Maximal implies cocentral or self-centralizing

Statement
A maximal subgroup of a group is either cocentral (i.e., its product with the center of the group equals the whole group) or self-centralizing (i.e., it contains its centralizer in the whole group).

Weaker facts

 * Maximal implies central factor or self-centralizing
 * Maximal implies cocentral or contains the center