Detecting low-diameter clusters is an important graph-based data mining technique used in social network analysis, bioinformatics and text-mining. Low pairwise distances within a cluster can facilitate fast communication or good reachability between vertices in the cluster. Formally, a subset of vertices that induce a subgraph of diameter at most $k$ is called a $k$-club. For low values of the parameter $k$, this model offers a graph-theoretic relaxation of the clique model that formalizes the notion of a low-diameter cluster. Using a combination of graph decomposition and model decomposition techniques, we demonstrate how the fundamental optimization problem of finding a maximum size $k$-club can be solved optimally on large-scale benchmark instances that are available in the public domain. Our approach circumvents the use of complicated formulations of the maximum $k$-club problem in favor of a simple relaxation based on necessary conditions, combined with canonical hypercube cuts introduced by Balas and Jeroslow.