  Hilbert space fragmentation is a novel type of ergodicity breaking in closed
quantum systems. Recently, an algebraic approach was utilized to provide a
definition of Hilbert space fragmentation characterizing \emph{families} of
Hamiltonian systems based on their (generalized) symmetries. In this paper, we
reveal a simple connection between the aforementioned classification of
physical systems and their coherence generation properties, quantified by the
coherence generating power (CGP). The maximum CGP (in the basis associated to
the algebra of each family of Hamiltonians) is exactly related to the number of
independent Krylov subspaces $K$, which is precisely the characteristic used in
the classification of the system. In order to gain further insight, we
numerically simulate paradigmatic models with both ordinary symmetries and
Hilbert space fragmentation, comparing the behavior of the CGP in each case
with the system dimension. More generally, allowing the time evolution to be
any unitary channel in a specified algebra, we show analytically that the
scaling of the Haar averaged value of the CGP depends only on $K$. These
results illustrate the intuitive relationship between coherence generation and
symmetry algebras.
