← #34444634
Subgroups — #34444634
Order 20224.
528 conjugacy classes of subgroups.
The subgroup lattice has 332 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #23773986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #23765549 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23765537 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23765511 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #23765445 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #23765437 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #23765156 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C79×#3169 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #19099120 | — | 2 | 1 | no | — |
| 10 | C2×#14579105 | C22 | 1 | 4 | yes | — |
| 11 | C2×#14579101 | C22 | 1 | 2 | yes | — |
| 12 | #19098968 | C22 | 1 | 1 | yes | characteristic |
| 13 | #19098943 | C22 | 1 | 1 | yes | characteristic |
| 14 | #19098914 | — | 2 | 1 | no | — |
| 15 | #19098804 | — | 2 | 1 | no | — |
| 16 | #19098804 | C22 | 1 | 2 | yes | — |
| 17 | C4×Dic316 | — | 2 | 1 | no | — |
| 18 | C4×Dic316 | C22 | 1 | 2 | yes | — |
| 19 | #19098388 | C22 | 1 | 4 | yes | — |
| 20 | C4⋊(C79⋊C16) | C22 | 1 | 2 | yes | — |
| 21 | C4⋊(C79⋊C16) | C22 | 1 | 2 | yes | — |
| 22 | #19098366 | C22 | 1 | 4 | yes | — |
| 23 | #19098364 | C22 | 1 | 4 | yes | — |
| 24 | #532×C79 | — | 2 | 1 | no | — |
| 25 | #500×C79 | C22 | 1 | 1 | yes | characteristic |
| 26 | #493×C79 | C22 | 1 | 1 | yes | characteristic |
| 27 | #486×C79 | — | 2 | 1 | no | — |
| 28 | #466×C79 | — | 2 | 1 | no | — |
| 29 | #466×C79 | C22 | 1 | 2 | yes | — |
| 30 | C4×D8×C79 | — | 2 | 1 | no | — |
| 31 | C4×D8×C79 | C22 | 1 | 2 | yes | — |
| 32 | C2×C4×C8×C79 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×Dic316 | — | 2 | 1 | no | — |
| 34 | C2×Dic316 | C23 | 1 | 2 | yes | — |
| 35 | #14579161 | — | 2 | 1 | no | — |
| 36 | #14579161 | — | 2 | 1 | no | — |
| 37 | #14579153 | — | 4 | 1 | no | — |
| 38 | #14579140 | — | 2 | 2 | no | — |
| 39 | #14579140 | — | 4 | 1 | no | — |
| 40 | C4×Dic158 | — | 2 | 2 | no | — |
| 41 | C4×Dic158 | — | 4 | 1 | no | — |
| 42 | #14579120 | — | 4 | 1 | no | — |
| 43 | #14579105 | — | 4 | 4 | no | — |
| 44 | #14579101 | — | 4 | 2 | no | — |
| 45 | C2×C79⋊C16 | C23 | 1 | 4 | yes | — |
| 46 | #14579095 | — | 2 | 1 | no | — |
| 47 | #14579095 | C23 | 1 | 2 | yes | — |
| 48 | #14579086 | — | 2 | 2 | no | — |
| 49 | #14579086 | — | 2 | 2 | no | — |
| 50 | C2×#41×C79 | — | 4 | 1 | no | — |
| 51 | C2×D8×C79 | — | 2 | 1 | no | — |
| 52 | C2×D8×C79 | C23 | 1 | 2 | yes | — |
| 53 | C22×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 54 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 55 | C4⋊Q8×C79 | — | 2 | 1 | no | — |
| 56 | #127×C79 | — | 2 | 1 | no | — |
| 57 | #124×C79 | — | 4 | 1 | no | — |
