  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.
