  We describe the finite subgraph $\mathfrak{M}$ of Conway's Big Picture
required to describe all $171$ genus zero groups appearing in monstrous
moonshine. We determine the local structure of $\mathfrak{M}$ and give a purely
group-theoretic description of this picture, based on powers of the conjugacy
classes $24J$ and $8C$. We expect similar results to hold for umbral moonshine
groups and give the details for the largest Mathieu group $M_{24}$.
