Why is the difference between a group and an organization?

For example,

We usually say gay rights groups, not gay rights organizations.

An organization is a group of people, that has a structure (i.e. division of power and responsiblities) and strives to achieve goals that individuals acting alone cannot reach.
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I guess that means calling gay rights groups gay rights organizations is also correct because they are likely to have leaders appointed to carry out the group's missions.
Yes! I believe so.
Is it correct to say then that an organization has 'structure', such as a constitution that must be followed, whereas a 'group' is mainly loosely formed by several individuals. Is there any way an 'organization' can be referred to as a 'group'?
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Group is to organization as hypernym is to hyponym.