On subgroups of finite exponent in groups

We investigate properties of groups with subgroups of finite exponent and prove that a non-perfect group $G$ of infinite exponent with all proper subgroups of finite exponent has the following properties:
$(1)$ $G$ is an indecomposable $p$-group,
$(2)$ if the derived subgroup $G'$ is non-perfect, then $G/G''$ is a group of Heineken-Mohamed type.
We also prove that a non-perfect indecomposable group $G$ with the non-perfect locally nilpotent derived subgroup $G'$ is a locally finite $p$-group.