  We investigate theoretically the topological properties of dimerized
quasi-one-dimensional (1D) lattice comprising of multi legs $(L)$ as well as
multi sublattices $(R)$. The system has main and subsidiary exchange
symmetries. In the basis of latter one, the system can be divided into $L$ 1D
subsystems each of which corresponds to a generalized $SSH_R$ model having $R$
sublattices and on-site potentials. Chiral symmetry is absent in all subsystems
except when the axis of main exchange symmetry coincides on the central chain.
We find that the system may host zero- and finite-energy topological edge
states. The existence of zero-energy edge state requires a certain relation
between the number of legs and sublattices. As such, different topological
phases, protected by subsystem symmetry, including zero-energy edge states in
the main gap, no zero-energy edge states, and zero-energy edge states in the
bulk states are characterized. Despite the classification symmetry of the
system belongs to $BDI$ but each subsystem falls in either $AI$ or $BDI$
symmetry class.
