  We generalize a theorem of Ogg on supersingular $j$-invariants to
supersingular elliptic curves with level. Ogg observed that the level one case
yields a characterization of the primes dividing the order of the monster. We
show that the corresponding analyses for higher levels give analogous
characterizations of the primes dividing the orders of other sporadic simple
groups (e.g., baby monster, Fischer's largest group). This situates Ogg's
theorem in a broader setting. More generally, we characterize, in terms of
supersingular elliptic curves with level, the primes arising as orders of
Fricke elements in centralizer subgroups of the monster. We also present a
connection between supersingular elliptic curves and umbral moonshine. Finally,
we present a procedure for explicitly computing invariants of supersingular
elliptic curves with level structure.
