The Monster group has order $2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^2 · 13^3 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 \approx 8*10^{53}$, and the primes that occur in its prime factorization are the supersingular primes.
Is there any finite simple group larger than the Monster, with a prime factorization only consisting of supersingular primes? Note the first prime not in the Monster is $37$; the largest alternating group satisfying the condition is $A_{36}$, having order around $10^{41}$, not quite there.