Frattini and related subgroups of mapping class groups

Let $\Gamma _{g,b}$ denote the orientation-preserving mapping class group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $\Phi _f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $\Phi _f(G)$ is nilpotent when $G$ is a finitely generated subgroup of $\Gamma _{g,b}$, in this paper we compute $\Phi _f(G)$ for certain subgroups of $\Gamma _{g,b}$. In particular, we answer Ivanov's question in the affirmative for these subgroups of $\Gamma _{g,b}$.