This post is part of a three-post set (1 2 3) about groups of order $pqr$, where $p<q<r$ are prime. While thinking about such $pqr$-groups, I've collected a bunch of theorems and lemmas into a little mental toolbox. An interesting commonality between a lot of these tools is that they tell us about the relationship between various different animals in the subgroup lattice; an interesting difference is that some of them go “upwards,” some of them go “downwards,”1 and some of them even g...
