Is there a classification of all connected semisimple Lie groups with infinite center?
I know there is the universal cover of $SL(2, \mathbb{R})$ and I know it is the only contractible semisimple Lie group (up to direct products etc.), so in particular its maximal compact subgroup is trivial.
Is there a connected semisimple Lie groups with infinite center and non-trivial compact subgroups? (other than things like $\widetilde{SL}(2,\mathbb{R}) \times SO(3, \mathbb{R})$, etc.)