Table 5: Partial results in groups of order 𝑣 by Criterion 3. Parameters with asterisk indicate new results.? means the number of groups of order 𝑣 is unknown.

( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 Factoring of 𝑝 in [ 𝜁 𝑞 ] No. of groups of order 𝑣 No. of groups ruled out Solutions in 𝐻

1(40, 13, 4) 3 3 3 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
14 10 3 + 𝐻 , | 𝐻 | = 1 0
2 (400, 57, 8) 7 7 7 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
221 166 7 + 5 𝐻 , | 𝐻 | = 1 0
3 (280, 63, 14) 7 7 7 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
40 30 7 + 7 𝐻 , | 𝐻 | = 1 0
4 (220, 73, 24) 7 7 7 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
15 7 7 + 8 𝐻 , | 𝐻 | = 1 0
5 (820, 91, 10) 9 3 3 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
20 7 9 + 1 0 𝐻 , | 𝐻 | = 1 0
6 (540, 99, 18) 9 3 3 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
119 56 9 + 9 𝐻 , | 𝐻 | = 1 0
7 (3876, 125, 4) 11 11 1 1 1 ( m o d 𝑏 )
𝑏 = 3 , 6 , 1 2
40 32 1 1 + 1 9 𝐻 , | 𝐻 | = 6
8 (1464, 133, 12) 11 11 1 1 2 1 ( m o d 6 1 ) 61 30 1 1 + 𝐻 , | 𝐻 | = 1 2 2
9 (988, 141, 30) 11 11 1 1 6 1 ( m o d 𝑏 )
𝑏 = 1 3 , 2 6
11 5 1 1 + 5 𝐻 , | 𝐻 | = 2 6
1 0 (756, 151, 30) 11 11 1 1 1 ( m o d 𝑏 )
𝑏 = 3 , 6 , 1 2
189 96 1 1 + 2 7 𝐻 , | 𝐻 | = 6
1 1 (2380, 183, 14) 13 13 1 3 2 1 ( m o d 1 7 ) 35 15 1 3 + 5 𝐻 , | 𝐻 | = 3 4
1 2 (1056, 211, 42) 13 13 1 3 5 1 ( m o d 1 1 ) 1028 995 1 3 + 9 𝐻 , | 𝐻 | = 2 2
1 3 (1456, 195, 26) 13 13 1 3 1 ( m o d 7 ) 179 171 1 3 + 1 3 𝐻 , | 𝐻 | = 1 4
1 4 (1380, 197, 28) 13 13 1 3 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
29 15 1 3 + 2 1 𝐻 , | 𝐻 | = 1 0
1 5 (1548, 273, 48) 15 3, 5 𝑎 2 1 1 ( m o d 4 3 )
𝑎 = 3 , 5
46 6 1 5 + 3 𝐻 , | 𝐻 | = 8 6
1 6 (1160, 305, 80) 15 3, 5 𝑎 1 ( m o d 2 9 )
𝑎 = 3 1 4 , 5 7
49 33 1 5 + 5 𝐻 , | 𝐻 | = 5 8
1 7 (1012, 337, 112) 15 3, 5 5 1 1 1 ( m o d 2 3 )
Remark 2.3
13 5 1 5 + 7 𝐻 , | 𝐻 | = 4 6
1 8 (1300, 433, 144) 17 17 1 7 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
50 16 1 7 + 4 5 𝐻 , | 𝐻 | = 1 0
