What is the difference between "The X Society" and "The Society of X"? [closed]

What is the difference between the following?

1. "The Gilbert Keith Chesterton Society" vs "The Society of Gilbert Keith Chesterton"
2. "The Last Mohican Society" vs "The Society of the Last Mohican"
3. "The Last of the Mohicans Society" vs "The Society of the Last of the Mohicans"

Which sounds better and why, in each pair?

What about the capitalization of "the last Mohican" / "the last of the Mohicans"?
Would you capitalize "the", "last", both or none?

First, when you are referring to an existing entity like a society you should use its exact name. If the society is called "The Gilbert Keith Chesterton Society" then you should call it exactly that. No variants.

When it comes to naming a society, then "The X Society" implies an interest or relationship to X. "The Society of X" implies composition of X.

To take an example "The Society of Painters" would be assumed to have members who were all painters. "The Painters Society" might be composed of painters, but it might also be composed of people who were interested in painters.

"The Society of The Last of the Mohicans" would be assumed to have members who were the last of the Mohicans. "The Last of the Mohicans Society" would be for people who were interested in The Last of the Mohicans.

The "Society of Gilbert Keith Chesterton" or "The Society of the Last Mohican" would have very limited membership indeed. If we are naming a society for people interested in GKC or LoM then use the first form.

1. While the "Society of " form is certainly common, the " Society" is more common in English, presumably because English puts adjectives before the noun they modify (the red door).
2. Proper names of organizations would normally follow title case, thus The Last of the Mohicans, with "the" capitalized because it's the initial word, or Return to Oz, with Oz capitalized even though it's a short word because it's a proper noun.
3. When referencing such a Society, make sure you know whether "The" is part of its name or not, and capitalize accordingly. For instance: It appears that The Gilbert Keith Chesterton Society includes the "the," but that the Society of St. Andrew does not. (If you're asking with an eye to making up your own new club... your pick.)

Now, what you didn't ask, but what you've got me wondering: Say you've got a club dedicated to the appreciation of the James Fenimore Cooper novel. Do you italicize the title within the society name? "Welcome to The Last of the Mohicans Society"? Should it get an extra "the": "Welcome to the The Last of the Mohicans Society"?