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Spoiler

Not accounting for rotations, there are 48 solutions to this problem.

If you account for the fact that this can be rotated 6 times, I see 8 unique solutions.

Below are the 8 unique solutions (with each of the 5 other equal rotations shown)...

Solution 1:    A=1, B=2, C=9, D=11, E=6, F=10, G=14, H=4, I=8, J=3, K=5, L=12, M=13
A=13, B=12, C=5, D=3, E=8, F=4, G=14, H=10, I=6, J=11, K=9, L=2, M=1
A=2, B=8, C=10, D=9, E=1, F=3, G=14, H=11, I=13, J=5, K=4, L=6, M=12
A=12, B=6, C=4, D=5, E=13, F=11, G=14, H=3, I=1, J=9, K=10, L=8, M=2
A=8, B=13, C=3, D=10, E=2, F=5, G=14, H=9, I=12, J=4, K=11, L=1, M=6
A=6, B=1, C=11, D=4, E=12, F=9, G=14, H=5, I=2, J=10, K=3, L=13, M=8

Solution 2:   A=1, B=3, C=8, D=12, E=5, F=10, G=14, H=4, I=9, J=2, K=6, L=11, M=13
A=13, B=11, C=6, D=2, E=9, F=4, G=14, H=10, I=5, J=12, K=8, L=3, M=1
A=3, B=9, C=10, D=8, E=1, F=2, G=14, H=12, I=13, J=6, K=4, L=5, M=11
A=11, B=5, C=4, D=6, E=13, F=12, G=14, H=2, I=1, J=8, K=10, L=9, M=3
A=9, B=13, C=2, D=10, E=3, F=6, G=14, H=8, I=11, J=4, K=12, L=1, M=5
A=5, B=1, C=12, D=4, E=11, F=8, G=14, H=6, I=3, J=10, K=2, L=13, M=9

Solution 3:    A=1, B=5, C=12, D=8, E=3, F=4, G=14, H=10, I=11, J=6, K=2, L=9, M=13
A=13, B=9, C=2, D=6, E=11, F=10, G=14, H=4, I=3, J=8, K=12, L=5, M=1
A=5, B=11, C=4, D=12, E=1, F=6, G=14, H=8, I=13, J=2, K=10, L=3, M=9
A=9, B=3, C=10, D=2, E=13, F=8, G=14, H=6, I=1, J=12, K=4, L=11, M=5
A=11, B=13, C=6, D=4, E=5, F=2, G=14, H=12, I=9, J=10, K=8, L=1, M=3
A=3, B=1, C=8, D=10, E=9, F=12, G=14, H=2, I=5, J=4, K=6, L=13, M=11

Solution 4:    A=1, B=6, C=11, D=9, E=2, F=4, G=14, H=10, I=12, J=5, K=3, L=8, M=13
A=13, B=8, C=3, D=5, E=12, F=10, G=14, H=4, I=2, J=9, K=11, L=6, M=1
A=6, B=12, C=4, D=11, E=1, F=5, G=14, H=9, I=13, J=3, K=10, L=2, M=8
A=8, B=2, C=10, D=3, E=13, F=9, G=14, H=5, I=1, J=11, K=4, L=12, M=6
A=12, B=13, C=5, D=4, E=6, F=3, G=14, H=11, I=8, J=10, K=9, L=1, M=2
A=2, B=1, C=9, D=10, E=8, F=11, G=14, H=3, I=6, J=4, K=5, L=13, M=12

Solution 5:    A=2, B=4, C=6, D=13, E=5, F=11, G=14, H=3, I=9, J=1, K=8, L=10, M=12
A=12, B=10, C=8, D=1, E=9, F=3, G=14, H=11, I=5, J=13, K=6, L=4, M=2
A=4, B=9, C=11, D=6, E=2, F=1, G=14, H=13, I=12, J=8, K=3, L=5, M=10
A=10, B=5, C=3, D=8, E=12, F=13, G=14, H=1, I=2, J=6, K=11, L=9, M=4
A=9, B=12, C=1, D=11, E=4, F=8, G=14, H=6, I=10, J=3, K=13, L=2, M=5
A=5, B=2, C=13, D=3, E=10, F=6, G=14, H=8, I=4, J=11, K=1, L=12, M=9

Solution 6:    A=2, B=5, C=13, D=6, E=4, F=3, G=14, H=11, I=10, J=8, K=1, L=9, M=12
A=12, B=9, C=1, D=8, E=10, F=11, G=14, H=3, I=4, J=6, K=13, L=5, M=2
A=5, B=10, C=3, D=13, E=2, F=8, G=14, H=6, I=12, J=1, K=11, L=4, M=9
A=9, B=4, C=11, D=1, E=12, F=6, G=14, H=8, I=2, J=13, K=3, L=10, M=5
A=10, B=12, C=8, D=3, E=5, F=1, G=14, H=13, I=9, J=11, K=6, L=2, M=4
A=4, B=2, C=6, D=11, E=9, F=13, G=14, H=1, I=5, J=3, K=8, L=12, M=10

Solution 7:    A=3, B=4, C=5, D=13, E=6, F=12, G=14, H=2, I=8, J=1, K=9, L=10, M=11
A=11, B=10, C=9, D=1, E=8, F=2, G=14, H=12, I=6, J=13, K=5, L=4, M=3
A=4, B=8, C=12, D=5, E=3, F=1, G=14, H=13, I=11, J=9, K=2, L=6, M=10
A=10, B=6, C=2, D=9, E=11, F=13, G=14, H=1, I=3, J=5, K=12, L=8, M=4
A=8, B=11, C=1, D=12, E=4, F=9, G=14, H=5, I=10, J=2, K=13, L=3, M=6
A=6, B=3, C=13, D=2, E=10, F=5, G=14, H=9, I=4, J=12, K=1, L=11, M=8

Solution 8:    A=3, B=6, C=13, D=5, E=4, F=2, G=14, H=12, I=10, J=9, K=1, L=8, M=11
A=11, B=8, C=1, D=9, E=10, F=12, G=14, H=2, I=4, J=5, K=13, L=6, M=3
A=6, B=10, C=2, D=13, E=3, F=9, G=14, H=5, I=11, J=1, K=12, L=4, M=8
A=8, B=4, C=12, D=1, E=11, F=5, G=14, H=9, I=3, J=13, K=2, L=10, M=6
A=10, B=11, C=9, D=2, E=6, F=1, G=14, H=13, I=8, J=12, K=5, L=3, M=4
A=4, B=3, C=5, D=12, E=8, F=13, G=14, H=1, I=6, J=2, K=9, L=11, M=10

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You forgot reflections. You're solution 4 is a reflection of solution 1. Likewise 3 of 2, 6 of 5, and 8 of 7.

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