← C2×H(5,𝔽2)×Dic59 #29495293
Order 15104.
12378 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | H(5,𝔽2)×Dic59 | C2 | 1 | 32 | yes | maximal, direct product factor |
| 2 | C2×#16837885 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | C2×#16837877 | C2 | 1 | 9 | yes | maximal, complemented |
| 4 | C2×#41×Dic59 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C22×D4×Dic59 | C2 | 1 | 9 | yes | maximal, complemented |
| 6 | C22×H(5,𝔽2)×C59 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C24×Dic59 | C22 | 1 | 6 | yes | complemented |
| 8 | #16837885 | C22 | 1 | 96 | yes | complemented |
| 9 | #16837877 | C22 | 1 | 144 | yes | complemented |
| 10 | #41×Dic59 | C22 | 1 | 96 | yes | complemented |
| 11 | C2×Q8×Dic59 | C22 | 1 | 2 | yes | complemented |
| 12 | C2×Q8×Dic59 | C22 | 1 | 6 | yes | complemented |
| 13 | C2×D4×Dic59 | C22 | 1 | 18 | yes | complemented |
| 14 | C2×D4×Dic59 | C22 | 1 | 36 | yes | complemented |
| 15 | C2×D4×Dic59 | C22 | 1 | 18 | yes | complemented |
| 16 | C2×D4×Dic59 | C22 | 1 | 144 | yes | complemented |
| 17 | C2×#14273686 | C22 | 1 | 18 | yes | complemented |
| 18 | C22×C4⋊Dic59 | C22 | 1 | 9 | yes | complemented |
| 19 | C22×C22⋊Dic59 | C22 | 1 | 18 | yes | complemented |
| 20 | C22×C4×Dic59 | C22 | 1 | 9 | yes | complemented |
| 21 | C2×H(5,𝔽2)×C59 | C4 | 1 | 32 | yes | complemented |
| 22 | C2×H(5,𝔽2)×C59 | C22 | 1 | 16 | yes | — |
| 23 | C22×#41×C59 | C22 | 1 | 6 | yes | — |
| 24 | C23×D4×C59 | C22 | 1 | 9 | yes | — |
| 25 | C23×Dic59 | — | 4 | 24 | no | — |
| 26 | C23×Dic59 | C23 | 1 | 9 | yes | — |
| 27 | C23×Dic59 | C23 | 1 | 18 | yes | — |
| 28 | C23×Dic59 | C23 | 1 | 9 | yes | — |
| 29 | C23×Dic59 | C23 | 1 | 48 | yes | complemented |
| 30 | Q8×Dic59 | C23 | 1 | 16 | yes | complemented |
| 31 | Q8×Dic59 | C23 | 1 | 48 | yes | complemented |
| 32 | D4×Dic59 | C23 | 1 | 144 | yes | complemented |
| 33 | D4×Dic59 | C23 | 1 | 288 | yes | complemented |
| 34 | D4×Dic59 | C23 | 1 | 144 | yes | complemented |
| 35 | #14273686 | C23 | 1 | 144 | yes | complemented |
| 36 | C2×C4⋊Dic59 | C23 | 1 | 18 | yes | — |
| 37 | C2×C4⋊Dic59 | C23 | 1 | 6 | yes | — |
| 38 | C2×C4⋊Dic59 | C23 | 1 | 72 | yes | complemented |
| 39 | C2×C22⋊Dic59 | C23 | 1 | 36 | yes | — |
| 40 | C2×C22⋊Dic59 | C23 | 1 | 144 | yes | complemented |
| 41 | C2×C4×Dic59 | C23 | 1 | 6 | yes | — |
