Theorem: Recurring Parts of Multiples of Reciprocal of 53

The multiples of $\dfrac 1 {53}$ from $\dfrac 1 {53}$ to $\dfrac {52} {53}$ can be divided into $4$ sets of equal size:
:one where the digits of the recurring part consists of a cyclic permutation of $01886 \, 79245 \, 283$
:one where the digits of the recurring part consists of a cyclic permutation of $03773 \, 58490 \, 566$
:one where the digits of the recurring part consists of a cyclic permutation of $07547 \, 16981 \, 132$
:one where the digits of the recurring part consists of a cyclic permutation of $09433 \, 96226 \, 415$.

Proof:
{{begin-eqn}}
{{eqn | l = 1 / 53
      | r = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3
      | c = 
}}
{{eqn | l = 2 / 53
      | r = 0 \cdotp \dot 03773 \, 58490 \, 56 \dot 6
      | c = 
}}
{{eqn | l = 3 / 53
      | r = 0 \cdotp \dot 05660 \, 37735 \, 84 \dot 9
      | c = 
}}
{{eqn | l = 4 / 53
      | r = 0 \cdotp \dot 07547 \, 16981 \, 13 \dot 2
      | c = 
}}
{{eqn | l = 5 / 53
      | r = 0 \cdotp \dot 09433 \, 96226 \, 41 \dot 5
      | c = 
}}
{{eqn | l = 6 / 53
      | r = 0 \cdotp \dot 11320 \, 75471 \, 69 \dot 8
      | c = 
}}
{{eqn | l = 7 / 53
      | r = 0 \cdotp \dot 13207 \, 54716 \, 98 \dot 1
      | c = 
}}
{{eqn | l = 8 / 53
      | r = 0 \cdotp \dot 15094 \, 33962 \, 26 \dot 4
      | c = 
}}
{{eqn | l = 9 / 53
      | r = 0 \cdotp \dot 16981 \, 13207 \, 54 \dot 7
      | c = 
}}
{{eqn | l = 10 / 53
      | r = 0 \cdotp \dot 18867 \, 92452 \, 83 \dot 0
      | c = 
}}
{{eqn | l = 11 / 53
      | r = 0 \cdotp \dot 20754 \, 71698 \, 11 \dot 3
      | c = 
}}
{{eqn | l = 12 / 53
      | r = 0 \cdotp \dot 22641 \, 50943 \, 39 \dot 6
      | c = 
}}
{{eqn | l = 13 / 53
      | r = 0 \cdotp \dot 24528 \, 30188 \, 67 \dot 9
      | c = 
}}
{{eqn | l = 14 / 53
      | r = 0 \cdotp \dot 26415 \, 09433 \, 96 \dot 2
      | c = 
}}
{{eqn | l = 15 / 53
      | r = 0 \cdotp \dot 28301 \, 88679 \, 24 \dot 5
      | c = 
}}
{{eqn | l = 16 / 53
      | r = 0 \cdotp \dot 30188 \, 67924 \, 52 \dot 8
      | c = 
}}
{{eqn | l = 17 / 53
      | r = 0 \cdotp \dot 32075 \, 47169 \, 81 \dot 1
      | c = 
}}
{{eqn | l = 18 / 53
      | r = 0 \cdotp \dot 33962 \, 26415 \, 09 \dot 4
      | c = 
}}
{{eqn | l = 19 / 53
      | r = 0 \cdotp \dot 35849 \, 05660 \, 37 \dot 7
      | c = 
}}
{{eqn | l = 20 / 53
      | r = 0 \cdotp \dot 37735 \, 84905 \, 66 \dot 0
      | c = 
}}
{{eqn | l = 21 / 53
      | r = 0 \cdotp \dot 39622 \, 64150 \, 94 \dot 3
      | c = 
}}
{{eqn | l = 22 / 53
      | r = 0 \cdotp \dot 41509 \, 43396 \, 22 \dot 6
      | c = 
}}
{{eqn | l = 23 / 53
      | r = 0 \cdotp \dot 43396 \, 22641 \, 50 \dot 9
