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>
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>
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.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<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>.
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>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<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.
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.
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.
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>.
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>.
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>.
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"/>).
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>.
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>.
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"/>).
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.
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"/>).
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.
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"/>).
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"/>).
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>.
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"/>).
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.
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>.
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"/>.
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>.
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.
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"/>),
and compute the maximal index of a group of order <M>n/4</M>;
the possible subgroups of <M>{}^2E_6(2)</M> to consider are the following
The indices of the solvable groups of maximal orders in the groups
<M>U_6(2)</M>, <M>O_8^-(2)</M>, <M>F_4(2)</M>, <M>L_3(4)</M>, and <M>Fi_{22}</M>
are larger than the bounds we get for <M>n</M>,
see <Cite Key="CCN85" Where="pp. 115, 89, 170, 23, 163"/>.
<P/>
It remains to consider the subgroups of the type <M>2^{9+16}S_8(2)</M>.
The group <M>S_8(2)</M> contains maximal subgroups of the type
<M>2^{3+8}:(S_3 \times S_6)</M> and of index <M>5\,355</M>
(see <Cite Key="CCN85" Where="p. 123"/>),
which contain solvable subgroups <M>S'</M> of index <M>10</M>.
This yields solvable subgroups of order
<M>2^{9+16+3+8} \cdot 6 \cdot 72 = n</M>.
There are no other solvable subgroups of larger or equal order in <M>S_8(2)</M>:
We would need solvable subgroups of index at most <M>446</M> in <M>O_8^-(2):2</M>,
<M>393</M> in <M>O_8^+(2):2</M>, <M>210</M> in <M>S_6(2)</M>, or <M>23</M> in <M>A_8</M>,
which is not the case by <Cite Key="CCN85" Where="pp. 89, 85, 46, 22"/>.
So the <M>2^{9+16}S_8(2)</M> type subgroups of <M>B</M> yield
solvable subgroups <M>S'</M>
of the type <M>2^{9+16}.2^{3+8}:(S_3 \times 3^2:D_8)</M>,
and of order <M>n</M>.
<P/>
We want to show that <M>S'</M> is a <M>B</M>-conjugate of <M>S</M>.
For that, we first show the following:
<P/>
Lemma:
<P/>
The group <M>B</M> contains exactly two conjugacy classes of Klein four groups
whose involutions lie in the class <C>2B</C>.
(We will call these Klein four groups <C>2B</C>-pure.)
Their normalizers in <M>B</M> have the orders
<M>22\,858\,846\,741\,463\,040</M> and
<M>292\,229\,574\,819\,840</M>, respectively.
<P/>
<E>Proof.</E>
Let <M>V</M> be a <C>2B</C>-pure Klein four group in <M>B</M>, and set <M>N = N_B(V)</M>.
Let <M>x \in V</M> be an involution and set <M>H = C_B(x)</M>,
then <M>H</M> is maximal in <M>B</M> and has the structure <M>2^{1+22}.Co_2</M>.
The index of <M>C = C_B(V) = C_H(V)</M> in <M>N</M> divides <M>6</M>,
and <M>C</M> stabilizes the central involution in <M>H</M>
and another <C>2B</C> involution.
The group <M>H</M> contains exactly four conjugacy classes of
<C>2B</C> elements.
The <M>B</M>-classes of <C>2B</C>-pure Klein four groups arise from those of these
classes <M>y^H \subset H</M> such that <M>x \neq y</M> holds
and <M>x y</M> is a <C>2B</C> element.
We compute this subset.
The <M>B</M>-conjugacy class of <M>V</M> has length
<M>[B:N] = [B:H] \cdot [H:C] / [N:C]</M>, where <M>[N:C]</M> divides <M>6</M>.
We see that <M>[N:C] = 6</M> in both cases.
<P/>
<Example><![CDATA[
gap> cand:= List( pos, i -> Size( b ) / SizesCentralizers( h )[i] / 6 );
[ 181758140654146875, 14217525668946600000 ]
gap> Sum( cand ) = n0;
true
]]></Example>
<P/>
The orders of the normalizers of the two classes of
<C>2B</C>-pure Klein four groups are as claimed.
<P/>
<Example><![CDATA[
gap> List( cand, x -> Size( b ) / x );
[ 22858846741463040, 292229574819840 ]
]]></Example>
<P/>
The subgroup <M>S</M> of order <M>n</M> is contained in a maximal subgroup
<M>M</M> of the type <M>2^{2+10+20}(M_{22}:2 \times S_3)</M> in <M>B</M>.
