Given an algebraic group $G$ is $G^0$ (connected component including the identity) unique?

Question

I have two basic questions regarding connected components including the identity of algebraic groups.

1. Given an algebraic group $G$ is $G^0$ (connected component including the identity) unique?

2. Let $H$ be an algebraic subgroup of $G$. Does it then always follow that $H^0$ is contained in $G^0$?

Thank you very much!