Intersection of maximal subgroups in prosolvable groups

Let $H$ be an open subgroup of a profinite group that can be expressed as intersection of maximal subgroups of $G\mathrm{.}$ Given a positive real number $\eta ,$ we say that $H$ is an $\eta$-intersection if there exists a family of maximal subgroups $M_1,\dots ,M_t$ such that $H=M_1\cap \dots \cap M_t$ and $\mathrm{\mid }G:M_1\mathrm{\mid }\cdots \mathrm{\mid }G:M_t\mathrm{\mid }\le \mathrm{\mid }G:H{\mathrm{\mid }}^{\eta }$. We investigate the meaning of this property and its influence on the group structure.