← #34260306
Subgroups — #34260306
Order 20224.
644 conjugacy classes of subgroups.
The subgroup lattice has 304 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #23773979 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23772466 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23762313 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23762228 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #23762223 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #23761816 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #23761689 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C79×#1564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #19099113 | C22 | 1 | 2 | yes | — |
| 10 | #19098968 | C4 | 1 | 1 | yes | characteristic |
| 11 | #19098940 | C4 | 1 | 1 | yes | characteristic |
| 12 | #19098910 | C4 | 1 | 2 | yes | — |
| 13 | #19098799 | C4 | 1 | 2 | yes | — |
| 14 | #19098799 | C4 | 1 | 2 | yes | — |
| 15 | C4×D632 | C22 | 1 | 2 | yes | — |
| 16 | C4×D632 | C22 | 1 | 2 | yes | — |
| 17 | #19098642 | — | 2 | 2 | no | — |
| 18 | #19098640 | C22 | 1 | 4 | yes | — |
| 19 | C8×D316 | — | 2 | 2 | no | — |
| 20 | #19098628 | C22 | 1 | 4 | yes | — |
| 21 | #19098616 | C22 | 1 | 1 | yes | characteristic |
| 22 | #19098528 | — | 2 | 4 | no | — |
| 23 | C4×#11929608 | C22 | 1 | 1 | yes | characteristic |
| 24 | #19098290 | C22 | 1 | 4 | yes | — |
| 25 | #19098290 | C22 | 1 | 4 | yes | — |
| 26 | #19098285 | C22 | 1 | 2 | yes | — |
| 27 | #19098284 | C22 | 1 | 2 | yes | — |
| 28 | #431×C79 | C22 | 1 | 2 | yes | — |
| 29 | C2×C4×C8×C79 | C22 | 1 | 1 | yes | characteristic |
| 30 | #335×C79 | C22 | 1 | 2 | yes | — |
| 31 | #334×C79 | C22 | 1 | 1 | yes | characteristic |
| 32 | #334×C79 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×#11929624 | — | 2 | 2 | no | — |
| 34 | C2×D632 | C2×C4 | 1 | 2 | yes | — |
| 35 | C2×D632 | C2×C4 | 1 | 2 | yes | — |
| 36 | C2×#11929608 | — | 2 | 1 | no | — |
| 37 | C2×#11929607 | — | 2 | 4 | no | — |
| 38 | C2×C8×D79 | — | 2 | 4 | no | — |
| 39 | #14579161 | C2×C4 | 1 | 2 | yes | — |
| 40 | #14579158 | C2×C4 | 1 | 2 | yes | — |
| 41 | #14579149 | — | 2 | 2 | no | — |
| 42 | #14579135 | — | 2 | 2 | no | — |
| 43 | #14579135 | C2×C4 | 1 | 4 | yes | — |
| 44 | C4×D316 | — | 2 | 2 | no | — |
| 45 | C4×D316 | C23 | 1 | 4 | yes | — |
| 46 | #14579095 | C2×C4 | 1 | 2 | yes | — |
| 47 | #14579095 | C2×C4 | 1 | 2 | yes | — |
| 48 | C4⋊(C79⋊C8) | D4 | 1 | 2 | yes | — |
| 49 | #14579083 | C2×C4 | 1 | 4 | yes | — |
| 50 | #14579083 | C2×C4 | 1 | 4 | yes | — |
| 51 | C22⋊(C79⋊C8) | D4 | 1 | 2 | yes | — |
| 52 | (C2×D79)⋊C8 | — | 2 | 4 | no | — |
| 53 | (C2×D79)⋊C8 | — | 2 | 4 | no | — |
| 54 | #14579072 | — | 2 | 1 | no | — |
| 55 | C4×C79⋊C8 | C23 | 1 | 2 | yes | — |
| 56 | C4×C79⋊C8 | C23 | 1 | 2 | yes | — |
| 57 | C22×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | C22×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C2×C42×C79 | C23 | 1 | 1 | yes | characteristic |
| 60 | C4⋊C8×C79 | C23 | 1 | 2 | yes | — |
| 61 | C4⋊C8×C79 | C23 | 1 | 2 | yes | — |
| 62 | C22⋊C8×C79 | — | 2 | 2 | no | — |
| 63 | C4×C8×C79 | — | 2 | 2 | no | — |
| 64 | C4×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 65 | C4×C8×C79 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 67 | C4×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 68 | #11929624 | — | 8 | 2 | no | — |
