Klein four-subgroup of alternating group:A5

Let $$G$$ be the alternating group:A5, i.e., the alternating group (the group of even permutations) on the set $$\{ 1,2,3,4,5 \}$$. $$G$$ has order $$5!/2 = 60$$.

Consider the subgroup:

$$H = H_5 := \{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$

This subgroup is isomorphic to the Klein four-group. There are five conjugates (including the subgroup itself) depending on which of the five points $$1,2,3,4,5$$ is fixed:

Effect of subgroup operators
In the table below, we provide values specific to $$H$$.

Sylow and corollaries
The subgroup is a 2-Sylow subgroup, so many properties follow as a corollary of that.