Results from the character theory of symmetric groups are used to obtain an asymptotic estimate for the subgroup growth of fundamental groups of closed 2-manifolds. The main result implies an affirmative answer, for the class of groups investigated, to a question of Lubotzky's concerning the relationship between the subgroup growth of a one-relator group and that of a free group of appropriately chosen rank. As byproducts, an interesting statistical property of commutators in symmetric groups and the fact that in a ‘large’ surface group almost all finite index subgroups are maximal are obtained, among other things. The approach requires an asymptotic estimate for the sum $\Sigma 1/(\chi_\lambda(1))^s$ taken over all partitions $\lambda$ of $n$ with fixed $s \ge 1$, which is also established.