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The cold dark matter (DM) model predicts that every galaxy contains thousands of DM subhalos; almost all other DM models include a physical process that smooths away the subhalos. The subhalos are invisible, but could be detected via strong gravitational lensing, if they lie on the line of sight to a multiply-imaged background source, and perturb its apparent shape. We present a predominantly automated strong lens analysis framework, and scan for DM subhalos in Hubble Space Telescope imaging of 54 strong lenses. We identify five DM subhalo candidates, including two especially compelling candidates (one previously known in SLACS0946+1006) where a subhalo is favoured after all of our tests for systematics. We find that the detectability of subhalos depends upon the assumed parametric form for the lens galaxy's mass distribution, especially its degree of azimuthal freedom. Using separate components for dark matter and stellar mass reveals two DM subhalo candidates and removes four false-positives compared to the single power-law mass model that is common in the literature. We identify 45 lenses without substructures, the number of which is key to statistical tests able to rule out models of e.g. warm or self-interacting DM. Our full analysis results are available at https://github.com/Jammy2211/autolens_subhalo.