However, a three-man FA panel has rejected their claims, meaning the 10-point penalty and £50,000 fine for irregular payments to agents will stand.
Luton were hit with the latest 20-point sanction after applying to come out of administration without a CVA (Company Voluntary Arrangement).
A statement from the FA read: "It is highly unfortunate for Luton Town FC and their loyal fans that shortly after the FA Regulatory Commission reached its decision, the Football League quite separately imposed a 20 point deduction for entirely different actions by Luton Town FC.
"The Football Association Appeal Board takes the view that the Football League's decision to deduct 20 points was entirely a matter for the Football League.
"The FA Regulatory Commission's responsibility was to decide the appropriate punishment for Luton Town's admitted breaches of The FA's own rules and regulations.
"The deduction of 10 points was a heavy sanction, as it was intended to be, but was not excessive as a reflection of the seriousness of the breaches and the need to deter such conduct within football clubs."