Consider the special unitary group $SU(5)$ and the unitary group $U(16)$.

Below I specify a specfic way to embed the $SU(5) \subset U(16)$:

1. First we can embed the spin group $Spin(10)\subset U(16)$. Here we choose the ${\bf 16}$-dimensional spinor representation of $Spin(10)$ to be also the ${\bf 16}$-dimensional fundamental representation of $U(16)$. Thus, the data for the representation: $${\bf 16} \text{ in } Spin(10) \text{ as } {\bf 16} \text{ in } U(16)$$ gives us an action of $Spin(10)$ and $U(16)$ on the complex vector space $\mathbb{C}^{16}$.

2. Then, we can embed the $SU(5)$ to $Spin(10)$ next, see the discussion of this embedding of $SU(n)$ to $Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\bf 5}$, the anti-symmetric representation ${\bf 10}$ and the 1-dimensional ${\bf 1}$ of $SU(5)$: $$\overline{\bf 5} \oplus {\bf 10} \oplus {\bf 1} \text{ in } SU(5) \text{ as } {\bf 16} \text{ in } Spin(10).$$ Again, the data for the representation gives us an action of $SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $SU(5)$ into $U(16)$.

question

Then my question is about the normalizer of this $SU(5)$ into $U(16)$, which certainly depends on the embedding that I provided above. So what is this normalizer of this $SU(5)$ into $U(16)$? My suspicious may be $U(5)$, but it could also be larger than $U(5)$ with extra more $U(1)$ factors if I was mistaken...