  Resetting a stochastic process is an important problem describing the
evolution of physical, biological and other systems which are continually
returned to their certain fixed point. We consider the motion of a subdiffusive
particle with a constant drift under Poissonian resetting. In this model the
stochastic process is Brownian motion subordinated by an inverse infinitely
divisible process (subordinator). Although this approach includes a wide class
of subdiffusive system with Poissonian resetting by using different
subordinators, each of such systems has a stationary state with the asymmetric
Laplace distribution in which the scale and asymmetric parameters depend on the
Laplace exponent of the subordinators used. Moreover, the mean time for the
particle to reach a target is finite and has a minimum, optimal with respect to
the resetting rate. Features of L\'evy motion under this resetting and the
effect of a linear potential are discussed.