| 42 | C2×C4×Dic59 | C23 | 1 | 18 | yes | — |
| 43 | C2×C4×Dic59 | C23 | 1 | 72 | yes | complemented |
| 44 | C25×C59 | C23 | 1 | 6 | yes | — |
| 45 | H(5,𝔽2)×C59 | C2×C4 | 1 | 256 | yes | complemented |
| 46 | C2×#41×C59 | C2×C4 | 1 | 96 | yes | complemented |
| 47 | C2×#41×C59 | C23 | 1 | 48 | yes | — |
| 48 | C22×Q8×C59 | C23 | 1 | 2 | yes | — |
| 49 | C22×D4×C59 | C2×C4 | 1 | 144 | yes | complemented |
| 50 | C22×D4×C59 | C23 | 1 | 18 | yes | — |
| 51 | C22×D4×C59 | C23 | 1 | 72 | yes | — |
| 52 | C23×C4×C59 | C23 | 1 | 9 | yes | — |
| 53 | C24×C59 | — | 4 | 12 | no | — |
| 54 | C24×C59 | — | 4 | 12 | no | — |
| 55 | C24×C59 | C22×C4 | 1 | 48 | yes | complemented |
| 56 | C24×C59 | C24 | 1 | 9 | yes | — |
| 57 | C24×C59 | C24 | 1 | 24 | yes | — |
| 58 | #41×C59 | C22×C4 | 1 | 384 | yes | complemented |
| 59 | C2×Q8×C59 | C22×C4 | 1 | 16 | yes | complemented |
| 60 | C2×Q8×C59 | C24 | 1 | 8 | yes | — |
| 61 | C2×D4×C59 | C22×C4 | 1 | 144 | yes | complemented |
| 62 | C2×D4×C59 | C22×C4 | 1 | 576 | yes | complemented |
| 63 | C2×D4×C59 | C24 | 1 | 72 | yes | — |
| 64 | C22×C4×C59 | C22×C4 | 1 | 72 | yes | complemented |
| 65 | C22×C4×C59 | C24 | 1 | 6 | yes | — |
| 66 | C22×C4×C59 | C24 | 1 | 36 | yes | — |
| 67 | C22×Dic59 | — | 2 | 36 | no | — |
| 68 | C22×Dic59 | — | 4 | 96 | no | — |
| 69 | C22×Dic59 | — | 4 | 36 | no | — |
| 70 | C22×Dic59 | — | 4 | 18 | no | — |
| 71 | C22×Dic59 | C24 | 1 | 6 | yes | — |
| 72 | C22×Dic59 | C24 | 1 | 9 | yes | — |
| 73 | C22×Dic59 | C24 | 1 | 36 | yes | — |
| 74 | C22×Dic59 | C24 | 1 | 36 | yes | — |
| 75 | C22×Dic59 | C24 | 1 | 72 | yes | — |
| 76 | C22×Dic59 | C24 | 1 | 1 | yes | characteristic |
| 77 | C22⋊Dic59 | C24 | 1 | 144 | yes | — |
| 78 | C4⋊Dic59 | C24 | 1 | 72 | yes | — |
| 79 | C4⋊Dic59 | C24 | 1 | 24 | yes | — |
| 80 | C4×Dic59 | C24 | 1 | 24 | yes | — |
| 81 | C4×Dic59 | C24 | 1 | 72 | yes | — |
| 82 | C23×C59 | — | 2 | 18 | no | — |
| 83 | C23×C59 | — | 2 | 18 | no | — |
| 84 | C23×C59 | — | 4 | 24 | no | — |
| 85 | C23×C59 | — | 4 | 96 | no | — |
| 86 | C23×C59 | — | 4 | 24 | no | — |
| 87 | C23×C59 | C23×C4 | 1 | 36 | yes | — |
| 88 | C23×C59 | C23×C4 | 1 | 96 | yes | complemented |
| 89 | C23×C59 | C25 | 1 | 18 | yes | — |
| 90 | C23×C59 | C25 | 1 | 1 | yes | characteristic |
| 91 | Q8×C59 | C23×C4 | 1 | 32 | yes | complemented |
| 92 | D4×C59 | C23×C4 | 1 | 288 | yes | complemented |
