largest group that fulfills a certain equality Let $S_n$ be the symmetric group of degree $n$ and let $G,H$ be subgroups of $S_n$. Is there a way to find the largest group $G'$ such that $GH = G'H$? What can we say about $G'$?

In particular im interested in the case $H=S_{M_1}\times S_{M_2} \times ... \times S_{M_k} \subseteq S_n$ for $M_1 \sqcup M_2 \sqcup ... \sqcup M_k = \{1, 2, ..., n\}$.
The largest subgroup $G'$ of $S_n$ for which $GH=G'H$ is just $GH$. Indeed, the equation $GH=GHH$ trivially holds, and if $K$ is another subgroup of $S_n$ for which $GH=KH$, then $K \leq KH=GH$ because $e \in H$.

Actually, the above argument would work for any group in place of $S_n$.
The largest subgroup $G'$ of $S_n$ for which $GH=G'H$ is just $GH$. Indeed, the equation $GH=GHH$ trivially holds, and if $K$ is another subgroup of $S_n$ for which $GH=KH$, then $K \leq KH=GH$ because $e \in H$.

Actually, the above argument would work for any group in place of $S_n$.