Subgroup structure of symmetric groups

This article gives specific information, namely, subgroup structure, about a family of groups, namely: symmetric group.
View subgroup structure of group families | View other specific information about symmetric group

The symmetric group on a set is the group, under multiplication, of permutations of that set. The symmetric group of degree $n$ is the symmetric group on a set of size $n$ . For convenience, we consider the set to be $\{1,2,\dots ,n\}$ .

This article discusses the element structure of the symmetric group of degree $n$ .

Here are links to more detailed information for small values of degree $n$ .

$n$	$n!$ (order of symmetric group)	symmetric group of degree $n$	subgroup structure page
0	1	trivial group	--
1	1	trivial group	--
2	2	cyclic group:Z2	--
3	6	symmetric group:S3	subgroup structure of symmetric group:S3
4	24	symmetric group:S4	subgroup structure of symmetric group:S4
5	120	symmetric group:S5	subgroup structure of symmetric group:S5
6	720	symmetric group:S6	subgroup structure of symmetric group:S6
7	5040	symmetric group:S7	subgroup structure of symmetric group:S7
8	40320	symmetric group:S8	subgroup structure of symmetric group:S8

Key statistics

$n$	$n!$ (order of symmetric group)	Symmetric group of degree $n$	Number of subgroups	Number of conjugacy classes of subgroups	Number of automorphism classes of subgroups	Number of normal subgroups	Number of characteristic subgroups
1	1	trivial group	1	1	1	1	1
2	2	cyclic group:Z2	2	2	2	2	2
3	6	symmetric group:S3	6	4	4	3	3
4	24	symmetric group:S4	30	11	11	4	4
5	120	symmetric group:S5	156	19	19	3	3
6	720	symmetric group:S6	1455	56	37	3	3
7	5040	symmetric group:S7	11300	96	96	3	3
8	40320	symmetric group:S8	151221	296	296	3	3
9	362880	symmetric group:S9	1694723	554	554	3	3
10	3628800	symmetric group:S10	29594446	1593	1593	3	3

Subgroups by orbit

Transitive subgroups

We first list key statistics on the transitive subgroups of $S_{n}$ for small values of $n$ . Note that $n$ must divide the order of any transitive subgroup of $S_{n}$ .

$n$	Symmetric group	Number of transitive subgroups	Number of conjugacy classes of transitive subgroups	List of conjugacy classes of transitive subgroups
1	trivial group	1	1	the whole group
2	cyclic group:Z2	1	1	the whole group
3	symmetric group:S3	2	2	the whole group, A3 in S3
4	symmetric group:S4	9	5	the whole group, Z4 in S4, normal Klein four-subgroup of symmetric group:S4, D8 in S4, and A4 in S4
5	symmetric group:S5	?	?	?

All subgroups in terms of partition of orbit sizes

For each partition into orbit sizes, the subgroups giving rise to such a partition are subdirect products of the transitive subgroups corresponding to the orbit sizes. The table below needs to be completed.

$n$	Symmetric group	Partition of $n$ given by orbit sizes	Number of subgroups	Number of conjugacy classes of subgroups	List of conjugacy classes of subgroups
1	trivial group	1	1	1	the whole group
2	cyclic group:Z2	2	1	1	the whole group
2	cyclic group:Z2	1 + 1	1	1	trivial subgroup
3	symmetric group:S3	3	2	2	the whole group, A3 in S3
3	symmetric group:S3	2 + 1	3	1	S2 in S3
3	symmetric group:S3	1 + 1 + 1	1	1	trivial subgroup
4	symmetric group:S4	4	9	5	the whole group, Z4 in S4, normal Klein four-subgroup of smymetric group:S4, D8 in S4, and A4 in S4

External links

Blog post on number of (conjugacy classes of) subgroups of symmetric groups