$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.

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

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

Then, we can embed $\SU(5)$ into $\Spin(10)$. See the discussion of this embedding of $\SU(n)$ into $\Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\mathbf 5}$, the anti-symmetric representation $\mathbf{10}$ and the 1-dimensional representation $\mathbf 1$ of $\SU(5)$: $$\text{$\overline{\mathbf 5} \oplus \mathbf{10} \oplus \mathbf 1$ in $\SU(5)$ as $\mathbf{16}$ in $\Spin(10)$}.$$ 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 $\SU(5)$ inside $\U(16)$, which certainly depends on the embedding that I provided above. So what is the *normalizer* of this $\SU(5)$ inside $\U(16)$? My suspicion is that it may be $\U(5)$, but it could also be larger than $\U(5)$ with more $\U(1)$ factors if I was mistaken ….