The $s$-clubs model cohesive social subgroups as vertex subsets that induce subgraphs of diameter at most $s$. In defender-attacker settings, for low values of $s$, they can represent tightly-knit communities whose operation is undesirable for the defender. For instance, in online social networks, large communities of malicious accounts can effectively propagate undesirable rumors. In this article we consider a defender that can disrupt vertices of the adversarial network to minimize its threat, which leads us to consider a maximum $s$-club interdiction problem where interdiction is penalized in the objective function. Using a new notion of $H$-heredity in $s$-clubs, we provide a mixed-integer linear programming formulation for this problem that uses far fewer constraints than the formulation based on standard techniques. We show that the linear programming relaxation of this formulation has no redundant constraints and identify facets of the convex hull of integral feasible solutions under special conditions. We further relate $H$-heredity to latency-$s$ connected dominating sets and design a decomposition branch-and-cut algorithm for the problem. Our implementation solves benchmark instances with more than 10,000 vertices in a matter of minutes and is orders of magnitude faster than algorithms based on the standard formulation.