Cyclic subgroup of maximum order is direct factor in finite abelian group

Statement
Suppose $$G$$ is a finite abelian group and $$H$$ is a  cyclic subgroup of $$G$$ such that the order of $$H$$ is greater than or equal to the order of any cyclic subgroup of $$G$$. Then, the following hold:


 * 1) The order of $$H$$ equals the exponent of $$G$$.
 * 2) $$H$$ is a  direct factor of the group $$G$$.