Let $S_n$ denote the symmetric group on $\lbrace 1,2,\ldots ,n\rbrace$, for $n\geq 2$, and let
$$ \begin{align} X&= \lbrace \sigma \in S_n \mid \sigma(1)=1 \Leftrightarrow \sigma(2)=2\rbrace \\ &= \lbrace \sigma \in S_n \mid\sigma(1)=1, \sigma(2)=2\rbrace \cup \lbrace \sigma \in S_n \ | \ \sigma(1) \neq 1, \sigma(2) \neq 2\rbrace \end{align}$$
Denote by $sg(X)$ the sets of all subgroups of $S_n$ contained in $X$.
Question. What are the maximal elements of $sg(X)$ for inclusion ?
Obviously, the stabilizer subgroup $M_1=\lbrace \sigma \in S_n \mid \sigma(\lbrace 1,2 \rbrace)= \lbrace 1,2 \rbrace \rbrace$ is one such maximal element. But (at least for large enough $n$) there are others : if $\tau$ is any product of two or more disjoint cycles of the same length, with one cycle containing $1$ and another containing $2$, then $\langle\tau\rangle \in sg(X)$, but $\tau\not\in M_1$, so any maximal element containing $\langle \tau\rangle $ will be different from $M_1$.