Let $G$ be a finite group. Consider the set $$X = \bigcup_{H \le G} G/H$$ which is a disjoint union of left cosets of subgroups $H$ of $G$. Then $G$ acts on $X$ by left multiplication, and the number $|X/G|$ of orbits is the number of subgroups of $G$. I want to apply Burnsides Lemma in this situation $$|X/G| = \frac{1}{|G|} \sum_{g \in G} |X^g|$$ where $X^g = \{ x \in X | g \cdot x = x\}$, to maybe get a "formula" for the number of subgroups of $G$. For this I need to "compute" $|X^g|$. Is there any other nice description of this quantity? Thanks for your help!

We have $|X^g| = |\{g'H \in X| g\cdot g' \cdot H = g' \cdot H\}|$, but how to proceed?

**Edit** The reason I suspect such a formula can be computed is the group $G=C_n$, for which we have:

$$\tau(n)=\frac{1}{n}\sum_{k=0}^{n-1}\sigma(\gcd(n,k))$$

Also, using the Lagarias inequality, one can show that an upper bound on $\tau(n)$ is equivalent to RH.

Related: https://math.stackexchange.com/questions/1315302/group-action-so-that-every-subgroup-is-a-stabilizer