| 69 | #11929624 | — | 8 | 2 | no | — |
| 70 | C2×Dic158 | — | 2 | 2 | no | — |
| 71 | C2×D316 | — | 2 | 2 | no | — |
| 72 | C2×D316 | — | 2 | 2 | no | — |
| 73 | C2×D316 | C22×C4 | 1 | 4 | yes | — |
| 74 | C2×C22⋊D79 | — | 2 | 4 | no | — |
| 75 | C2×C4×D79 | — | 2 | 4 | no | — |
| 76 | D632 | — | 2 | 4 | no | — |
| 77 | D632 | — | 4 | 2 | no | — |
| 78 | #11929608 | — | 4 | 2 | no | — |
| 79 | #11929607 | — | 8 | 4 | no | — |
| 80 | C2×C79⋊C8 | — | 2 | 2 | no | — |
| 81 | C2×C79⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 82 | C8×D79 | — | 8 | 4 | no | — |
| 83 | C4⋊Dic79 | — | 2 | 2 | no | — |
| 84 | C4⋊Dic79 | C22×C4 | 1 | 4 | yes | — |
| 85 | (C2×D79)⋊C4 | — | 2 | 4 | no | — |
| 86 | (C2×D79)⋊C4 | — | 2 | 4 | no | — |
| 87 | C22×C4×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C22×C4×C79 | #41 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C79 | — | 2 | 2 | no | — |
| 91 | C2×C8×C79 | — | 2 | 2 | no | — |
| 92 | C2×C8×C79 | — | 2 | 2 | no | — |
| 93 | C2×C8×C79 | — | 4 | 1 | no | — |
| 94 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 95 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C8×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | C42×C79 | C22×C4 | 1 | 1 | yes | characteristic |
| 99 | C42×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C42×C79 | #41 | 1 | 1 | yes | characteristic |
| 101 | C42×C79 | C24 | 1 | 1 | yes | characteristic |
| 102 | C23×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 103 | C22×D79 | — | 2 | 4 | no | — |
| 104 | C2×C4×C79 | — | 2 | 1 | no | — |
| 105 | C2×C4×C79 | — | 2 | 1 | no | — |
| 106 | C2×C4×C79 | — | 2 | 2 | no | — |
| 107 | C2×C4×C79 | — | 2 | 2 | no | — |
| 108 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 109 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C79 | C4×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C79 | C23×C4 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C79 | C22×D4 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C79 | C2×#41 | 1 | 1 | yes | characteristic |
| 114 | C22⋊D79 | — | 8 | 4 | no | — |
| 115 | C2×Dic79 | — | 2 | 4 | no | — |
| 116 | D316 | — | 2 | 4 | no | — |
| 117 | D316 | — | 2 | 4 | no | — |
| 118 | D316 | — | 4 | 4 | no | — |
| 119 | D316 | — | 8 | 2 | no | — |
| 120 | C4×D79 | — | 4 | 4 | no | — |
| 121 | C4×D79 | — | 4 | 4 | no | — |
| 122 | Dic158 | — | 8 | 2 | no | — |
| 123 | C8×C79 | — | 2 | 1 | no | — |
| 124 | C8×C79 | — | 2 | 1 | no | — |
| 125 | C8×C79 | — | 2 | 1 | no | — |
| 126 | C8×C79 | — | 4 | 2 | no | — |
| 127 | C8×C79 | — | 4 | 2 | no | — |
| 128 | C8×C79 | #131 | 1 | 2 | yes | — |
| 129 | C79⋊C8 | — | 2 | 4 | no | — |
| 130 | C79⋊C8 | — | 4 | 2 | no | — |
| 131 | C22×C79 | — | 2 | 1 | no | — |
| 132 | C22×C79 | — | 2 | 1 | no | — |
| 133 | C22×C79 | — | 2 | 1 | no | — |
| 134 | C22×C79 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 135 | C2×D79 | — | 4 | 4 | no | — |
| 136 | C2×D79 | — | 4 | 4 | no | — |
| 137 | C2×D79 | — | 4 | 4 | no | — |
| 138 | C4×C79 | — | 2 | 1 | no | — |
| 139 | C4×C79 | — | 2 | 1 | no | — |
| 140 | C4×C79 | — | 2 | 1 | no | — |
| 141 | C4×C79 | #444 | 1 | 2 | yes | — |
| 142 | C4×C79 | C2×#131 | 1 | 1 | yes | characteristic |
| 143 | C4×C79 | #568 | 1 | 1 | yes | characteristic |