| 93 | C2×C4×C59 | C23×C4 | 1 | 24 | yes | — |
| 94 | C2×C4×C59 | C23×C4 | 1 | 144 | yes | complemented |
| 95 | C2×C4×C59 | C25 | 1 | 12 | yes | — |
| 96 | C2×Dic59 | — | 2 | 18 | no | — |
| 97 | C2×Dic59 | — | 2 | 12 | no | — |
| 98 | C2×Dic59 | — | 2 | 72 | no | — |
| 99 | C2×Dic59 | — | 4 | 72 | no | — |
| 100 | C2×Dic59 | — | 4 | 36 | no | — |
| 101 | C2×Dic59 | H(5,𝔽2) | 1 | 4 | yes | direct product factor |
| 102 | C2×Dic59 | C25 | 1 | 12 | yes | — |
| 103 | C2×Dic59 | C25 | 1 | 18 | yes | — |
| 104 | C2×Dic59 | C25 | 1 | 2 | yes | — |
| 105 | C2×C4×H(5,𝔽2) | — | 59 | 1 | no | maximal, Sylow |
| 106 | C22×C59 | — | 2 | 18 | no | — |
| 107 | C22×C59 | — | 2 | 18 | no | — |
| 108 | C22×C59 | — | 2 | 72 | no | — |
| 109 | C22×C59 | — | 4 | 96 | no | — |
| 110 | C22×C59 | C24×C4 | 1 | 36 | yes | — |
| 111 | C22×C59 | C24×C4 | 1 | 2 | yes | — |
| 112 | C22×C59 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 113 | C22×C59 | #585 | 1 | 2 | yes | — |
| 114 | C22×C59 | C26 | 1 | 1 | yes | characteristic |
| 115 | C4×C59 | C24×C4 | 1 | 24 | yes | — |
| 116 | Dic59 | — | 2 | 18 | no | — |
| 117 | Dic59 | — | 2 | 12 | no | — |
| 118 | Dic59 | C2×H(5,𝔽2) | 1 | 4 | yes | direct product factor |
| 119 | C22×H(5,𝔽2) | D59 | 1 | 1 | yes | characteristic |
| 120 | C4×H(5,𝔽2) | — | 59 | 32 | no | — |
| 121 | C2×#520 | — | 59 | 6 | no | — |
| 122 | C2×#518 | — | 59 | 9 | no | — |
| 123 | C2×C4×#41 | — | 59 | 6 | no | — |
| 124 | C22×C4×D4 | — | 59 | 9 | no | — |
| 125 | C2×C59 | — | 2 | 36 | no | — |
| 126 | C2×C59 | C4×H(5,𝔽2) | 1 | 4 | yes | complemented |
| 127 | C2×C59 | C25×C4 | 1 | 1 | yes | characteristic |
| 128 | C2×C59 | C22×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 129 | C2×C59 | C2×#585 | 1 | 1 | yes | characteristic |
| 130 | C2×H(5,𝔽2) | Dic59 | 1 | 32 | yes | direct product factor |
| 131 | C2×H(5,𝔽2) | C2×D59 | 1 | 16 | yes | — |
| 132 | C22×#41 | C2×D59 | 1 | 6 | yes | — |
| 133 | C23×D4 | C2×D59 | 1 | 9 | yes | — |
| 134 | C24×C4 | — | 59 | 6 | no | — |
| 135 | #520 | — | 59 | 96 | no | — |
| 136 | #518 | — | 59 | 144 | no | — |
| 137 | C4×#41 | — | 59 | 96 | no | — |
| 138 | C2×C4×Q8 | — | 59 | 2 | no | — |
| 139 | C2×C4×Q8 | — | 59 | 6 | no | — |
| 140 | C2×C4×D4 | — | 59 | 18 | no | — |
| 141 | C2×C4×D4 | — | 59 | 36 | no | — |
| 142 | C2×C4×D4 | — | 59 | 144 | no | — |
