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 $H\neq G$ satisfying **at least one of the following conditions**:


  $|G:H|$ divides $a$,  $|G:H|$ divides $b$,  $a$ divides $|H|$,  $b$ 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 $\mathcal{C}$ (if there are)


**(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.