| 144 | C4×C79 | #576 | 1 | 2 | yes | — |
| 145 | Dic79 | — | 4 | 4 | no | — |
| 146 | #18091 | — | 79 | 1 | no | maximal, Sylow |
| 147 | C2×C79 | — | 4 | 1 | no | — |
| 148 | C2×C79 | #2935 | 1 | 1 | yes | characteristic |
| 149 | C2×C79 | #2948 | 1 | 1 | yes | characteristic |
| 150 | C2×C79 | C2×#444 | 1 | 1 | yes | characteristic |
| 151 | D79 | — | 8 | 4 | no | — |
| 152 | #3169 | — | 79 | 1 | no | — |
| 153 | #2979 | — | 79 | 2 | no | — |
| 154 | #1594 | — | 79 | 2 | no | — |
| 155 | #1581 | — | 79 | 4 | no | — |
| 156 | #1580 | — | 79 | 2 | no | — |
| 157 | #1564 | D79 | 1 | 1 | yes | characteristic, complemented |
| 158 | #1484 | — | 79 | 2 | no | — |
| 159 | #1451 | — | 79 | 1 | no | — |
| 160 | C79 | #18091 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 161 | #532 | — | 79 | 2 | no | — |
| 162 | #500 | — | 79 | 1 | no | — |
| 163 | #493 | — | 79 | 1 | no | — |
| 164 | #486 | — | 79 | 2 | no | — |
| 165 | #466 | — | 79 | 2 | no | — |
| 166 | #466 | — | 79 | 2 | no | — |
| 167 | C4×D8 | — | 79 | 2 | no | — |
| 168 | C4×D8 | — | 79 | 2 | no | — |
| 169 | #436 | — | 79 | 4 | no | — |
| 170 | #436 | — | 158 | 2 | no | — |
| 171 | C8×D4 | — | 79 | 4 | no | — |
| 172 | C8×D4 | — | 158 | 2 | no | — |
| 173 | #431 | — | 79 | 1 | no | — |
| 174 | #431 | C2×D79 | 1 | 2 | yes | — |
| 175 | #408 | — | 158 | 4 | no | — |
| 176 | C4×M16 | — | 79 | 1 | no | — |
| 177 | C2×C4×C8 | C2×D79 | 1 | 1 | yes | characteristic |
| 178 | #335 | C2×D79 | 1 | 2 | yes | — |
| 179 | #334 | C2×D79 | 1 | 1 | yes | characteristic |
| 180 | #334 | C2×D79 | 1 | 1 | yes | characteristic |
| 181 | D4⋊C8 | — | 79 | 4 | no | — |
| 182 | D4⋊C8 | — | 79 | 4 | no | — |
| 183 | #322 | — | 79 | 2 | no | — |
| 184 | #322 | — | 79 | 2 | no | — |
| 185 | C2×#41 | — | 158 | 2 | no | — |
| 186 | C2×D8 | — | 79 | 2 | no | — |
| 187 | C2×D8 | — | 79 | 2 | no | — |
| 188 | C2×M16 | — | 158 | 4 | no | — |
| 189 | C2×M16 | — | 158 | 1 | no | — |
| 190 | C22×C8 | — | 158 | 4 | no | — |
| 191 | C22×C8 | C4×D79 | 1 | 1 | yes | characteristic |
| 192 | C22×C8 | C4×D79 | 1 | 1 | yes | characteristic |
| 193 | C4⋊Q8 | — | 79 | 2 | no | — |
| 194 | #127 | — | 79 | 2 | no | — |
| 195 | #124 | — | 158 | 2 | no | — |
| 196 | #121 | — | 79 | 4 | no | — |
| 197 | #121 | — | 158 | 2 | no | — |
| 198 | C4×D4 | — | 79 | 4 | no | — |
| 199 | C4×D4 | — | 158 | 2 | no | — |
| 200 | C2×C42 | C22×D79 | 1 | 1 | yes | characteristic |
| 201 | #107 | — | 79 | 2 | no | — |
| 202 | #107 | — | 79 | 2 | no | — |
| 203 | C4⋊C8 | — | 79 | 2 | no | — |
| 204 | C4⋊C8 | C22×D79 | 1 | 2 | yes | — |
| 205 | C4⋊C8 | C22×D79 | 1 | 2 | yes | — |
| 206 | #102 | — | 79 | 4 | no | — |
| 207 | #102 | — | 79 | 4 | no | — |
| 208 | C22⋊C8 | — | 2 | 2 | no | — |
| 209 | C22⋊C8 | — | 79 | 2 | no | — |
| 210 | C22⋊C8 | — | 158 | 4 | no | — |
| 211 | C22⋊C8 | — | 158 | 4 | no | — |
| 212 | #97 | — | 158 | 1 | no | — |
| 213 | C4×C8 | — | 2 | 2 | no | — |
| 214 | C4×C8 | — | 79 | 2 | no | — |
| 215 | C4×C8 | — | 79 | 2 | no | — |
| 216 | C4×C8 | C4×D79 | 1 | 1 | yes | characteristic |
| 217 | C4×C8 | C4×D79 | 1 | 1 | yes | characteristic |
| 218 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 219 | C4×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 220 | #41 | — | 632 | 2 | no | — |
| 221 | #41 | — | 632 | 2 | no | — |
