Table 5: Partial results in groups of order by Criterion 3. Parameters with asterisk indicate new results.? means the number of groups of order is unknown.
 ( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 Factoring of 𝑝 in ℤ [ 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 ) 28 16 − 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 ) 36 15 − 1 9 + 1 9 𝐻 , | 𝐻 | = 2 2 2 7 ∗ (2508, 437, 76) 19 19 1 9 5 ≡ − 1 ( m o d 1 1 ) 34 20 1 9 + 1 9 𝐻 , | 𝐻 | = 2 2 2 8 ∗ (1976, 475, 114) 19 19 1 9 6 ≡ − 1 ( m o d 1 3 ) 39 30 − 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 ) 56 30 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 0 178 147 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 35 15 − 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 ) 118 50 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