  We predict and examine various topological states on a two-dimensional (2D)
triangular kagome lattice (TKL) using the tight-binding (TB) models and theory
of topological quantum chemistry (TQC). Firstly, on the basis of TQC, we
diagnose band structures with fragile topology and calculate Wilson-loop
spectra and Hofstadter butterfly spectra to confirm their non-trivial nature.
Secondly, we examine the bulk-boundary correspondence and find that an
obstructed-atomic-limit (OAL) insulator hosts fractional corner states without
being accompanied by fragile topological band structures, which implies that
the presence of OALs and corner states is not a sufficient condition to fragile
topology. Last but not least, we predict a topological phase transition from a
second-order topological phase to a first-order topological phase that can be
realized in the TKL under the action of a magnetic field.
