Getting a chance to lead one of the four major broadcast networks — NBC, CBS, ABC or Fox — is rare. A woman has never had the CEO spot at any of them.

That should change, and CBS Corp. should be the company that changes it.

CBS started a search for a CEO after Les Moonves departed following sexual misconduct allegations.

Joe Ianniello, Moonves' chief operating officer, is acting as the company's interim CEO and Dick Parsons, the company's interim chairman, had been leading the effort to decide on a full-time replacement until his sudden departure was announced Sunday night, citing health concerns.

CBS has spoken with executive search firms including Heidrick & Struggles, Korn Ferry and Spencer Stuart to help with the search, which will kick off in earnest in the next few weeks, according to people familiar with the matter who declined to identify themselves because the process is private. They spoke before the Parsons' resignation was announced. A CBS spokesman declined to give a timeline on the search process, though people familiar with the matter expect a decision by early 2019.

The search will be methodical, as many of CBS' directors are new and have to figure out what the company should do next. Controlling shareholder Shari Redstone named six new independent directors in September after the Moonves scandal. These board members need to familiarize themselves with the company and determine a strategic direction: Should CBS stay as a stand-alone company or buyer of smaller media companies, or should it look to sell? The answer will inform the CEO search.

Whichever way they go, selecting a female executive for the job would have obvious and immediate positive effects on a company culture that has been mired in sexual misconduct scandals.

In addition to Moonves, "CBS This Morning" co-host Charlie Rose lost his job after The Washington Post reported eight women said he sexually harassed them. Three women have subsequently sued him and CBS News, which they claim knew about Rose's behavior for decades. Jeff Fager, the former chairman of CBS News and former executive producer of "60 Minutes," also lost his job after threatening a reporter for looking into allegations of sexual harassment against him.

Hiring a woman would also break a particularly resistant glass ceiling. Very few media companies are run by women.

There are a number of viable candidates — too many to list here. But CNBC spoke with some current media executives about whom they'd recommend for the job: