Citadel LLC, which also has a hedge fund, executed just over 1.1 billion shares in its brokerage the week of April 4, according to the Financial Industry Regulatory Authority.

At that scale, if Citadel was a public stock exchange, it would be the seventh-largest out of 13, ahead of some exchanges run by Intercontinental Exchange Inc's (>> Intercontinental Exchange Inc) NYSE Group, Nasdaq Inc (>> Nasdaq Inc), and Bats Global Markets , which operate multiple trading platforms, according to data from Bats.

Citadel runs a type of off-exchange trading platform, or "dark pool," known as an internalizer, in which it sends retail stock orders that it has purchased as part of its market making business and matches them against its institutional orders, as well as with orders from client firms.

Nearly all retail "mom-and-pop" stock orders bypass public exchanges. Instead, retail brokerages such as Charles Schwab Corp (>> Charles Schwab Corp), TD Ameritrade (>> TD Ameritrade Holding Corp.), and E*Trade (>> E*TRADE Financial Corp), sell their clients' orders to retail market makers like Citadel.

Critics of what is called "payment for order flow" say it harms the market because it prevents firms that trade on exchanges from accessing retail orders. Proponents of the practice say the payments allow retail brokerages to offer their clients cheap trading and services.

The FINRA report, which will be released weekly going forward with a three-week lag in the data, was significant because it was the first time Wall Street's self-funded watchdog made the trading statistics of internalizers and stock wholesalers public.

"By putting out this data, it gives us a sense of who is doing what and it gives us a better understanding of how stock trades take place," said James Angel, a finance professor at Georgetown University.

Other firms atop the list included market making firms KCG (>> KCG Holdings Inc) and Susquehanna's G1 Execution Services, as well as bulge-bracket banks such as Goldman Sachs (>> Goldman Sachs Group Inc) and UBS .

The data also gives insight into what retail investors are trading, said Chris Nagy, who heads trading-analytics firm KOR Group.

For instance, more than 58 million shares of Facebook Inc (>> Facebook Inc) and nearly 47 million shares of Apple Inc (>> Apple Inc.) were traded that week on the private platforms, which do not include dark pools known as Alternative Trading Systems.

(Reporting by John McCrank; Editing by Bill Trott)

By John McCrank