Suppose $G=S_n$ is permutation group in n letters and $7\leq n\leq 10$. Also consider subgroup $H_1,H_2$ such that $H_1$ is isomorphic with $S_k$ and $H_2$ is isomorphic with $S_{n-k}$ and every element of $H_1$ commute with every element of $H_2$. Is it true that $H_1,H_2$ are stabilizer of $n-k$ and $k$ letters respectively?

Suppose $G=S_n$ is the permutation group in $n$ letters and $7\leq n\leq 10$. Also consider the subgroups $H_1,H_2$ such that $H_1$ is isomorphic with $S_k$ and $H_2$ is isomorphic with $S_{n-k}$ and every element of $H_1$ commutes with every element of $H_2$. Is it true that $H_1,H_2$ are the stabilizers of $n-k$ and $k$ letters respectively?