| 58 | #121×C79 | — | 2 | 2 | no | — |
| 59 | #121×C79 | — | 4 | 1 | no | — |
| 60 | C4×D4×C79 | — | 2 | 2 | no | — |
| 61 | C4×D4×C79 | — | 4 | 1 | no | — |
| 62 | C2×C42×C79 | D4 | 1 | 1 | yes | characteristic |
| 63 | #107×C79 | — | 2 | 1 | no | — |
| 64 | #107×C79 | C23 | 1 | 2 | yes | — |
| 65 | #102×C79 | — | 2 | 2 | no | — |
| 66 | #102×C79 | — | 2 | 2 | no | — |
| 67 | C4×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 68 | C4×C8×C79 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 71 | C2×Dic158 | — | 2 | 2 | no | — |
| 72 | C2×Dic158 | — | 4 | 1 | no | — |
| 73 | Dic316 | — | 2 | 2 | no | — |
| 74 | Dic316 | — | 2 | 2 | no | — |
| 75 | Dic316 | — | 4 | 2 | no | — |
| 76 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 77 | C4⋊Dic79 | — | 4 | 1 | no | — |
| 78 | C4⋊Dic79 | — | 4 | 1 | no | — |
| 79 | C4⋊Dic79 | — | 4 | 1 | no | — |
| 80 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 81 | Dic79⋊C4 | — | 4 | 2 | no | — |
| 82 | C22⋊Dic79 | — | 4 | 2 | no | — |
| 83 | C4×Dic79 | — | 4 | 2 | no | — |
| 84 | C79⋊C16 | — | 2 | 4 | no | — |
| 85 | #41×C79 | — | 8 | 4 | no | — |
| 86 | C2×Q8×C79 | — | 4 | 1 | no | — |
| 87 | C2×D4×C79 | — | 2 | 2 | no | — |
| 88 | C2×D4×C79 | — | 4 | 2 | no | — |
| 89 | C2×D4×C79 | — | 4 | 1 | no | — |
| 90 | C2×D4×C79 | — | 4 | 1 | no | — |
| 91 | C22×C4×C79 | — | 4 | 2 | no | — |
| 92 | C22×C4×C79 | D8 | 1 | 1 | yes | characteristic |
| 93 | C22×C4×C79 | D8 | 1 | 1 | yes | characteristic |
| 94 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | D8×C79 | — | 2 | 2 | no | — |
| 96 | D8×C79 | — | 2 | 2 | no | — |
| 97 | D8×C79 | — | 4 | 2 | no | — |
| 98 | C2×C8×C79 | — | 2 | 1 | no | — |
| 99 | C2×C8×C79 | — | 2 | 1 | no | — |
| 100 | C2×C8×C79 | — | 4 | 1 | no | — |
| 101 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C2×C8×C79 | C24 | 1 | 1 | yes | characteristic |
| 105 | C4⋊C4×C79 | — | 2 | 2 | no | — |
| 106 | C4⋊C4×C79 | — | 4 | 1 | no | — |
| 107 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 108 | C22⋊C4×C79 | — | 4 | 2 | no | — |
| 109 | C42×C79 | D8 | 1 | 1 | yes | characteristic |
| 110 | C42×C79 | D8 | 1 | 1 | yes | characteristic |
| 111 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | C23×C79 | — | 4 | 2 | no | — |
| 114 | C23×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 115 | Q8×C79 | — | 8 | 2 | no | — |
| 116 | D4×C79 | — | 4 | 2 | no | — |
| 117 | D4×C79 | — | 4 | 2 | no | — |
| 118 | D4×C79 | — | 8 | 2 | no | — |
| 119 | D4×C79 | — | 8 | 4 | no | — |
| 120 | D4×C79 | — | 8 | 2 | no | — |
| 121 | C2×C4×C79 | — | 2 | 1 | no | — |
| 122 | C2×C4×C79 | — | 2 | 1 | no | — |
| 123 | C2×C4×C79 | — | 2 | 2 | no | — |
| 124 | C2×C4×C79 | — | 2 | 2 | no | — |
| 125 | C2×C4×C79 | — | 4 | 2 | no | — |
