We summarize the algebraic structure of spherically symmetric Yang–Mills potentials for a general compact gauge group, and investigate the particular case of gauge groups with Lie algebra su(n) in detail. We develop techniques that lead to a complete classification of the possible spherical symmetry ansätze, including descriptions of the reduced gauge group 𝒵, the space of magnetic potentials ℋ, and for those ansätze that admit extensions across the symmetry axis, a description of the space of vacuum potentials H0 and its little group Z0. These results are illustrated by listing all irreducible models for su(n), n⩽6.