  We first give complete characterizations of the structure of finite group $G$
in which every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is
a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of
$|G|$. Furthermore, we prove that every self-centralizing subgroup (or
non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or
subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$ if and only
if every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$
is a TI-subgroup or subnormal or has $p'$-order. Based on these results, we
obtain the structure of finite group $G$ in which every self-centralizing
subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup
or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$.
