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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.