# Subgroup structure of alternating group:A4

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A4.

View subgroup structure of particular groups | View other specific information about alternating group:A4

The alternating group on is a group of order 12.

There is no subgroup of order . The alternating group of degree four is the group of smallest possible order (in this case ) *not* having subgroups of all orders dividing the group order.

## 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 | 10 |

Number of conjugacy classes of subgroups | 5 |

Number of automorphism classes of subgroups | 5 |

### Table classifying subgroups up to automorphism

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

trivial subgroup | trivial group | 1 | 12 | 1 | 1 | 1 | alternating group:A4 | 1 | trivial | |

subgroup generated by double transposition in A4 | , , | cyclic group:Z2 | 2 | 6 | 1 | 3 | 3 | -- | 2 | |

V4 in A4 | Klein four-group | 4 | 3 | 1 | 1 | 1 | cyclic group:Z3 | 1 | 2-Sylow, minimal normal, maximal | |

A3 in A4 | , , , | cyclic group:Z3 | 3 | 4 | 1 | 4 | 4 | -- | -- | 3-Sylow, maximal |

whole group | all elements | alternating group:A4 | 12 | 1 | 1 | 1 | 1 | trivial group | 1 | whole |

Total (5 rows) | -- | -- | -- | -- | 5 | -- | 10 | -- | -- | -- |

### 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 | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|---|---|

trivial group | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

cyclic group:Z2 | 2 | 1 | 3 | 1 | 1 | 0 | 0 |

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

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

alternating group:A4 | 12 | 3 | 1 | 1 | 1 | 1 | 1 |

Total | -- | -- | 10 | 5 | 5 | 3 | 3 |

### Table listing numbers of subgroups by group property

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

cyclic group | 8 | 3 | 3 | 1 | 1 |

abelian group | 9 | 4 | 4 | 2 | 2 |

nilpotent group | 9 | 4 | 4 | 2 | 2 |

solvable group | 10 | 5 | 5 | 3 | 3 |