← #34187615
Subgroups — #34187615
Order 20224.
514 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #23773828 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23762148 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #23761894 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23761798 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C8×C4⋊C4×C79 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14579183 | — | 2 | 1 | no | — |
| 7 | C2×#14579183 | — | 2 | 1 | no | — |
| 8 | #19099113 | C22 | 1 | 2 | yes | — |
| 9 | #19098947 | — | 2 | 1 | no | — |
| 10 | #19098947 | — | 2 | 1 | no | — |
| 11 | #19098943 | C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | #19098940 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | #19098936 | C4 | 1 | 2 | yes | complemented |
| 14 | #19098616 | C22 | 1 | 2 | yes | — |
| 15 | #19098316 | — | 2 | 1 | no | — |
| 16 | #19098314 | — | 2 | 1 | no | — |
| 17 | #19098296 | — | 2 | 4 | no | — |
| 18 | #19098290 | — | 2 | 4 | no | — |
| 19 | C2×C4⋊C8×C79 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×C8×C79 | — | 2 | 1 | no | — |
| 21 | C2×C4×C8×C79 | C22 | 1 | 1 | yes | characteristic |
| 22 | C4×C4⋊C4×C79 | C22 | 1 | 1 | yes | characteristic |
| 23 | #336×C79 | — | 2 | 1 | no | — |
| 24 | C2×#11929624 | — | 2 | 2 | no | — |
| 25 | #14579183 | — | 4 | 2 | no | — |
| 26 | #14579183 | — | 4 | 2 | no | — |
| 27 | #14579183 | — | 4 | 2 | no | — |
| 28 | #14579183 | — | 4 | 2 | no | — |
| 29 | C2×Dic316 | — | 2 | 1 | no | — |
| 30 | C2×Dic316 | — | 2 | 1 | no | — |
| 31 | C2×#11929613 | — | 2 | 2 | no | — |
| 32 | C2×#11929613 | — | 2 | 2 | no | — |
| 33 | C2×D632 | — | 2 | 1 | no | — |
| 34 | C2×D632 | — | 2 | 1 | no | — |
| 35 | #14579161 | C2×C4 | 1 | 2 | yes | — |
| 36 | #14579158 | C2×C4 | 1 | 2 | yes | — |
| 37 | #14579149 | — | 2 | 2 | no | — |
| 38 | #14579135 | — | 2 | 2 | no | — |
| 39 | #14579135 | — | 2 | 2 | no | — |
| 40 | C4×D316 | — | 2 | 2 | no | — |
| 41 | C4×D316 | — | 2 | 2 | no | — |
| 42 | C4⋊(C79⋊C8) | — | 2 | 2 | no | — |
| 43 | C22⋊(C79⋊C8) | — | 2 | 2 | no | — |
| 44 | C4×C79⋊C8 | — | 2 | 2 | no | — |
| 45 | C22×C8×C79 | — | 2 | 1 | no | — |
| 46 | C22×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 47 | C22×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 48 | C2×C4⋊C4×C79 | — | 2 | 1 | no | — |
| 49 | C2×C42×C79 | — | 2 | 1 | no | — |
| 50 | C2×C42×C79 | C23 | 1 | 1 | yes | characteristic |
| 51 | C4⋊C8×C79 | — | 2 | 1 | no | — |
| 52 | C4⋊C8×C79 | D4 | 1 | 2 | yes | — |
| 53 | C4×C8×C79 | — | 4 | 2 | no | — |
| 54 | C4×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C4×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C4×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 57 | C4×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 58 | #95×C79 | D4 | 1 | 2 | yes | — |
