This post is a generalization of [Uniqueness of the direct product decomposition of finite groups][1]. 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][2] 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][3]) 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][4]). 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][5]. It generalizes into the [Kurosh-Ore theorem][6] in the general theory of [modular lattices][7], with a specific relevant additional result if the lattice is [distributive][8]. Perhaps we can use this theorem for answering the question. [1]: https://math.stackexchange.com/q/908033/84284 [2]: https://math.stackexchange.com/q/908033/84284 [3]: http://www.gap-system.org/Datalib/trans.html [4]: https://math.stackexchange.com/q/913510/84284 [5]: http://en.wikipedia.org/wiki/Krull%E2%80%93Schmidt_theorem [6]: http://planetmath.org/kuroshoretheorem [7]: http://en.wikipedia.org/wiki/Modular_lattice [8]: http://en.wikipedia.org/wiki/Distributive_lattice