# Subgroup structure of cyclic group:Z4

From Groupprops

This article gives specific information, namely, subgroup structure, about a particular group, namely: cyclic group:Z4.

View subgroup structure of particular groups | View other specific information about cyclic group:Z4

### Table classifying subgroups up to automorphism

Note that since the group is a cyclic group, all subgroups are characteristic subgroups (see cyclic implies every subgroup is characteristic) and in particular normal subgroups.

Automorphism class of subgroups | List of subgroups (power notation, generator ) | List of subgroups ( notation) | 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 | cyclic group:Z4 | 1 | 0 | ||

Z2 in Z4 | cyclic group:Z2 | 2 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 1 | 1 | ||

whole group | cyclic group:Z4 | 4 | 1 | 1 | 1 | 1 | trivial group | 1 | 1 | ||

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