Let $G$ be a group. We know that the subgroup generated by a family of normal subgroups is again normal, hence for each subgroup $H$ of $G$, there is a maximal subgroup of $G$ contained in $H$ that is normal in $G$. I was interested in knowing if there is a notion for this normal subgroup. In other words, for a subset $S$ of $G$, what is called the subgroup generated by normal subgroups of $G$ contained in $S$?
This question arises naturally when one considers Galois correspondence: for a field extension $L/F/K$ with $L/K$ being Galois, what is the relationship between $\operatorname{Gal}(L/E)$ and $\operatorname{Gal}(L/F)$, where $E/K$ is the Galois closure of $F/K$ within $L$?
Thank you in advance.