      | c = 
}}
{{eqn | l = 24 / 53
      | r = 0 \cdotp \dot 45283 \, 01886 \, 79 \dot 2
      | c = 
}}
{{eqn | l = 25 / 53
      | r = 0 \cdotp \dot 47169 \, 81132 \, 07 \dot 5
      | c = 
}}
{{eqn | l = 26 / 53
      | r = 0 \cdotp \dot 49056 \, 60377 \, 35 \dot 8
      | c = 
}}
{{eqn | l = 27 / 53
      | r = 0 \cdotp \dot 50943 \, 39622 \, 64 \dot 1
      | c = 
}}
{{eqn | l = 28 / 53
      | r = 0 \cdotp \dot 52830 \, 18867 \, 92 \dot 4
      | c = 
}}
{{eqn | l = 29 / 53
      | r = 0 \cdotp \dot 54716 \, 98113 \, 20 \dot 7
      | c = 
}}
{{eqn | l = 30 / 53
      | r = 0 \cdotp \dot 56603 \, 77358 \, 49 \dot 0
      | c = 
}}
{{eqn | l = 31 / 53
      | r = 0 \cdotp \dot 58490 \, 56603 \, 77 \dot 3
      | c = 
}}
{{eqn | l = 32 / 53
      | r = 0 \cdotp \dot 60377 \, 35849 \, 05 \dot 6
      | c = 
}}
{{eqn | l = 33 / 53
      | r = 0 \cdotp \dot 62264 \, 15094 \, 33 \dot 9
      | c = 
}}
{{eqn | l = 34 / 53
      | r = 0 \cdotp \dot 64150 \, 94339 \, 62 \dot 2
      | c = 
}}
{{eqn | l = 35 / 53
      | r = 0 \cdotp \dot 66037 \, 73584 \, 90 \dot 5
      | c = 
}}
{{eqn | l = 36 / 53
      | r = 0 \cdotp \dot 67924 \, 52830 \, 18 \dot 8
      | c = 
}}
{{eqn | l = 37 / 53
      | r = 0 \cdotp \dot 69811 \, 32075 \, 47 \dot 1
      | c = 
}}
{{eqn | l = 38 / 53
      | r = 0 \cdotp \dot 71698 \, 11320 \, 75 \dot 4
      | c = 
}}
{{eqn | l = 39 / 53
      | r = 0 \cdotp \dot 73584 \, 90566 \, 03 \dot 7
      | c = 
}}
{{eqn | l = 40 / 53
      | r = 0 \cdotp \dot 75471 \, 69811 \, 32 \dot 0
      | c = 
}}
{{eqn | l = 41 / 53
      | r = 0 \cdotp \dot 77358 \, 49056 \, 60 \dot 3
      | c = 
}}
{{eqn | l = 42 / 53
      | r = 0 \cdotp \dot 79245 \, 28301 \, 88 \dot 6
      | c = 
}}
{{eqn | l = 43 / 53
      | r = 0 \cdotp \dot 81132 \, 07547 \, 16 \dot 9
      | c = 
}}
{{eqn | l = 44 / 53
      | r = 0 \cdotp \dot 83018 \, 86792 \, 45 \dot 2
      | c = 
}}
{{eqn | l = 45 / 53
      | r = 0 \cdotp \dot 84905 \, 66037 \, 73 \dot 5
      | c = 
}}
{{eqn | l = 46 / 53
      | r = 0 \cdotp \dot 86792 \, 45283 \, 01 \dot 8
      | c = 
}}
{{eqn | l = 47 / 53
      | r = 0 \cdotp \dot 88679 \, 24528 \, 30 \dot 1
      | c = 
}}
{{eqn | l = 48 / 53
      | r = 0 \cdotp \dot 90566 \, 03773 \, 58 \dot 4
      | c = 
}}
{{eqn | l = 49 / 53
      | r = 0 \cdotp \dot 92452 \, 83018 \, 86 \dot 7
      | c = 
}}
{{eqn | l = 50 / 53
      | r = 0 \cdotp \dot 94339 \, 62264 \, 15 \dot 0
      | c = 
}}
{{eqn | l = 51 / 53
      | r = 0 \cdotp \dot 96226 \, 41509 \, 43 \dot 3
      | c = 
}}
{{eqn | l = 52 / 53
      | r = 0 \cdotp \dot 98113 \, 20754 \, 71 \dot 6
      | c = 
}}
{{end-eqn}}
{{qed}}