← C2×H(3,𝔽4)×D59 #29495560
Order 15104.
10246 conjugacy classes of subgroups.
The subgroup lattice has 264 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#16838145 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C2×#16838144 | C2 | 1 | 9 | yes | maximal, complemented |
| 3 | H(3,𝔽4)×D59 | C2 | 1 | 32 | yes | maximal, direct product factor |
| 4 | C2×#124×D59 | C2 | 1 | 9 | yes | maximal, complemented |
| 5 | C2×#120×D59 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×C59×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C25×D59 | C22 | 1 | 2 | yes | complemented |
| 8 | C22×Q8×D59 | C22 | 1 | 3 | yes | complemented |
| 9 | C22×D4×D59 | C22 | 1 | 9 | yes | — |
| 10 | #16838145 | C22 | 1 | 96 | yes | complemented |
| 11 | #16838144 | C22 | 1 | 144 | yes | complemented |
| 12 | C2×#14273716 | C22 | 1 | 9 | yes | complemented |
| 13 | C2×#14273715 | C22 | 1 | 9 | yes | complemented |
| 14 | C2×#14273714 | C22 | 1 | 9 | yes | complemented |
| 15 | C2×#14273713 | C22 | 1 | 36 | yes | complemented |
| 16 | #124×D59 | C22 | 1 | 144 | yes | complemented |
| 17 | C2×#14273697 | C22 | 1 | 18 | yes | complemented |
| 18 | C2×#14273696 | C22 | 1 | 6 | yes | complemented |
| 19 | C2×#14273695 | C22 | 1 | 18 | yes | complemented |
| 20 | #120×D59 | C22 | 1 | 96 | yes | complemented |
| 21 | C2×C22⋊C4×D59 | C22 | 1 | 18 | yes | complemented |
| 22 | C2×C42×D59 | C22 | 1 | 3 | yes | complemented |
| 23 | C59×H(3,𝔽4) | C22 | 1 | 16 | yes | complemented |
| 24 | C2×#124×C59 | C22 | 1 | 9 | yes | complemented |
| 25 | C2×#120×C59 | C22 | 1 | 6 | yes | complemented |
| 26 | C24×D59 | — | 4 | 24 | no | — |
| 27 | C24×D59 | C23 | 1 | 6 | yes | — |
| 28 | C24×D59 | C23 | 1 | 16 | yes | complemented |
| 29 | C24×D59 | C23 | 1 | 6 | yes | complemented |
| 30 | C22×Dic118 | C23 | 1 | 9 | yes | complemented |
| 31 | C2×Q8×D59 | — | 4 | 12 | no | — |
| 32 | C2×Q8×D59 | C23 | 1 | 24 | yes | complemented |
| 33 | C22×D236 | C23 | 1 | 9 | yes | — |
| 34 | C22×C22⋊D59 | C23 | 1 | 18 | yes | — |
| 35 | C2×D4×D59 | — | 4 | 36 | no | — |
| 36 | C2×D4×D59 | C23 | 1 | 72 | yes | — |
| 37 | C22×C4×D59 | C23 | 1 | 9 | yes | — |
| 38 | #14273716 | C23 | 1 | 72 | yes | complemented |
| 39 | #14273715 | C23 | 1 | 72 | yes | complemented |
| 40 | #14273714 | C23 | 1 | 72 | yes | complemented |
| 41 | #14273713 | C23 | 1 | 288 | yes | complemented |
| 42 | #14273697 | C23 | 1 | 144 | yes | complemented |
| 43 | #14273696 | C23 | 1 | 48 | yes | complemented |
