The interplay between lattice topology, frustration, and spin quantum number, $s$, is explored for the Heisenberg antiferromagnet (HAFM) on the eleven two-dimensional Archimedean lattices (square, honeycomb, CaVO, SHD, SrCuBO, triangle, bounce, trellis, maple-leaf, star, and kagome). We show the CCM provides consistently accurate results when compared to the results of other approximate methods. The $\sqrt{3}\times\sqrt{3}$ model state provides lower ground-state energies than those of the $q=0$ model state for the kagome and star lattices for most values of $s$. The $q=0$ model state provides lower ground-state energies only for $s=1/2$ for the kagome lattice and $s=1/2$ and $s=1$ for the star lattice. The kagome and star lattices demonstrate the least amount of magnetic ordering and the unfrustrated lattices (square, honeycomb, SHD, and CaVO) demonstrate the most magnetic ordering for all values of $s$. The SrCuBO and triangular lattices also demonstrate high levels of magnetic ordering, while the remaining lattices (bounce, maple-leaf, and trellis) tend to lie between these extremes, again for all values of $s$. These results also clearly reflect the strong increase in magnetic order with increasing spin quantum number $s$ for all lattices. The ground-state energy, $E_g/(NJs^2)$, scales with $s^{-1}$ to first order, as expected from spin-wave theory, although the order parameter, $M/s$, scales with $s^{-1}$ for most of the lattices only. Self-consistent spin-wave theory calculations indicated previously that $M/s$ scales with $s^{-2/3}$ for the kagome lattice HAFM, whereas previous CCM results (replicated here also) suggested that $M/s$ scales with $s^{-1/2}$. By using similar arguments, we find here also that $M/s$ scales with $s^{-1/3}$ on the star lattice and with $s^{-2/3}$ on the SrCuBO lattice.

29 pages, 4 figures, 4 tables