We adapt techniques used to investigate the divisibility of class numbers in families of algebraic number fields to study related topics in three particular families of number fields. Let $r$ be a positive integer. We prove that the quartic family contains infinitely many distinct fields whose class group contains a cyclic subgroup of order $r$. We further show that when $r$ is odd, an ideal class generating this subgroup does not come from the field's unique quadratic subfield. When $r$ is odd, we show that the family of sextics contains infinitely many distinct fields whose class group contains a cyclic subgroup of order $r$ generated by an ideal class which does not come from either the quadratic or cubic subfields of the sextic field. Finally, we extend and modify the techniques to handle a family of non-Galois cubic extensions of $\mathbb{Q}.$ We prove that here too the result holds; this family contains infinitely many distinct fields whose class group contains a cyclic subgroup of order $r.$