group theory - Is the Standard Model an invariant subgroup of $SU(5)$?

It is well known that the Standard Model (SM) gauge group is a subgroup of $SU(5)$:

$$\begin{equation} SU(3) \times SU(2)\times U(1) ~\subset~SU(5) \end{equation}$$ This can be easily checked using the method of Dynkin diagrams. Is this subgroup an invariant subgroup such that, $$\begin{equation} g _{SU(5)} g _{SM} g _{SU(5)} = g _{SM} ' \,, \end{equation}$$ where $g_{SU(5)}$ ($g_{SM}$) is an element of $SU(5)$ ($SM$)?

Background: The reason I'm interested in this is because then its necessarily true that the non-SM gauge group generators of $SU(5)$ can be written as solely off-diagonal matrices and the SM as solely diagonal (this is easy to see by writing the matrices in block diagonal form), which simplifies calculations.