| 44 | #14273695 | C23 | 1 | 144 | yes | complemented |
| 45 | C2×C22⋊Dic59 | C23 | 1 | 18 | yes | complemented |
| 46 | C2×(C2×D59)⋊C4 | C23 | 1 | 36 | yes | complemented |
| 47 | C22⋊C4×D59 | C23 | 1 | 144 | yes | complemented |
| 48 | C2×C4×Dic59 | C23 | 1 | 9 | yes | complemented |
| 49 | C42×D59 | C23 | 1 | 24 | yes | complemented |
| 50 | C25×C59 | C23 | 1 | 2 | yes | complemented |
| 51 | C22×Q8×C59 | C23 | 1 | 3 | yes | complemented |
| 52 | C22×D4×C59 | C23 | 1 | 9 | yes | — |
| 53 | #124×C59 | C23 | 1 | 72 | yes | complemented |
| 54 | #120×C59 | C23 | 1 | 48 | yes | complemented |
| 55 | C2×C22⋊C4×C59 | C23 | 1 | 18 | yes | complemented |
| 56 | C2×C42×C59 | C23 | 1 | 3 | yes | complemented |
| 57 | C24×C59 | — | 4 | 12 | no | — |
| 58 | C24×C59 | C24 | 1 | 6 | yes | — |
| 59 | C24×C59 | C24 | 1 | 8 | yes | complemented |
| 60 | C23×D59 | — | 2 | 72 | no | — |
| 61 | C23×D59 | — | 4 | 96 | no | — |
| 62 | C23×D59 | — | 4 | 36 | no | — |
| 63 | C23×D59 | — | 4 | 128 | no | — |
| 64 | C23×D59 | C24 | 1 | 24 | yes | — |
| 65 | C23×D59 | C24 | 1 | 6 | yes | — |
| 66 | C23×D59 | C24 | 1 | 1 | yes | characteristic |
| 67 | C23×D59 | C24 | 1 | 24 | yes | complemented |
| 68 | C2×Q8×C59 | — | 4 | 6 | no | — |
| 69 | C2×Q8×C59 | C24 | 1 | 12 | yes | complemented |
| 70 | C2×D4×C59 | — | 4 | 18 | no | — |
| 71 | C2×D4×C59 | C24 | 1 | 36 | yes | — |
| 72 | C22×C4×C59 | C24 | 1 | 9 | yes | — |
| 73 | C2×C22⋊D59 | — | 4 | 36 | no | — |
| 74 | C2×C22⋊D59 | C24 | 1 | 72 | yes | — |
| 75 | C22×Dic59 | C24 | 1 | 9 | yes | — |
| 76 | Q8×D59 | — | 4 | 48 | no | — |
| 77 | D4×D59 | — | 4 | 144 | no | — |
| 78 | C2×D236 | — | 4 | 18 | no | — |
| 79 | C2×D236 | C24 | 1 | 36 | yes | — |
| 80 | C2×C4×D59 | — | 2 | 36 | no | — |
| 81 | C2×C4×D59 | C24 | 1 | 36 | yes | — |
| 82 | C2×Dic118 | — | 4 | 18 | no | — |
| 83 | C2×Dic118 | C24 | 1 | 36 | yes | complemented |
| 84 | C22⋊C4×C59 | C24 | 1 | 72 | yes | complemented |
| 85 | C42×C59 | C24 | 1 | 12 | yes | complemented |
| 86 | C22⋊Dic59 | C24 | 1 | 72 | yes | complemented |
| 87 | (C2×D59)⋊C4 | C24 | 1 | 144 | yes | complemented |
| 88 | C4×Dic59 | C24 | 1 | 36 | yes | complemented |
| 89 | C23×C59 | — | 2 | 36 | no | — |
| 90 | C23×C59 | — | 4 | 24 | no | — |
| 91 | C23×C59 | — | 4 | 32 | no | — |
| 92 | C23×C59 | C25 | 1 | 12 | yes | — |
| 93 | C23×C59 | C25 | 1 | 1 | yes | characteristic |
| 94 | C22×D59 | — | 2 | 36 | no | — |
| 95 | C22×D59 | — | 2 | 144 | no | — |
| 96 | C22×D59 | — | 4 | 72 | no | — |
