What are examples (general features) of the finite groups $G$, such that every irrep (irreducible representation) contained (as constituent) in the representation induced from trivial representation of some non-trivial (not identity) subgroup ?

Remark: if induce from the subgroup = {identity}, we will get regular representation of whole group, and every irrep lives there. So I should not allow this trivial subgroup.

Example 1: S_n satisfy this property (there is some comment by David Speyer which I cannot find now, which says (as far as I remember) that induction from $S_{k1}\times S_{k2} \times ... \times S_{kl}$ will do the job. I have another argument but it is more complicated.

Example 2: Take Z/pZ for prime p - it does NOT satisfy my requirement - the only subgroups are {e} and Z/pZ itself, first one is forbidden by the rules of the game induction from the second will give trivial irrep.

Example 3: If we take PSL(2,7)=GL(3,2) and induce from the 3-Sylow subgroup - it will contain all irreps as constituents (see MO 104939)

In particular: what about GL(n,F_p) and A_n ?