| 59 | #11929624 | — | 4 | 2 | no | — |
| 60 | #11929624 | — | 4 | 2 | no | — |
| 61 | #11929624 | — | 4 | 2 | no | — |
| 62 | #11929624 | — | 4 | 2 | no | — |
| 63 | C2×Dic158 | — | 2 | 2 | no | — |
| 64 | C2×D316 | — | 2 | 2 | no | — |
| 65 | C2×D316 | — | 2 | 2 | no | — |
| 66 | C2×D316 | — | 2 | 2 | no | — |
| 67 | C2×C22⋊D79 | — | 4 | 2 | no | — |
| 68 | C2×C4×D79 | — | 4 | 2 | no | — |
| 69 | Dic316 | — | 4 | 2 | no | — |
| 70 | Dic316 | — | 4 | 2 | no | — |
| 71 | #11929613 | — | 4 | 4 | no | — |
| 72 | #11929613 | — | 4 | 4 | no | — |
| 73 | D632 | — | 4 | 2 | no | — |
| 74 | D632 | — | 4 | 2 | no | — |
| 75 | C2×C79⋊C8 | — | 4 | 2 | no | — |
| 76 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 77 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 78 | (C2×D79)⋊C4 | — | 4 | 2 | no | — |
| 79 | (C2×D79)⋊C4 | — | 4 | 2 | no | — |
| 80 | C22×C4×C79 | — | 2 | 2 | no | — |
| 81 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C22×C4×C79 | #41 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C79 | — | 2 | 2 | no | — |
| 85 | C2×C8×C79 | — | 2 | 1 | no | — |
| 86 | C2×C8×C79 | — | 2 | 1 | no | — |
| 87 | C2×C8×C79 | — | 4 | 1 | no | — |
| 88 | C2×C8×C79 | — | 4 | 1 | no | — |
| 89 | C2×C8×C79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 91 | C2×C8×C79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C79 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C79 | D8 | 1 | 2 | yes | — |
| 94 | C2×C8×C79 | SD16 | 1 | 2 | yes | — |
| 95 | C4⋊C4×C79 | — | 2 | 4 | no | — |
| 96 | C42×C79 | — | 4 | 1 | no | — |
| 97 | C42×C79 | — | 4 | 1 | no | — |
| 98 | C42×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 99 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C42×C79 | #41 | 1 | 1 | yes | characteristic |
| 102 | C23×C79 | #121 | 1 | 1 | yes | characteristic |
| 103 | C22×D79 | — | 4 | 2 | no | — |
| 104 | C2×C4×C79 | — | 2 | 2 | no | — |
| 105 | C2×C4×C79 | — | 2 | 4 | no | — |
| 106 | C2×C4×C79 | — | 4 | 1 | no | — |
| 107 | C2×C4×C79 | — | 4 | 2 | no | — |
| 108 | C2×C4×C79 | — | 4 | 2 | no | — |
| 109 | C2×C4×C79 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C79 | #121 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C79 | #124 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C79 | #127 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C79 | C2×SD16 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C79 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 119 | C22⋊D79 | — | 8 | 2 | no | — |
| 120 | C22⋊D79 | — | 8 | 2 | no | — |
| 121 | C2×Dic79 | — | 4 | 2 | no | — |
| 122 | D316 | — | 4 | 2 | no | — |
| 123 | D316 | — | 4 | 2 | no | — |
| 124 | D316 | — | 4 | 4 | no | — |
| 125 | D316 | — | 8 | 2 | no | — |
| 126 | C4×D79 | — | 8 | 2 | no | — |
| 127 | C4×D79 | — | 8 | 2 | no | — |
| 128 | Dic158 | — | 4 | 2 | no | — |
