A4 in 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:

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

$$H$$ is the alternating group on the set $$\{ 1,2,3,4 \}$$ fixing the point 5.

$$H$$ has five conjugate subgroups (including the subgroup itself) in $$G$$, based on the choice of fixed point:

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

Intermediate subgroups
There are no intermediate subgroups, since the subgroup is a maximal subgroup.