The group <M>M</M> is the normalizer of a <C>2B</C>-pure Klein four group
in <M>B</M>,
and the other class of normalizers of <C>2B</C>-pure Klein four groups
does not contain subgroups of order <M>n</M>.
Thus the conjugates of <M>S</M> are uniquely determined by <M>|S|</M> and the
property that they normalize <C>2B</C>-pure Klein four groups.
<P/>
<Example><![CDATA[
gap> m:= mx[7];
CharacterTable( "2^(2+10+20).(M22.2xS3)" )
gap> Size( m );
22858846741463040
gap> nsg:= ClassPositionsOfMinimalNormalSubgroups( m );
[ [ 1, 2 ] ]
gap> SizesConjugacyClasses( m ){ nsg[1] };
[ 1, 3 ]
gap> GetFusionMap( m, b ){ nsg[1] };
[ 1, 3 ]
gap> List( cand, x -> Size( b ) / ( n * x ) );
[ 770, 315/32 ]
]]></Example>
<P/>
Now consider the subgroup <M>S'</M> of order <M>n</M>, which is contained in
a maximal subgroup of the type <M>2^{9+16}S_8(2)</M> in <M>B</M>.
In order to prove that <M>S'</M> is <M>B</M>-conjugate to <M>S</M>,
it is enough to show that <M>S'</M> normalizes a <C>2B</C>-pure
Klein four group.
<P/>
The unique minimal normal subgroup <M>V</M> of <M>2^{9+16}S_8(2)</M>
has order <M>2^8</M>.
Its involutions lie in the class <C>2B</C> of <M>B</M>.
<P/>
<Example><![CDATA[
gap> m:= mx[4];
CharacterTable( "2^(9+16).S8(2)" )
gap> nsg:= ClassPositionsOfMinimalNormalSubgroups( m );
[ [ 1, 2 ] ]
gap> SizesConjugacyClasses( m ){ nsg[1] };
[ 1, 255 ]
gap> GetFusionMap( m, b ){ nsg[1] };
[ 1, 3 ]
]]></Example>
<P/>
The group <M>V</M> is central in the normal subgroup <M>W = 2^{9+16}</M>,
since all nonidentity elements of <M>V</M> lie in one conjugacy class
of odd length.
As a module for <M>S_8(2)</M>,
<M>V</M> is the unique irreducible eight-dimensional module
in characteristic two.
Hence we are done if the restriction of the <M>S_8(2)</M>-action on <M>V</M>
to <M>S'/W</M> leaves a two-dimensional subspace of <M>V</M> invariant.
In fact we show that already the restriction of the <M>S_8(2)</M>-action
on <M>V</M> to the maximal subgroups of the structure
<M>2^{3+8}:(S_3 \times S_6)</M> has a two-dimensional submodule.
<P/>
These maximal subgroups have index <M>5\,355</M> in <M>S_8(2)</M>.
The primitive permutation representation of degree <M>5\,355</M> of
<M>S_8(2)</M>
and the irreducible eight-dimensional matrix representation of <M>S_8(2)</M>
over the field with two elements are available via the &GAP; package
<Package>AtlasRep</Package>, see <Cite Key="AtlasRep"/>.
We compute generators for an index <M>5\,355</M> subgroup in the matrix
group via an isomorphism to the permutation group.
<P/>
<Example><![CDATA[
gap> permg:= AtlasGroup( "S8(2)", NrMovedPoints, 5355 );
<permutation group of size 47377612800 with 2 generators>
gap> matg:= AtlasGroup( "S8(2)", Dimension, 8 );
<matrix group of size 47377612800 with 2 generators>
gap> hom:= GroupHomomorphismByImagesNC( matg, permg,
> GeneratorsOfGroup( matg ), GeneratorsOfGroup( permg ) );;
gap> max:= PreImages( hom, Stabilizer( permg, 1 ) );;
]]></Example>
<P/>
These generators define the action of the index <M>5\,355</M> subgroup
of <M>S_8(2)</M> on the eight-dimensional module.
We compute the dimensions of the factors of an ascending composition series
of this module.
