Minimal normal subgroup with order greater than index is characteristic

For an arbitrary group
A fact about::minimal normal subgroup of a group, such that the order of the subgroup is greater than its index, must be a fact about::characteristic subgroup.

For a finite group
A minimal normal subgroup of a finite group, whose order is more than the squareroot of the order of the group, is characteristic.

Related facts

 * 1) Minimal normal subgroup with order not dividing index is characteristic
 * 2) Simple normal subgroup with order not dividing index is fully characteristic