# Subgroup structure of symmetric groups

## Contents

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  ? 554 554 3 3
10 3628800 symmetric group:S10  ? 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