Let $G$ be a compact Lie group of dimension $n$. Then we can embed $G$ (topologically) into a connected compact Lie group $H$. (One may choose $H=U(m)$, the unitary group, for example.)

The question is: Are there are any (non trivial) upper bounds on the dimensions of possible groups $H$? (In the case $H=U(m)$, how compares a minimal $m$ to $n$?)

If for arbitrary compact Lie groups $G$ such an answer seems not possible, are there subclasses of the class of compact Lie groups, where one can obtain upper bounds in the above sense? (The class of connected groups excluded...)

I imagine some upper bounds, which for example may utilize the number of components, the rank of $G$, or other Lie group data. But as I am not so experienced with Lie groups my imagination might prove unrealistic.