Aryaman · I wasn't

Order 60

Let G be a group of order 60.
Then, G may or may not be simple.

For example, G=Z/60Z is an example of a non-simple group of order 60. This can be easily verified by considering the subgroup H={0,30}G. This is clearly normal as G is abelian.

On the other hand, consider G=A5, the alternating group of degree 5. It is known that An is simple for n5. Thus, G is simple and moreover, it has order 5!/2=60.

Interestingly, it can be shown that A5 is the only simple group (up to isomorphism) of order 60.