  Subgraphs reveal information about the geometry and functionalities of
complex networks. For scale-free networks with unbounded degree fluctuations,
  we obtain the asymptotics of the number of times a small connected graph
occurs as a subgraph or as an induced subgraph. We obtain these results by
analyzing the configuration model with degree exponent $\tau\in(2,3)$ and
introducing a novel class of optimization problems. For any given subgraph, the
unique optimizer describes the degrees of the vertices that together span the
subgraph.
  We find that subgraphs typically occur between vertices with specific degree
ranges. In this way, we can count and characterize {\it all} subgraphs. We
refrain from double counting in the case of multi-edges, essentially counting
the subgraphs in the {\it erased} configuration model.
