  Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class
group of a surface of genus $g$ with $n$ boundary components. We prove that
${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up
to conjugation, a unique subgroup of index $2^{g-1}(2^{g}+1)$ up to
conjugation, and the other proper subgroups of ${\mathcal{M}}_{g,n}$ are of
index greater than $2^{g-1}(2^{g}+1)$. In particular, the minimum index for a
proper subgroup of ${\mathcal{M}}_{g,n}$ is $2^{g-1}(2^{g}-1)$.
