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Right: The imposing exterior of the Melbourne Club

Background information

(The following information is drawn from an analysis written by Malcom Knox and published in The Monthly No 50 in October, 2009.  This material has been abridged.  The full text of the article, titled 'In Retreat', can be found at https://www.themonthly.com.au/issue/2009/october/1268634164/malcolm-knox/retreat)
NOTE: The above link will take you to The Monthly, but you may need to use the site's internal search to find the item.

Australia has approximately 30 single-sex clubs. The oldest is Sydney's The Australian Club, founded in 1838, followed by the Melbourne Club, which was founded a year later.
These clubs and their imitators had a similar founding principle: a cocoon for the landed gentry. Melbourne gained its Savage and Athenaeum clubs, as well as its own branch of The Australian Club (established 1878), while the less clubbable Perth, Hobart, Brisbane and Adelaide subsequently formed their own exclusively male clubs.
Sydney's Union Club was founded in 1857 by men from The Australian Club who opposed the overuse of the blackball within the ranks of the original club.  The Union Club has since merged with the Universities and Schools Club and now allows female 'associate' members. As a response to its two gender membership policy, the Union Club lost reciprocal privileges with two London gentlemen's clubs.
The Queen's and Women's clubs, in Sydney, maintain a ladies-only policy, as do the Alexandra and Lyceum clubs in Melbourne and the Moreton Club in Brisbane. Many members have spouses at the men's clubs. In Sydney's Palm Beach, the exclusive clubs for former members of the surf club are divided by sex. The Cabbage Tree Club, for men, and the Pacific Club, for women, sit on the beach front side by side like a monastery and a convent.
With most of these clubs, money is not the primary basis for entry. The barrier to entry is not high annual membership fees (which range mostly from $500 to $2000), but rather the hurdle of sponsoring and refereeing and vetting: the months and sometimes years of interviews and checking before a nominee is accepted for membership. The clubs are said to primarily select by type or class, not wealth.