# Subgroup structure of M16

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: M16.

View subgroup structure of particular groups | View other specific information about M16

To describe subgroups, we use the defining presentation given at the beginning:

.

## Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (group of prime power order)

Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so all subgroups have prime power orders)|order of quotient group divides order of group (and equals index of corresponding normal subgroup, so all quotients have prime power orders)

prime power order implies not centerless | prime power order implies nilpotent | prime power order implies center is normality-large

size of conjugacy class of subgroups divides index of center

congruence condition on number of subgroups of given prime power order: The total number of subgroups of any fixed prime power order is congruent to 1 mod the prime.

### 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(=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 | Nilpotency class |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 16 | 1 | 1 | 1 | M16 | 1 | 0 | |

derived subgroup of M16 | cyclic group:Z2 | 2 | 8 | 1 | 1 | 1 | direct product of Z4 and Z2 | 1 | 1 | |

other subgroups of order two | , | cyclic group:Z2 | 2 | 8 | 1 | 2 | 2 | -- | 2 | 1 |

center of M16 | cyclic group:Z4 | 4 | 4 | 1 | 1 | 1 | Klein four-group | 1 | 1 | |

other cyclic subgroup of order 4 | cyclic group:Z4 | 4 | 4 | 1 | 1 | 1 | cyclic group:Z4 | 1 | 1 | |

V4 in M16 | Klein four-group | 4 | 4 | 1 | 1 | 1 | cyclic group:Z4 | 1 | 1 | |

Z8 in M16 | |
cyclic group:Z8 | 8 | 2 | 2 | 1 | 2 | cyclic group:Z2 | 1 | 1 |

direct product of Z4 and Z2 in M16 | direct product of Z4 and Z2 | 8 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 1 | 1 | |

whole group | all elements | M16 | 16 | 1 | 1 | 1 | 1 | trivial group | 1 | 1 |

Total (9 rows) | -- | -- | -- | -- | 10 | -- | 11 | -- | -- | -- |

### Table classifying isomorphism types of subgroups

Group name | GAP ID | 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 | |

Cyclic group:Z2 | 3 | 2 | 2 | 1 | 1 | |

Cyclic group:Z4 | 2 | 2 | 2 | 2 | 2 | |

Klein four-group | 1 | 1 | 1 | 1 | 1 | |

Cyclic group:Z8 | 2 | 2 | 1 | 2 | 0 | |

Direct product of Z4 and Z2 | 1 | 1 | 1 | 1 | 1 | |

M16 | 1 | 1 | 1 | 1 | 1 | |

Total | -- | 11 | 10 | 9 | 9 | 7 |

### Table listing number of subgroups by order

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

1 | 1 | 1 | 1 | 1 | 1 |

2 | 3 | 2 | 2 | 1 | 1 |

4 | 3 | 3 | 3 | 3 | 3 |

8 | 3 | 3 | 2 | 3 | 1 |

16 | 1 | 1 | 1 | 1 | 1 |

Total | 11 | 10 | 9 | 9 | 7 |