Let $G$ be a topological group and $H\leq G$ be a subgroup.
Let $p\;\colon G\to\pi_0\left(G\right)$ be the quotient map, $G_0$ be the identity path component of $G$ and $\left(G/H\right)_0$ be the path component of $G/H$ containing the coset $H$. Then I think these are true, but I can't seem to prove them: $$ \pi_0\left(G/H\right)\cong\pi_0\left(G\right)/p\left(H\right)\\ \left(G/H\right)_0\cong G_0/\left(G_0\cap H\right)$$