= Solution
The <complexification of a Lie algebra> in the <Adjoint representation> of <SU(5)> consists of traceless endomorphisms of $V=A\oplus B$. Splitting an endomorphism into its four blocks gives traceless endomorphisms of $A$ and $B$, one relative block-scalar direction, and the two off-diagonal spaces $\operatorname{Hom}(B,A)$ and $\operatorname{Hom}(A,B)$.
The first three pieces transform as $(\mathbf3,\mathbf1)_0$, $(\mathbf1,\mathbf8)_0$ and $(\mathbf1,\mathbf1)_0$. The off-diagonal pieces have <U(1) charges> equal to output minus input charge, namely $3-(-2)=5$ and $-2-3=-5$. Thus
$$
\boxed{\mathbf{24}\downarrow=(\mathbf3,\mathbf1)_0\oplus(\mathbf1,\mathbf8)_0\oplus(\mathbf1,\mathbf1)_0
\oplus(\mathbf2,\overline{\mathbf3})_5\oplus(\mathbf2,\mathbf3)_{-5}.}
$$
The dimensions are $3+8+1+6+6=24$. The first three summands are precisely the <gauge fields> of the local <gauge group> $H\times U_c(1)$. The extra <gauge fields> of the <SU(5)> <gauge theory> transform as
$$
\boxed{(\mathbf2,\overline{\mathbf3})_5\oplus(\mathbf2,\mathbf3)_{-5}.}
$$
Hermiticity relates the two off-diagonal blocks, so these describe twelve real <gauge fields>, or six complex fields and their conjugates. Globally, the subgroup generated by $H$ and $U_c(1)$ inside <SU(5)> is $(SU(2)\times SU(3)\times U(1))/\mathbb Z_6$, since the parametrization $(U_2,U_3,e^{i\theta})\mapsto\operatorname{diag}(e^{3i\theta}U_2,e^{-2i\theta}U_3)$ has a six-element <kernel>. This finite quotient does not alter the stated <branching of a representation>.
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