From wikipedia: $\mathbb{CP}^n$ is a [Symmetric space of type AIII][1] for $p=n$, $q=1$.  There are embeddings of both $U(1)$ and $SU(n)$ into $S(U(n) \times U(1)) \subset SU(n+1)$ which give you your quotient by right multiplication.

  [1]: http://en.wikipedia.org/wiki/Symmetric_space#Classification_result