| 97 | C22×D59 | — | 4 | 96 | no | — |
| 98 | C22×D59 | — | 4 | 256 | no | — |
| 99 | C22×D59 | — | 4 | 96 | no | — |
| 100 | C22×D59 | H(5,𝔽2) | 1 | 12 | yes | — |
| 101 | C22×D59 | C25 | 1 | 12 | yes | — |
| 102 | C22×D59 | C25 | 1 | 2 | yes | — |
| 103 | Q8×C59 | — | 4 | 12 | no | — |
| 104 | D4×C59 | — | 4 | 36 | no | — |
| 105 | C2×C4×C59 | — | 2 | 18 | no | — |
| 106 | C2×C4×C59 | C25 | 1 | 18 | yes | — |
| 107 | C22⋊D59 | — | 4 | 72 | no | — |
| 108 | C2×Dic59 | — | 2 | 18 | no | — |
| 109 | C2×Dic59 | C25 | 1 | 18 | yes | — |
| 110 | D236 | — | 4 | 36 | no | — |
| 111 | C4×D59 | — | 2 | 72 | no | — |
| 112 | Dic118 | — | 4 | 36 | no | — |
| 113 | C22×H(3,𝔽4) | — | 59 | 1 | no | maximal, Sylow |
| 114 | C22×C59 | — | 2 | 36 | no | — |
| 115 | C22×C59 | — | 4 | 32 | no | — |
| 116 | C22×C59 | — | 4 | 24 | no | — |
| 117 | C22×C59 | C2×H(5,𝔽2) | 1 | 6 | yes | — |
| 118 | C22×C59 | C26 | 1 | 1 | yes | characteristic |
| 119 | C2×D59 | — | 2 | 36 | no | — |
| 120 | C2×D59 | — | 4 | 96 | no | — |
| 121 | C2×D59 | — | 4 | 24 | no | — |
| 122 | C2×D59 | — | 4 | 96 | no | — |
| 123 | C2×D59 | H(3,𝔽4) | 1 | 16 | yes | direct product factor |
| 124 | C2×D59 | C2×H(5,𝔽2) | 1 | 12 | yes | — |
| 125 | C4×C59 | — | 2 | 18 | no | — |
| 126 | Dic59 | — | 2 | 18 | no | — |
| 127 | C2×H(3,𝔽4) | — | 59 | 32 | no | — |
| 128 | C2×H(3,𝔽4) | — | 59 | 9 | no | — |
| 129 | C2×H(3,𝔽4) | — | 59 | 6 | no | — |
| 130 | C2×H(3,𝔽4) | D59 | 1 | 1 | yes | characteristic, direct product factor |
| 131 | C22×#124 | — | 59 | 9 | no | — |
| 132 | C22×#120 | — | 59 | 6 | no | — |
| 133 | C2×C59 | — | 4 | 12 | no | — |
| 134 | C2×C59 | C2×H(3,𝔽4) | 1 | 4 | yes | complemented |
| 135 | C2×C59 | C22×H(5,𝔽2) | 1 | 3 | yes | — |
| 136 | D59 | — | 4 | 12 | no | — |
| 137 | D59 | C2×H(3,𝔽4) | 1 | 8 | yes | direct product factor |
| 138 | C26 | — | 59 | 2 | no | — |
| 139 | C23×Q8 | — | 59 | 3 | no | — |
| 140 | C23×D4 | — | 59 | 9 | no | — |
| 141 | H(3,𝔽4) | — | 59 | 144 | no | — |
| 142 | H(3,𝔽4) | — | 59 | 96 | no | — |
| 143 | H(3,𝔽4) | C2×D59 | 1 | 16 | yes | direct product factor |
| 144 | C2×#124 | — | 59 | 9 | no | — |
| 145 | C2×#124 | — | 59 | 9 | no | — |
| 146 | C2×#124 | — | 59 | 144 | no | — |
| 147 | C2×#124 | — | 59 | 36 | no | — |
| 148 | C2×#124 | — | 59 | 9 | no | — |
| 149 | C2×#124 | C2×D59 | 1 | 9 | yes | complemented |
| 150 | C2×#120 | — | 59 | 6 | no | — |
| 151 | C2×#120 | — | 59 | 18 | no | — |
| 152 | C2×#120 | — | 59 | 96 | no | — |
