Number of equivalence classes under rational conjuacy

Definition
Let $$G$$ be a group. The number of equivalence classes under rational conjugacy, also termed the number of conjugacy classes of cyclic subgroups, is defined as follows:


 * 1) It is the number of equivalence classes of elements in $$G$$ under the equivalence relation: $$x \sim y \iff \langle x \rangle$$ is conjugate to $$\langle y \rangle$$.
 * 2) It is the number of equivalence classes of cyclic subgroups of $$G$$ under the equivalence relation of being conjugate subgroups.
 * 3) (For a finite group): It is the number of representations of $$G$$ over a splitting field of characteristic zero that have rational-valued characters and that have no proper nonzero subrepresentation with rational-valued character.
 * 4) (For a finite group): It is the number of irreducible representations of $$G$$ over $$\mathbb{Q}$$, the field of rational numbers. Here, irreducible simply means irreducible over $$\mathbb{Q}$$, not necessarily absolutely irreducible.