$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
My question concerns all the possible ways to embed SU(5) to U(16) by specifying the 16-dimensional complex representation, denoted as ${\bf 16}$ in U(16). (The ${\bf 16}$ is also a box in Young tableau as the fundamental representation of U(16).)
Warm up
For example, if we consider the embedding: $$ \U(16)\supset \Spin(10) \supset \SU(5), $$ then there are at least two ways of embedding: $$\text{$\mathbf{16}$ in $\U(16)$ as $\mathbf{16}$ in $\Spin(10)$, decomposed as $ \mathbf 1 \oplus \overline{\mathbf 5} \oplus \mathbf{10} $ in $\SU(5).$ }$$ $$\text{$\mathbf{16}$ in $\U(16)$ as $\mathbf{16}$ in $\Spin(10)$, decomposed as $\mathbf 1 \oplus {\mathbf 5} \oplus \overline{\mathbf{10}} $ in $\SU(5).$ }$$
Question
Now consider the embedding: $$ \U(16)\supset \SU(5) $$ via specifying $$ \text{Representation ${\bf 16}$ of U(16) is decomposed as the representation ? of SU(5).} $$ What are the possible ways of embeddings and decompositions?
Since the dimension of representation of SU(5) smaller than 16-dimensional ${\bf 16}$ involves ${\mathbf 1}$, $\mathbf{5}$, $\mathbf{10}$, $\mathbf{15}$, and $\overline{\mathbf{5}}$, $\overline{\mathbf{10}}$, $\overline{\mathbf{15}}$.
For example, do we have new embedding via new decompositions: $$\text{$\mathbf{16}$ in $\U(16)$ decomposed as $\mathbf 1 \oplus {\mathbf 5} \oplus {\mathbf{10}} $ in $\SU(5)?$ }$$ $$\text{$\mathbf{16}$ in $\U(16)$ decomposed as $\mathbf 1 \oplus \overline{\mathbf 5} \oplus \overline{\mathbf{10}} $ in $\SU(5)?$ }$$
Are all the combinatory allowed? for example $$\text{$\mathbf{16}$ in $\U(16)$ decomposed as $a \mathbf{1} \oplus b {\mathbf 5} \oplus c \overline{\mathbf 5} \oplus d \mathbf{10} \oplus e \overline{\mathbf{10}} \oplus f \mathbf{15} \oplus g \overline{\mathbf{15}} $ in $\SU(5)?$ }$$ What are the constraints on $a,b,c,d,e,f,g$ other than $$a+5b+5c+10d+10e+15f+15g=16?$$ Other constraints?