| 126 | C2×C4×C79 | — | 8 | 4 | no | — |
| 127 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C79 | C2×D8 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C79 | C22×D4 | 1 | 1 | yes | characteristic |
| 133 | C2×Dic79 | — | 4 | 2 | no | — |
| 134 | C2×Dic79 | — | 4 | 2 | no | — |
| 135 | Dic158 | — | 4 | 2 | no | — |
| 136 | Dic158 | — | 4 | 2 | no | — |
| 137 | Dic158 | — | 8 | 2 | no | — |
| 138 | C8×C79 | — | 2 | 1 | no | — |
| 139 | C8×C79 | — | 2 | 1 | no | — |
| 140 | C8×C79 | — | 2 | 1 | no | — |
| 141 | C8×C79 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 142 | C8×C79 | 2−1+4 | 1 | 1 | yes | characteristic |
| 143 | C22×C79 | — | 2 | 1 | no | — |
| 144 | C22×C79 | — | 2 | 1 | no | — |
| 145 | C22×C79 | — | 2 | 1 | no | — |
| 146 | C22×C79 | — | 8 | 2 | no | — |
| 147 | C22×C79 | — | 8 | 2 | no | — |
| 148 | C22×C79 | — | 8 | 2 | no | — |
| 149 | C22×C79 | C22×D8 | 1 | 1 | yes | characteristic |
| 150 | C4×C79 | — | 2 | 1 | no | — |
| 151 | C4×C79 | — | 2 | 1 | no | — |
| 152 | C4×C79 | — | 2 | 1 | no | — |
| 153 | C4×C79 | — | 8 | 2 | no | — |
| 154 | C4×C79 | #509 | 1 | 2 | yes | — |
| 155 | C4×C79 | #510 | 1 | 2 | yes | — |
| 156 | C4×C79 | #537 | 1 | 1 | yes | characteristic |
| 157 | C4×C79 | #575 | 1 | 1 | yes | characteristic |
| 158 | Dic79 | — | 8 | 2 | no | — |
| 159 | Dic79 | — | 8 | 2 | no | — |
| 160 | #33820 | — | 79 | 1 | no | maximal, Sylow |
| 161 | C2×C79 | — | 4 | 1 | no | — |
| 162 | C2×C79 | — | 16 | 2 | no | — |
| 163 | C2×C79 | #3245 | 1 | 1 | yes | characteristic |
| 164 | C2×C79 | C2×#509 | 1 | 1 | yes | characteristic |
| 165 | C2×C79 | C2×#510 | 1 | 1 | yes | characteristic |
| 166 | #3170 | — | 79 | 1 | no | — |
| 167 | #3169 | D79 | 1 | 1 | yes | characteristic |
| 168 | #2227 | — | 79 | 2 | no | — |
| 169 | #2224 | — | 79 | 2 | no | — |
| 170 | #2220 | — | 79 | 4 | no | — |
| 171 | #2208 | — | 79 | 2 | no | — |
| 172 | #2207 | — | 79 | 2 | no | — |
| 173 | #2150 | — | 79 | 1 | no | — |
| 174 | C79 | #33820 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 175 | #533 | — | 158 | 1 | no | — |
| 176 | #532 | — | 2 | 1 | no | — |
| 177 | C2×SD32 | — | 79 | 4 | no | — |
| 178 | C2×M32 | — | 79 | 2 | no | — |
| 179 | #500 | — | 79 | 1 | no | — |
| 180 | #500 | C2×D79 | 1 | 1 | yes | characteristic |
| 181 | #494 | — | 79 | 1 | no | — |
| 182 | #493 | C2×D79 | 1 | 1 | yes | characteristic |
| 183 | #487 | — | 158 | 1 | no | — |
| 184 | #486 | — | 2 | 1 | no | — |
| 185 | #467 | — | 79 | 2 | no | — |
| 186 | #467 | — | 158 | 1 | no | — |
| 187 | #466 | — | 2 | 1 | no | — |
| 188 | #466 | C2×D79 | 1 | 2 | yes | — |
| 189 | C4×Q16 | — | 79 | 2 | no | — |
| 190 | C4×Q16 | — | 158 | 1 | no | — |
| 191 | C4×D8 | — | 2 | 1 | no | — |
| 192 | C4×D8 | C2×D79 | 1 | 2 | yes | — |
| 193 | C2×C4×C8 | C2×D79 | 1 | 1 | yes | characteristic |
| 194 | #366 | — | 79 | 4 | no | — |
| 195 | C4⋊C16 | — | 79 | 2 | no | — |
| 196 | C4⋊C16 | — | 79 | 2 | no | — |
| 197 | #358 | — | 79 | 4 | no | — |
| 198 | #357 | — | 79 | 4 | no | — |
| 199 | C2×#41 | — | 4 | 1 | no | — |
| 200 | C2×Q16 | — | 79 | 2 | no | — |
| 201 | C2×Q16 | — | 158 | 1 | no | — |
| 202 | C2×D8 | — | 2 | 1 | no | — |
| 203 | C2×D8 | C22×D79 | 1 | 2 | yes | — |
| 204 | C22×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 205 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 206 | C4⋊Q8 | — | 2 | 1 | no | — |
| 207 | C4⋊Q8 | — | 158 | 1 | no | — |
| 208 | C4⋊Q8 | — | 158 | 1 | no | — |
| 209 | #127 | — | 2 | 1 | no | — |
| 210 | #125 | — | 316 | 1 | no | — |
| 211 | #124 | — | 4 | 1 | no | — |
| 212 | C22⋊Q8 | — | 316 | 1 | no | — |
| 213 | C22⋊Q8 | — | 158 | 2 | no | — |
| 214 | #121 | — | 2 | 2 | no | — |
| 215 | #121 | — | 4 | 1 | no | — |
| 216 | C4×Q8 | — | 316 | 1 | no | — |
| 217 | C4×Q8 | — | 158 | 2 | no | — |
| 218 | C4×D4 | — | 2 | 2 | no | — |
| 219 | C4×D4 | — | 4 | 1 | no | — |
| 220 | #117 | — | 316 | 1 | no | — |
| 221 | C2×C42 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 222 | SD32 | — | 316 | 4 | no | — |
| 223 | M32 | — | 316 | 2 | no | — |
| 224 | C2×C16 | — | 79 | 4 | no | — |
| 225 | #107 | — | 2 | 1 | no | — |
| 226 | #107 | — | 79 | 2 | no | — |
| 227 | #107 | — | 158 | 1 | no | — |
| 228 | #107 | C22×D79 | 1 | 2 | yes | — |
| 229 | #103 | — | 158 | 2 | no | — |
| 230 | #103 | — | 158 | 2 | no | — |
| 231 | #102 | — | 2 | 2 | no | — |
| 232 | #102 | — | 2 | 2 | no | — |
| 233 | C4×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 234 | C4×C8 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 235 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 236 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 237 | #41 | — | 8 | 4 | no | — |
| 238 | C2×Q8 | — | 4 | 1 | no | — |
| 239 | C2×Q8 | — | 316 | 1 | no | — |
| 240 | C2×Q8 | — | 158 | 2 | no | — |
| 241 | C2×D4 | — | 2 | 2 | no | — |
| 242 | C2×D4 | — | 4 | 2 | no | — |
| 243 | C2×D4 | — | 4 | 1 | no | — |
| 244 | C2×D4 | — | 4 | 1 | no | — |
| 245 | C22×C4 | — | 4 | 2 | no | — |
| 246 | C22×C4 | #11929609 | 1 | 1 | yes | characteristic |
| 247 | C22×C4 | #11929609 | 1 | 1 | yes | characteristic |
| 248 | C22×C4 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 249 | Q16 | — | 316 | 2 | no | — |
| 250 | Q16 | — | 158 | 2 | no | — |
| 251 | Q16 | — | 158 | 2 | no | — |
| 252 | D8 | — | 2 | 2 | no | — |
| 253 | D8 | — | 2 | 2 | no | — |
| 254 | D8 | — | 4 | 2 | no | — |
| 255 | C2×C8 | — | 2 | 1 | no | — |
| 256 | C2×C8 | — | 2 | 1 | no | — |
| 257 | C2×C8 | — | 4 | 1 | no | — |
| 258 | C2×C8 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 261 | C2×C8 | C23×D79 | 1 | 1 | yes | characteristic |
