Hely's side sit at the foot of the Ligue 1 table after taking just two points from their opening seven games.
Sochaux are already four points adrift of safety at this early point in 2013-14, with their heavy defeat in Brittany undoubtedly the lowest point of what has been a dismal campaign thus far.
Hely took over from Mehmed Bazdarevic in March 2012, steering them to survival before repeating the feat on the final day of last season.
However, the 48-year-old appears to have decided the task is beyond him this time around.
"For the last 20 months, I have given everything along with my technical staff, the players and the board," he said.
"I have saved the club twice but, in the interests of the club, I have offered my resignation to the board."
Club president Laurent Pernet has yet to accept Hely's resignation, however, insisting that he would "take time" before coming to a final decision on the offer.
Sochaux will aim to secure their first win of the season when they welcome Valenciennes to the Stade Auguste Bonal on Saturday.
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