This order is by no means ideal; but it may help to provide some framework for a re-editing of the main topic. Nor are the entries below considered a definitive set of topics; rather they reflect the current state of group theory articles on Wikipedia. Feel free to add or subtract from this list as appropriate.
History, major contributors
Famous problems
Applications
Definitions in common with other mathematical ideas
Mathematical objects which have (or make use of) a group operation
Category theory etc.
Abstract algebra
Other mathematical disciplines which make great use of groups
Basic properties of groups and subgroups
Group homomorphisms
Basic group structures and operations
Basic building block of groups, and classifications
Simple groups and their classification
Understanding groups as permutation groups