**Definition** : Let $\sim$ be the equivalence relation on inclusions of finite groups, **generated** by :

$(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and with $ker(\phi) \subset H$.

(See the appendix below)

Let $(H \subset G)$ be a **maximal** inclusion of finite groups:

Question: Is there an inclusion of finite groups $(A \subset B)$ having one and only one non-trivial intermediate subgroup $(A \subset P \subset B)$, and with $(A \subset P) \sim (P \subset B) \sim (H \subset G)$ ?

(Unicity of $(A \subset B)$ up to $\sim$ ?)

**Remark**: obviously, if $H$ is a normal subgroup of $G$ then $(H \subset G) \sim (\{ 1 \} \subset G/H)$ and by maximality, $G/H \simeq \mathbb{Z}_p$ with $p$ prime, so, we can choose $(A \subset B) = (\{ 1 \} \subset \mathbb{Z}_{p^2})$.

Then, we can restrict to $(H \subset G)$ such that $H$ does not contain any non-trivial normal subgroup of $G$.

Examplesof $(A \subset B)$ as above, with $ (H \subset G)\not\sim (\{ 1 \} \subset \mathbb{Z}_{p})$, are also welcome (if they exist).

**Application**: the theory of group-subgroup subfactors (see here).

Let $(H \subset G)$ be a maximal inclusion of finite groups, and let $n$ be a positive integer.

Generalization: Is there an inclusion of finite groups $(A \subset B)$ whose lattice of intermediate subgroups is a single chain $A = P_0 \subset P_1 \subset ... \subset P_n = B$ with $(P_i \subset P_{i+1}) \sim (H \subset G)$? (Unicity of $(A \subset B)$ up to $\sim$ ?)

**Remark**: We can write a remark as above, with $\mathbb{Z}_{p^n}$

Appendixon the equivalence relation $\sim$ :

Let $\sim_1$ generated by $(H \subset G) \sim_1 (H/K \subset G/K)$ with $K \subset H$ a normal subgroup of $G$.

Let $\sim_2$ generated by $(H \subset G) \sim_2 (\phi(H) \subset L)$ with $ \phi: G \to L$ isomorphism.

Let $\sim_3$ gen. by $(H_1 \subset G_1) \sim_3 (H_2 \subset G_2)$, $\phi \in Aut(G_1 \times G_2)$, $\phi(H_1 \times G_2) = G_1 \times H_2$.

Obviously:

($\sim_1$ et $\sim_2$) $\Leftrightarrow$ $\sim$

$\sim_3$ $\Rightarrow$ $ \sim$

$\sim$ $\not\Rightarrow$ $ \sim_3 $.