The identification of electronic nematicity across series of iron-based superconductors raises the question of its relationship with superconductivity and other ordered states. Here, we report a systematic elastoresistivity study on LaFe$_{1-x}$Co$_x$AsO single crystals, which have well separated structural and magnetic transition lines. All crystals show Curie-Weiss-like nematic susceptibility in the tetragonal phase. The extracted nematic temperature is monotonically suppressed upon cobalt doping, and changes sign around the optimal doping level, indicating a possible nematic quantum critical point beneath the superconducting dome. The amplitude of nematic susceptibility shows a peculiar double-peak feature. This could be explained by a combined effect of different contributions to the nematic susceptibility, which are amplified at separated doping levels of LaFe$_{1-x}$Co$_x$AsO.
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