It's an old maxim of Aristotelian metaphysics that substances do not have substantial proper parts. Here's an argument for it, in the case of material substances. Suppose a material substance A has a substance B as a proper part. Now, arguably A is wholly composed of two parts: the matter M and the form F.
Now the form G of B cannot overlap M, as then the form would be partly material. So G must be a part of F. But forms of substances are simples. So G must be all of F. But then we have two substances with the numerically same form, and that seems absurd.
A central assumption in the argument is that forms are simples. There may be a way of making an argument without that assumption. Suppose we say that G, the form of B, is a proper part of F, the form of A. Now if any proper part of a substance is a substance, then my heart is a substance--it's nicely delineated, and one of the best candidates. But I can survive the destruction of my heart (I would just need a machine to circulate the blood). And surely if my heart is destroyed, its form is destroyed as well. But my form doesn't seem to be intrinsically changed by the destruction of my heart. Yet if the form of the heart were a part of my form, then my form would be intrinsically changed by the destruction of the heart.