| 153 | C2×#120 | — | 59 | 18 | no | — |
| 154 | C2×#120 | C2×D59 | 1 | 6 | yes | complemented |
| 155 | C22×C22⋊C4 | — | 59 | 18 | no | — |
| 156 | C22×C42 | — | 59 | 3 | no | — |
| 157 | C59 | C22×H(3,𝔽4) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 158 | C25 | — | 59 | 6 | no | — |
| 159 | C25 | — | 59 | 6 | no | — |
| 160 | C25 | — | 59 | 16 | no | — |
| 161 | C25 | — | 236 | 24 | no | — |
| 162 | C25 | C22×D59 | 1 | 2 | yes | complemented |
| 163 | C22×Q8 | — | 59 | 24 | no | — |
| 164 | C22×Q8 | — | 59 | 9 | no | — |
| 165 | C22×Q8 | — | 236 | 12 | no | — |
| 166 | C22×Q8 | C22×D59 | 1 | 3 | yes | complemented |
| 167 | C22×D4 | — | 59 | 18 | no | — |
| 168 | C22×D4 | — | 59 | 72 | no | — |
| 169 | C22×D4 | — | 59 | 9 | no | — |
| 170 | C22×D4 | — | 236 | 36 | no | — |
| 171 | C22×D4 | C22×D59 | 1 | 9 | yes | — |
| 172 | C23×C4 | — | 59 | 9 | no | — |
| 173 | #124 | — | 59 | 72 | no | — |
| 174 | #124 | — | 59 | 72 | no | — |
| 175 | #124 | — | 59 | 288 | no | — |
| 176 | #124 | — | 59 | 72 | no | — |
| 177 | #124 | C22×D59 | 1 | 72 | yes | complemented |
| 178 | #120 | — | 59 | 48 | no | — |
| 179 | #120 | — | 59 | 144 | no | — |
| 180 | #120 | — | 59 | 144 | no | — |
| 181 | #120 | C22×D59 | 1 | 48 | yes | complemented |
| 182 | C2×C22⋊C4 | — | 59 | 18 | no | — |
| 183 | C2×C22⋊C4 | — | 59 | 36 | no | — |
| 184 | C2×C22⋊C4 | — | 59 | 144 | no | — |
| 185 | C2×C22⋊C4 | C22×D59 | 1 | 18 | yes | complemented |
| 186 | C2×C42 | — | 59 | 9 | no | — |
| 187 | C2×C42 | — | 59 | 24 | no | — |
| 188 | C2×C42 | C22×D59 | 1 | 3 | yes | complemented |
| 189 | C24 | — | 4 | 12 | no | — |
| 190 | C24 | — | 59 | 1 | no | — |
| 191 | C24 | — | 59 | 24 | no | — |
| 192 | C24 | — | 59 | 6 | no | — |
| 193 | C24 | — | 59 | 24 | no | — |
| 194 | C24 | — | 118 | 72 | no | — |
| 195 | C24 | — | 236 | 36 | no | — |
| 196 | C24 | — | 236 | 96 | no | — |
| 197 | C24 | — | 236 | 128 | no | — |
| 198 | C24 | C23×D59 | 1 | 6 | yes | — |
| 199 | C24 | C23×D59 | 1 | 8 | yes | complemented |
| 200 | C2×Q8 | — | 4 | 6 | no | — |
| 201 | C2×Q8 | — | 59 | 36 | no | — |
| 202 | C2×Q8 | — | 236 | 18 | no | — |
| 203 | C2×Q8 | — | 236 | 48 | no | — |
| 204 | C2×Q8 | C23×D59 | 1 | 12 | yes | complemented |
| 205 | C2×D4 | — | 4 | 18 | no | — |
| 206 | C2×D4 | — | 59 | 72 | no | — |
| 207 | C2×D4 | — | 59 | 36 | no | — |
| 208 | C2×D4 | — | 236 | 18 | no | — |
| 209 | C2×D4 | — | 236 | 144 | no | — |
| 210 | C2×D4 | — | 236 | 36 | no | — |
| 211 | C2×D4 | C23×D59 | 1 | 36 | yes | — |
