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In this paper, our aims are to study the stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects. Using Lyapunov function, nonnegative semi-martingale convergence theorem, we obtain the almost sure stability with general decay rate, including the exponential stability and the polynomial stability. Moreover, we give an upper bound of each coefficient at any mode according to the theory of M-matrix. Finally, one example is given to show the effectiveness of the obtained theory.