Classification of finite simple groups whose order has at most five prime factors counting multiplicities

Statement
Suppose $$G$$ is a finite simple non-abelian group (i.e., a finite simple group that is not cyclic of prime order) with the property that the sum of exponents on the prime divisors of the order of $$G$$ is at most $$5$$. Then, $$G$$ must be a fact about::projective special linear group of degree two over a field of size 5, 7, 11, or 13, i.e,. $$G$$ is one of the groups $$PSL(2,5)$$, $$PSL(2,7)$$, $$PSL(2,11)$$, and $$PSL(2,13)$$. In other words, the only possibilities for $$G$$ up to isomorphism are as follows:

Related facts

 * Order has only two prime factors implies solvable
 * Neumann's open problem on number of projective special linear groups whose order has exactly six prime factors counting multiplicities