<p>This paper is concerned with the fixed-time synchronization (FXS) problems of Caputo/conformable fractional-order inertial Cohen-Grossberg neural networks (FOICGNNs) by one/two-phase hybrid impulsive control (HIC) through event-triggered update strategies. By utilizing the properties of fractional calculus, several novel inequalities regarding the fixed-time convergence of hybrid impulsive systems (HIS) are obtained. We especially discuss and compare the cases of Caputo and conformable fractional order to gain deep insight into fractional calculus. By applying the Lyapunov stability theory, two hybrid controllers, which consist of event-triggered continuous controllers and impulsive controllers, are designed to realize the FXS of FOICGNNs. It’s worth pointing out that, we unprecedentedly study and compare the differences of the one-phase HIC and two-phase HIC, where a novel nonlinear impulsive controller is proposed and designed to obtain fixed-time convergence in the impulsive control phase. In addition, the exclusion of Zeno behavior is proved for the designed event-triggered strategy. Finally, several numerical examples are provided to illustrate the feasibility of the proposed control approach and the correctness of the theoretical results.</p>