Subgroups of $S_6$

What are all the subgroups of $S_6$ up to isomorphism? I have been able to find a few listings but not one I can be sure is complete.

Solution 1:

The following line requests that GAP compute all conjugacy classes of subgroups of $S_6$, take a representative of each, take its isomorphism class (ignoring duplicates), construct a platonic example of such a group, and then describe its structure.

gap> List( List( Set( List( ConjugacyClassesSubgroups(SymmetricGroup(6)),
> Representative), IdGroup), SmallGroup), StructureDescription);
[ "1", "C2", "C3", "C4", "C2 x C2", "C5", "S3", "C6", "C4 x C2", "D8", 
  "C2 x C2 x C2", "C3 x C3", "D10", "A4", "D12", "C2 x D8", "C3 x S3", 
  "(C3 x C3) : C2", "C5 : C4", "S4", "C2 x A4", "(C3 x C3) : C4", 
  "S3 x S3", "C2 x S4", "A5", "(S3 x S3) : C2", "S5", "A6", "S6" ]

The groups with ":" in their descriptions are not uniquely defined by their descriptions amongst all finite groups, but are amongst the subgroups of $S_6$.

"(C3 x C3) : C2" describes the wreath product $C_3 \wr S_2$, "C5 : C4" describes the Frobenius group of order 20 (the normalizer of a Sylow 5-subgroup), "(C3 x C3) : C4" describes $\langle (1,2,3),(3,6)(1,4,2,5) \rangle$, and "(S3 x S3) : C2" describes the wreath product $S_3 \wr S_2$.

C$n$ is the cyclic group of order $n$, D$n$ is the dihedral group of order $n$, S$n$ is the symmetric group on $n$ points, A$n$ is the alternating group on $n$ points.

Solution 2:

The GroupProps Subwiki is a decent resource for information about small groups (up to isomorphism) and their subgroups.

Drilling down a level we learn that $S_6$ is unique among the symmetric groups in having a different number of conjugacy classes (56) from automorphism classes (37) of subgroups. That is, there are subgroups of $S_6$ which are isomorphic under an outer automorphism of $S_6$ but not under an inner automorphism (conjugacy).

Unfortunately the latter Wiki page has a very incomplete "Table classifying subgroups up to conjugacy", listing only 4 of the 56 conjugacy classes. Of course equivalence up to isomorphism is even coarser than equivalence up to automorphism (of the whole group $S_6$). A paper at arxiv.org lists 29 such subgroups up to isomorphism (in an applied context):

$$ D_6, D_3, D_2, Z_6, Z_3, Z_2, D_3\times D_3, Z_3×Z_3, (Z_3\times Z_3)⋊ Z_2, D_3 \times Z_3, Z_2 \times Z_2 \times Z_2, S_3 ≀ Z_2, Z_4, Z_5, D_4, Z_2 \times Z_4, D_4, D_5, A_4, D_4 \times Z_2, Z_5 ⋊ Z_4, A_4 \times Z_2, S_4, (Z_3 \times Z_3) ⋊ Z_4, S_4 \times Z_2, A_5, S_5, A_6, S_6 $$

and references a PDF that lists by order/index the 56 conjugacy classes of $S_6$ subgroups from this Web site. There is an error in this list, as $D_4$ appears twice and the trivial group $Z_1$ is omitted.

For comparison with Jack Schmidt's Answer, the $Z_n$ here are the cyclic groups denoted $C_n$ by GAP. As Derek Holt points out, the notation here for dihedral group of order $2n$ is $D_n$, where GAP uses $D_{2n}$. There's a redundancy in that $D_3 \cong S_3$, yet here both are used.