| 212 | C22×C4 | — | 59 | 9 | no | — |
| 213 | C22×C4 | — | 59 | 36 | no | — |
| 214 | C22×C4 | — | 118 | 36 | no | — |
| 215 | C22×C4 | C23×D59 | 1 | 9 | yes | — |
| 216 | C22⋊C4 | — | 59 | 72 | no | — |
| 217 | C22⋊C4 | — | 59 | 144 | no | — |
| 218 | C22⋊C4 | C23×D59 | 1 | 72 | yes | complemented |
| 219 | C42 | — | 59 | 36 | no | — |
| 220 | C42 | C23×D59 | 1 | 12 | yes | complemented |
| 221 | C23 | — | 2 | 36 | no | — |
| 222 | C23 | — | 4 | 24 | no | — |
| 223 | C23 | — | 4 | 32 | no | — |
| 224 | C23 | — | 59 | 12 | no | — |
| 225 | C23 | — | 59 | 2 | no | — |
| 226 | C23 | — | 59 | 12 | no | — |
| 227 | C23 | — | 118 | 144 | no | — |
| 228 | C23 | — | 118 | 36 | no | — |
| 229 | C23 | — | 236 | 72 | no | — |
| 230 | C23 | — | 236 | 96 | no | — |
| 231 | C23 | — | 236 | 256 | no | — |
| 232 | C23 | — | 236 | 96 | no | — |
| 233 | C23 | C24×D59 | 1 | 12 | yes | — |
| 234 | C23 | C24×D59 | 1 | 1 | yes | characteristic |
| 235 | Q8 | — | 4 | 12 | no | — |
| 236 | Q8 | — | 236 | 36 | no | — |
| 237 | D4 | — | 4 | 36 | no | — |
| 238 | D4 | — | 236 | 36 | no | — |
| 239 | D4 | — | 236 | 72 | no | — |
| 240 | C2×C4 | — | 2 | 18 | no | — |
| 241 | C2×C4 | — | 59 | 18 | no | — |
| 242 | C2×C4 | — | 118 | 72 | no | — |
| 243 | C2×C4 | — | 118 | 18 | no | — |
| 244 | C2×C4 | C24×D59 | 1 | 18 | yes | — |
| 245 | C22 | — | 2 | 36 | no | — |
| 246 | C22 | — | 4 | 32 | no | — |
| 247 | C22 | — | 4 | 24 | no | — |
| 248 | C22 | — | 59 | 16 | no | — |
| 249 | C22 | — | 59 | 12 | no | — |
| 250 | C22 | — | 118 | 36 | no | — |
| 251 | C22 | — | 236 | 96 | no | — |
| 252 | C22 | — | 236 | 96 | no | — |
| 253 | C22 | — | 236 | 24 | no | — |
| 254 | C22 | H(5,𝔽2)×D59 | 1 | 6 | yes | — |
| 255 | C22 | C25×D59 | 1 | 1 | yes | characteristic |
| 256 | C4 | — | 2 | 18 | no | — |
| 257 | C4 | — | 118 | 18 | no | — |
| 258 | C2 | — | 4 | 12 | no | — |
| 259 | C2 | — | 59 | 8 | no | — |
| 260 | C2 | — | 236 | 12 | no | — |
| 261 | C2 | H(3,𝔽4)×D59 | 1 | 4 | yes | direct product factor |
| 262 | C2 | C2×H(5,𝔽2)×D59 | 1 | 3 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C2×#16838145 | C2 | semidirect |
| 2. C2×#16838144 | C2 | semidirect |
| 3. H(3,𝔽4)×D59 | C2 | direct (N × H) |
| 4. C2×#124×D59 | C2 | semidirect |
| 5. C2×#120×D59 | C2 | semidirect |
| 6. C2×C59×H(3,𝔽4) | C2 | semidirect |
| 7. C25×D59 | C22 | semidirect |
| 8. C22×Q8×D59 | C22 | semidirect |
| 10. #16838145 | C22 | semidirect |
| 11. #16838144 | C22 | semidirect |
