*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](https://math.stackexchange.com/q/908033/84284) 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](http://www.gap-system.org/Datalib/trans.html)) 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](https://math.stackexchange.com/q/914095/84284). Are there others examples beyond the maximal case? 


**Question**: What's the classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable? 



Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable
If I have understood the definition of an indecomposable inclusion correctly, then any core-free subgroup $H$ of a direct product $G=G\_1 \times G\_2$ in which $H$ is indecomposable, and is not contained in any direct factor of $G$ isomorphic to $G\_1$ and $G\_2$ will be an example.


There are many such examples, such as $G=S\_3 \times S\_3$ with $H$ a diagonal subgroup of order $2$, $3$, or $6$.


