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We first give complete characterizations of the structure of finite group $G$in which every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) isa TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of$|G|$. Furthermore, we prove that every self-centralizing subgroup (ornon-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup orsubnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$ if and onlyif 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, weobtain the structure of finite group $G$ in which every self-centralizingsubgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroupor subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$.

Finite groups in which every self-centralizing subgroup is a TI-subgroup or subnormal or has $p'$-order