Characteristic subgroup

Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.

Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic.

Every subgroup that is fully characteristic is certainly strictly characteristic and characteristic; but a characteristic or even strictly characteristic subgroup need not be fully characteristic.

The relationship amongst these subgroup properties can be expressed as: