So while I've known about the classification theorem of finite simple groups for a while, I saw the orders of the sporadic groups factored and noticed that all of them have very many small prime factors. Given the importance of (and rareness of large) "smooth numbers" in various aspects of number theory, this seems to be fairly significant!
After some searches on Stack Exchange, Overflow, and Google (not digging too deep), I couldn't find anything specifically addressing this phenomenon, and having seen the Sylow Theorems and things like Burnside's Theorem, it is clear that the prime factorization is closely related to a group having/not having normal subgroups in some way.
So I wanted to ask, "Is there a (big-picture) group-theoretic explanation for why the orders of sporadic groups have no large prime factors, and how (if at all) is this at all useful in the study of these groups or other places where these groups are found?"
But, of course, the classification of finite simple groups is a several-thousand page endeavor and this question is rather broad, so I might expect a satisfactory answer to either require hundreds of pages of mathematics, or be so soft that it may be frowned upon.
So I wanted to make this post on meta to see if this question would be received well if I asked it on main.