| 222 | C2×Q8 | — | 158 | 2 | no | — |
| 223 | C2×D4 | — | 79 | 4 | no | — |
| 224 | C2×D4 | — | 158 | 4 | no | — |
| 225 | C2×D4 | — | 158 | 2 | no | — |
| 226 | C2×D4 | — | 158 | 2 | no | — |
| 227 | C22×C4 | — | 158 | 4 | no | — |
| 228 | C22×C4 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 229 | C22×C4 | D4×D79 | 1 | 1 | yes | characteristic |
| 230 | C22×C4 | #11929625 | 1 | 1 | yes | characteristic |
| 231 | D8 | — | 316 | 2 | no | — |
| 232 | D8 | — | 158 | 4 | no | — |
| 233 | M16 | — | 316 | 2 | no | — |
| 234 | M16 | — | 632 | 4 | no | — |
| 235 | C2×C8 | — | 2 | 2 | no | — |
| 236 | C2×C8 | — | 2 | 2 | no | — |
| 237 | C2×C8 | — | 2 | 2 | no | — |
| 238 | C2×C8 | — | 4 | 1 | no | — |
| 239 | C2×C8 | — | 79 | 4 | no | — |
| 240 | C2×C8 | — | 632 | 4 | no | — |
| 241 | C2×C8 | — | 158 | 2 | no | — |
| 242 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 246 | C4⋊C4 | — | 79 | 4 | no | — |
| 247 | C4⋊C4 | — | 158 | 2 | no | — |
| 248 | C22⋊C4 | — | 158 | 4 | no | — |
| 249 | C22⋊C4 | — | 158 | 4 | no | — |
| 250 | C42 | C2×C4×D79 | 1 | 1 | yes | characteristic |
| 251 | C42 | D4×D79 | 1 | 1 | yes | characteristic |
| 252 | C42 | #11929625 | 1 | 1 | yes | characteristic |
| 253 | C42 | C23×D79 | 1 | 1 | yes | characteristic |
| 254 | C23 | — | 158 | 4 | no | — |
| 255 | C23 | #14579123 | 1 | 1 | yes | characteristic |
| 256 | Q8 | — | 632 | 2 | no | — |
| 257 | D4 | — | 316 | 4 | no | — |
| 258 | D4 | — | 632 | 2 | no | — |
| 259 | D4 | — | 632 | 4 | no | — |
| 260 | D4 | — | 158 | 4 | no | — |
| 261 | D4 | — | 158 | 4 | no | — |
| 262 | C2×C4 | — | 2 | 2 | no | — |
| 263 | C2×C4 | — | 2 | 1 | no | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 316 | 4 | no | — |
| 267 | C2×C4 | — | 316 | 4 | no | — |
| 268 | C2×C4 | — | 158 | 4 | no | — |
| 269 | C2×C4 | #14579123 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #14579123 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | #14579123 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | C22×C4×D79 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C2×D4×D79 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | C2×#11929625 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 2 | 1 | no | — |
| 277 | C8 | — | 2 | 1 | no | — |
| 278 | C8 | — | 4 | 2 | no | — |
| 279 | C8 | — | 4 | 2 | no | — |
| 280 | C8 | — | 316 | 2 | no | — |
| 281 | C8 | — | 158 | 4 | no | — |
| 282 | C8 | #14579167 | 1 | 2 | yes | — |
| 283 | C22 | — | 2 | 1 | no | — |
| 284 | C22 | — | 2 | 1 | no | — |
| 285 | C22 | — | 2 | 1 | no | — |
| 286 | C22 | — | 316 | 4 | no | — |
| 287 | C22 | — | 316 | 4 | no | — |
| 288 | C22 | — | 316 | 4 | no | — |
| 289 | C22 | C2×#14579123 | 1 | 1 | yes | characteristic |
| 290 | C4 | — | 2 | 1 | no | — |
| 291 | C4 | — | 2 | 1 | no | — |
| 292 | C4 | — | 2 | 1 | no | — |
| 293 | C4 | — | 316 | 4 | no | — |
| 294 | C4 | #19098684 | 1 | 2 | yes | — |
| 295 | C4 | C2×#14579167 | 1 | 1 | yes | characteristic |
| 296 | C4 | #19099329 | 1 | 1 | yes | characteristic |
| 297 | C4 | #19099365 | 1 | 2 | yes | — |
| 298 | C2 | — | 4 | 1 | no | — |
| 299 | C2 | — | 632 | 4 | no | — |
| 300 | C2 | #23772143 | 1 | 1 | yes | characteristic |
| 301 | C2 | #23772251 | 1 | 1 | yes | characteristic |
| 302 | C2 | C2×#19098684 | 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