| 12. C2×#14273716 | C22 | semidirect |
| 13. C2×#14273715 | C22 | semidirect |
| 14. C2×#14273714 | C22 | semidirect |
| 15. C2×#14273713 | C22 | semidirect |
| 16. #124×D59 | C22 | semidirect |
| 17. C2×#14273697 | C22 | semidirect |
| 18. C2×#14273696 | C22 | semidirect |
| 19. C2×#14273695 | C22 | semidirect |
| 20. #120×D59 | C22 | semidirect |
| 21. C2×C22⋊C4×D59 | C22 | semidirect |
| 22. C2×C42×D59 | C22 | semidirect |
| 23. C59×H(3,𝔽4) | C22 | semidirect |
| 24. C2×#124×C59 | C22 | semidirect |
| 25. C2×#120×C59 | C22 | semidirect |
| 28. C24×D59 | C23 | semidirect |
| 29. C24×D59 | C23 | semidirect |
| 30. C22×Dic118 | C23 | semidirect |
| 32. C2×Q8×D59 | C23 | semidirect |
| 38. #14273716 | C23 | semidirect |
| 39. #14273715 | C23 | semidirect |
| 40. #14273714 | C23 | semidirect |
| 41. #14273713 | C23 | semidirect |
| 42. #14273697 | C23 | semidirect |
| 43. #14273696 | C23 | semidirect |
| 44. #14273695 | C23 | semidirect |
| 45. C2×C22⋊Dic59 | C23 | semidirect |
| 46. C2×(C2×D59)⋊C4 | C23 | semidirect |
| 47. C22⋊C4×D59 | C23 | semidirect |
| 48. C2×C4×Dic59 | C23 | semidirect |
| 49. C42×D59 | C23 | semidirect |
| 50. C25×C59 | C23 | semidirect |
| 51. C22×Q8×C59 | C23 | semidirect |
| 53. #124×C59 | C23 | semidirect |
| 54. #120×C59 | C23 | semidirect |
| 55. C2×C22⋊C4×C59 | C23 | semidirect |
| 56. C2×C42×C59 | C23 | semidirect |
| 59. C24×C59 | C24 | semidirect |
| 67. C23×D59 | C24 | semidirect |
| 69. C2×Q8×C59 | C24 | semidirect |
| 83. C2×Dic118 | C24 | semidirect |
| 84. C22⋊C4×C59 | C24 | semidirect |
| 85. C42×C59 | C24 | semidirect |
| 86. C22⋊Dic59 | C24 | semidirect |
| 87. (C2×D59)⋊C4 | C24 | semidirect |
| 88. C4×Dic59 | C24 | semidirect |
| 123. C2×D59 | H(3,𝔽4) | direct (N × H) |
| 130. C2×H(3,𝔽4) | D59 | direct (N × H) |
| 134. C2×C59 | C2×H(3,𝔽4) | semidirect |
| 137. D59 | C2×H(3,𝔽4) | direct (N × H) |
| 143. H(3,𝔽4) | C2×D59 | direct (N × H) |
| 149. C2×#124 | C2×D59 | semidirect |
| 154. C2×#120 | C2×D59 | semidirect |
| 157. C59 | C22×H(3,𝔽4) | semidirect |
| 162. C25 | C22×D59 | semidirect |
| 166. C22×Q8 | C22×D59 | semidirect |
| 177. #124 | C22×D59 | semidirect |
| 181. #120 | C22×D59 | semidirect |
| 185. C2×C22⋊C4 | C22×D59 | semidirect |
| 188. C2×C42 | C22×D59 | semidirect |
| 199. C24 | C23×D59 | semidirect |
| 204. C2×Q8 | C23×D59 | semidirect |
| 218. C22⋊C4 | C23×D59 | semidirect |
| 220. C42 | C23×D59 | semidirect |
| 261. C2 | H(3,𝔽4)×D59 | 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