Subgraph Graph Neural Networks (Subgraph GNNs) enhance the expressivity of message-passing GNNs by representing graphs as sets of subgraphs. They have shown impressive performance on several tasks, but their complexity limits applications to larger graphs. Previous approaches suggested processing only subsets of subgraphs, selected either randomly or via learnable sampling. However, they make suboptimal subgraph selections or can only cope with very small subset sizes, inevitably incurring performance degradation. This paper introduces a new Subgraph GNNs framework to address these issues. We employ a graph coarsening function to cluster nodes into super-nodes with induced connectivity. The product between the coarsened and the original graph reveals an implicit structure whereby subgraphs are associated with specific sets of nodes. By running generalized message-passing on such graph product, our method effectively implements an efficient, yet powerful Subgraph GNN. Controlling the coarsening function enables meaningful selection of any number of subgraphs while, contrary to previous methods, being fully compatible with standard training techniques. Notably, we discover that the resulting node feature tensor exhibits new, unexplored permutation symmetries. We leverage this structure, characterize the associated linear equivariant layers and incorporate them into the layers of our Subgraph GNN architecture. Extensive experiments on multiple graph learning benchmarks demonstrate that our method is significantly more flexible than previous approaches, as it can seamlessly handle any number of subgraphs, while consistently outperforming baseline approaches.