Subgroup structure of 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 $a,b,c$ all of order two.

We can realize this Klein four-group as $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, in which case we can set $a = (1,0), b = (0,1), c = (1,1), e = (0,0)$. For more, see element structure of Klein four-group.

Tables for quick information

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 $\{ e \}$ trivial group 1 4 1 1 1 Klein four-group 1 0
Z2 in V4 $\{ e, a\}, \{ e,b \}, \{ e,c \}$ cyclic group:Z2 2 2 3 1 3 cyclic group:Z2 1 1
whole group $\{ e,a,b,c\}$ Klein four-group 4 1 1 1 1 trivial group 0 1
Total (3 rows) -- -- -- -- 5 -- 5 -- -- --