\,408\,000</M></Item>
<Item><M>0.3523</M></Item>
<Item><M>177</M></Item>
</Row>
<Row>
<Item><M>Fi_{23}</M></Item>
<Item><M>3\,265\,173\,504</M></Item>
<Item><M>1\,252\,451\,200</M></Item>
<Item><M>0.5111</M></Item>
<Item><M>177</M></Item>
</Row>
<Row>
<Item><M>Co_1</M></Item>
<Item><M>84\,934\,656</M></Item>
<Item><M>48\,952\,653\,750</M></Item>
<Item><M>0.4258</M></Item>
<Item><M>183</M></Item>
</Row>
<Row>
<Item><M>J_4</M></Item>
<Item><M>28\,311\,552</M></Item>
<Item><M>3\,065\,023\,459\,190</M></Item>
<Item><M>0.3737</M></Item>
<Item><M>190</M></Item>
</Row>
<Row>
<Item><M>Fi_{24}
'
<Item><M>29\,386\,561\,536</M></Item>
<Item><M>42\,713\,595\,724\,800</M></Item>
<Item><M>0.4343</M></Item>
<Item><M>207</M></Item>
</Row>
<Row>
<Item><M>Fi_{24}
'.2
<Item><M>58\,773\,123\,072</M></Item>
<Item><M>42\,713\,595\,724\,800</M></Item>
<Item><M>0.4413</M></Item>
<Item><M>207</M></Item>
</Row>
<Row>
<Item><M>B</M></Item>
<Item><M>29\,686\,813\,949\,952</M></Item>
<Item><M>139\,953\,768\,303\,693\,093\,750</M></Item>
<Item><M>0.4007</M></Item>
<Item><M>217</M></Item>
</Row>
<Row>
<!-- value in column 3 does not fit into one line -->
<Item><M>M</M></Item>
<Item><M>2\,849\,934\,139\,195\,392</M></Item>
<Item><M>283\,521\,437\,805\,098\,363\,752</M></Item>
<Item></Item>
<Item></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>344\,287\,234\,566\,406\,250</M></Item>
<Item><M>0.2866</M></Item>
<Item><M>234</M></Item>
</Row>
<HorLine/>
</Table>
<P/>
<Table Align=
"|l|l|l|rl|l|">
<Caption>Table II: Solvable subgroups of maximal order – structures and overgroups</Caption>
<HorLine/>
<Row>
<Item><M>G</M></Item>
<Item><M>S</M></Item>
<Item>Max. overgroups</Item>
<Item><Cite Key=
"CCN85"/></Item>
<Item></Item>
<Item>see</Item>
</Row>
<HorLine/>
<HorLine/>
<Row>
<Item><M>M_{11}</M></Item>
<Item><M>3^2:Q_8.2</M></Item>
<Item><M>S</M></Item>
<Item>18</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>M_{12}</M></Item>
<Item><M>3^2:2S_4</M></Item>
<Item><M>S</M></Item>
<Item>33</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item></Item>
<Item><M>3^2:2S_4</M></Item>
<Item><M>S</M></Item>
<Item>33</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>M_{12}.2</M></Item>
<Item><M>3^2:2S_4</M></Item>
<Item><M>M_{12}</M></Item>
<Item>33</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>J_1</M></Item>
<Item><M>2^3:7:3</M></Item>
<Item><M>S</M></Item>
<Item>36</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>M_{22}</M></Item>
<Item><M>2^4:3^2:4</M></Item>
<Item><M>2^4:A_6</M></Item>
<Item>39</Item>
<Item>(4)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>M_{22}.2</M></Item>
<Item><M>2^4:3^2:D_8</M></Item>
<Item><M>2^4:S_6</M></Item>
<Item>39</Item>
<Item>(4)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>J_2</M></Item>
<Item><M>2^{2+4}:(3 \times S_3)</M></Item>
<Item><M>S</M></Item>
<Item>42</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>J_2.2</M></Item>
<Item><M>2^{2+4}:(S_3 \times S_3)</M></Item>
<Item><M>S</M></Item>
<Item>42</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>M_{23}</M></Item>
<Item><M>2^4:(3 \times A_4):2</M></Item>
<Item><M>2^4:(3 \times A_5):2</M>,</Item>
<Item>71</Item>
<Item>(2)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>2^4:A_7</M></Item>
<Item></Item>
<Item>(10)</Item>
<Item></Item>
</Row>
<Row>
<Item><M>HS</M></Item>
<Item><M>5^{1+2}_+:8:2</M></Item>
<Item><M>U_3(5).2</M></Item>
<Item>80</Item>
<Item>(34)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>U_3(5).2</M></Item>
<Item></Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>HS.2</M></Item>
<Item><M>5^{1+2}_+:[2^5]</M></Item>
<Item><M>S</M></Item>
<Item>80</Item>
<Item>(34)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>J_3</M></Item>
<Item><M>3^2.3^{1+2}_+:8</M></Item>
<Item><M>S</M></Item>
<Item>82</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>J_3.2</M></Item>
<Item><M>3^2.3^{1+2}_+:QD_{16}</M></Item>
<Item><M>S</M></Item>
<Item>82</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>M_{24}</M></Item>
<Item><M>2^6:3^{1+2}_+:D_8</M></Item>
<Item><M>2^6:3.S_6</M></Item>
<Item>96</Item>
<Item>(4)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>McL</M></Item>
<Item><M>3^{1+4}_+:2S_4</M></Item>
<Item><M>3^{1+4}_+:2S_5</M>,</Item>
<Item>100</Item>
<Item>(2)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>U_4(3)</M></Item>
<Item></Item>
<Item>(52)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>McL.2</M></Item>
<Item><M>3^{1+4}_+:4S_4</M></Item>
<Item><M>3^{1+4}_+:4S_5</M>,</Item>
<Item>100</Item>
<Item>(2)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>U_4(3).2_3</M></Item>
<Item></Item>
<Item>(52)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>He</M></Item>
<Item><M>2^6:3^{1+2}_+:D_8</M></Item>
<Item><M>2^6:3.S_6</M></Item>
<Item>104</Item>
<Item>(4)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item></Item>
<Item><M>2^6:3^{1+2}_+:D_8</M></Item>
<Item><M>2^6:3.S_6</M></Item>
<Item></Item>
<Item>(4)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>He.2</M></Item>
<Item><M>2^{4+4}.(S_3 \times S_3).2</M></Item>
<Item><M>S</M></Item>
<Item>104</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item><M>Ru</M></Item>
<Item><M>2.2^{4+6}:S_4</M></Item>
<Item><M>2^{3+8}:L_3(2)</M>,</Item>
<Item>126</Item>
<Item>(3)</Item>
<Item><Ref Subsect=
"sect:Ru"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>2.2^{4+6}:S_5</M></Item>
<Item></Item>
<Item>(2)</Item>
<Item></Item>
</Row>
<Row>
<Item></Item>
<Item><M>2^{3+8}:S_4</M></Item>
<Item><M>2^{3+8}:L_3(2)</M>,</Item>
<Item></Item>
<Item>(3)</Item>
<Item><Ref Subsect=
"sect:Ru"/></Item>
</Row>
<Row>
<Item><M>Suz</M></Item>
<Item><M>3^{2+4}:2(A_4 \times 2^2).2</M></Item>
<Item><M>S</M></Item>
<Item>131</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:Suz"/></Item>
</Row>
<Row>
<Item><M>Suz.2</M></Item>
<Item><M>3^{2+4}:2(S_4 \times D_8)</M></Item>
<Item><M>S</M></Item>
<Item>131</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:Suz"/></Item>
</Row>
<Row>
<Item><M>O
'N
<Item><M>3^4:2^{1+4}_-D_{10}</M></Item>
<Item><M>S</M></Item>
<Item>132</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:ON"/></Item>
</Row>
<Row>
<Item><M>O
'N.2
<Item><M>3^4:2^{1+4}_-.(5:4)</M></Item>
<Item><M>S</M></Item>
<Item>132</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:ON"/></Item>
</Row>
<Row>
<Item><M>Co_3</M></Item>
<Item><M>3^{1+4}_+:4.3^2:D_8</M></Item>
<Item><M>3^{1+4}_+:4S_6</M></Item>
<Item>134</Item>
<Item>(4)</Item>
<Item><Ref Subsect=
"sect:EASY"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>3^5:(2 \times M_{11})</M></Item>
<Item></Item>
<Item>(18)</Item>
<Item></Item>
</Row>
<Row>
<Item><M>Co_2</M></Item>
<Item><M>2^{4+10}(S_4 \times S_3)</M></Item>
<Item><M>2^{1+8}_+:S_6(2)</M>,</Item>
<Item>154</Item>
<Item>(46)</Item>
<Item><Ref Subsect=
"sect:Co2"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>2^{4+10}(S_5 \times S_3)</M></Item>
<Item></Item>
<Item>(2)</Item>
<Item></Item>
</Row>
<Row>
<Item><M>Fi_{22}</M></Item>
<Item><M>3^{1+6}_+:2^{3+4}:3^2:2</M></Item>
<Item><M>S</M></Item>
<Item>163</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:Fi22"/></Item>
</Row>
<Row>
<Item><M>Fi_{22}.2</M></Item>
<Item><M>3^{1+6}_+:2^{3+4}:(S_3 \times S_3)</M></Item>
<Item><M>S</M></Item>
<Item>163</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:Fi22"/></Item>
</Row>
<Row>
<Item><M>HN</M></Item>
<Item><M>5^{1+4}_+:2^{1+4}_-.5.4</M></Item>
<Item><M>S</M></Item>
<Item>166</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:HN"/></Item>
</Row>
<Row>
<Item><M>HN.2</M></Item>
<!-- replace Y by \Ydown for central product -->
<Item><M>5^{1+4}_+:(4 Y 2^{1+4}_-.5.4)</M></Item>
<Item><M>S</M></Item>
<Item>166</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:HN"/></Item>
</Row>
<Row>
<Item><M>Ly</M></Item>
<Item><M>5^{1+4}_+:4.3^2:D_8</M></Item>
<Item><M>5^{1+4}_+:4S_6</M></Item>
<Item>174</Item>
<Item>(4)</Item>
<Item><Ref Subsect=
"sect:Ly"/></Item>
</Row>
<Row>
<Item><M>Th</M></Item>
<Item><M>[3^9].2S_4</M></Item>
<Item><M>S</M></Item>
<Item>177</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:Th"/></Item>
</Row>
<Row>
<Item></Item>
<Item><M>3^2.[3^7].2S_4</M></Item>
<Item><M>S</M></Item>
<Item></Item>
<Item></Item>
<Item></Item>
</Row>
<Row>
<Item><M>Fi_{23}</M></Item>
<Item><M>3^{1+8}_+.2^{1+6}_-.3^{1+2}_+.2S_4</M></Item>
<Item><M>S</M></Item>
<Item>177</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:Fi23"/></Item>
</Row>
<Row>
<Item><M>Co_1</M></Item>
<Item><M>2^{4+12}.(S_3 \times 3^{1+2}_+:D_8)</M></Item>
<Item><M>2^{4+12}.(S_3 \times 3S_6)</M></Item>
<Item>183</Item>
<Item></Item>
<Item><Ref Subsect=
"sect:Co1"/></Item>
</Row>
<HorLine/>
</Table>
<P/>
<Table Align=
"|l|l|l|rl|l|">
<Caption>Table II: Solvable subgroups of maximal order – structures and overgroups (continued)<
/Caption>
<HorLine/>
<Row>
<Item><M>G</M></Item>
<Item><M>S</M></Item>
<Item>Max. overgroups</Item>
<Item><Cite Key="CCN85"/></Item>
<Item></Item>
<Item>see</Item>
</Row>
<HorLine/>
<HorLine/>
<Row>
<Item><M>J_4</M></Item>
<Item><M>2^{11}:2^6:3^{1+2}_+:D_8</M></Item>
<Item><M>2^{11}:M_{24}</M>,</Item>
<Item>190</Item>
<Item>(96)</Item>
<Item><Ref Subsect="sect:J4"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>2^{1+12}_+.3M_{22}:2</M></Item>
<Item></Item>
<Item>(39)</Item>
<Item></Item>
</Row>
<Row>
<Item><M>Fi_{24}'
<Item><M>3^{1+10}_+:2^{1+6}_-:3^{1+2}_+:2S_4</M></Item>
<Item><M>3^{1+10}_+:U_5(2):2</M></Item>
<Item>207</Item>
<Item>(73)</Item>
<Item><Ref Subsect="sect:F3+"/></Item>
</Row>
<Row>
<Item><M>Fi_{24}'.2
<Item><M>3^{1+10}_+:(2 \times 2^{1+6}_-:3^{1+2}_+:2S_4)</M></Item>
<Item><M>3^{1+10}_+:(2 \times U_5(2):2)</M></Item>
<Item>207</Item>
<Item>(73)</Item>
<Item><Ref Subsect="sect:F3+"/></Item>
</Row>
<Row>
<Item><M>B</M></Item>
<Item><M>2^{2+10+20}(2^4:3^2:D_8 \times S_3)</M></Item>
<Item><M>2^{2+10+20}(M_{22}:2 \times S_3)</M>,</Item>
<Item>217</Item>
<Item>(39)</Item>
<Item><Ref Subsect="sect:B"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>2^{9+16}S_8(2)</M></Item>
<Item></Item>
<Item>(123)</Item>
<Item></Item>
</Row>
<Row>
<Item><M>M</M></Item>
<Item><M>2^{1+2+6+12+18}.(S_4 \times 3^{1+2}_+:D_8)</M></Item>
<Item><M>2^{[39]}.(L_3(2) \times 3S_6)</M>,</Item>
<Item>234</Item>
<Item>(3, 4)</Item>
<Item><Ref Subsect="sect:M"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>2^{1+24}_+.Co_1</M></Item>
<Item></Item>
<Item>(183)</Item>
<Item></Item>
</Row>
<Row>
<Item></Item>
<Item><M>2^{2+1+6+12+18}.(S_4 \times 3^{1+2}_+:D_8)</M></Item>
<Item><M>2^{[39]}.(L_3(2) \times 3S_6)</M>,</Item>
<Item></Item>
<Item>(3, 4)</Item>
<Item><Ref Subsect="sect:M"/></Item>
</Row>
<Row>
<Item></Item>
<Item></Item>
<Item><M>2^{2+11+22}.(M_{24} \times S_3)</M></Item>
<Item></Item>
<Item>(96)</Item>
<Item></Item>
</Row>
<HorLine/>
</Table>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:approach">
<Heading>The Approach</Heading>
We combine the information in the &ATLAS; <Cite Key="CCN85"/>
with explicit computations using the &GAP; system <Cite Key="GAP"/>,
in particular its Character Table Library <Cite Key="CTblLib"/>
and its library of Tables of Marks <Cite Key="TomLib"/>.
