# Quiz:Subgroup structure of special linear group:SL(2,3)

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See subgroup structure of special linear group:SL(2,3) for more information.

### Basic stuff

has order 24. Summary table on the structure of subgroups: [SHOW MORE]### Quick summary

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

number of subgroups | 15 |

number of conjugacy classes of subgroups | 7 |

number of automorphism classes of subgroups | 7 |

isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers | 2-Sylow: quaternion group (order 8) as Q8 in SL(2,3) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 1 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 4 |

Hall subgroups | The order has only two prime divisors, so no possibility of Hall subgroups other than trivial subgroup, whole group, and Sylow subgroups |

maximal subgroups | There are maximal subgroups of orders 6 (Z6 in SL(2,3)) and 8 (2-Sylow subgroup of special linear group:SL(2,3)) |

normal subgroups | There are two proper nontrivial normal subgroups: center of special linear group:SL(2,3) and 2-Sylow subgroup of special linear group:SL(2,3) |

### Table classifying subgroups up to automorphism

Note that, in the matrices, -1 can be written as 2 since elements are taken modulo 3.

Automorphism class of subgroups | Representative subgroup | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes (=1 iff automorph-conjugate subgroup) | Size of each conjugacy class (=1 iff normal subgroup) | Total number of subgroups (=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) | Note |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 24 | 1 | 1 | 1 | special linear group:SL(2,3) | 1 | trivial | |

center of special linear group:SL(2,3) | cyclic group:Z2 | 2 | 12 | 1 | 1 | 1 | alternating group:A4 | 1 | ||

Z4 in SL(2,3) | cyclic group:Z4 | 4 | 6 | 1 | 3 | 3 | -- | 2 | ||

2-Sylow subgroup of special linear group:SL(2,3) | quaternion group | 8 | 3 | 1 | 1 | 1 | cyclic group:Z3 | 1 | 2-Sylow | |

Z3 in SL(2,3) | cyclic group:Z3 | 3 | 8 | 1 | 4 | 4 | -- | -- | 3-Sylow | |

Z6 in SL(2,3) | cyclic group:Z6 | 6 | 4 | 1 | 4 | 4 | -- | -- | 3-Sylow normalizer | |

whole group | all elements | special linear group:SL(2,3) | 24 | 1 | 1 | 1 | 1 | trivial group | 0 | |

Total (7 rows) | -- | -- | -- | -- | 7 | -- | 15 | -- | -- | -- |