  The abstract Cauchy problem for the fractional evolution equation with the
Caputo derivative of order $\beta\in(0,1)$ and operator $-A^\alpha$,
$\alpha\in(0,1)$, is considered, where $-A$ generates a strongly continuous
one-parameter semigroup on a Banach space. Subordination formulae for the
solution operator are derived, which are integral representations containing a
subordination kernel (a scalar probability density function) and a
$C_0$-semigroup of operators. Some properties of the subordination kernel are
established and representations in terms of Mainardi function and L\'evy
extremal stable densities are derived. Applications of the subordination
formulae are given with a special focus on the multi-dimensional space-time
fractional diffusion equation for some special values of the parameters.
