# Subgroup structure of symmetric group:S4

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This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S4.

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The symmetric group of degree four has many subgroups.

Note that since is a complete group, every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.

## Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)

Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)

Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order

Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugateMINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

### Quick summary

Item | Value |
---|---|

Number of subgroups | 30 |

Number of conjugacy classes of subgroups | 11 |

Number of automorphism classes of subgroups | 11 |

Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems | 2-Sylow: dihedral group:D8 (order 8), Sylow number is 3, fusion system is non-inner non-simple fusion system for dihedral group:D8 3-Sylow: cyclic group:Z3, Sylow number is 4, fusion system is non-inner fusion system for cyclic group:Z3 |

Hall subgroups | Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups |

maximal subgroups | maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4). |

normal subgroups | There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4. |

### Table classifying subgroups up to automorphisms

Automorphism class of subgroups | Representative | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes | Size of each conjugacy class | Number of subgroups | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) | Note |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 24 | 1 | 1 | 1 | symmetric group:S4 | 1 | ||

S2 in S4 | cyclic group:Z2 | 2 | 12 | 1 | 6 | 6 | -- | -- | ||

subgroup generated by double transposition in S4 | cyclic group:Z2 | 2 | 12 | 1 | 3 | 3 | -- | 2 | ||

A3 in S4 | cyclic group:Z3 | 3 | 8 | 1 | 4 | 4 | -- | -- | 3-Sylow | |

Z4 in S4 | cyclic group:Z4 | 4 | 6 | 1 | 3 | 3 | -- | -- | ||

normal Klein four-subgroup of S4 | |
Klein four-group | 4 | 6 | 1 | 1 | 1 | symmetric group:S3 | 1 | 2-core |

non-normal Klein four-subgroups of S4 | Klein four-group | 4 | 6 | 1 | 3 | 3 | -- | -- | ||

S3 in S4 | symmetric group:S3 | 6 | 4 | 1 | 4 | 4 | -- | -- | ||

D8 in S4 | dihedral group:D8 | 8 | 3 | 1 | 3 | 3 | -- | -- | 2-Sylow | |

A4 in S4 | alternating group:A4 | 12 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 1 | ||

whole group | symmetric group:S4 | 24 | 1 | 1 | 1 | 1 | trivial group | 0 | ||

Total (11 rows) | -- | -- | -- | -- | 11 | -- | 30 | -- | -- | -- |

### Table classifying isomorphism types of subgroups

Group name | Order | Second part of GAP ID (first part is order) | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|---|

Trivial group | 1 | 1 | 1 | 1 | 1 | 1 |

Cyclic group:Z2 | 2 | 1 | 9 | 2 | 0 | 0 |

Cyclic group:Z3 | 3 | 1 | 4 | 1 | 0 | 0 |

Cyclic group:Z4 | 4 | 1 | 3 | 1 | 0 | 0 |

Klein four-group | 4 | 2 | 4 | 2 | 1 | 1 |

Symmetric group:S3 | 6 | 1 | 4 | 1 | 0 | 0 |

Dihedral group:D8 | 8 | 3 | 3 | 1 | 0 | 0 |

Alternating group:A4 | 12 | 3 | 1 | 1 | 1 | 1 |

Symmetric group:S4 | 24 | 12 | 1 | 1 | 1 | 1 |

Total | -- | -- | 30 | 11 | 4 | 4 |

### Table listing number of subgroups by order

These numbers satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order for a fixed nonnegative integer is congruent to 1 mod . For , this means the number is odd, and for , this means the number is congruent to 1 mod 3 (so it is among 1,4,7,...)

Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

1 | 1 | 1 | 1 | 1 |

2 | 9 | 2 | 0 | 0 |

3 | 4 | 1 | 0 | 0 |

4 | 7 | 3 | 1 | 1 |

6 | 4 | 1 | 0 | 0 |

8 | 3 | 1 | 0 | 0 |

12 | 1 | 1 | 1 | 1 |

24 | 1 | 1 | 1 | 1 |

Total | 30 | 11 | 4 | 4 |

### Table listing numbers of subgroups by group property

Group property | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

Cyclic group | 17 | 5 | 1 | 1 |

Abelian group | 21 | 7 | 2 | 2 |

Nilpotent group | 24 | 8 | 2 | 2 |

Solvable group | 30 | 11 | 4 | 4 |

### Table listing numbers of subgroups by subgroup property

Subgroup property | Occurences as subgroup | Conjugacy classes of occurrences as subgroup | Automorphism classes of occurrences as subgroup |
---|---|---|---|

Subgroup | 30 | 11 | 11 |

Normal subgroup | 4 | 4 | 4 |

Characteristic subgroup | 4 | 4 | 4 |

## Subgroup structure viewed as symmetric group

### Classification based on partition given by orbit sizes

For any subgroup of , the natural action on induces a partition of the set into orbits, which in turn induces an unordered integer partition of the number 4. Below, we classify this information for the subgroups.

Conjugacy class of subgroups | Size of conjugacy class | Induced partition of 4 | Direct product of transitive subgroups on each orbit? | Illustration with representative |
---|---|---|---|---|

trivial subgroup | 1 | 1 + 1 + 1 + 1 | Yes | The subgroup fixes each point, so the orbits are singleton subsets. |

S2 in S4 | 6 | 2 + 1 + 1 | Yes | has orbits |

subgroup generated by double transposition in S4 | 3 | 2 + 2 | No | has orbits |

A3 in S4 | 4 | 3 + 1 | Yes | has orbits |

Z4 in S4 | 3 | 4 | Yes | The action is a transitive group action, so only one orbit. |

normal Klein four-subgroup of S4 | 1 | 4 | Yes | The action is a transitive group action, so only one orbit. |

non-normal Klein four-subgroups of S4 | 3 | 2 + 2 | Yes | has orbits |

S3 in S4 | 4 | 3 + 1 | Yes | has orbits |

D8 in S4 | 3 | 4 | Yes | The action is a transitive group action, so only one orbit. |

A4 in S4 | 1 | 4 | Yes | The action is a transitive group action, so only one orbit. |

whole group | 1 | 4 | Yes | The action is a transitive group action, so only one orbit. |