# Subgroups of order 4 in groups of order 16

From Groupprops

This article contains summary information about all subgroups of order 4 inside groups of order 16. See also groups of order 16 | groups of order 4 | subgroup structure of groups of order 16 | supergroups of groups of order 4

This article describes the occurrence of groups of order 4 as subgroups inside groups of order 16.

There are two groups of order 4: cyclic group:Z4 and Klein four-group. There are 14 groups of order 16.

Note the following:

- A subgroup of order 4 in a group of order 16 is also a subgroup of index 4, because .
- A subgroup of order 4 in a group of order 16 need not be a normal subgroup. However, it has subnormal depth at most 2 so even if it is not normal, it is a 2-subnormal subgroup. If it is not normal, its normal core is a subgroup of order 2 and index 8, and its normal closure is a subgroup of order 8 and index 2.
- There are a few cases where there are multiple automorphism classes of isomorphic subgroups of order 4 in a given group of order 16.

## Numerical information on counts of subgroups

The table below presents information on counts of subgroups of order 4 in groups of order 16. Note the following:

General assertion | Implication for the counts in this case |
---|---|

congruence condition on number of subgroups of given prime power order | The number of subgroups of order 4 is odd. The number of normal subgroups of order 4 is odd. The number of 2-core-automorphism-invariant subgroups of order 4 is odd. |

Group | Second part of GAP ID | Hall-Senior number | Hall-Senior symbol | Nilpotency class | Minimum size of generating set | Number of subgroups of order 4 (must be odd) |
Number of normal subgroups of order 4 (must be odd) |
Number of 2-core-automorphism-invariant subgroups of order 4 (must be odd) |
Number of 2-automorphism-invariant subgroups of order 4 | Number of characteristic subgroups of order 4 |
---|---|---|---|---|---|---|---|---|---|---|

cyclic group:Z16 | 1 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

direct product of Z4 and Z4 | 2 | 3 | 1 | 2 | 7 | 7 | 1 | 1 | 1 | |

SmallGroup(16,3) | 3 | 9 | 2 | 2 | 11 | 3 | 1 | 1 | 1 | |

nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 2 | 2 | 7 | 3 | 1 | 1 | 1 | |

direct product of Z8 and Z2 | 5 | 4 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | |

M16 | 6 | 11 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | |

dihedral group:D16 | 7 | 12 | 3 | 2 | 5 | 1 | 1 | 1 | 1 | |

semidihedral group:SD16 | 8 | 13 | 3 | 2 | 5 | 1 | 1 | 1 | 1 | |

generalized quaternion group:Q16 | 9 | 14 | 3 | 2 | 5 | 1 | 1 | 1 | 1 | |

direct product of Z4 and V4 | 10 | 2 | 1 | 3 | 11 | 11 | 3 | 0 | 0 | |

direct product of D8 and Z2 | 11 | 6 | 2 | 3 | 15 | 7 | 1 | 1 | 1 | |

direct product of Q8 and Z2 | 12 | 7 | 2 | 3 | 7 | 7 | 1 | 1 | 1 | |

central product of D8 and Z4 | 13 | 8 | 2 | 3 | 7 | 7 | 7 | 1 | 1 | |

elementary abelian group:E16 | 14 | 1 | 1 | 4 | 35 | 35 | 35 | 0 | 0 |