For a (finitely generated) subgroup of a (finitely generated) group, we will consider two properties of this subgroup: the separability of this group and its subgroup distortion. The separability of a subgroup measures whether the property of an element not lying in this subgroup is visible by taking some finite quotient. We will give a characterization on whether a subgroup of a 3-manifold group is separable. The subgroup distortion compares the intrinsic and extrinsic geometry of a subgroup. For an arbitrary subgroup of a 3-manifold group, we prove that the subgroup distortion can only be linear, quadratic, exponential and double exponential. It turns out the subgroup separability and subgroup distortion are closed related for subgroups of 3-manifold groups. The subgroup distortion part is joint work with Hoang Nguyen.