List of groups with specific divisors

Question

I want to find list of finite simple nonabelian groups which their orders divisor lies in specific set of primes for example the set $\{2,3,5,7,11\}$.

Is there a method to do this? Would Zsigmondy's theorem help?

Answer 1

The ATLAS of finite groups is a large, red book, and was written by John Conway, Robert Curtis, Simon Norton, Richard Parker, and Robert Wilson. It lists information about $93$ finite simple groups, including their orders.

All the sporadic simple groups are covered in the ATLAS. Simple groups which are not covered are the large ones which are part of a family, so the familial (that's a nice word!) information can be used to find the order of such groups (for example, $A_{n}$ for $n\geq 11$ contains all your listed prime divisors).

There is an online continuation of the ATLAS, which can be found here.