This is an exercise from Tom Leinster's Basic Category Theory. It asks:

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> 1) What are the subcategories of an ordered set? Which are full?
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> 2) What are the subcategories of a group? Which are full?
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I'm self studying and I couldn't discuss this with someone, so I'm curious for what the solution is.

For the second one I think, if you have a one object group category, no subcategories are full, as they all consist of the group object with fewer morphisms. Also all of them are monoids, but only the ones that contain the inverse of every morphism remain groups, with less elements than the original.

Also a related question: 

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> Is there a functor $Z:\mathbf{Grp} \to \mathbf {Grp}$ so that $Z(G)$ is the center of the group $G$?
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(I'm guessing there is one, and also that the center of $G$ is a full subcategory of $G$, and also an Abelian group.)