SPb. Saint Petersburg State University . - 42s. Some well-known and recently discovered unexpected types of motion of a simple rigid pendulum with an oscillating suspension. are considered. A visual physical explanation of the phenomenon of dynamic stabilization of the inverted pendulum is given . The conditions for the occurrence of subharmonic resonances are calculated, their relationship with dynamic stabilization is discussed, and an updated criterion for stabilization. is obtained. The relationship of the upper limit of stability of the inverted pendulum with the loss of stability of the lower position . Computer modeling and analytical research complement and enrich each other, contributing to an understanding of the complex behavior of the pendulum.