# Subgroup structure of nontrivial semidirect product of Z4 and Z4

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: nontrivial semidirect product of Z4 and Z4.

View subgroup structure of particular groups | View other specific information about nontrivial semidirect product of Z4 and Z4

This article describes the subgroup structure of nontrivial semidirect product of Z4 and Z4, given explicitly by the following presentation, where denotes the identity element:

The elements are:

## 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 automorphisms

In case a single equivalence class of subgroups under automorphisms comprises multiple conjugacy classes of subgroups, outer curly braces are used to bucket the conjugacy classes.