# Subgroup structure of symmetric groups

From Groupprops

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 is the symmetric group on a set of size . For convenience, we consider the set to be .

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

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

(order of symmetric group) | symmetric group of degree | 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

(order of symmetric group) | Symmetric group of degree | 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 for small values of . Note that must divide the order of any transitive subgroup of .

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.*

Symmetric group | Partition of 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 |