| 262 | C4⋊C4 | — | 2 | 2 | no | — |
| 263 | C4⋊C4 | — | 4 | 1 | no | — |
| 264 | C4⋊C4 | — | 316 | 2 | no | — |
| 265 | C4⋊C4 | — | 316 | 2 | no | — |
| 266 | C4⋊C4 | — | 316 | 1 | no | — |
| 267 | C4⋊C4 | — | 316 | 1 | no | — |
| 268 | C4⋊C4 | — | 316 | 1 | no | — |
| 269 | C4⋊C4 | — | 158 | 2 | no | — |
| 270 | C22⋊C4 | — | 4 | 2 | no | — |
| 271 | C22⋊C4 | — | 4 | 2 | no | — |
| 272 | C22⋊C4 | — | 316 | 2 | no | — |
| 273 | C42 | — | 316 | 2 | no | — |
| 274 | C42 | #11929609 | 1 | 1 | yes | characteristic |
| 275 | C42 | #11929609 | 1 | 1 | yes | characteristic |
| 276 | C42 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 277 | C42 | C2×C22⋊D79 | 1 | 1 | yes | characteristic |
| 278 | C16 | — | 158 | 4 | no | — |
| 279 | C23 | — | 4 | 2 | no | — |
| 280 | C23 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 8 | 2 | no | — |
| 282 | Q8 | — | 316 | 2 | no | — |
| 283 | Q8 | — | 316 | 2 | no | — |
| 284 | Q8 | — | 632 | 2 | no | — |
| 285 | D4 | — | 4 | 2 | no | — |
| 286 | D4 | — | 4 | 2 | no | — |
| 287 | D4 | — | 8 | 2 | no | — |
| 288 | D4 | — | 8 | 4 | no | — |
| 289 | D4 | — | 8 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 1 | no | — |
| 291 | C2×C4 | — | 2 | 2 | no | — |
| 292 | C2×C4 | — | 2 | 2 | no | — |
| 293 | C2×C4 | — | 2 | 1 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 8 | 4 | no | — |
| 296 | C2×C4 | — | 316 | 2 | no | — |
| 297 | C2×C4 | — | 316 | 2 | no | — |
| 298 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | C2×#11929609 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C22×C22⋊D79 | 1 | 1 | yes | characteristic |
| 304 | C8 | — | 2 | 1 | no | — |
| 305 | C8 | — | 2 | 1 | no | — |
| 306 | C8 | — | 2 | 1 | no | — |
| 307 | C8 | #14579207 | 1 | 1 | yes | characteristic |
| 308 | C8 | #14579208 | 1 | 1 | yes | characteristic |
| 309 | C22 | — | 2 | 1 | no | — |
| 310 | C22 | — | 2 | 1 | no | — |
| 311 | C22 | — | 2 | 1 | no | — |
| 312 | C22 | — | 8 | 2 | no | — |
| 313 | C22 | — | 8 | 2 | no | — |
| 314 | C22 | — | 8 | 2 | no | — |
| 315 | C22 | C22×#11929609 | 1 | 1 | yes | characteristic |
| 316 | C4 | — | 2 | 1 | no | — |
| 317 | C4 | — | 2 | 1 | no | — |
| 318 | C4 | — | 2 | 1 | no | — |
| 319 | C4 | — | 8 | 2 | no | — |
| 320 | C4 | — | 632 | 2 | no | — |
| 321 | C4 | — | 632 | 2 | no | — |
| 322 | C4 | #19098999 | 1 | 2 | yes | — |
| 323 | C4 | #19099003 | 1 | 2 | yes | — |
| 324 | C4 | #19099141 | 1 | 1 | yes | characteristic |
| 325 | C4 | #19099361 | 1 | 1 | yes | characteristic |
| 326 | C2 | — | 4 | 1 | no | — |
| 327 | C2 | — | 16 | 2 | no | — |
| 328 | C2 | #23774707 | 1 | 1 | yes | characteristic |
| 329 | C2 | C2×#19098999 | 1 | 1 | yes | characteristic |
| 330 | C2 | C2×#19099003 | 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