# 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

TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted lexicographically based on the powers of 3 and 2 in the order. They arenotsorted by order. To sort by order, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.

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 | ||

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 | -- | -- | ||

D8 in S4 | dihedral group:D8 | 8 | 3 | 1 | 3 | 3 | -- | -- | 2-Sylow, fusion system is non-inner non-simple fusion system for dihedral group:D8 | |

A3 in S4 | cyclic group:Z3 | 3 | 8 | 1 | 4 | 4 | -- | -- | 3-Sylow, fusion system is non-inner fusion system for cyclic group:Z3 | |

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

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. |