  We demonstrate that a plethora of higher order topological phases emerge in
magnet-superconductor hybrid (MSH) systems through the interplay of a stacked
magnetic structure and an underlying triangular surface lattice; the latter
being of great current experimental interest. Such lattices offer the ability
to create three main types of edge terminations -- called x-, y- and y'-edges
-- of MSH islands that, in turn, give rise to a complex phase diagrams
exhibiting various regions of HOTSC phases. We identify the single adatom
chain, as well as a pair of adjacent adatom chains (called a double-chain), as
the basic building blocks for the emergence of HOTSC phases. Of particular
interest are those HOTSC phase which arise from a competition between the
topology of single and double-chain blocks, which are absent for square
latices.