1 9 (5220, 307, 18) 17 17 1 7 2 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
113 50 1 7 + 2 9 𝐻 , | 𝐻 | = 1 0
2 0 (5220, 307, 18) 17 17 1 7 2 1 ( m o d 2 9 ) 113 50 1 7 + 5 𝐻 , | 𝐻 | = 5 8
2 1 (5220, 307, 18) 17 17 1 7 1 ( m o d 𝑏 )
𝑏 = 3 , 6
113 50 1 7 + 2 7 𝐻 , | 𝐻 | = 1 2
2 2 (33216, 365, 4) 19 19 1 9 8 6 1 ( m o d 1 7 3 ) ?? 1 9 + 𝐻 , | 𝐻 | = 3 4 6
2 3 (11564, 373, 12) 19 19 1 9 3 1 ( m o d 7 ) 2816 1 9 + 2 8 𝐻 , | 𝐻 | = 1 4
2 4 (7240, 381, 20) 19 19 1 9 2 1 ( m o d 1 8 1 ) ?? 1 9 + 𝐻 , | 𝐻 | = 3 6 2
2 5 (4368, 397, 36) 19 19 1 9 3 1 ( m o d 7 ) ?? 1 9 + 2 7 𝐻 , | 𝐻 | = 1 4
2 6 (4180, 399, 38) 19 19 1 9 5 1 ( m o d 1 1 ) 3615 1 9 + 1 9 𝐻 , | 𝐻 | = 2 2
2 7 (2508, 437, 76) 19 19 1 9 5 1 ( m o d 1 1 ) 3420 1 9 + 1 9 𝐻 , | 𝐻 | = 2 2
2 8 (1976, 475, 114) 19 19 1 9 6 1 ( m o d 1 3 ) 3930 1 9 + 1 9 𝐻 , | 𝐻 | = 2 6
2 9 (1624, 541, 180) 19 19 1 9 1 4 1 ( m o d 2 9 ) 5630 1 9 + 9 𝐻 , | 𝐻 | = 5 8
3 0 (1520, 589, 228) 19 19 1 9 3 1 ( m o d 𝑏 )
𝑏 = 5 , 1 0
178147 1 9 + 5 7 𝐻 , | 𝐻 | = 1 0
3 1 (97904, 443, 2) 21 3, 7 𝑎 1 0 5 1 ( m o d 2 1 1 )
𝑎 = 3 , 7
? ? 2 1 + 𝐻 , | 𝐻 | = 4 2 2
3 2 (11680, 459, 18) 21 3, 7 𝑎 1 ( m o d 7 3 )
𝑎 = 3 6 , 7 1 2
? ? 2 1 + 3 𝐻 , | 𝐻 | = 1 4 6
3 3 (9724, 463, 22) 21 3, 7 𝑎 8 1 ( m o d 1 7 )
𝑎 = 3 , 7
35 15 2 1 + 1 3 𝐻 , | 𝐻 | = 3 4
3 4 (7840, 469, 28) 21 3, 7 𝑎 2 1 ( m o d 5 )
𝑎 = 3 , 7
? ? 2 1 + 4 9 𝐻 , | 𝐻 | = 1 0
3 5 (7380, 471, 30) 21 3, 7 𝑎 2 1 ( m o d 5 )
𝑎 = 3 , 7
149 66 2 1 + 4 5 𝐻 , | 𝐻 | = 1 0
3 6 (3128, 531, 90) 21 3, 7 𝑎 8 1 ( m o d 1 7 )
𝑎 = 3 , 7
? ? 2 1 + 1 5 𝐻 , | 𝐻 | = 3 4
3 7 (2756, 551, 110) 21 3, 7 𝑎 1 ( m o d 5 3 )
𝑎 = 3 2 6 , 7 1 3
20 5 2 1 + 5 𝐻 , | 𝐻 | = 1 0 6
3 8 (2296, 595, 154) 21 3, 7 𝑎 1 ( m o d 4 1 )
𝑎 = 3 4 , 7 2 0
? ? 2 1 + 7 𝐻 , | 𝐻 | = 8 2
3 9 (1904, 693, 252) 21 3, 7 𝑎 8 1 ( m o d 1 7 )
𝑎 = 3 , 7
186 147 2 1 + 2 1 𝐻 , | 𝐻 | = 3 4
4 0 (1836, 735, 294) 21 3, 7 𝑎 1 ( m o d 1 7 )
𝑎 = 3 2 , 7 2
117 56 2 1 + 2 1 𝐻 , | 𝐻 | = 3 4
4 1 (1820, 749, 308) 21 3, 7 𝑎 1 ( m o d 5 )
𝑎 = 3 2 , 7 2
3515 2 1 + 7 7 𝐻 , | 𝐻 | = 1 0
