Washington (CNN) A new poll from Quinnipiac University finds Donald Trump with a significant lead over his closest rivals for the Republican nomination for president, and provides further evidence the field of 16 candidates vying for the GOP nomination has a clear top tier.

Trump tops the field at 20% in the poll, Quinnipiac's first since the businessman announced his run for the presidency in June. He is followed not by former Florida Gov. Jeb Bush, but by Wisconsin Gov. Scott Walker at 13%. Bush follows Walker at 10%. This trio comprises the only Republican candidates in double-digits in each of the last four publicly released national polls.

Behind them, Quinnipiac finds a group of four tied at 6%: retired neurosurgeon Ben Carson, former Arkansas Gov. Mike Huckabee, Kentucky Sen. Rand Paul and Florida Sen. Marco Rubio.Texas Sen. Ted Cruz and Ohio Gov. John Kasich are just a shade behind at 5%.

But Trump's success with some comes with resounding negatives from others. He tops the list of candidates Republican voters say they definitely could not support: 30% say they would not back him, up from 21% saying so in Quinnipiac's late-May survey. That figure is even higher among Republican women, 37% of whom say they would never vote for Trump.

And in general election match-ups, Trump lags far behind three potential Democratic opponents: He trails both Democratic frontrunner Hillary Clinton and Vice President Joe Biden in head-to-head matchups by 12 points, while Vermont Sen. Bernie Sanders holds an 8-point lead over Trump.

Bush and Walker each run about even with Clinton and Biden in their head-to-heads, while Sanders narrowly trails the two Republicans.

On the Democratic side, Quinnipiac's poll finds Clinton solidly ahead, with support from 55% of Democratic and Democratic-leaning registered voters. That puts her nearly 40 points ahead of her closest competitor, Sanders at 17%.

Adding the Quinnipiac poll to the CNN Poll of Polls, we find that across the last five national telephone polls, Trump averages 19% support, followed by Bush at 13% and Walker at 12%. Paul and Rubio stand at 6% each, Carson, Cruz and Huckabee at 5%, with New Jersey Gov. Chris Christie and Kasich rounding out the top 10 at 3%.

Beyond the horse race, the Quinnipiac University poll finds Trump's favorability rating tilts sharply negative among all registered voters (59% unfavorable to 27% favorable) but more positive among Republicans (50% favorable to 33% unfavorable).

Nearly 6-in-10 say Trump is not honest and trustworthy (58%) and slightly more (63%) say he does not care about the needs and problems of people like you. Still, Trump's appeal lies more in his assertiveness than empathy: 58% say he has strong leadership qualities.

The Quinnipiac University survey has a margin of sampling error of plus or minus 3.7 percentage points for results among 710 Republican and Republican-leaning registered voters. For results among the full sample of 1,644 registered voters, it is plus or minus 2.4 points.