Assume that $N\subsetneq M $ are two indecomposable modules of finite length. Further, assume that if there is no indecomposable submodule in between them, i.e., if $I$ is indecomposable such that $N\subseteq I\subseteq M$ then either $I=N$ or $I=M$.
Can you please provide an example when we have two such modules, but $M/N$ isn't indecomposable?
Intuitively, there is no need for factor $M/N$ to be indecomposable. But I can't find any example.
My particular area of interest is finite-dimensional representations over fields. So I would be especially interested in an example there. I have already ruled out that there is no such example with thin representations or representations over Dynkin diagram $D_4$ or for modules over Kronecker 2-algebra.
I don't really know how to approach searching for this counter-example except for trying random rings and their modules.