4 2 (1800, 771, 330) 21 3, 7 𝑎 1 ( m o d 5 )
𝑎 = 3 2 , 7 2
749 412 2 1 + 7 5 𝐻 , | 𝐻 | = 1 0
4 3 (47616, 535, 6) 23 23 2 3 1 ( m o d 3 ) ? ? 2 3 + 9 3 𝐻 , | 𝐻 | = 6
4 4 (35980, 537, 8) 23 23 2 3 3 2 1 ( m o d 2 5 7 ) 35 15 2 3 + 𝐻 , | 𝐻 | = 5 1 4
4 5 (12720, 553, 24) 23 23 2 3 2 1 ( m o d 5 ) ? ? 2 3 + 5 3 𝐻 , | 𝐻 | = 1 0
4 6 (12720, 553, 24) 23 23 2 3 2 1 ( m o d 5 3 ) ? ? 2 3 + 5 𝐻 ,
| 𝐻 | = 1 0 6
4 7 (7176, 575, 46) 23 23 2 3 3 1 ( m o d 1 3 ) ? ? 2 3 + 2 3 𝐻 , | 𝐻 | = 2 6
4 8 (4320, 617, 88) 23 23 2 3 1 ( m o d 3 ) ? ? 2 3 + 9 9 𝐻 , | 𝐻 | = 6
4 9 (3220, 667, 138) 23 23 2 3 2 1 ( m o d 5 ) 27 15 2 3 + 6 9 𝐻 , | 𝐻 | = 1 0
5 0 (2760, 713, 184) 23 23 2 3 1 ( m o d 3 ) ? ? 2 3 + 1 1 5 𝐻 , | 𝐻 | = 6
5 1 (2760, 713, 184) 23 23 2 3 2 1 ( m o d 5 ) ? ? 2 3 + 6 9 𝐻 , | 𝐻 | = 1 0
5 2 (2380, 793, 264) 23 23 2 3 8 1 ( m o d 1 7 ) 35 15 2 3 + 2 4 𝐻 , | 𝐻 | = 3 4
5 3 (2380, 793, 264) 23 23 2 3 2 1 ( m o d 5 ) 35 15 2 3 + 7 7 𝐻 , | 𝐻 | = 1 0
5 4 (196252, 627, 2) 25 5 5 3 1 ( m o d 7 ) ? ? 2 5 + 4 3 𝐻 , | 𝐻 | = 1 4
5 5 (196252, 627, 2) 25 5 5 2 1 1 ( m o d 4 3 ) ? ? 2 5 + 7 𝐻 , | 𝐻 | = 8 6
5 6 (40260, 635, 10) 25 5 5 1 5 1 ( m o d 6 1 ) 370 66 2 5 + 5 𝐻 , | 𝐻 | = 1 2 2
5 7 (16276, 651, 26) 25 5 5 4 1 ( m o d 3 1 3 ) 20 5 2 5 + 𝐻 , | 𝐻 | = 6 2 6
5 8 (14280, 655, 30) 25 5 5 3 1 ( m o d 7 ) ? ? 2 5 + 4 5 𝐻 , | 𝐻 | = 1 4
5 9 (11040, 665, 40) 25 5 5 1 ( m o d 3 ) ? ? 2 5 + 1 1 5 𝐻 , | 𝐻 | = 6
6 0 (9100, 675, 50) 25 5 5 2 1 ( m o d 1 3 ) 11850 2 5 + 2 5 𝐻 , | 𝐻 | = 2 6
6 1 (6328, 703, 78) 25 5 5 5 6 1 ( m o d 1 1 3 ) ? ? 2 5 + 3 𝐻 , | 𝐻 | = 2 2 6
6 2 (4620, 745, 120) 25 5 5 3 1 ( m o d 7 ) 140 66 2 5 + 5 5 𝐻 , | 𝐻 | = 1 4
6 3 (4380, 755, 130) 25 5 5 3 6 1 ( m o d 7 3 ) 53 15 2 5 + 5 𝐻 , | 𝐻 | = 1 4 6
6 4 (3400, 825, 200) 25 5 5 8 1 ( m o d 1 7 ) ? ? 2 5 + 2 5 𝐻 , | 𝐻 | = 3 4
6 5 (3060, 875, 250) 25 5 5 8 1 ( m o d 1 7 ) 113 50 2 5 + 2 5 𝐻 , | 𝐻 | = 3 4
6 6 (2640, 1015, 390) 25 5 5 1 ( m o d 3 ) ? ? 2 5 + 1 6 5 𝐻 , | 𝐻 | = 6
6 7 (2520, 1145, 520) 25 5 5 1 ( m o d 3 ) ? ? 2 5 + 1 9 5 𝐻 , | 𝐻 | = 6