Notation for subgroup properties

Notations for subgroup properties

• $\le ,\ge$ denotes the tautology subgroup property, viz $H\le G$ or $G\ge H$ means that $H$ is a subgroup of $G$ (with no further restrictions)
• $◃,▹$ denotes the subgroup property of being a normal subgroup, viz $H◃G$ or $G▹H$ means that $H$ is a normal subgroup of $G$.
• $char$ denotes the subgroup property of being a characteristic subgroup, viz $HcharG$ means that $H$ is a characteristic subgroup of $G$.
• $◃◃,▹▹$ denotes the subgroup property of being a subnormal subgroup, viz $H◃◃G$ or $G▹▹H$ means that $H$ is a subnormal subgroup of $G$.
• ${\le }_{max},{\ge }_{max}$ denotes the subgroup property of being a maximal subgroup, viz $H{\le }_{max}G,H{\ge }_{max}H$ iff $H$ is a maximal subgroup of $G$