First we load these two packages.
<P/>
<Example><![CDATA[
gap> LoadPackage( "CTblLib", "1.2", false );
true
gap> LoadPackage( "TomLib", false );
true
]]></Example>
<P/>
The orders of solvable subgroups of maximal order will be collected in a
global record <C>MaxSolv</C>.
<P/>
<Example><![CDATA[
gap> MaxSolv:= rec();;
]]></Example>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Use the Table of Marks">
<Heading>Use the Table of Marks</Heading>
If the &GAP; library of Tables of Marks <Cite Key="TomLib"/> contains
the table of marks of a group <M>G</M>
then we can easily inspect all conjugacy classes of subgroups of <M>G</M>.
The following small &GAP; function can be used for that.
It returns <K>false</K> if the table of marks of the group with the name
<C>name</C> is not available,
and the list <C>[ name, n, super ]</C> otherwise,
where <C>n</C> is the maximal order of solvable subgroups of <M>G</M>,
and <C>super</C> is a list of lists;
for each conjugacy class of solvable subgroups <M>S</M> of order <C>n</C>,
<C>super</C> contains the list of orders of representatives <M>M</M>
of the classes of maximal subgroups of <M>G</M> such that <M>M</M>
contains a conjugate of <M>S</M>.
<P/>
Note that a subgroup in the <M>i</M>-th class of a table of marks contains
a subgroup in the <M>j</M>-th class if and only if the entry in the position
<M>(i,j)</M> of the table of marks is nonzero.
For tables of marks objects in &GAP;,
this is the case if and only if <M>j</M> is contained in the <M>i</M>-th row of
the list that is stored as the value of the attribute <C>SubsTom</C> of the
table of marks object;
for this test, one need not unpack the matrix of marks.
<P/>
<Ignore Remark="gapfilecomments">
</Ignore>
<Example><![CDATA[
gap> MaximalSolvableSubgroupInfoFromTom:= function( name )
> local tom, # table of marks for `name'
> n, # maximal order of a solvable subgroup
> maxsubs, # numbers of the classes of subgroups of order `n'
> orders, # list of orders of the classes of subgroups
> i, # loop over the classes of subgroups
> maxes, # list of positions of the classes of max. subgroups
> subs, # `SubsTom' value
> cont; # list of list of positions of max. subgroups
>
> tom:= TableOfMarks( name );
> if tom = fail then
> return false;
> fi;
> n:= 1;
> maxsubs:= [];
> orders:= OrdersTom( tom );
> for i in [ 1 .. Length( orders ) ] do
> if IsSolvableTom( tom, i ) then
> if orders[i] = n then
> Add( maxsubs, i );
> elif orders[i] > n then
> n:= orders[i];
> maxsubs:= [ i ];
> fi;
> fi;
> od;
> maxes:= MaximalSubgroupsTom( tom )[1];
> subs:= SubsTom( tom );
> cont:= List( maxsubs, j -> Filtered( maxes, i -> j in subs[i] ) );
>
> return [ name, n, List( cont, l -> orders{ l } ) ];
> end;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Use Information from the Character Table Library">
<Heading>Use Information from the Character Table Library</Heading>
The &GAP; Character Table Library contains the character tables of all
maximal subgroups of sporadic simple groups,
except for the Monster group.
This information can be used as follows.
<P/>
We start, for a sporadic simple group <M>G</M>,
with a known solvable subgroup of order <M>n</M>, say, in <M>G</M>.
In order to show that <M>G</M> contains no solvable subgroup of larger order,
it suffices to show that no maximal subgroup of <M>G</M> contains
a larger solvable subgroup.
<P/>
The point is that usually the orders of the maximal subgroups of <M>G</M>
are not much larger than <M>n</M>,
and that a maximal subgroup <M>M</M> contains a solvable subgroup
of order <M>n</M> only if the factor group of <M>M</M>
by its largest solvable normal subgroup <M>N</M>
contains a solvable subgroup of order <M>n/|N|</M>.
This reduces the question to relatively small groups.
<P/>
What we can check <E>automatically</E> from the character table of <M>M/N</M>
is whether <M>M/N</M> can contain subgroups (solvable or not) of indices
between five and <M>|M|/n</M>,
by computing possible permutation characters of these degrees.
(Note that a solvable subgroup of a nonsolvable group has index
at least five.
This lower bound could be improved for example by considering the
smallest degree of a nontrivial character, but this is not an issue here.)
<P/>
Then we are left with a –hopefully short– list
of maximal subgroups of <M>G</M>,
together with upper bounds on the indices of possible solvable subgroups;
excluding these possibilities then yields that the initially chosen
solvable subgroup of <M>G</M> is indeed the largest one.
<P/>
The following &GAP; function can be used to compute this information
for the character table <C>tblM</C> of <M>M</M> and a given order
<C>minorder</C>.
It returns <K>false</K> if <M>M</M> cannot contain a solvable subgroup
of order at least <C>minorder</C>,
otherwise a list <C>[ tblM, m, k ]</C> where <C>m</C> is the maximal index
of a subgroup that has order at least <C>minorder</C>,
and <C>k</C> is the minimal index of a possible subgroup of <M>M</M>
(a proper subgroup if <M>M</M> is nonsolvable),
according to the &GAP; function <Ref Func="PermChars" BookName="ref"/>.
<P/>
<Ignore Remark="gapfilecomments">
</Ignore>
<Example><![CDATA[
gap> SolvableSubgroupInfoFromCharacterTable:= function( tblM, minorder )
> local maxindex, # index of subgroups of order `minorder'
> N, # class positions describing a solvable normal subgroup
> fact, # character table of the factor by `N'
> classes, # class sizes in `fact'
> nsg, # list of class positions of normal subgroups
> i; # loop over the possible indices
>
> maxindex:= Int( Size( tblM ) / minorder );
> if maxindex = 0 then
> return false;
> elif IsSolvableCharacterTable( tblM ) then
> return [ tblM, maxindex, 1 ];
> elif maxindex < 5 then
> return false;
> fi;
>
> N:= [ 1 ];
> fact:= tblM;
> repeat
> fact:= fact / N;
> classes:= SizesConjugacyClasses( fact );
> nsg:= Difference( ClassPositionsOfNormalSubgroups( fact ), [ [ 1 ] ] );
> N:= First( nsg, x -> IsPrimePowerInt( Sum( classes{ x } ) ) );
> until N = fail;
>
> for i in Filtered( DivisorsInt( Size( fact ) ),
> d -> 5 <= d and d <= maxindex ) do
> if Length( PermChars( fact, rec( torso:= [ i ] ) ) ) > 0 then
> return [ tblM, maxindex, i ];
> fi;
> od;
>
> return false;
> end;;
]]></Example>
</Subsection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:EASY">
<Heading>Cases where the Table of Marks is available in &GAP;</Heading>
For twelve sporadic simple groups,
the &GAP; library of Tables of Marks knows the tables of marks,
so we can use <C>MaximalSolvableSubgroupInfoFromTom</C>.
<P/>
<Example><![CDATA[
gap> solvinfo:= Filtered( List(
> AllCharacterTableNames( IsSporadicSimple, true,
> IsDuplicateTable, false ),
> MaximalSolvableSubgroupInfoFromTom ), x -> x <> false );;
gap> for entry in solvinfo do
> MaxSolv.( entry[1] ):= entry[2];
> od;
gap> for entry in solvinfo do
> Print( String( entry[1], 5 ), String( entry[2], 7 ),
> String( entry[3], 28 ), "\n" );
> od;
Co3 69984 [ [ 3849120, 699840 ] ]
HS 2000 [ [ 252000, 252000 ] ]
He 13824 [ [ 138240 ], [ 138240 ] ]
J1 168 [ [ 168 ] ]
J2 1152 [ [ 1152 ] ]
J3 1944 [ [ 1944 ] ]
M11 144 [ [ 144 ] ]
M12 432 [ [ 432 ], [ 432 ] ]
M22 576 [ [ 5760 ] ]
M23 1152 [ [ 40320, 5760 ] ]
M24 13824 [ [ 138240 ] ]
McL 11664 [ [ 3265920, 58320 ] ]
]]></Example>
<P/>
We see that for <M>J_1</M>, <M>J_2</M>, <M>J_3</M>, <M>M_{11}</M>,
and <M>M_{12}</M>, the subgroup <M>S</M> is maximal.
For <M>M_{12}</M> and <M>He</M>, there are two classes of subgroups <M>S</M>.
For the other groups, the class of subgroups <M>S</M> is unique,
and there are one or two classes of maximal subgroups
of <M>G</M> that contain <M>S</M>.
From the shown orders of these maximal subgroups, their structures can
be read off from the &ATLAS;,
on the pages listed in Table II.
<P/>
Similarly,
the &ATLAS; tells us about the extensions of the subgroups <M>S</M>
in Aut<M>(G)</M>.
In particular,
<List>
<Item>
the order <M>2\,000</M> subgroups of <M>HS</M> are contained in maximal
subgroups of the type <M>U_3(5).2</M> (two classes) which do not extend to
<M>HS.2</M>, but there are novelties of the type <M>5^{1+2}_+:[2^5]</M>
and of the order <M>4\,000</M>,
so the solvable subgroups of maximal order in <M>HS</M>
do in fact extend to <M>HS.2</M>.
</Item>
<Item>
the order <M>13\,824</M> subgroups of <M>He</M> are contained in maximal
subgroups of the type <M>2^6:3S_6</M> (two classes) which do not extend to
<M>He.2</M>,
but there are novelties of the type <M>2^{4+4}.(S_3 \times S_3).2</M>
and of the order <M>18\,432</M>.
(So the solvable subgroups <M>S</M> of maximal order in <M>He</M>
do not extend to <M>He.2</M>
but there are larger solvable subgroups in <M>He.2</M>.)
<P/>
We inspect the maximal subgroups of <M>He.2</M> in order to show
that these are in fact the solvable subgroups of maximal order
(see <Cite Key="CCN85" Where="p. 104"/>):
Any other solvable subgroup of order at least <M>n</M> in <M>He.2</M>
must be contained in a subgroup of one of the types
<M>S_4(4).4</M> (of index at most <M>212</M>),
<M>2^2.L_3(4).D_{12}</M> (of index at most <M>52</M>),
or
<M>2^{1+6}_+.L_3(2).2</M> (of index at most <M>2</M>).
By <Cite Key="CCN85" Where="pp. 44, 23, 3"/>, this is not the case.
</Item>
<Item>
the maximal subgroups of order <M>1\,152</M> in <M>J_2</M> extend
to subgroups of order <M>2\,304</M> in <M>J_2.2</M>.
</Item>
<Item>
the maximal subgroups of order <M>1\,944</M> in <M>J_3</M> extend
to subgroups of the type <M>3^2.3^{1+2}_+:8.2</M>
and of order <M>3888</M> in <M>J_3.2</M>.
(The structure stated in <Cite Key="CCN85" Where="p. 82"/>
is not correct, see <Cite Key="BN95"/>.)
</Item>
<Item>
the maximal subgroups of order <M>432</M> in <M>M_{12}</M> (two classes)
do <E>not</E> extend in <M>M_{12}.2</M>,
and we see from the table of marks of <M>M_{12}.2</M> that there are no
larger solvable subgroups in this group, i. e.,
the solvable subgroups of maximal order in <M>M_{12}.2</M> lie in
<M>M_{12}</M>.
</Item>
<Item>
the order <M>576</M> subgroups of <M>M_{22}</M> are contained in maximal
subgroups of the type <M>2^4:A_6</M> which extend to subgroups of the type
<M>2^4:S_6</M> in <M>M_{22}.2</M>,
so the solvable subgroups of maximal order in <M>M_{22}.2</M>
have the type <M>2^4:3^2:D_8</M> and the order <M>1\,152</M>.
In fact the structure is <M>S_4 \wr S_2</M>.
</Item>
<Item>
the order <M>11\,664</M> subgroups of <M>McL</M> are contained in maximal
subgroups of the type <M>3^{1+4}_+:2S_5</M> which extend to subgroups
of the type <M>3^{1+4}:4S_5</M> in <M>McL.2</M>,
so the solvable subgroups of maximal order in <M>McL.2</M>
have the type <M>3^{1+4}:4S_4</M> and the order <M>23\,328</M>.
</Item>
</List>
<P/>
<Example><![CDATA[
gap> MaxSolv.( "HS.2" ):= 2 * MaxSolv.( "HS" );;
gap> n:= 2^(4+4) * ( 6 * 6 ) * 2; MaxSolv.( "He.2" ):= n;;
18432
gap> List( [ Size( CharacterTable( "S4(4).4" ) ),
> Factorial( 5 )^2 * 2,
> Size( CharacterTable( "2^2.L3(4).D12" ) ),
> 2^7 * Size( CharacterTable( "L3(2)" ) ) * 2,
> 7^2 * 2 * Size( CharacterTable( "L2(7)" ) ) * 2,
> 3 * Factorial( 7 ) * 2 ], i -> Int( i / n ) );
[ 212, 1, 52, 2, 1, 1 ]
gap> MaxSolv.( "J2.2" ):= 2 * MaxSolv.( "J2" );;
gap> MaxSolv.( "J3.2" ):= 2 * MaxSolv.( "J3" );;
gap> info:= MaximalSolvableSubgroupInfoFromTom( "M12.2" );
[ "M12.2", 432, [ [ 95040 ] ] ]
gap> MaxSolv.( "M12.2" ):= info[2];;
gap> MaxSolv.( "M22.2" ):= 2 * MaxSolv.( "M22" );;
gap> MaxSolv.( "McL.2" ):= 2 * MaxSolv.( "McL" );;
]]></Example>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:Cases where the Table of Marks is not available in GAP">
<Heading>Cases where the Table of Marks is not available in &GAP;</Heading>
We use the &GAP; function <C>SolvableSubgroupInfoFromCharacterTable</C>,
and individual arguments.
In several cases, information about smaller sporadic simple groups is needed,
so we deal with the groups in increasing order.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Ru">
<Heading><M>G = Ru</M></Heading>
The group <M>Ru</M> contains exactly two conjugacy classes of nonisomorphic
solvable subgroups of order <M>n = 49\,152</M>,
and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Ru" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 49152;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "2^3+8:L3(2)" ), 7, 7 ],
[ CharacterTable( "2.2^4+6:S5" ), 5, 5 ] ]
]]></Example>
<P/>
The maximal subgroups of the structure <M>2.2^{4+6}:S_5</M> in <M>Ru</M>
contain one class of solvable subgroups of order <M>n</M>
and with the structure <M>2.2^{4+6}:S_4</M>,
see <Cite Key="CCN85" Where="p. 126, p. 2"/>.
<P/>
The maximal subgroups of the structure <M>2^{3+8}:L_3(2)</M> in <M>Ru</M>
contain two classes of solvable subgroups of order <M>n</M>
and with the structure <M>2^{3+8}:S_4</M>,
see <Cite Key="CCN85" Where="p. 126, p. 3"/>.
These groups are the stabilizers of vectors and two-dimensional subspaces,
respectively, in the three-dimensional submodule;
note that each <M>2^{3+8}:L_3(2)</M> type subgroup <M>H</M> of <M>Ru</M>
is the normalizer of an elementary abelian group of order eight
all of whose involutions are in the
<M>Ru</M>-class <C>2A</C> and are conjugate in <M>H</M>.
Since the <M>2.2^{4+6}:S_5</M> type subgroups of <M>Ru</M>
are the normalizers of
<C>2A</C>-elements in <M>Ru</M>, the groups in one of the two classes in question
coincide with the largest solvable subgroups in the <M>2.2^{4+6}:S_5</M> type
subgroups.
The groups in the other class do not centralize a <C>2A</C>-element in <M>Ru</M>
and are therefore not isomorphic with the <M>2.2^{4+6}:S_4</M> type groups.
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Ru" ):= n;;
gap> s:= info[1][1];;
gap> cls:= SizesConjugacyClasses( s );;
gap> nsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),
> x -> Sum( cls{ x } ) = 2^3 );
[ [ 1, 2 ] ]
gap> cls{ nsg[1] };
[ 1, 7 ]
gap> GetFusionMap( s, t ){ nsg[1] };
[ 1, 2 ]
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Suz">
<Heading><M>G = Suz</M></Heading>
The group <M>Suz</M> contains a unique conjugacy class of solvable subgroups
of order <M>n = 139\,968</M>, and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Suz" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 139968;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "G2(4)" ), 1797, 416 ],
[ CharacterTable( "3_2.U4(3).2_3'" ), 140, 72 ],
[ CharacterTable( "3^5:M11" ), 13, 11 ],
[ CharacterTable( "2^4+6:3a6" ), 7, 6 ],
[ CharacterTable( "3^2+4:2(2^2xa4)2" ), 1, 1 ] ]
]]></Example>
<P/>
The maximal subgroups <M>S</M> of the structure
<M>3^{2+4}:2(A_4 \times 2^2).2</M>
in <M>Suz</M> are solvable and have order <M>n</M>,
see <Cite Key="CCN85" Where="p. 131"/>.
<P/>
In order to show that <M>Suz</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>G_2(4)</M> of index at most <M>1\,797</M>
(see <Cite Key="CCN85" Where="p. 97"/>),
in <M>U_4(3).2_3^{\prime}</M> of index at most <M>140</M>
(see <Cite Key="CCN85" Where="p. 52"/>),
in <M>M_{11}</M> of index at most <M>13</M>
(see <Cite Key="CCN85" Where="p. 18"/>), and
in <M>A_6</M> of index at most <M>7</M>
(see <Cite Key="CCN85" Where="p. 4"/>).
<P/>
The group <M>S</M> extends to a group of the structure
<M>3^{2+4}:2(S_4 \times D_8)</M>
in the automorphism group <M>Suz.2</M>.
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Suz" ):= n;;
gap> MaxSolv.( "Suz.2" ):= 2 * n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:ON">
<Heading><M>G = ON</M></Heading>
The group <M>ON</M> contains a unique conjugacy class of solvable subgroups
of order <M>25\,920</M>, and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "ON" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 25920;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "L3(7).2" ), 144, 114 ],
[ CharacterTable( "ONM2" ), 144, 114 ],
[ CharacterTable( "3^4:2^(1+4)D10" ), 1, 1 ] ]
]]></Example>
<P/>
The maximal subgroups <M>S</M> of the structure <M>3^4:2^{1+4}_-D_{10}</M>
in <M>ON</M> are solvable and have order <M>n</M>,
see <Cite Key="CCN85" Where="pp. 132"/>.
<P/>
In order to show that <M>ON</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>L_3(7).2</M> of index at most <M>144</M>
(see <Cite Key="CCN85" Where="p. 50"/>);
note that the groups in the second class of maximal subgroups of <M>ON</M>
are isomorphic with <M>L_3(7).2</M>.
<P/>
The group <M>S</M> extends to a group of order <M>|S.2|</M>
in the automorphism group <M>ON.2</M>.
<P/>
<Example><![CDATA[
gap> MaxSolv.( "ON" ):= n;;
gap> MaxSolv.( "ON.2" ):= 2 * n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Co2">
<Heading><M>G = Co_2</M></Heading>
The group <M>Co_2</M> contains a unique conjugacy class of solvable subgroups
of order <M>2\,359\,296</M>, and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Co2" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 2359296;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "U6(2).2" ), 7796, 672 ],
[ CharacterTable( "2^10:m22:2" ), 385, 22 ],
[ CharacterTable( "McL" ), 380, 275 ],
[ CharacterTable( "2^1+8:s6f2" ), 315, 28 ],
[ CharacterTable( "2^1+4+6.a8" ), 17, 8 ],
[ CharacterTable( "U4(3).D8" ), 11, 8 ],
[ CharacterTable( "2^(4+10)(S5xS3)" ), 5, 5 ] ]
]]></Example>
<P/>
The maximal subgroups of the structure <M>2^{4+10}(S_5 \times S_3)</M>
in <M>Co_2</M>
contain solvable subgroups <M>S</M> of order <M>n</M> and with the structure
<M>2^{4+10}(S_4 \times S_3)</M>,
see <Cite Key="CCN85" Where="p. 154"/>.
<P/>
The subgroups <M>S</M> are contained also in the maximal subgroups
of the type <M>2^{1+8}_+:S_6(2)</M>;
note that the <M>2^{1+8}_+:S_6(2)</M> type subgroups are described
as normalizers of elements in the <M>Co_2</M>-class <C>2A</C>,
and <M>S</M> normalizes an elementary abelian group of order <M>16</M>
containing an <M>S</M>-class of length five that is contained in the
<M>Co_2</M>-class <C>2A</C>.
<P/>
<Example><![CDATA[
gap> s:= info[7][1];
CharacterTable( "2^(4+10)(S5xS3)" )
gap> cls:= SizesConjugacyClasses( s );;
gap> nsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),
> x -> Sum( cls{ x } ) = 2^4 );
[ [ 1 .. 3 ] ]
gap> cls{ nsg[1] };
[ 1, 5, 10 ]
gap> GetFusionMap( s, t ){ nsg[1] };
[ 1, 2, 3 ]
]]></Example>
<P/>
The stabilizers of these involutions in <M>2^{4+10}(S_5 \times S_3)</M>
have index five,
they are solvable, and they are contained in <M>2^{1+8}_+:S_6(2)</M> type
subgroups,
so they are <M>Co_2</M>-conjugates of <M>S</M>.
(The corresponding subgroups of <M>S_6(2)</M> are maximal and have the type
<M>2.[2^6]:(S_3 \times S_3)</M>.)
