We consider the problem of minimizing the sum of a convex function and a sparsity-inducing group regularizer. Problems involving such regularizers arise in modern machine learning applications often for the purpose of obtaining models that are easier to interpret and that have higher predictive accuracy. We present a new second-order algorithm for solving such problems that uses subspace acceleration, domain decomposition, and support identification. Our analysis shows that our algorithm has strong complexity properties, and preliminary numerical results for binary classification based on regularized logistic regression show that our approach is efficient and robust.