  This paper examines the structure of the largest subgraphs of the
Erd\H{o}s-R\'enyi random graph, $G_{n,p}$, with a given matching number. This
extends a result of Erd\H{o}s and Gallai who, in 1959, gave a classification of
the structures of the largest subgraphs of $K_n$ with a given matching number.
We show that their result extends to $G_{n,p}$ with high probability when $p\ge
\frac{8 \ln n}{n}$ or $p \ll \frac{1}{n}$, but that it does not extend (again
with high probability) when $\frac{4\ln(2e)}{n} < p< \frac{\ln n}{3n}$.
