Finite-dominating subgroup

Symbol-free definition
A subgroup of a group is said to be finite-dominating if every finite subgroup of the whole group, is conjugate to a finite subgroup within the subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be finite-dominating if for every finite subgroup $$K$$ of $$G$$, there exists $$g \in G$$ such that $$gKg^{-1} \subset H$$.

Incomparable properties

 * Conjugate-dense subgroup

Examples
An example is $$O(n,\R) \le GL(n,\R)$$. Any finite subgroup (and more generally any compact subgroup) of $$GL(n,\R)$$ can be conjugated to a subgroup inside $$O(n,\R)$$, by finding an invariant symmetric positive definite bilinear form using the method of averages.

Note that $$O(n,\R)$$ is not conjugate-dense.