# Finite Groups with a subgroup of every possible index

Suppose $G$ is a finite group, with $|G|=n$. Suppose also that for every positive integer $m\mid n$, $G$ has a subgroup of index $m$. Are there any general statements (structural or otherwise) I can make about such $G$?

For example, all such groups will be solvable; but as $S_4$ shows, they need not be supersolvable. The collection is also strictly smaller than solvable groups, as $A_4$ shows.

Thanks!

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In this paper, Jing mentions a variant: a group is a $\mathscr{G}$-group if and only if for each subgroup $H$ of $G$ and each prime factor $p$ of $[G:H]$, there is a subgroup $K$, $H\leq K\leq G$ such that $[K:H]=p$. He describes all values of $n$ such that every group $G$ with $|G|=n$ is a $\mathscr{G}$-group.
$\mathscr{G}$-groups have been characterized in sundry ways; the paper by Jing cites a book by Bray, Deskins, Johnson, Humphreys, Puttaswamaiah, Venke, Walls, and Weinstein in which it is shown that $G$ is a $\mathscr{G}$-group if and only if there is a normal Hall subgroup $N$ in $G$ such that $N$ and $G/N$ are nilpotent, and for each $H\leq N$ we have $G=N\cdot N_{G}(H)$.
Thanks for the pointer to $\mathscr{G}$-groups. I'm gonna take a look at the Jing paper. – user641 Sep 22 '11 at 11:59