Does the special unitary Lie group $SU(5)$ contains $SU(3) \times SU(2) \times U(1)$ as a subgroup?
Can one show which of the following embeddings are possible rigorously: $$SU(5) \supset SU(3) \times SU(2) \times U(1)?$$ $$SU(5) \supset \frac{SU(3) \times SU(2) \times U(1)}{\mathbb{Z}_2}?$$ $$SU(5) \supset \frac{SU(3) \times SU(2) \times U(1)}{\mathbb{Z}_3}?$$ $$SU(5) \supset \frac{SU(3) \times SU(2) \times U(1)}{\mathbb{Z}_3 \times \mathbb{Z}_2}?$$
Here $\mathbb{Z}_n$ means the cyclic group of order $n$.
it looks to me that the first one is impossible, can we prove it is a no go?
(p.s. $SO(10)$ and $SU(5)$ are different groups - how can the two questions be duplicate to each other, without knowing the way of embedding?)