| 143 | C2×C4×D4 | — | 59 | 18 | no | — |
| 144 | C2×#117 | — | 59 | 18 | no | — |
| 145 | C22×C4⋊C4 | — | 59 | 9 | no | — |
| 146 | C22×C22⋊C4 | — | 59 | 18 | no | — |
| 147 | C22×C42 | — | 59 | 9 | no | — |
| 148 | C59 | C2×C4×H(5,𝔽2) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 149 | C25 | C22×D59 | 1 | 6 | yes | — |
| 150 | H(5,𝔽2) | C2×Dic59 | 1 | 256 | yes | direct product factor |
| 151 | C2×#41 | C2×Dic59 | 1 | 96 | yes | complemented |
| 152 | C2×#41 | C22×D59 | 1 | 48 | yes | — |
| 153 | C22×Q8 | C22×D59 | 1 | 2 | yes | — |
| 154 | C22×D4 | C2×Dic59 | 1 | 144 | yes | complemented |
| 155 | C22×D4 | C22×D59 | 1 | 18 | yes | — |
| 156 | C22×D4 | C22×D59 | 1 | 72 | yes | — |
| 157 | C23×C4 | — | 59 | 9 | no | — |
| 158 | C23×C4 | — | 59 | 9 | no | — |
| 159 | C23×C4 | — | 59 | 18 | no | — |
| 160 | C23×C4 | — | 59 | 48 | no | — |
| 161 | C23×C4 | — | 236 | 24 | no | — |
| 162 | C23×C4 | C22×D59 | 1 | 9 | yes | — |
| 163 | C4×Q8 | — | 59 | 16 | no | — |
| 164 | C4×Q8 | — | 59 | 48 | no | — |
| 165 | C4×D4 | — | 59 | 144 | no | — |
| 166 | C4×D4 | — | 59 | 288 | no | — |
| 167 | C4×D4 | — | 59 | 144 | no | — |
| 168 | #117 | — | 59 | 144 | no | — |
| 169 | C2×C4⋊C4 | — | 59 | 18 | no | — |
| 170 | C2×C4⋊C4 | — | 59 | 72 | no | — |
| 171 | C2×C4⋊C4 | — | 59 | 6 | no | — |
| 172 | C2×C22⋊C4 | — | 59 | 36 | no | — |
| 173 | C2×C22⋊C4 | — | 59 | 144 | no | — |
| 174 | C2×C42 | — | 59 | 18 | no | — |
| 175 | C2×C42 | — | 59 | 6 | no | — |
| 176 | C2×C42 | — | 59 | 72 | no | — |
| 177 | C24 | — | 4 | 12 | no | — |
| 178 | C24 | — | 4 | 12 | no | — |
| 179 | C24 | C22×Dic59 | 1 | 48 | yes | complemented |
| 180 | C24 | C23×D59 | 1 | 9 | yes | — |
| 181 | C24 | C23×D59 | 1 | 24 | yes | — |
| 182 | #41 | C22×Dic59 | 1 | 384 | yes | complemented |
| 183 | C2×Q8 | C22×Dic59 | 1 | 16 | yes | complemented |
| 184 | C2×Q8 | C23×D59 | 1 | 8 | yes | — |
| 185 | C2×D4 | C22×Dic59 | 1 | 144 | yes | complemented |
| 186 | C2×D4 | C22×Dic59 | 1 | 576 | yes | complemented |
| 187 | C2×D4 | C23×D59 | 1 | 72 | yes | — |
| 188 | C22×C4 | — | 59 | 1 | no | — |
| 189 | C22×C4 | — | 59 | 36 | no | — |
| 190 | C22×C4 | — | 59 | 6 | no | — |
| 191 | C22×C4 | — | 59 | 9 | no | — |
| 192 | C22×C4 | — | 59 | 36 | no | — |
| 193 | C22×C4 | — | 59 | 72 | no | — |
| 194 | C22×C4 | — | 118 | 36 | no | — |
| 195 | C22×C4 | — | 236 | 96 | no | — |
