  We study the magnetic structures and their connections with topological
superconductivity due to proximity effect for coupled magnetic atomic chains
deposited on a superconductor. Several magnetic phases are self-consistently
determined, including both the coplanar and non-coplanar ones. For a $N$-chain
triangular atomic ladder, topologically nontrivial superconducting states can
always be realized, but strongly depend on its magnetic structure and the
number of atomic chains. When $N$ is even, the topologically nontrivial states
with noncoplanar structures are characterized by $\mathbb{Z}_{2}$ invariants,
while the topologically nontrivial noncoplanar states with an odd $N$ are
characterized by integer $\mathbb{Z}$ invariants, due to the presence of a new
chiral symmetry. The new chiral symmetry for the noncoplanar states is found to
be robust against the on-site disorder, as long as the crystal reflection
symmetry is respected.
