Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$.
We can generalize the notion of indecomposable from groups to inclusion of groups as follows:
Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.
Remark: The equivalence class of $(A \subset B)$ is the same that the conjugacy class of a transitive permutation group $G$ (see this GAP Data Library) with $(A \subset B) \sim (G_1 \subset G)$.
Definition: An inclusion of groups $(H \subset G)$ is indecomposable if (for $H_i \le G_i)$: $(H \subset G) \sim (H_1 \times H_2 \subset G_1 \times G_2)$ $\Rightarrow$ $\exists i \ H_i = G_i$
Examples: The maximal inclusions are indecomposable.
In the maximal (finite index) case, $(H \subset G)$ is indecomposable and $G$ decomposable if and only if
$(H \subset G) \sim (D_S \subset S \times S)$ with $S$ a nonabelian finite simple group and $D_S$ the diagonal subgroup.
It's proved by the answer here. Are there others examples beyond the maximal case?
Question: What's the classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable?