Normal subgroups orbits

Question

Let $G$ be a topological group acting transitively and effectively on the space $X$ and let $J,K$ be two normal subgroups of $G$ such that $G=J\cdot K$ and $J\cap K\not =\{e\}$. Let $Gx_0$ be the orbit of $x_0\in X$ with isotropy subgroup $H$, can we decompose this orbit by $J$ and $K$ orbits, for example something like $G/H=J/(J\cap H)\times K/(K\cap H)$? What about the intersection $J\cap K$ action?