We perform a systematic study of the antiferromagnetic Heisenberg model on the Husimi lattice using numerical tensor-network methods based on Projected Entangled Simplex States (PESS). The nature of the ground state varies strongly with the spin quantum number, S. For S = 1/2, it is an algebraic (gapless) quantum spin liquid. For S = 1, it is a gapped, non-magnetic state with spontaneous breaking of triangle symmetry (a trimerized simplex-solid state). For S = 2, it is a simplex-solid state with a spin gap and no symmetry-breaking; both integer-spin simplex-solid states are characterized by specific degeneracies in the entanglement spectrum. For S = 3/2, and indeed for all spin values S ≥ 5/2, the ground states have 120-degree antiferromagnetic order. In a finite magnetic field, we find that, irrespective of the value of S, there is always a plateau in the magnetization at m = 1/3.