# Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups.
Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups.

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 $H_i \le G_i)$: "$(H \subset G) \sim (H_1 \times H_2 \subset G_1 \times G_2)$" $\Rightarrow$ "$\exists i$ such that $H_i = G_i$"

Examples: The maximal inclusions are indecomposable.
Warning: $(H \subset G)$ maximal (a fortiori indecomposable) $\not \Rightarrow$ $G$ indecomposable (see here).

An inclusion of finite groups decomposes into a direct product of indecomposable inclusions: $$(H \subset G) \sim (\prod_i H_i \subset \prod_i G_i)$$ with $(H_i \subset G_i)$ an indecomposable inclusion of finite groups $\forall i$.

Question: Is this decomposition unique (up to permutation and equivalence $\sim$)?

Remark: The finite group case comes from the the Krull–Schmidt theorem. It generalizes into the Kurosh-Ore theorem in the general theory of modular lattices, with a specific relevant additional result if the lattice is distributive. Perhaps we can use this theorem for answering the question.