  We consider the situation in which a finite group acts on an
infinite-dimensional graded module in such a way that the graded trace
functions are weakly holomorphic modular forms. Under a mild hypothesis we
completely describe the asymptotic module structure of the homogeneous
subspaces. As a consequence we find that moonshine for a group gives rise to
partial orderings on its irreducible representations. This serves as a first
answer to a question posed by Griess. In particular, we show that our
hypothesis holds for umbral moonshine and for automorphism groups of certain
vertex operator algebras.
