  We study finite groups $G$ with the property that for any subgroup $M$
maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$
is supersolvable. We show that any nonabelian simple group can occur as a
composition factor of such a group and that, if $G$ is solvable, then the
nilpotency length and the rank are arbitrarily large. On the other hand, for
every prime $p$, the $p$-length of such a group is at most $1$. This answers
questions proposed by V. Monakhov in The Kourovka Notebook.
