A one-dimensional quantum charge pump transfers a quantized charge in each pumping cycle. This quantization is topologically robust being analogous to the quantum Hall effect. The charge transferred in a fraction of the pumping period is instead generally unquantized. We show, however, that with specific symmetries in parameter space the charge transferred at well-defined fractions of the pumping period is quantized as integer fractions of the Chern number. We illustrate this in a one-dimensional Harper-Hofstadter model and show that the fractional quantization of the topological charge pumping is independent of the specific boundary conditions taken into account. We further discuss the relevance of this phenomenon for cold atomic gases in optical superlattices. ; Comment: 8 pages, 7 figures, new material added