What are examples (general features) of the finite groups $G$, such that every irrep (irreducible representation) is contained (as constituent) in the representation induced from  trivial representation of some non-trivial subgroup? (I allow subgroups to vary - I mean take all subgroups, induce reps from trivial characters and require that any irrep sits in this set).

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

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

*Example 2*: Take $\mathbb{Z}/p\mathbb{Z}$ for prime p - it does NOT satisfy my requirement - the only subgroups are $\lbrace 1\rbrace$ and $\mathbb{Z}/p\mathbb{Z}$ itself; first one is forbidden by the rules of the game, and induction from the second will give trivial irrep.

*Example 3*: If we take [PSL(2,7)=GL(3,2)][1] and induce from the 3-Sylow subgroup - it will contain all irreps as constituents (see [MO 104939][2]).

In particular:  what about $GL(n,\mathbb{F}_p)$ and $A_n$?

  [1]: http://en.wikipedia.org/wiki/PSL(2,7)
  [2]: https://mathoverflow.net/questions/104939/induction-from-cyclic-sylow-subgroup-are-there-any-nice-properties