# How to prove that a group is simple only from its class equation

Can someone give me a hint/solution how to prove that a group is simple if its class equation is $60=1+15+20+12+12$ ?

-

A normal subgroup is a union of conjugacy classes. And of course any subgroup contains the identity, and has order dividing the order of the group. So it's enough to consider all possible submultisets of $\{1,15,20,12,12\}$ containing $1$ and check that none (apart from the trivial ones) is a factor of $60$.