<P/>
<Example><![CDATA[
gap> m:= GModuleByMats( GeneratorsOfGroup( max ), GF(2) );;
gap> comp:= MTX.CompositionFactors( m );;
gap> List( comp, r -> r.dimension );
[ 2, 4, 2 ]
]]></Example>
<!-- % Do the two classes have the same permutation character? --> <!-- % --> <!-- % n:= 29686813949952; --> <!-- % b:= CharacterTable( "B" ); --> <!-- % Maxes( b ); --> <!-- % s1:= CharacterTable( "2^(9+16).S8(2)" ); --> <!-- % s2:= CharacterTable( "2^(2+10+20).(M22.2xS3)" ); --> <!-- % Size( s1 ) / n; --> <!-- % # 53550 --> <!-- % Size( s2 ) / n; --> <!-- % # 770 --> <!-- % --> <!-- % ker2:= ClassPositionsOfSolvableRadical( s2 );; --> <!-- % f2:= s2 / ker2;; --> <!-- % pi2:= PermChars( f2, rec( torso:= [ 770 ] ) );; --> <!-- % List( pi2, ValuesOfClassFunction ); --> <!-- % Size( f2 ) / 770; --> <!-- % # 1152 --> <!-- % # -> only one of the five candidates comes from a subgroup: --> <!-- % # u:= CharacterTable( "M22.2" ); --> <!-- % # tom:= TableOfMarks( u ); --> <!-- % # PermCharsTom( u, tom ); --> <!-- % # Filtered( last, x -> x[1] = 770 ); --> <!-- % # -> and this subgroup is of course MOSS( M22.2 ) --> <!-- % pi2:= Filtered( pi2, x -> 34 in x );; --> <!-- % rest:= RestrictedClassFunctions( pi2, s2 );; --> <!-- % ind2:= InducedClassFunctions( rest, b ); --> <!-- % --> <!-- % ker1:= ClassPositionsOfSolvableRadical( s1 ); --> <!-- % f1:= s1 / ker1; --> <!-- % # PermChars( f1, rec( torso:= [ 53550 ] ) ); -> too slow ... --> <!-- % u:= CharacterTable( "2^(3+8):(S3xS6)" ); --> <!-- % ufusf1:= PossibleClassFusions( u, f1 );; --> <!-- % Length( last ); # 4 --> <!-- % upi:= PermChars( u, rec( torso:= [10]));; --> <!-- % Length( upi ); # 1 --> <!-- % ind:= List( ufusf1, map -> InducedClassFunctionsByFusionMap( u, f1, upi, map )[1] );; --> <!-- % Length( Set( ind ) ); # 1 --> <!-- % rest1:= RestrictedClassFunctions( Set( ind ), s1 );; --> <!-- % ind1:= InducedClassFunctions( rest1, b );; --> <!-- % ind1 = ind2; # true -->
The group <M>M</M> contains exactly two conjugacy classes of solvable
subgroups of order <M>n = 2\,849\,934\,139\,195\,392</M>,
and no larger solvable subgroups.
<P/>
The maximal subgroups of the structure <M>2^{1+24}_+.Co_1</M>
in the group <M>M</M>
contain solvable subgroups <M>S</M> of order <M>n</M> and with the structure
<M>2^{1+24}_+.2^{4+12}.(S_3 \times 3^{1+2}_+:D_8)</M>,
see <Cite Key="CCN85" Where="p. 234"/>
and Section <Ref Subsect="sect:Co1"/>.
The solvable subgroups of maximal order in groups of the types
<M>2^{2+11+22}.(M_{24} \times S_3)</M> and
<M>2^{[39]}.(L_3(2) \times 3S_6)</M> have order <M>n</M>.
For inspecting the other maximal subgroups of <M>M</M>,
we use the description from <Cite Key="NW12"/>,
which lists <M>44</M> classes of maximal subgroups of <M>G</M>,
and states that any possible other maximal subgroup of <M>G</M>
has socle isomorphic to one of
<M>L_2(13)</M>, <M>Sz(8)</M>, <M>U_3(4)</M>, <M>U_3(8)</M>;
so these maximal subgroups are isomorphic to subgroups of the automorphism
groups of these groups
– the maximum of these group orders is smaller than <M>n</M>,
hence we may ignore these possible subgroups.
The classes of maximal subgroups of <M>G</M> are classified in
<Cite Key="DLP25"/>.
As a consequence,
The result is that there are no maximal subgroups with socle
<M>Sz(8)</M> or <M>U_3(8)</M>,
there is one class of maximal subgroups of each of the isomorphism types
<M>L_2(13).2</M> and <M>U_3(4).4</M>,
and there are no subgroups o the type <M>L_2(59)</M>
(hence the subgroups of the type <M>59:29</M> are maximal).
<P/>
Thus only the following maximal subgroups of <M>M</M> have order
bigger than <M>|S|</M>.
