  A partial order on prime knots can be defined by declaring $J\ge K$ if there
exists an epimorphism from the knot group of $J$ onto the knot group of $K$.
Suppose that $J$ is a 2-bridge knot that is strictly greater than $m$ distinct,
nontrivial knots. In this paper we determine a lower bound on the crossing
number of $J$ in terms of $m$. Using this bound we answer a question of Suzuki
regarding the 2-bridge epimorphism number $\mbox{EK}(n)$ which is the maximum
number of nontrivial knots which are strictly smaller than some 2-bridge knot
with crossing number $n$. We establish our results using techniques associated
to parsings of a continued fraction expansion of the defining fraction of a
2-bridge knot.
