Subgroup structure of 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.

Table classifying subgroups up to automorphism
Note that because abelian implies every subgroup is normal, all the subgroups are normal subgroups.