A5 is the alternating group on 5 elements. Since the order of S5 is 120, the order of A5 is half the size at 60. Here is the Simplified Hasse Diagram of the structure of the subgroups.
The red circles indicate the normal subgroups, which in this case are just the whole group and the identity. having just two normal subgroups means this is a simple group.
Red arrows can only originate in normal subgroups, and in this case, the identity is a normal subgroup of every subgroup of prime order.
There are 15 subgroups of order 2, all of the generators of the form (ab)(cd).
There are 10 subgroups of order 3, all of generators of the form (abc).
The 6 subgroups of order 5 have generators of the form (abcde).
For the Klein-4 subgroups, we have the identity and three double transpositions that all keep the same element fixed. For example, (12)(34), (13)(24) and (14)(23) all leave 5 untouched. Likewise, the five copies of A4 all leave one element fixed.
The subgroups of order 5, 6 and 10 do not leave any element fixed, so they are not subgroups of A4. Of those three orders, only 6 divides 12, but the groups isomorphic to S3 are generated by a 3-cycle (abc) and a double transposition (ab)(de), so no element is fixed.
Every purple arrow goes from a subgroup of order n to a subgroup of order 2n. These subgroups are not normal, which is indicated by the blue rectangles, nor are they subnormal, since subnormality only occurs if a subgroup is part of a chain of subgroups, each normal in the next subgroup up in the chain and terminating in the entire group. A simple group that has any proper subgroup that is not the identity cannot contain a subnormal subgroup, so these subgroups that have purple arrows coming out of them are normalish, the phrase I am using until I learn the proper term.
The sum of all the numbers in red circles and blue rectangles is 59, the total number of subgroups.
The next Simplified Hasse diagram will be of S5, which will be a much larger undertaking with 156 total subgroups.
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