| 129 | Dic158 | — | 4 | 2 | no | — |
| 130 | C8×C79 | — | 2 | 1 | no | — |
| 131 | C8×C79 | — | 2 | 1 | no | — |
| 132 | C8×C79 | — | 4 | 2 | no | — |
| 133 | C8×C79 | #102 | 1 | 2 | yes | — |
| 134 | C8×C79 | #102 | 1 | 2 | yes | — |
| 135 | C79⋊C8 | — | 8 | 2 | no | — |
| 136 | C22×C79 | — | 2 | 1 | no | — |
| 137 | C22×C79 | — | 2 | 1 | no | — |
| 138 | C22×C79 | #390 | 1 | 1 | yes | characteristic |
| 139 | C22×C79 | #459 | 1 | 1 | yes | characteristic |
| 140 | C22×C79 | #460 | 1 | 1 | yes | characteristic |
| 141 | C2×D79 | — | 8 | 2 | no | — |
| 142 | C2×D79 | — | 8 | 2 | no | — |
| 143 | C2×D79 | — | 8 | 2 | no | — |
| 144 | C4×C79 | — | 2 | 1 | no | — |
| 145 | C4×C79 | — | 2 | 1 | no | — |
| 146 | C4×C79 | — | 2 | 1 | no | — |
| 147 | C4×C79 | — | 4 | 4 | no | — |
| 148 | C4×C79 | C2×#102 | 1 | 1 | yes | characteristic |
| 149 | C4×C79 | #418 | 1 | 1 | yes | characteristic |
| 150 | C4×C79 | #443 | 1 | 2 | yes | — |
| 151 | C4×C79 | #459 | 1 | 1 | yes | characteristic |
| 152 | C4×C79 | #460 | 1 | 1 | yes | characteristic |
| 153 | Dic79 | — | 8 | 2 | no | — |
| 154 | #9750 | — | 79 | 1 | no | maximal, Sylow |
| 155 | C2×C79 | — | 2 | 1 | no | — |
| 156 | C2×C79 | — | 2 | 1 | no | — |
| 157 | C2×C79 | #1974 | 1 | 1 | yes | characteristic |
| 158 | C2×C79 | #1976 | 1 | 1 | yes | characteristic |
| 159 | C2×C79 | #1982 | 1 | 1 | yes | characteristic |
| 160 | D79 | — | 16 | 2 | no | — |
| 161 | #3147 | — | 79 | 1 | no | — |
| 162 | C8×C4⋊C4 | D79 | 1 | 1 | yes | characteristic, complemented |
| 163 | #1564 | — | 79 | 1 | no | — |
| 164 | #1502 | — | 79 | 2 | no | — |
| 165 | #1480 | — | 79 | 2 | no | — |
| 166 | C79 | #9750 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 167 | C2×#135 | — | 158 | 1 | no | — |
| 168 | C2×#135 | — | 158 | 1 | no | — |
| 169 | #532 | — | 79 | 2 | no | — |
| 170 | #495 | — | 158 | 1 | no | — |
| 171 | #495 | — | 158 | 1 | no | — |
| 172 | #494 | — | 79 | 1 | no | — |
| 173 | #493 | — | 79 | 1 | no | — |
| 174 | #492 | — | 79 | 2 | no | — |
| 175 | #431 | — | 79 | 2 | no | — |
| 176 | C2×C4⋊C8 | C2×D79 | 1 | 1 | yes | characteristic |
| 177 | C2×C4×C8 | — | 2 | 1 | no | — |
| 178 | C2×C4×C8 | C2×D79 | 1 | 1 | yes | characteristic |
| 179 | C4×C4⋊C4 | C2×D79 | 1 | 1 | yes | characteristic |
| 180 | #336 | — | 2 | 1 | no | — |
| 181 | #335 | — | 158 | 1 | no | — |
| 182 | #334 | — | 158 | 1 | no | — |
| 183 | #327 | — | 158 | 4 | no | — |
| 184 | D4⋊C8 | — | 158 | 4 | no | — |
| 185 | C2×#41 | — | 158 | 2 | no | — |
| 186 | #135 | — | 316 | 2 | no | — |
| 187 | #135 | — | 316 | 2 | no | — |
| 188 | #135 | — | 316 | 2 | no | — |
| 189 | #135 | — | 316 | 2 | no | — |
| 190 | C2×Q16 | — | 158 | 1 | no | — |
| 191 | C2×Q16 | — | 158 | 1 | no | — |
| 192 | C2×SD16 | — | 158 | 2 | no | — |
| 193 | C2×SD16 | — | 158 | 2 | no | — |
