Can you please list all of the $H$ such that $H<A_5$?
I know that $A_4$ has following subgroups:
-one trivial subgroup (every group has it)
-the full $A_4$
-the normal Klein four group
-four cyclic subgroups of order $3$
-three cyclic subgroups of order $2$.
But $A_5$ clearly has a more complicated subgroup structure...
There are:
-one trivial subgroup
-$15$ cyclic subgroups of order $2$ generated by double transpositions
-$5$ Klein four groups generated by pairs of double transpositions, $10$ cyclic subgroups of order $3$ generated by $3$-cycles
-$10$ subgroups isomorphic to $S_3$ generated by a $3$-cycle and a double transposition
-$5$ subgroups isomorphic to $A_4$
-$6$ subgroups isomorphic to $C_5$
-$6$ subgroups isomorphic to $D_5$
-the whole group
Total $59$ subgroups.