I figured this deserves at least one answer before getting shut down. As many of you know, Jordan proved that a finite subgroup of $GL_n(\mathbb{C})$ contains an abelian normal subgroup of index, say $C(n)$ depending only on $n$. One can find a proof in Curtis and Reiner for instance. In the paper by B. Weisfeiler called "Post-classification version of Jordan’s theorem on ﬁnite linear groups" he uses classification to sharpen the existing bound on $C(n)$. There are some extensions to positive characteristic fields as well.