| 194 | C2×D8 | — | 158 | 1 | no | — |
| 195 | C2×D8 | — | 158 | 1 | no | — |
| 196 | C22×C8 | — | 2 | 1 | no | — |
| 197 | C22×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 198 | C22×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 199 | C4⋊Q8 | — | 79 | 2 | no | — |
| 200 | #127 | — | 79 | 2 | no | — |
| 201 | #124 | — | 158 | 2 | no | — |
| 202 | #121 | — | 158 | 2 | no | — |
| 203 | #121 | — | 158 | 2 | no | — |
| 204 | C4×D4 | — | 158 | 2 | no | — |
| 205 | C4×D4 | — | 158 | 2 | no | — |
| 206 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 207 | C2×C42 | — | 2 | 1 | no | — |
| 208 | C2×C42 | C22×D79 | 1 | 1 | yes | characteristic |
| 209 | C4⋊C8 | — | 2 | 1 | no | — |
| 210 | C4⋊C8 | — | 158 | 2 | no | — |
| 211 | C4⋊C8 | D316 | 1 | 2 | yes | — |
| 212 | C22⋊C8 | — | 158 | 2 | no | — |
| 213 | C4×C8 | — | 4 | 2 | no | — |
| 214 | C4×C8 | — | 158 | 2 | no | — |
| 215 | C4×C8 | C4×D79 | 1 | 1 | yes | characteristic |
| 216 | C4×C8 | C4×D79 | 1 | 1 | yes | characteristic |
| 217 | C4×C8 | D316 | 1 | 1 | yes | characteristic |
| 218 | C4×C8 | D316 | 1 | 1 | yes | characteristic |
| 219 | #95 | D316 | 1 | 2 | yes | — |
| 220 | #41 | — | 316 | 2 | no | — |
| 221 | #41 | — | 316 | 2 | no | — |
| 222 | #41 | — | 316 | 2 | no | — |
| 223 | #41 | — | 316 | 2 | no | — |
| 224 | C2×Q8 | — | 158 | 2 | no | — |
| 225 | C2×D4 | — | 316 | 2 | no | — |
| 226 | C2×D4 | — | 158 | 2 | no | — |
| 227 | C2×D4 | — | 158 | 2 | no | — |
| 228 | C2×D4 | — | 158 | 2 | no | — |
| 229 | C22×C4 | — | 2 | 2 | no | — |
| 230 | C22×C4 | — | 316 | 2 | no | — |
| 231 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 232 | C22×C4 | C2×D316 | 1 | 1 | yes | characteristic |
| 233 | C22×C4 | #11929624 | 1 | 1 | yes | characteristic |
| 234 | Q16 | — | 316 | 2 | no | — |
| 235 | Q16 | — | 316 | 2 | no | — |
| 236 | SD16 | — | 316 | 4 | no | — |
| 237 | SD16 | — | 316 | 4 | no | — |
| 238 | D8 | — | 316 | 2 | no | — |
| 239 | D8 | — | 316 | 2 | no | — |
| 240 | C2×C8 | — | 2 | 1 | no | — |
| 241 | C2×C8 | — | 2 | 1 | no | — |
| 242 | C2×C8 | — | 2 | 2 | no | — |
| 243 | C2×C8 | — | 4 | 1 | no | — |
| 244 | C2×C8 | — | 4 | 1 | no | — |
| 245 | C2×C8 | — | 316 | 2 | no | — |
| 246 | C2×C8 | (C2×D79)⋊C4 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | (C2×D79)⋊C4 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | (C2×D79)⋊C4 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | (C2×D79)⋊C4 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | #11929609 | 1 | 2 | yes | — |
| 251 | C2×C8 | #11929611 | 1 | 2 | yes | — |
| 252 | C4⋊C4 | — | 2 | 4 | no | — |
| 253 | C4⋊C4 | — | 158 | 2 | no | — |
| 254 | C4⋊C4 | — | 158 | 2 | no | — |
| 255 | C22⋊C4 | — | 316 | 2 | no | — |
| 256 | C22⋊C4 | — | 316 | 2 | no | — |
| 257 | C42 | — | 4 | 1 | no | — |
