Abstract: This paper studies stabilities of the solution of stochastic differential equations (SDE) driven by time-changed Lévy noise, in both probability and moment sense. This provides more flexibility in modeling schemes in application areas, including physics, biology, engineering, finance and hydrology. Necessary conditions for the solution of time-changed SDE to be stable in different senses will be established. Connection between the stability of solution to time-changed SDE and that to the corresponding original SDE will be disclosed. Examples related to different stabilities will be given. We study SDEs with time-changed Lévy noise, where the  time-change processes  are the inverse of general Lévy subordinators. These results are important generalizations of the results in Wu (2016).