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Z. Arad, W. Herfort:
"Maximal cyclic subgroups and prime divisors in finite groups";
Archiv der Mathematik (invited), 85 (2005), 31 - 36.

Let $G$ be a finite group with no chief factor simple
of Lie type $E_8(q)$ and $C$ a cyclic
subgroup of largest order in $G$. It is shown that at most
two primes
in the open interval $([|C|/2],|C|)$ divide $|G|$.

http://aleph.ub.tuwien.ac.at/F?base=tuw01&func=find-c&ccl_term=AC05937632