# Groups of order 40

From Groupprops

This article gives information about, and links to more details on, groups of order 40

See pages on algebraic structures of order 40| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 40. The prime factorization of 40 is .

## Statistics at a glance

Quantity | Value |
---|---|

Total number of groups | 14 |

Number of abelian groups | 3 |

Number of nilpotent groups | 5 |

Number of solvable groups | 14 |

Number of simple groups | 0 |

## The list

There are 14 groups of order 40:

Group | Second part of GAP ID (GAP ID is (40,second part)) | Abelian? | Nilpotent? |
---|---|---|---|

semidirect product of Z5 and Z8 via inverse map | 1 | No | No |

cyclic group:Z40 | 2 | Yes | Yes |

semidirect product of Z5 and Z8 via square map | 3 | No | No |

nontrivial semidirect product of Z5 and Q8 | 4 | No | No |

direct product of D10 and Z4 | 5 | No | No |

dihedral group:D40 | 6 | No | No |

? | 7 | No | No |

? | 8 | No | No |

direct product of Z20 and Z2 (also direct product of Z10 and Z4) | 9 | Yes | Yes |

direct product of D8 and Z5 | 10 | No | Yes |

direct product of Q8 and Z5 | 11 | No | Yes |

direct product of GA(1,5) and Z2 | 12 | No | No |

direct product of D10 and V4 | 13 | No | No |

direct product of E8 and Z5 | 14 | Yes | Yes |

## Sylow subgroups

### 5-Sylow subgroups

Combining the congruence condition on Sylow numbers and the divisibility condition on Sylow numbers, we see that the number of 5-Sylow subgroups must be congruent to 1 modulo 5 and also must be a divisor of 8. The only possibility for both of these to hold simultaneously is that there is exactly one 5-Sylow subgroup, and hence it is a normal Sylow subgroup and the 2-Sylow subgroups are its permutable complements. In particular, this means that the whole group is a semidirect product with normal subgroup equal to the 5-Sylow subgroup and quotient/complement of order 8.