<P/>
In order to show that <M>G</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>U_6(2)</M> of index at most <M>7\,796</M>
(see <Cite Key="CCN85" Where="p. 115"/>),
in <M>M_{22}.2</M> of index at most <M>385</M>
(see <Cite Key="CCN85" Where="p. 39"/>
or Section <Ref Subsect="sect:EASY"/>),
in <M>McL</M> of index at most <M>380</M>
(see <Cite Key="CCN85" Where="p. 100"/>
or Section <Ref Subsect="sect:EASY"/>),
in <M>A_8</M> of index at most <M>17</M>
(see <Cite Key="CCN85" Where="p. 20"/>),
and
in <M>U_4(3).D_8</M> of index at most <M>11</M>
(see <Cite Key="CCN85" Where="p. 52"/>).
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Co2" ):= n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Fi22">
<Heading><M>G = Fi_{22}</M></Heading>
The group <M>Fi_{22}</M> contains a unique conjugacy class of
solvable subgroups of order <M>5\,038\,848</M>,
and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Fi22" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 5038848;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "2.U6(2)" ), 3650, 672 ],
[ CharacterTable( "O7(3)" ), 910, 351 ],
[ CharacterTable( "Fi22M3" ), 910, 351 ],
[ CharacterTable( "O8+(2).3.2" ), 207, 6 ],
[ CharacterTable( "2^10:m22" ), 90, 22 ],
[ CharacterTable( "3^(1+6):2^(3+4):3^2:2" ), 1, 1 ] ]
]]></Example>
<P/>
The maximal subgroups <M>S</M> of the structure <M>3^{1+6}:2^{3+4}:3^2:2</M>
in <M>Fi_{22}</M> are solvable and have order <M>n</M>,
see <Cite Key="CCN85" Where="p. 163"/>.
<P/>
In order to show that <M>Fi_{22}</M> contains no other solvable subgroups
of order larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>U_6(2)</M> of index at most <M>3\,650</M>
(see <Cite Key="CCN85" Where="p. 115"/>),
in <M>O_7(3)</M> of index at most <M>910</M>
(see <Cite Key="CCN85" Where="p. 109"/>),
in <M>O_8^+(2).S_3</M> of index at most <M>207</M>
(see <Cite Key="CCN85" Where="p. 85"/>),
and
in <M>M_{22}.2</M> of index at most <M>90</M>
(see <Cite Key="CCN85" Where="p. 39"/>
or Section <Ref Subsect="sect:EASY"/>);
note that the groups in the third class of maximal subgroups of <M>Fi_{22}</M>
are isomorphic with <M>O_7(3)</M>.
<P/>
The group <M>S</M> extends to a group of order <M>|S.2|</M>
in the automorphism group <M>Fi_{22}.2</M>.
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Fi22" ):= n;;
gap> MaxSolv.( "Fi22.2" ):= 2 * n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:HN">
<Heading><M>G = HN</M></Heading>
The group <M>HN</M> contains a unique conjugacy class of solvable subgroups
of order <M>2\,000\,000</M>, and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "HN" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 2000000;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "A12" ), 119, 12 ],
[ CharacterTable( "5^(1+4):2^(1+4).5.4" ), 1, 1 ] ]
]]></Example>
<P/>
The maximal subgroups <M>S</M> of the structure <M>5^{1+4}:2^{1+4}.5.4</M>
in <M>HN</M> are solvable and have order <M>n</M>,
see <Cite Key="CCN85" Where="p. 166"/>.
<P/>
In order to show that <M>HN</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>A_{12}</M> of index at most <M>119</M>
(see <Cite Key="CCN85" Where="p. 91"/>).
<P/>
The group <M>S</M> extends to a group of order <M>|S.2|</M>
in the automorphism group <M>HN.2</M>.
<P/>
<Example><![CDATA[
gap> MaxSolv.( "HN" ):= n;;
gap> MaxSolv.( "HN.2" ):= 2 * n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Ly">
<Heading><M>G = Ly</M></Heading>
The group <M>Ly</M> contains a unique conjugacy class of solvable subgroups
of order <M>900\,000</M>, and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Ly" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 900000;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "G2(5)" ), 6510, 3906 ],
[ CharacterTable( "3.McL.2" ), 5987, 275 ],
[ CharacterTable( "5^3.psl(3,5)" ), 51, 31 ],
[ CharacterTable( "2.A11" ), 44, 11 ],
[ CharacterTable( "5^(1+4):4S6" ), 10, 6 ] ]
]]></Example>
<P/>
The maximal subgroups of the structure <M>5^(1+4):4S6</M> in <M>Ly</M>
contain solvable subgroups <M>S</M> of order <M>n</M> and with the structure
<M>5^{1+4}:4.3^2.D_8</M>, see <Cite Key="CCN85" Where="p. 174"/>.
<P/>
In order to show that <M>Ly</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>G_2(5)</M> of index at most <M>6\,510</M>
(see <Cite Key="CCN85" Where="p. 114"/>),
in <M>McL.2</M> of index at most <M>5\,987</M>
(see <Cite Key="CCN85" Where="p. 100"/>
or Section <Ref Subsect="sect:EASY"/>),
in <M>L_3(5)</M> of index at most <M>51</M>
(see <Cite Key="CCN85" Where="p. 38"/>),
and
in <M>A_{11}</M> of index at most <M>44</M>
(see <Cite Key="CCN85" Where="p. 75"/>).
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Ly" ):= n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Th">
<Heading><M>G = Th</M></Heading>
The group <M>Th</M> contains exactly two conjugacy classes of nonisomorphic
solvable subgroups of order <M>n = 944\,784</M>,
and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Th" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 944784;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "2^5.psl(5,2)" ), 338, 31 ],
[ CharacterTable( "2^1+8.a9" ), 98, 9 ],
[ CharacterTable( "U3(8).6" ), 35, 6 ],
[ CharacterTable( "ThN3B" ), 1, 1 ],
[ CharacterTable( "ThM7" ), 1, 1 ] ]
]]></Example>
<P/>
The maximal subgroups <M>S</M> of the structures <M>[3^9].2S_4</M> and <M>3^2.[3^7].2S_4</M>
in <M>Th</M> are solvable and have order <M>n</M>,
see <Cite Key="CCN85" Where="p. 177"/>.
<P/>
In order to show that <M>Th</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>L_5(2)</M> of index at most <M>338</M>
(see <Cite Key="CCN85" Where="p. 70"/>),
in <M>A_9</M> of index at most <M>98</M>
(see <Cite Key="CCN85" Where="p. 37"/>),
and
in <M>U_3(8).6</M> of index at most <M>35</M>
(see <Cite Key="CCN85" Where="p. 66"/>).
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Th" ):= n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Fi23">
<Heading><M>G = Fi_{23}</M></Heading>
The group <M>Fi_{23}</M> contains a unique conjugacy class of
solvable subgroups of order <M>n = 3\,265\,173\,504</M>,
and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Fi23" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 3265173504;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "2.Fi22" ), 39545, 3510 ],
[ CharacterTable( "O8+(3).3.2" ), 9100, 6 ],
[ CharacterTable( "3^(1+8).2^(1+6).3^(1+2).2S4" ), 1, 1 ] ]
]]></Example>
<P/>
The maximal subgroups <M>S</M> of the structure <M>3^{1+8}_+.2^{1+6}_-.3^{1+2}_+.2S_4</M>
in <M>Fi_{23}</M> are solvable and have order <M>n</M>,
see <Cite Key="CCN85" Where="p. 177"/>.
<P/>
In order to show that <M>Fi_{23}</M> contains no other solvable subgroups
of order larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>Fi_{22}</M> of index at most <M>39\,545</M>
(see Section <Ref Subsect="sect:Fi22"/>)
and in <M>O_8^+(3).S_3</M> of index at most <M>9\,100</M>
(see <Cite Key="CCN85" Where="p. 140"/>).
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Fi23" ):= n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:Co1">
<Heading><M>G = Co_1</M></Heading>
The group <M>Co_1</M> contains a unique conjugacy class of solvable subgroups
of order <M>n = 84\,934\,656</M>,
and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Co1" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 84934656;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "Co2" ), 498093, 2300 ],
[ CharacterTable( "3.Suz.2" ), 31672, 1782 ],
[ CharacterTable( "2^11:M24" ), 5903, 24 ],
[ CharacterTable( "Co3" ), 5837, 276 ],
[ CharacterTable( "2^(1+8)+.O8+(2)" ), 1050, 120 ],
[ CharacterTable( "U6(2).3.2" ), 649, 6 ],
[ CharacterTable( "2^(2+12):(A8xS3)" ), 23, 8 ],
[ CharacterTable( "2^(4+12).(S3x3S6)" ), 10, 6 ] ]
]]></Example>
<P/>
The maximal subgroups of the structure <M>2^{4+12}.(S_3 \times 3S_6)</M> in <M>Co_1</M>
contain solvable subgroups <M>S</M> of order <M>n</M> and with the structure
<M>2^{4+12}.(S_3 \times 3^{1+2}_+:D_8)</M>,
see <Cite Key="CCN85" Where="p. 183"/>.
<P/>
In order to show that <M>Co_1</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>Co_2</M> of index at most <M>498\,093</M>
(see Section <Ref Subsect="sect:Co2"/>),
in <M>Suz.2</M> of index at most <M>31\,672</M>
(see Section <Ref Subsect="sect:Suz"/>),
in <M>M_{24}</M> of index at most <M>5\,903</M>
(see Section <Ref Subsect="sect:EASY"/>),
in <M>Co_3</M> of index at most <M>5\,837</M>
(see <Cite Key="CCN85" Where="p. 134"/>
or Section <Ref Subsect="sect:EASY"/>),
in <M>O_8^+(2)</M> of index at most <M>1\,050</M>
(see <Cite Key="CCN85" Where="p. 185"/>),
in <M>U_6(2).S_3</M> of index at most <M>649</M>
(see <Cite Key="CCN85" Where="p. 115"/>),
and
in <M>A_8</M> of index at most <M>23</M>
(see <Cite Key="CCN85" Where="p. 22"/>).
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Co1" ):= n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:J4">
<Heading><M>G = J_4</M></Heading>
The group <M>J_4</M> contains a unique conjugacy class of solvable subgroups
of order <M>28\,311\,552</M>,
and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "J4" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 28311552;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "mx1j4" ), 17710, 24 ],
[ CharacterTable( "c2aj4" ), 770, 22 ],
[ CharacterTable( "2^10:L5(2)" ), 361, 31 ],
[ CharacterTable( "J4M4" ), 23, 5 ] ]
]]></Example>
<P/>
The maximal subgroups of the structure <M>2^{11}:M_{24}</M> in <M>J_4</M>
contain solvable subgroups <M>S</M> of order <M>n</M> and with the structure
<M>2^{11}:2^6:3^{1+2}_+:D_8</M>,
see Section <Ref Subsect="sect:EASY"/>
and <Cite Key="CCN85" Where="p. 190"/>.
<P/>
(The subgroups in the first four classes of maximal subgroups of <M>J_4</M>
have the structures <M>2^{11}:M_{24}</M>, <M>2^{1+12}_+.3M_{22}:2</M>,
<M>2^{10}:L_5(2)</M>, and <M>2^{3+12}.(S_5 \times L_3(2))</M>,
in this order.)
<P/>
The subgroups <M>S</M> are contained also in the maximal subgroups of the type
<M>2^{1+12}_+.3M_{22}:2</M>;
note that these subgroups are described
as normalizers of elements in the <M>J_4</M>-class <C>2A</C>,
and <M>S</M> normalizes an elementary abelian group of order <M>2^{11}</M>
containing an <M>S</M>-class of length <M>1\,771</M> that is contained in the
<M>J_4</M>-class <C>2A</C>.
<P/>
<Example><![CDATA[
gap> s:= info[1][1];
CharacterTable( "mx1j4" )
gap> cls:= SizesConjugacyClasses( s );;
gap> nsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),
> x -> Sum( cls{ x } ) = 2^11 );
[ [ 1 .. 3 ] ]
gap> cls{ nsg[1] };
[ 1, 276, 1771 ]
gap> GetFusionMap( s, t ){ nsg[1] };
[ 1, 3, 2 ]
]]></Example>
<P/>
The stabilizers of these involutions in <M>2^{11}:M_{24}</M> have index <M>1\,771</M>,
they have the structure <M>2^{11}:2^6:3.S_6</M>,
and they are contained in <M>2^{1+12}_+.3M_{22}:2</M> type subgroups;
so also <M>S</M>, which has index <M>10</M> in <M>2^{11}:2^6:3.S_6</M>,
is contained in <M>2^{1+12}_+.3M_{22}:2</M>.
(The corresponding subgroups of <M>M_{22}:2</M> are of course
the solvable groups of maximal order described in
Section <Ref Subsect="sect:EASY"/>.)
<P/>
In order to show that <M>G</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>L_5(2)</M> of index at most <M>361</M>
(see <Cite Key="CCN85" Where="p. 70"/>)
and
in <M>S_5 \times L_3(2)</M> of index at most <M>23</M>
(see <Cite Key="CCN85" Where="pp. 2, 3"/>).
<P/>
<Example><![CDATA[
gap> MaxSolv.( "J4" ):= n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:F3+">
<Heading><M>G = Fi_{24}^{\prime}</M></Heading>
The group <M>Fi_{24}^{\prime}</M> contains a unique conjugacy class of solvable
subgroups
of order <M>29\,386\,561\,536</M>, and no larger solvable subgroups.
<P/>
<Example><![CDATA[
gap> t:= CharacterTable( "Fi24'" );;
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> n:= 29386561536;;
gap> info:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;
gap> info:= Filtered( info, IsList );
[ [ CharacterTable( "Fi23" ), 139161244, 31671 ],
[ CharacterTable( "2.Fi22.2" ), 8787, 3510 ],
[ CharacterTable( "(3xO8+(3):3):2" ), 3033, 6 ],
[ CharacterTable( "O10-(2)" ), 851, 495 ],
[ CharacterTable( "3^(1+10):U5(2):2" ), 165, 165 ],
[ CharacterTable( "2^2.U6(2).3.2" ), 7, 6 ] ]
]]></Example>
<P/>
The maximal subgroups of the structure <M>3^{1+10}_+:U5(2):2</M> in
<M>Fi_{24}^{\prime}</M>
contain solvable subgroups <M>S</M> of order <M>n</M> and with the structure
<M>3^{1+10}_+:2^{1+6}_-:3^{1+2}_+:2S_4</M>,
see <Cite Key="CCN85" Where="p. 73, p. 207"/>.
<P/>
In order to show that <M>G</M> contains no other solvable subgroups of order
larger than or equal to <M>|S|</M>,
we check that there are no solvable subgroups
in <M>Fi_{23}</M> of order at least <M>n</M> (see Section <Ref Subsect="sect:Fi23"/>),
in <M>Fi_{22}.2</M> of order at least <M>n</M> (see Section <Ref Subsect="sect:Fi22"/>),
in <M>O_8^+(3).S_3</M> of index at most <M>3\,033</M>
(see <Cite Key="CCN85" Where="p. 140"/>),
in <M>O_{10}^-(2)</M> of index at most <M>851</M>
(see <Cite Key="CCN85" Where="p. 147"/>),
and
in <M>U_6(2).S_3</M> of index at most <M>7</M>
(see <Cite Key="CCN85" Where="p. 115"/>).
<P/>
The group <M>S</M> extends to a group of order <M>|S.2|</M>
in the automorphism group <M>Fi_{24}</M>.
<P/>
<Example><![CDATA[
gap> MaxSolv.( "Fi24'" ):= n;;
gap> MaxSolv.( "Fi24'.2" ):= 2 * n;;
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="sect:B">
<Heading><M>G = B</M></Heading>
The group <M>B</M> contains a unique conjugacy class of solvable subgroups
of order <M>n = 29\,686\,813\,949\,952</M>, and no larger solvable subgroups.
<P/>
The maximal subgroups of the structure <M>2^{2+10+20}(M_{22}:2 \times S_3)</M>
in <M>B</M>
contain solvable subgroups <M>S</M> of order <M>n</M> and with the structure
<M>2^{2+10+20}(2^4:3^2:D_8 \times S_3)</M>,
see <Cite Key="CCN85" Where="p. 217"/>
and Section <Ref Subsect="sect:EASY"/>.
<P/>
<Example><![CDATA[
gap> n:= 29686813949952;;
gap> n = 2^(2+10+20) * 2^4 * 3^2 * 8 * 6;
true
gap> n = 2^(2+10+20) * MaxSolv.( "M22.2" ) * 6;
true
]]></Example>
<P/>
By <Cite Key="Wil99" Where="Table 1"/>,
the only maximal subgroups of <M>B</M> of order bigger than <M>|S|</M> have
the following structures.
<P/>
<Table Align="llll">
<Row>
<Item><M>2.{}^2E_6(2).2</M></Item>
<Item><M>2^{1+22}.Co_2</M></Item>
<Item><M>Fi_{23}</M></Item>
<Item><M>2^{9+16}S_8(2)</M></Item>
</Row>
<Row>
<Item><M>Th</M></Item>
<Item><M>(2^2 \times F_4(2)):2</M></Item>
<Item><M>2^{2+10+20}(M_{22}:2 \times S_3)</M></Item>
<Item><M>2^{5+5+10+10}L_5(2)</M></Item>
</Row>
<Row>
<Item><M>S_3 \times Fi_{22}:2</M></Item>
<Item><M>2^{[35]}(S_5 \times L_3(2))</M></Item>
<Item><M>HN:2</M></Item>
<Item><M>O_8^+(3):S_4</M></Item>
</Row>
</Table>
<P/>
(The character tables of the maximal subgroups of <M>B</M> are meanwhile
available in &GAP;.)
<P/>
<Example><![CDATA[
gap> b:= CharacterTable( "B" );;
gap> mx:= List( Maxes( b ), CharacterTable );;
gap> Filtered( mx, x -> Size( x ) >= n );
[ CharacterTable( "2.2E6(2).2" ), CharacterTable( "2^(1+22).Co2" ),
CharacterTable( "Fi23" ), CharacterTable( "2^(9+16).S8(2)" ),
CharacterTable( "Th" ), CharacterTable( "(2^2xF4(2)):2" ),
CharacterTable( "2^(2+10+20).(M22.2xS3)" ),
CharacterTable( "[2^30].L5(2)" ), CharacterTable( "S3xFi22.2" ),
CharacterTable( "[2^35].(S5xL3(2))" ), CharacterTable( "HN.2" ),
CharacterTable( "O8+(3).S4" ) ]
]]></Example>
<P/>
For the subgroups <M>2^{1+22}.Co_2</M>, <M>Fi_{23}</M>, <M>Th</M>, <M>S_3 \times Fi_{22}:2</M>,
and <M>HN:2</M>,
the solvable subgroups of maximal order are known from the previous sections
or can be derived from known values, and are smaller than <M>n</M>.
<P/>
<Example><![CDATA[
gap> List( [ 2^(1+22) * MaxSolv.( "Co2" ),
> MaxSolv.( "Fi23" ),
> MaxSolv.( "Th" ),
> 6 * MaxSolv.( "Fi22.2" ),
> MaxSolv.( "HN.2" ) ], i -> Int( i / n ) );
[ 0, 0, 0, 0, 0 ]
]]></Example>
<P/>
If one of the remaining maximal groups <M>U</M> from the above list
has a solvable subgroup of order at least <M>n</M>
then the index of this subgroup in <M>U</M> is bounded as follows.
<P/>
<Example><![CDATA[
gap> List( [ Size( CharacterTable( "2.2E6(2).2" ) ),
> 2^(9+16) * Size( CharacterTable( "S8(2)" ) ),
> 2^3 * Size( CharacterTable( "F4(2)" ) ),
> 2^(2+10+20) * Size( CharacterTable( "M22.2" ) ) * 6,
> 2^30 * Size( CharacterTable( "L5(2)" ) ),
> 2^35 * Factorial(5) * Size( CharacterTable( "L3(2)" ) ),
> Size( CharacterTable( "O8+(3)" ) ) * 24 ],
> i -> Int( i / n ) );
[ 10311982931, 53550, 892, 770, 361, 23, 4 ]
]]></Example>
<P/>
The group <M>O_8^+(3):S_4</M> is nonsolvable, and its order is less than <M>5 n</M>,
thus its solvable subgroups have orders less than <M>n</M>.
<P/>
The largest solvable subgroup of <M>S_5 \times L_3(2)</M> has index <M>35</M>,
thus the solvable subgroups of <M>2^{[35]}(S_5 \times L_3(2))</M> have orders
less than <M>n</M>.
<P/>
The groups of type <M>2^{5+5+10+10}L_5(2)</M> cannot contain solvable subgroups
of order at least <M>n</M> because <M>L_5(2)</M> has no solvable subgroup of index
up to <M>361</M> –such a subgroup would be contained in <M>2^4:L_4(2)</M>,
of index at most <M>\lfloor 361/31 \rfloor = 11</M>
(see <Cite Key="CCN85" Where="p. 70"/>),
and <M>L_4(2) \cong A_8</M> does not have such subgroups
(see <Cite Key="CCN85" Where="p. 22"/>).
<P/>
The largest proper subgroup of <M>F_4(2)</M> has index <M>69\,615</M>
(see <Cite Key="CCN85" Where="p. 170"/>),
which excludes solvable subgroups of order
at least <M>n</M> in <M>(2^2 \times F_4(2)):2</M>.
<P/>
Ruling out the group <M>2.{}^2E_6(2).2</M> is more involved.
We consider the list of maximal subgroups of <M>{}^2E_6(2)</M>
in <Cite Key="CCN85" Where="p. 191"/>
(which is complete, see <Cite Key="BN95"/>),
--> --------------------
--> maximum size reached
--> --------------------
