TITLE: Why do we care about two subgroups being conjugate?
QUESTION [40 upvotes]: In classifications of the subgroups of a given group, results are often stated up to conjugacy.  I would like to know why this is.
More generally, I don't understand why "conjugacy" is an equivalence relation we care about, beyond the fact that it is stronger than "abstractly isomorphic."
My vague understanding is that while "abstractly isomorphic" is the correct "intrinsic" notion of isomorphism, so "conjugate" is the correct "extrinsic" notion.  But why have we designated this notion of equivalence, and not some other one?
To receive a satisfactory answer, let me be slightly more precise:

Question: Given two subgroups $H_1, H_2$ of a given group $G$, what properties are preserved under conjugacy that may break under general abstract isomorphism?

For example, is it true that $G/H_1 \cong G/H_2$ iff $H_1$ is conjugate to $H_2$?  Or, is it true that two subgroups $H_1, H_2 \leq \text{GL}(V)$ are conjugate iff their representations are isomorphic?  I'm sure these are easy questions to answer -- admittedly, I haven't thought fully about either -- but I raise them by way of example.  What are other such equivalent characterizations?

REPLY [2 votes]: It has already been mentioned by arctic tern mainly by giving examples and is implicit in the answers of others, but just to make it more explicit. There is a close connection between the action of a group $G$ on a set $\Omega$ and its point stabilizers $G_{\alpha} = \{g \in G: \alpha^g = \alpha\}$ for $\alpha \in \Omega$. The action on each orbit is equivalent to the action of a point stabilizer on that orbit and we have that the point stabilizers on each orbit are conjugate and $G_{\alpha}^g = G_{\alpha^g}$,  If $G$ acts transitive, this is key to the observation that for a point $\alpha$ the other points fixed by $G_{\alpha}$ are $\alpha^{N_G(G_{\alpha})}$ and by orbit-stabilizer we have $|N_G(G_{\alpha}) : G_{\alpha}|$ points that are fixed by $G_{\alpha}$ and each element that acts on the fixed point of $G_{\alpha}$ normalizes $G_{\alpha}$, hence conjugation and fixed points are closely related. Taking this further, if we have a single $g \in G_{\alpha}$, the total number of points fixed by $g$ is $|N_G(G_{\alpha}) : G_{\alpha}| \cdot k$, where $k$ denotes the number of distinct conjugates of $G_{\alpha}$ that contain $g$. If for example different point stabilizers intersect trivially (a situation that is not that uncommon), $\alpha^{N_G(G_{\alpha})}$ is precisely the set fixed by every single $g \in G_{\alpha}$. This is sometimes useful if you look at groups on which several restrictions on the fixed points of nontrivial elements are imposed.
Also as mentioned by Qiaochu Yuan two subgroups are conjugate iff the action on the cosets are equivalent. Further two transitive actions on two sets $\Omega, \Gamma$ of $G$ are equivalent if every point stabilizer of the action on $\Omega$ is also a point stabilizer of the action on $\Gamma$, and vice versa. Which is just another way to say this by the fundamental relation between conjugate point stabilizers and the points fixed by elements of $G$.