Ranks of Homotopy Groups of Homogeneous Spaces

A simple way to evaluate the ranks of homotopy groups $\pi_j(M)$ is indicated for homogeneous spaces of the form $M=G/H$, where $G$ is a compact connected Lie group and $H$ is a connected regular subgroup or a subgroup of maximal rank in $G$. A classification of the spaces whose Onishchik ranks are equal to 3 is obtained. The transitive actions on the products of homogeneous spaces of the form $G/H$ are also described, where $G$ and $H$ are simple and $H$ is a subgroup of corank 1 in $G$ and the defect of the space $G/H$ is equal to 1.