  In a recent paper, Duncan, Griffin and Ono provide exact formulas for the
coefficients of McKay-Thompson series and use them to find asymptotic
expressions for the distribution of irreducible representations in the
moonshine module $V^\natural = \bigoplus_n V_n^\natural$. Their results show
that as $n$ tends to infinity, $V_n^\natural$ is dominated by direct sums of
copies of the regular representation. That is, if we view $V_n^\natural$ as a
module over the group ring $\mathbb{Z}[\mathbb{M}]$, the free-part dominates. A
natural problem, posed at the end of the aforementioned paper, is to
characterize the distribution of irreducible representations in the non-free
part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson
series to answer this question. We arrive at an ordering of the series by the
magnitude of their coefficients, which corresponds to various contributions to
the distribution. In particular, we show how the asymptotic distribution of the
non-free part is dictated by the column for conjugacy class 2A in the monster's
character table. We find analogous results for the other monster modules
$V^{(-m)}$ and $W^\natural$ studied by Duncan, Griffin, and Ono.