| 258 | C42 | — | 4 | 1 | no | — |
| 259 | C42 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 260 | C42 | C2×D316 | 1 | 1 | yes | characteristic |
| 261 | C42 | C2×D316 | 1 | 1 | yes | characteristic |
| 262 | C42 | #11929624 | 1 | 1 | yes | characteristic |
| 263 | C23 | — | 316 | 2 | no | — |
| 264 | C23 | #14579135 | 1 | 1 | yes | characteristic |
| 265 | Q8 | — | 316 | 2 | no | — |
| 266 | Q8 | — | 316 | 2 | no | — |
| 267 | D4 | — | 316 | 4 | no | — |
| 268 | D4 | — | 316 | 2 | no | — |
| 269 | D4 | — | 316 | 2 | no | — |
| 270 | D4 | — | 632 | 2 | no | — |
| 271 | D4 | — | 632 | 2 | no | — |
| 272 | D4 | — | 632 | 2 | no | — |
| 273 | C2×C4 | — | 2 | 2 | no | — |
| 274 | C2×C4 | — | 2 | 4 | no | — |
| 275 | C2×C4 | — | 4 | 1 | no | — |
| 276 | C2×C4 | — | 4 | 2 | no | — |
| 277 | C2×C4 | — | 4 | 2 | no | — |
| 278 | C2×C4 | — | 316 | 2 | no | — |
| 279 | C2×C4 | — | 632 | 2 | no | — |
| 280 | C2×C4 | — | 632 | 2 | no | — |
| 281 | C2×C4 | C2×(C2×D79)⋊C4 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C4×D316 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | C4×D316 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #14579135 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #14579149 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #14579158 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C2×#11929611 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | #14579186 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #14579186 | 1 | 1 | yes | characteristic |
| 291 | C8 | — | 2 | 1 | no | — |
| 292 | C8 | — | 2 | 1 | no | — |
| 293 | C8 | — | 4 | 2 | no | — |
| 294 | C8 | — | 632 | 2 | no | — |
| 295 | C8 | #14579081 | 1 | 2 | yes | — |
| 296 | C8 | #14579081 | 1 | 2 | yes | — |
| 297 | C22 | — | 2 | 1 | no | — |
| 298 | C22 | — | 2 | 1 | no | — |
| 299 | C22 | — | 632 | 2 | no | — |
| 300 | C22 | — | 632 | 2 | no | — |
| 301 | C22 | — | 632 | 2 | no | — |
| 302 | C22 | #19098465 | 1 | 1 | yes | characteristic |
| 303 | C22 | #19098758 | 1 | 1 | yes | characteristic |
| 304 | C22 | #19098763 | 1 | 1 | yes | characteristic |
| 305 | C4 | — | 2 | 1 | no | — |
| 306 | C4 | — | 2 | 1 | no | — |
| 307 | C4 | — | 2 | 1 | no | — |
| 308 | C4 | — | 4 | 4 | no | — |
| 309 | C4 | — | 632 | 2 | no | — |
| 310 | C4 | C2×#14579081 | 1 | 1 | yes | characteristic |
| 311 | C4 | #19098567 | 1 | 1 | yes | characteristic |
| 312 | C4 | #19098681 | 1 | 2 | yes | — |
| 313 | C4 | #19098758 | 1 | 1 | yes | characteristic |
| 314 | C4 | #19098763 | 1 | 1 | yes | characteristic |
| 315 | C2 | — | 2 | 1 | no | — |
| 316 | C2 | — | 2 | 1 | no | — |
| 317 | C2 | — | 1264 | 2 | no | — |
| 318 | C2 | #23764288 | 1 | 1 | yes | characteristic |
| 319 | C2 | #23764298 | 1 | 1 | yes | characteristic |
| 320 | C2 | #23764327 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta