Allen then announced a new power structure, saying the team's next coach will not have the all-encompassing role held by Shanahan. Allen said he has an "open list" as he begins the search for a replacement, including NFL and college coaches, but that he will assume of the duties of having final say over the roster.
"That power will be with me," Allen said.
Shanahan went 24-40 in four seasons in Washington and had one year remaining on his five-year, $35 million contract.
Snyder is seeking his eighth head coach for his 16th season as an NFL owner a span that includes just four winning seasons, two playoff victories and seven last-place finishes in the NFC East.
But Allen said Snyder is "still one of the great fans" and isn't the source of the team's problems.
"It's not Dan calling the plays, it's not Dan picking the plays. It's the people he's hired," Allen said. "It's our job to actually turn this team into a winner."
Shortly after his meeting with Snyder, Shanahan made a five-minute statement thanking fans, players, reporters and Snyder. Shanahan did not take questions, and he defended his efforts in rebuilding the team while repeating his assertion that an NFL-levied salary cap penalty hindered his ability to improve the roster even more.
"We're better off today than we were four years ago," Shanahan said.
Shanahan's career regular-season record is 170-138 over 20 seasons with the Los Angeles Raiders, Denver Broncos and Washington, but his two worst years have come in Washington 5-11 in 2011 and this year's 3-13. He captured Super Bowls titles with quarterback John Elway and the Broncos after the 1997 and 1998 seasons, but he won only one playoff game over his final 10 years in Denver and was fired after the 2008 season.
The selection of Heisman Trophy winner Griffin with the No. 2 overall draft pick and a season-ending seven-game winning streak propelled Washington to a 10-6 record in 2012, their first division title in 13 years.
But Griffin was injured in the playoff loss to the Seattle Seahawks and required major knee surgery days later, setting the stage for a year of conflict as the quarterback vowed to return in record time and felt empowered enough to openly challenge some of his Shanahan's decisions.
Griffin returned for Week 1 of the regular season just as he said he would but he wasn't the same dynamic player who won the NFL's Offensive Rookie of the Year award in 2012.
Griffin also declined to answer reporters' questions Monday. In brief remarks, he said that Shanahan "has taught me a lot in just two years."
Washington also struggled on defense and special teams, with Shanahan repeatedly citing the handicap of the two-year, $36 million salary cap penalty imposed by the league for the way Washington restructured contracts during the uncapped year of 2010.
Shanahan eventually benched Griffin for the final three games of the season, a move that made Griffin clearly unhappy.
Also dismissed Monday were several assistant coaches, including offensive coordinator Kyle Shanahan, the head coach's son.
Snyder's search for a new coach presents plenty of intrigue. He's tried nearly every angle: the hot college coach with no NFL experience (Steve Spurrier), the franchise icon (Joe Gibbs), the promising youngish coordinator (Jim Zorn) and the established demand-control-over-everything big names (Marty Schottenheimer and Shanahan).
Shanahan demanded and received contractual control over all football matters when he joined Washington, and he repeatedly emphasized the need to run a disciplined organization with a sense of decorum. Snyder met Shanahan's requests to upgrade the team's facility, spending millions on a new practice bubble and other amenities.
Shanahan weeded out the disgruntled players most notably Albert Haynesworth but ultimately was unable to stymie what he called the "circus atmosphere" that has permeated Washington under Snyder.
Leaks, rumors and power struggles were just as bad as before, as were the losses. Washington's 2013 record was its worst since 1994, and the season-ending eight-game losing streak is its longest in more than 50 years. Shanahan leaves with the same regular-season winning percentage (.375) in Washington as Spurrier and Zorn.
"Any coach, any organization knows in pro sports you need to eliminate distractions," Allen said. "Instead, we created our own distractions. And it distracted from our play on the field."
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