Table 2: Partial results in groups of order 𝑣 by Criterion 1. 𝐶 | 𝐺 / 𝑁 | = 𝑥 and parameters with asterisk indicate new results. ? means the number of groups of order 𝑣 is unknown.

( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐺 / 𝑁 | Factoring of 𝑝 in [ 𝜁 | 𝐺 / 𝑁 | ] No. of groups of order 𝑣 No. of groups ruled outSolutions in 𝐺 / 𝑁

1(171, 51, 15)6 2,319 𝑎 9 1 ( m o d 1 9 )
𝑎 = 2 , 3
52 6 + 3 𝑥
2(155, 56, 20)6 2,331 3 1 5 1 ( m o d 3 1 ) 2 factors trivially [20]21 6 + 2 𝑥
3(231, 70, 21)7 777 7 5 1 ( m o d 1 1 ) 7 factors trivially in [ 𝜁 𝑏 ] , 𝑏 = 1 1 , 7 7 21 7 + 7 𝑥 in 𝐶 7 ; None in 𝐶 7 7
4 (2325, 84, 3)9 331 3 1 5 1 ( m o d 3 1 ) 103 9 + 3 𝑥 ,
5 (10101, 101, 1)10 2,537 𝑎 1 8 1 ( m o d 3 7 )
𝑎 = 2 , 3
145 1 0 + 3 𝑥
6(715, 154, 33)11 11143 1 1 6 1 ( m o d 1 3 ) 11 factors trivially in [ 𝜁 𝑏 ] , 𝑏 = 1 3 , 1 4 3 21 1 1 + 1 1 𝑥 in 𝐶 1 3 ; None in 𝐶 1 4 3
7 (7155, 147, 3)12 2,353 𝑎 2 6 1 ( m o d 5 3 )
𝑎 = 2 , 3
?? 1 2 + 3 𝑥
8 (38613, 197, 1)14 2,7211 𝑎 1 0 5 1 ( m o d 2 1 1 )
𝑎 = 2 , 7
52 1 4 + 𝑥
9 (5859, 203, 7)14 2,731 7 1 5 1 ( m o d 3 1 )
2 factors trivially [20]
?? 1 4 + 7 𝑥
1 0 (903, 287, 91)14 2,743 𝑎 1 ( m o d 4 3 )
𝑎 = 2 7 , 7 3
72 1 4 + 7 𝑥
1 1 (2255, 392, 68)18 2,341 𝑎 1 ( m o d 4 1 )
𝑎 = 2 1 0 , 3 4
72 1 8 + 1 0 𝑥
1 2 (160401, 401, 1)20 2,5421 𝑎 1 ( m o d 4 2 1 )
𝑎 = 2 2 1 0 , 5 1 0 5
52 2 0 + 𝑥
1 3 (23607, 407, 7)20 2,561 𝑎 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 5 1 5
115 2 0 + 7 𝑥
1 4 (22451, 450, 9)21 3,7157 𝑎 1 ( m o d 1 5 7 )
𝑎 = 3 3 9 , 7 2 6
21 2 1 + 3 𝑥
1 5 (2619, 561, 120)21 3,797 𝑎 1 ( m o d 9 7 )
𝑎 = 3 2 4 , 7 4 8
135 2 1 + 6 𝑥
16(2211, 715, 231)22 2,11 67 𝑎 3 3 1 ( m o d 6 7 )
𝑎 = 2 , 1 1
41 2 2 + 1 1 𝑥
17(7450, 573, 44)23 23 149 2 3 7 4 1 ( m o d 1 4 9 ) 105 2 3 + 4 𝑥
18(111555, 579, 3)24 2,3 67 𝑎 1 ( m o d 6 7 )
𝑎 = 2 3 3 , 3 1 1
?? 2 4 + 9 𝑥
19(37961, 585, 9)24 2,3 29 𝑎 1 4 1 ( m o d 2 9 )
𝑎 = 2 , 3
21 2 4 + 2 1 𝑥
20(23247, 591, 15)24 2,3 41 𝑎 1 ( m o d 4 1 )
𝑎 = 2 1 0 , 3 4
?? 2 4 + 1 5 𝑥
21(25641, 641, 16)25 5 37 5 1 8 1 ( m o d 3 7 ) 144 2 5 + 1 8 𝑥