| 196 | C22×C4 | — | 236 | 36 | no | — |
| 197 | C22×C4 | — | 236 | 18 | no | — |
| 198 | C22×C4 | C22×Dic59 | 1 | 72 | yes | complemented |
| 199 | C22×C4 | C23×D59 | 1 | 6 | yes | — |
| 200 | C22×C4 | C23×D59 | 1 | 36 | yes | — |
| 201 | C4⋊C4 | — | 59 | 72 | no | — |
| 202 | C4⋊C4 | — | 59 | 24 | no | — |
| 203 | C22⋊C4 | — | 59 | 144 | no | — |
| 204 | C42 | — | 59 | 24 | no | — |
| 205 | C42 | — | 59 | 72 | no | — |
| 206 | C23 | — | 2 | 18 | no | — |
| 207 | C23 | — | 2 | 18 | no | — |
| 208 | C23 | — | 4 | 24 | no | — |
| 209 | C23 | — | 4 | 96 | no | — |
| 210 | C23 | — | 4 | 24 | no | — |
| 211 | C23 | C23×Dic59 | 1 | 36 | yes | — |
| 212 | C23 | C23×Dic59 | 1 | 96 | yes | complemented |
| 213 | C23 | C24×D59 | 1 | 18 | yes | — |
| 214 | C23 | C24×D59 | 1 | 1 | yes | characteristic |
| 215 | Q8 | C23×Dic59 | 1 | 32 | yes | complemented |
| 216 | D4 | C23×Dic59 | 1 | 288 | yes | complemented |
| 217 | C2×C4 | — | 59 | 4 | no | — |
| 218 | C2×C4 | — | 59 | 2 | no | — |
| 219 | C2×C4 | — | 59 | 12 | no | — |
| 220 | C2×C4 | — | 59 | 18 | no | — |
| 221 | C2×C4 | — | 118 | 72 | no | — |
| 222 | C2×C4 | — | 118 | 18 | no | — |
| 223 | C2×C4 | — | 118 | 12 | no | — |
| 224 | C2×C4 | — | 236 | 36 | no | — |
| 225 | C2×C4 | — | 236 | 72 | no | — |
| 226 | C2×C4 | C23×Dic59 | 1 | 24 | yes | — |
| 227 | C2×C4 | C23×Dic59 | 1 | 144 | yes | complemented |
| 228 | C2×C4 | C24×D59 | 1 | 12 | yes | — |
| 229 | C22 | — | 2 | 72 | no | — |
| 230 | C22 | — | 2 | 18 | no | — |
| 231 | C22 | — | 2 | 18 | no | — |
| 232 | C22 | — | 4 | 96 | no | — |
| 233 | C22 | C24×Dic59 | 1 | 36 | yes | — |
| 234 | C22 | C24×Dic59 | 1 | 2 | yes | — |
| 235 | C22 | H(5,𝔽2)×D59 | 1 | 2 | yes | — |
| 236 | C22 | #16838237 | 1 | 2 | yes | — |
| 237 | C22 | C25×D59 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 59 | 4 | no | — |
| 239 | C4 | — | 118 | 18 | no | — |
| 240 | C4 | — | 118 | 12 | no | — |
| 241 | C4 | C24×Dic59 | 1 | 24 | yes | — |
| 242 | C2 | — | 2 | 36 | no | — |
| 243 | C2 | H(5,𝔽2)×Dic59 | 1 | 4 | yes | direct product factor |
| 244 | C2 | C25×Dic59 | 1 | 1 | yes | characteristic |
| 245 | C2 | C2×H(5,𝔽2)×D59 | 1 | 1 | yes | characteristic |
| 246 | C2 | C2×#16838237 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. H(5,𝔽2)×Dic59 | C2 | direct (N × H) |
| 2. C2×#16837885 | C2 | semidirect |
| 3. C2×#16837877 | C2 | semidirect |