For the subgroups <M>2.B</M>, <M>3.Fi_{24}</M>, <M>3^{1+12}_+.2Suz.2</M>,
<M>S_3 \times Th</M>, and <M>(D_{10} \times HN).2</M>,
the solvable subgroups of maximal order are smaller than <M>n</M>.
The subgroup <M>2^2.{}^2E_6(2):S_3</M> can be excluded by the fact that
this group is only six times larger than the subgroup <M>2.{}^2E_6(2):2</M>
of <M>B</M>,
but <M>n</M> is <M>96</M> times larger than the maximal solvable subgroup
in <M>B</M>.
<P/>
<Example><![CDATA[
gap> n / MaxSolv.( "B" );
96
]]></Example>
<P/>
The group <M>3^8.O_8^-(3).2_3</M> can be excluded by the fact that
a solvable subgroup of order at least <M>n</M> would imply the existence of
a solvable subgroup of index at most <M>46</M> in <M>O_8^-(3).2_3</M>,
which is not the case
(see <Cite Key="CCN85" Where="p. 141"/>).
Similarly, the existence of a solvable subgroup of order at least <M>n</M>
in <M>2^{5+10+20}.(S_3 \times L_5(2))</M> would imply the existence of a
solvable subgroup of index at most <M>723</M> in <M>L_5(2)</M>
and in turn of a solvable subgroup of index at most <M>23</M> in
<M>L_4(2)</M>, which is not the case
(see <Cite Key="CCN85" Where="p. 70"/>).
It remains to exclude the subgroup <M>2^{10+16}.O_{10}^+(2)</M>,
which means to show that <M>O_{10}^+(2)</M> does not contain
a solvable subgroup of index at most <M>553\,350</M>.
If such a subgroup would exist then it would be contained in
one of the following maximal subgroups of <M>O_{10}^+(2)</M>
(see <Cite Key="CCN85" Where="p. 146"/>):
in <M>S_8(2)</M> (of index at most <M>1\,115</M>),
in <M>2^8:O_8^+(2)</M> (of index at most <M>1\,050</M>),
in <M>2^{10}:L_5(2)</M> (of index at most <M>241</M>),
in <M>(3 \times O_8^-(2)):2</M> (of index at most <M>27</M>),
in <M>(2^{1+12}_+:(S_3 \times A_8)</M> (of index at most <M>23</M>),
or
in <M>2^{3+12}:(S_3 \times S_3 \times L_3(2))</M> (of index at most <M>4</M>).
By <Cite Key="CCN85" Where="pp. 123, 85, 70, 89, 22"/>,
this is not the case.
In order to prove the statement about the conjugacy of subgroups of order
<M>n</M> in <M>M</M>, we first show the following.
<P/>
Lemma:
<P/>
The group <M>M</M> contains exactly three conjugacy classes of
<C>2B</C>-pure Klein four groups.
Their normalizers in <M>M</M> have the orders
<M>50\,472\,333\,605\,150\,392\,320</M>,
<M>259\,759\,622\,062\,080</M>, and
<M>9\,567\,039\,651\,840</M>, respectively.
<P/>
<E>Proof.</E>
The idea is the same as for the Baby Monster group,
see Section <Ref Subsect="sect:B"/>.
Let <M>V</M> be a <C>2B</C>-pure Klein four group in <M>M</M>,
and set <M>N = N_M(V)</M>.
Let <M>x \in V</M> be an involution and set <M>H = C_M(x)</M>,
then <M>H</M> is maximal in <M>M</M> and has the structure
<M>2^{1+24}_+.Co_1</M>.
The index of <M>C = C_M(V) = C_H(V)</M> in <M>N</M> divides <M>6</M>,
and <M>C</M> stabilizes the central involution in <M>H</M>
and another <C>2B</C> involution.
<P/>
The group <M>H</M> contains exactly five conjugacy classes of
<C>2B</C> elements,
three of them consist of elements that generate a <C>2B</C>-pure
Klein four group together with <M>x</M>.
Next we compute the number <M>n_0</M> of <C>2B</C>-pure Klein four groups
in <M>M</M>.
<P/>
<Example><![CDATA[
gap> nr:= NrPolyhedralSubgroups( m, 3, 3, 3 );
rec( number := 87569110066985387357550925521828244921875, type := "V4" )
gap> n0:= nr.number;;
]]></Example>
<P/>
The <M>M</M>-conjugacy class of <M>V</M> has length
<M>[M:N] = [M:H] \cdot [H:C] / [N:C]</M>, where <M>[N:C]</M> divides
<M>6</M>.
We see that <M>[N:C] = 6</M> in both cases.
<P/>
<Example><![CDATA[
gap> cand:= List( pos, i -> Size( m ) / SizesCentralizers( h )[i] / 6 );
[ 16009115629875684006343550944921875,
3110635203347364905168577322802100000000,
84458458854522392576698341855475200000000 ]
gap> Sum( cand ) = n0;
true
]]></Example>
<P/>
The orders of the normalizers of the three classes of
<C>2B</C>-pure Klein four groups are as claimed.
<P/>
<Example><![CDATA[
gap> List( cand, x -> Size( m ) / x );
[ 50472333605150392320, 259759622062080, 9567039651840 ]
]]></Example>
<P/>
As we have seen above, the group <M>M</M> contains exactly the following
(solvable) subgroups of order <M>n</M>.
<P/>
<Enum>
<Item>
One class in <M>2^{1+24}_+.Co_1</M> type subgroups,
</Item>
<Item>
one class in <M>2^{2+11+22}.(M_{24} \times S_3)</M> type subgroups, and
</Item>
<Item>
two classes in <M>2^{[39]}.(L_3(2) \times 3S_6)</M> type subgroups.
</Item>
</Enum>
<P/>
Note that <M>2^{[39]}.(L_3(2) \times 3S_6)</M> contains an elementary
abelian normal subgroup of order eight whose involutions lie in the class
<C>2B</C>, see <Cite Key="CCN85" Where="p. 234"/>.
As a module for the group <M>L_3(2)</M>, this normal subgroup is irreducible,
and the restriction of the action to the two classes of <M>S_4</M> type subgroups
fixes a one- and a two-dimensional subspace, respectively.
Hence we have one class of subgroups of order <M>n</M> that centralize a <C>2B</C> element and one class of subgroups of order <M>n</M> that normalize a
<C>2B</C>-pure Klein four group.
Clearly the subgroups in the first class coincide with the subgroups
of order <M>n</M> in <M>2^{1+24}_+.Co_1</M> type subgroups.
By the above classification of <C>2B</C>-pure Klein four groups in <M>M</M>,
the subgroups in the second class coincide with the subgroups of order <M>n</M>
in <M>2^{2+11+22}.(M_{24} \times S_3)</M> type subgroups.
<P/>
It remains to show that the subgroups of order <M>n</M> do <E>not</E> stabilize
both a <C>2B</C> element <E>and</E> a <C>2B</C>-pure Klein four group.
We do this by direct computations with a <M>2^{2+11+22}.(M_{24} \times S_3)</M> type group, which is available via the <Package>AtlasRep</Package> package,
see <Cite Key="AtlasRep"/>.
<P/>
First we fetch the group, and factor out the largest solvable normal
subgroup, by suitable actions on blocks.
Now we compute an isomorphism from the factor group of type <M>M_{24}</M>
to the group that belongs to &GAP;'s table of marks.
Then we use the information from the table of marks to compute a
solvable subgroup of maximal order in <M>M_{24}</M> (which is <M>13\,824</M>),
and take the preimage under the isomorphism.
Finally, we take the preimage of this group in the original group.
<P/>
<Example><![CDATA[
gap> tom:= TableOfMarks( "M24" );;
gap> tomgroup:= UnderlyingGroup( tom );;
gap> iso:= IsomorphismGroups( act3, tomgroup );;
gap> pos:= Positions( OrdersTom( tom ), 13824 );
[ 1508 ]
gap> sub:= RepresentativeTom( tom, pos[1] );;
gap> pre:= PreImages( iso, sub );;
gap> pre:= PreImages( hom3, pre );;
gap> pre:= PreImages( hom2, pre );;
gap> pre:= PreImages( hom1, pre );;
gap> Size( pre ) = n;
true
]]></Example>
<P/>
The subgroups stabilizes a Klein four group.
It does not stabilize a <C>2B</C> element because its centre is trivial.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:corollary">
<Heading>Proof of the Corollary</Heading>
With the computations in the previous sections,
we have collected the information that is needed to show the corollary
stated in Section <Ref Sect="sect:result"/>.
<P/>
<Example><![CDATA[
gap> Filtered( Set( RecNames( MaxSolv ) ),
> x -> MaxSolv.( x )^2 >= Size( CharacterTable( x ) ) );
[ "Fi23", "J2", "J2.2", "M11", "M12", "M22.2" ]
]]></Example>
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