# Subgroup structure of Klein four-group

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: Klein four-group.

View subgroup structure of particular groups | View other specific information about Klein four-group

We use here a Klein four-group with identity element and three non-identity elements all of order two.

We can realize this Klein four-group as , in which case we can set . For more, see element structure of Klein four-group.

## 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.

### Quick summary

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

Number of subgroups | 5 As elementary abelian group of prime-square order for prime : |

Number of conjugacy classes of subgroups | 5 (same as number of subgroups, because the group is an abelian group |

Number of automorphism classes of subgroups | 3 As elementary abelian group of order : |

Isomorphism classes of subgroups | trivia group (1 time), cyclic group:Z2 (3 times, all in the same automorphism class), Klein four-group (1 time). |

### Table classifying subgroups up to automorphism

Note that because abelian implies every subgroup is normal, all the subgroups are normal subgroups.

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 | 4 | 1 | 1 | 1 | Klein four-group | 1 | 0 | |

Z2 in V4 | cyclic group:Z2 | 2 | 2 | 3 | 1 | 3 | cyclic group:Z2 | 1 | 1 | |

whole group | Klein four-group | 4 | 1 | 1 | 1 | 1 | trivial group | 0 | 1 | |

Total (3 rows) | -- | -- | -- | -- | 5 | -- | 5 | -- | -- | -- |