| 4. C2×#41×Dic59 | C2 | semidirect |
| 5. C22×D4×Dic59 | C2 | semidirect |
| 7. C24×Dic59 | C22 | semidirect |
| 8. #16837885 | C22 | semidirect |
| 9. #16837877 | C22 | semidirect |
| 10. #41×Dic59 | C22 | semidirect |
| 11. C2×Q8×Dic59 | C22 | semidirect |
| 12. C2×Q8×Dic59 | C22 | semidirect |
| 13. C2×D4×Dic59 | C22 | semidirect |
| 14. C2×D4×Dic59 | C22 | semidirect |
| 15. C2×D4×Dic59 | C22 | semidirect |
| 16. C2×D4×Dic59 | C22 | semidirect |
| 17. C2×#14273686 | C22 | semidirect |
| 18. C22×C4⋊Dic59 | C22 | semidirect |
| 19. C22×C22⋊Dic59 | C22 | semidirect |
| 20. C22×C4×Dic59 | C22 | semidirect |
| 21. C2×H(5,𝔽2)×C59 | C4 | semidirect |
| 29. C23×Dic59 | C23 | semidirect |
| 30. Q8×Dic59 | C23 | semidirect |
| 31. Q8×Dic59 | C23 | semidirect |
| 32. D4×Dic59 | C23 | semidirect |
| 33. D4×Dic59 | C23 | semidirect |
| 34. D4×Dic59 | C23 | semidirect |
| 35. #14273686 | C23 | semidirect |
| 38. C2×C4⋊Dic59 | C23 | semidirect |
| 40. C2×C22⋊Dic59 | C23 | semidirect |
| 43. C2×C4×Dic59 | C23 | semidirect |
| 45. H(5,𝔽2)×C59 | C2×C4 | semidirect |
| 46. C2×#41×C59 | C2×C4 | semidirect |
| 49. C22×D4×C59 | C2×C4 | semidirect |
| 55. C24×C59 | C22×C4 | semidirect |
| 58. #41×C59 | C22×C4 | semidirect |
| 59. C2×Q8×C59 | C22×C4 | semidirect |
| 61. C2×D4×C59 | C22×C4 | semidirect |
| 62. C2×D4×C59 | C22×C4 | semidirect |
| 64. C22×C4×C59 | C22×C4 | semidirect |
| 88. C23×C59 | C23×C4 | semidirect |
| 91. Q8×C59 | C23×C4 | semidirect |
| 92. D4×C59 | C23×C4 | semidirect |
| 94. C2×C4×C59 | C23×C4 | semidirect |
| 101. C2×Dic59 | H(5,𝔽2) | direct (N × H) |
| 118. Dic59 | C2×H(5,𝔽2) | direct (N × H) |
| 126. C2×C59 | C4×H(5,𝔽2) | semidirect |
| 130. C2×H(5,𝔽2) | Dic59 | direct (N × H) |
| 148. C59 | C2×C4×H(5,𝔽2) | semidirect |
| 150. H(5,𝔽2) | C2×Dic59 | direct (N × H) |
| 151. C2×#41 | C2×Dic59 | semidirect |
| 154. C22×D4 | C2×Dic59 | semidirect |
| 179. C24 | C22×Dic59 | semidirect |
| 182. #41 | C22×Dic59 | semidirect |
| 183. C2×Q8 | C22×Dic59 | semidirect |
| 185. C2×D4 | C22×Dic59 | semidirect |
| 186. C2×D4 | C22×Dic59 | semidirect |
| 198. C22×C4 | C22×Dic59 | semidirect |
| 212. C23 | C23×Dic59 | semidirect |
| 215. Q8 | C23×Dic59 | semidirect |
| 216. D4 | C23×Dic59 | semidirect |
| 227. C2×C4 | C23×Dic59 | semidirect |
| 243. C2 | H(5,𝔽2)×Dic59 | direct (N × H) |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta