Let $\varphi:G \to G$ be an automorphism (a one-to-one, onto, homomorphism from $G$ to itself). It's easy to show that the image of a subgroup is itself a subgroup (this is true for any homomorphism), so if $H$ is a subgroup of $G$ then so is $\varphi(H)$.
When $G$ is finite, we have: $H$ is a Sylow $p$-subgroup, say of order $p^k$, implies that $\varphi(H)$ is a subgroup of order $p^k$ (because $\varphi$ is one-to-one) and thus it is also a Sylow $p$-subgroup. So automorphisms send Sylow $p$-subgroups to Sylow $p$-subgroups.
Conjugation by $g \in G$ is an automorphism, so $gHg^{-1}$ is a Sylow $p$-subgroup whenever $H$ is a Sylow $p$-subgroup.
Now to more directly address your question, most subgroups get moved around by automorphisms. If $\varphi(H)=H$ for all $\varphi \in \mathrm{Aut}(G)$, then $H$ is called a characteristic subgroup (the commutator subgroup G' which is generated by elements of the form $aba^{-1}b^{-1}$ is one such example). If $H$ is fixed by all inner-automorphisms (i.e. $gHg^{-1}=H$ for all $g\in G$), then $H$ is normal!
In other words, if $H$ is a non-normal subgroup of $G$, then there must exist some inner-automorphism which sends $H$ to a different subgroup (i.e. there exists some $g\in G$ such that $gHg^{-1} \not= H$). So automorphisms tend to shuffle the subgroups around.