Let $G$ be a group. Let $S \subset G$. Consider the partially ordered set of subgroups $K < G$ such that $k \cdot S = S$ for all $k \in K$.
If $K_{i}$ is a totally ordered set of subgroups of $G$ with this property, then $\bigcup_{i} K_{i}$ is also a subgroup with this property. Therefore, by Zorn's Lemma, the partially ordered set of subgroups of $G$ preserving $S$ by left-multiplication has a maximal element.
Furthermore, this maximal element is unique: if $M$ and $K$ are subgroups preserving $S$ by left-multiplication, then so is their join $<M, K>$.
What is this unique largest subgroup of $G$ preserving $S$ under left-multiplication called? (As for the plural used in the title, there is an analogous subgroup for right-multiplication.)