# Non-normal subgroups of dihedral group:D8

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This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) dihedral group:D8 (see subgroup structure of dihedral group:D8).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

## Contents

## Definition

Suppose is the dihedral group of order eight (degree four) given by the presentation:

.

Then, we are interested in the following four subgroups:

.

and are conjugate subgroups (via , for instance). and are conjugate subgroups (via , for instance). and are not conjugate but are related by an outer automorphism that fixes and sends to . Thus, all four subgroups are automorphic subgroups. These are the only non-normal subgroups of and they are all 2-subnormal subgroups.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order of whole group | 8 | |

order of subgroup | 2 | |

index | 2 | |

size of conjugacy class | 2 | |

number of conjugacy classes in automorphism class | 2 | |

size of automorphism class | 2 |

## Effect of subgroup operators

Specific values (in the second column) are for .

Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|

normalizer | Klein four-subgroups of dihedral group:D8 | Klein four-group | |

centralizer | Klein four-subgroups of dihedral group:D8 | Klein four-group | |

normal core | -- | trivial group | |

normal closure | Kein four-subgroups of dihedral group:D8 | Klein four-group | |

characteristic core | -- | trivial group | |

characteristic closure | , i.e., | -- | dihedral group:D8 |

## Related subgroups

### Intermediate subgroups

We use here.

Value of intermediate subgroup (descriptive) | Isomorphism class of intermediate subgroup | Small subgroup in intermediate subgroup | Intermediate subgroup in big group |
---|---|---|---|

Klein four-group | Z2 in V4 | Klein four-subgroups of dihedral group:D8 |