  Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the
structure of finite groups with a large number of $p$-regular conjugacy classes
or, equivalently, a large number of irreducible $p$-modular representations. We
prove sharp lower bounds for this number in terms of $p$ and the $p'$-part of
the order of $G$ which ensure that $G$ is $p$-solvable. A bound for the
$p$-length is obtained which is sharp for odd primes $p$. We also prove a new
best possible criterion for the existence of a normal Sylow $p$-subgroup in
terms of these quantities.
