Women fighting for a broader presence in the upper levels of management face at least one very personal obstacle: Most workers don’t want them there. Only one-fifth of people surveyed by Gallup this week said they preferred a female boss over a man. One-third preferred a male boss, and the rest had no preference.

The survey, which collected responses from 1,032 adults living in the U.S., found women were more likely than men to want a male boss: 39 percent of women wanted to be led by a man, compared with 26 percent of men.

In the 60 years that Gallup has conducted this survey, women have never preferred a female boss.

Results for surveyed women

The lack of faith in female leadership among some women may be partly related to widespread negative perceptions of women in charge, documented in scientific studies and at length in Facebook Chief Operating Officer Sheryl Sandberg’s book, Lean In.

A growing pile of evidence suggests that women mistrust, and can undermine, one another at work. In a 2010 survey of 142 legal secretaries, not one preferred to work for a woman partner (about 47 percent had no preference). An HBR blog post points out that men still dominate the legal industry, and “if you’re aiming to hitch your wagon to a shooting star, men are a better bet.”

A 2011 survey found that 95 percent of working women felt they were undercut by another woman at least once during their professional life. In a 2008 study, women working for female supervisors experienced more stress than those who had male supervisors, reported the Wall Street Journal.

One reason women may be wary of their own is that they may well face penalties for helping each other out. A July study of hundreds of executives found that senior women who work to advance the careers of other women get more negative performance reviews from their superiors.

It may be tempting to assume that the trend will change as a younger generation climbs the ranks. The Gallup data suggest, though, that many in the world-rocking millennial generation would still rather not have a woman in charge.