A Shimura datum $(G,X)$ is of *Hodge type* if there exists an injective morphism of Shimura data $(G,X) \hookrightarrow (\mathrm{GSp}_{2g}, \mathfrak{H}^{\pm})$, for some integer $g$.

Let $(G',X')$ be an *adjoint* Shimura datum; that is, $(G',X') = (G^{'\text{ad}},X^{'\text{ad}})$. Consider the class $\mathcal{C}$ consisting of all *simple* Shimura data of Hodge type $(G,X)$ for which $(G^{\text{ad}},X^{\text{ad}}) = (G',X')$. (The condition that $(G,X)$ is simple is there to exclude irrelevant torus factors coming from products with CM abelian varieties.)

By considering morphisms of Shimura data over $(G',X')$ we can endow $\mathcal{C}$ with a poset structure.

Assume that $(G',X')$ is of *abelian type*, which implies that $\mathcal{C}$ is not empty.

I would like to get a better feeling for $\mathcal{C}$. Here are a couple of concrete questions:

- Are there bounds on the rank of $G$, for $(G,X)$ in $\mathcal{C}$?
- Is the set of isomorphism classes in $\mathcal{C}$ finite?
- Are there other interesting things to be said about the poset $\mathcal{C}$?
- Is there (up to isomorphism) a canonical/natural Shimura datum $(G,X)$ in $\mathcal{C}$?

I do not have a good feeling for what kind of structure I should expect on $\mathcal{C}$. Maybe there is somehow a `minimal' element, or something like that. I have tried to put the classification by Deligne [section 1.3, Corvalis, 1979] to use; but I did manage to get something useful.

(*After the first comment, by Keerthi Madapusi Pera, the question got a slight facelift. Initially I only asked the final question, about a natural Shimura datum $(G,X)$ in $\mathcal{C}$.*)