During my researches, I've obtained a class of finite groups as follows.

Let $\mathcal{C}$ be the class of all finite groups $G$ such that for every factorization $|G|=ab$ there exists a subgroup $H\neq G$ such that $|G:H|$ divides $a$ or $b$ (equivalentely, $a$ or $b$ divides $|H|$).

We are looking for some classes of groups (resp. special groups) inside (resp. outside) $\mathcal{C}$.

For example, the class of all finite solvable groups, and also groups $G$ such that for every divisor $d$ of $|G|$ there is a subgroup of the order or index $d$ (containing all CLT and supersolvable groups) is a sub-class of $\mathcal{C}$ (see https://math.stackexchange.com/questions/961921/a-gap-code-for-a-class-of-small-groups).

Now, my questions are:

(1) What is the minimum of sizes of all groups outside $\mathcal{C}$?

(2) Is the class of all finite simple groups a sub-class of $\mathcal{C}$?

(3) Does anybody know some other vast sub-classes of $\mathcal{C}$?

Thanks in advance.