We report holographic generation of higher-order Laguerre-Gaussian (LG) beams using a liquid crystal on silicon spatial light modulator (LCOS-SLM) device. In our experimental set-up, a flat-top light beam was projected on the LCOS-SLM to generate LG beams of various mode indices without changes of the optical system. Additionally, the size of the holographic phase pattern was optimized for each beam to maximize the mode purity of the obtained beam. Holographic generation of LG beams is easily influenced by a distortion of the optical system and deviation of the phase setting from an ideal one. Nevertheless, we obtained high-quality LG beams with an additional phase pattern on the LCOS-SLM for canceling the distortion of the optical system and with calibration of the phase control voltage for precise expression of the phase patterns. Numerical analyses are also performed for two-dimensional beam profiles to verify the quality of the obtained beams. Through fitting the obtained profiles to theoretical ones, we calculate the correlation coefficients R between the observed and fitted profiles to find that R > 0.95 for all beams and that the correlation coefficients behave similarly to the theoretically estimated mode purities, facts indicating that the quality of the obtained LG beams is close to the theoretical limit in our experiments.