We will say that two subgroups $H,K\subset S_n$ of the symmetric group are a commuting pair if $hk=kh$ for all $h\in H, k\in K.$ For example, $S_m$ and $S_{m'}$ can be made into a commuting pair in $S_{m+m'}.$ In fact it is a "maximal" commuting pair, in the sense that neither $S_m$ nor $S_{m'}$ can be enlarged.
Question: is every maximal commuting pair of this form? More precisely, can every commuting pair $(H,K)$ in $S_n$ be enlarged to a commuting pair $(S_m,S_{m'})$ in $S_n$, where $S_m$ and $S_{m'}$ are symmetric groups on disjoint subsets of $\{1,...,n\}$ of cardinality $m$ and $m'$ respectively?
EDIT: For my question above assume that $H\cap K=\{e\}$ (since otherwise, the answer is obviously No -- just take $K=H$ an abelian subgroup). More generally, can all maximal pairs $(H,K)$ be characterized (not assuming $H\cap K=\{e\}$)?
EDIT: Captain Lamma answered my question and I intend to accept to his answer, but I would like to wait a bit first and see if anyone might be able to provide something towards a potential classification of maximal commuting pairs.
The answer is no, because you can have subgroups with intersect trivially and commute, but do not act on disjoint subsets.
For instance, in $S_4$, take $H=\{id, (12)(34)\}$ and $K=\{id, (13)(24)\}$.