  Let $D$ be an infinite discrete set of measurable cardinals. It is shown that
generalized Prikry forcing to add a countable sequence to each cardinal in $D$
is subcomplete. To do this it is shown that a simplified version of generalized
Prikry forcing which adds a point below each cardinal in $D$, called
generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized
diagonal Prikry forcing associated to $D$ is subcomplete above $\mu$, where
$\mu$ is any regular cardinal below the first limit point of $D$.
