Let $S$ be a finite set, $O(s)$ the orbit by $s\in S$ of the action of $\operatorname{Sym}(S)$ on $S$ $(\sigma \cdot s:=\sigma(s))$, and $Stab(s):=\lbrace \sigma \in \operatorname{Sym}(S) \mid \sigma(s)=s \rbrace$. By the Orbit-Stabilizer Theorem, we get:
$$Stab(t) \subseteq Stab(s) \Rightarrow |O(s)| \le |O(t)|$$
Conversely, can we state anything on $Stab(t)$ and $Stab(s)$ (inclusion/intersection), once we knew that $|O(s)| \le |O(t)|$?
In the situation you're describing, Sym$(S)$ acts transitively on $S$, i.e. $O(s) = S$ for any $s \in S$. So the orbit sizes are all the same, but all stabilisers are distinct index $n = |S|$ subgroups.
In more general situations, where we have some group $G$ acting on some set $X$, we can also not conclude anything from the orbit sizes alone. Of course there is the fact $|Stab(s)| \leq |Stab(t)| \iff |O(t)| \leq |O(s)|$, but one can find examples where $|O(s)| \leq |O(t)|$ where the stabilisers intersect trivially and where the stabiliser of $t$ is contained in the one of $s$.