← #34306752
Subgroups — #34306752
Order 20224.
846 conjugacy classes of subgroups.
The subgroup lattice has 330 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #23772276 | C2 | 1 | 2 | yes | maximal |
| 2 | #23771995 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23764186 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23763635 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #23763564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #23763459 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #23763279 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#443×C79 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #19099046 | C22 | 1 | 4 | yes | — |
| 10 | #19098664 | — | 2 | 4 | no | — |
| 11 | Q16×Dic79 | — | 2 | 2 | no | — |
| 12 | D8×Dic79 | — | 2 | 2 | no | — |
| 13 | C2×#14579097 | C22 | 1 | 1 | yes | characteristic |
| 14 | #19098605 | — | 2 | 4 | no | — |
| 15 | #19098605 | C22 | 1 | 4 | yes | — |
| 16 | #19098601 | — | 2 | 2 | no | — |
| 17 | #19098601 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#14579095 | C22 | 1 | 2 | yes | — |
| 19 | C2×#14579093 | — | 2 | 2 | no | — |
| 20 | C2×#14579093 | C22 | 1 | 2 | yes | — |
| 21 | #19098564 | C22 | 1 | 4 | yes | — |
| 22 | #19098518 | C22 | 1 | 2 | yes | — |
| 23 | #19098429 | C22 | 1 | 2 | yes | — |
| 24 | #19098331 | C22 | 1 | 4 | yes | — |
| 25 | #19098323 | C22 | 1 | 2 | yes | — |
| 26 | C2×#135×C79 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#131×C79 | C22 | 1 | 2 | yes | — |
| 28 | #443×C79 | C4 | 1 | 8 | yes | complemented |
| 29 | C2×#108×C79 | C22 | 1 | 1 | yes | characteristic |
| 30 | C2×#104×C79 | C22 | 1 | 2 | yes | — |
| 31 | C2×C4×C8×C79 | C22 | 1 | 1 | yes | characteristic |
| 32 | C2×#11929608 | C23 | 1 | 2 | yes | — |
| 33 | Q8×Dic79 | — | 2 | 4 | no | — |
| 34 | D4×Dic79 | — | 2 | 4 | no | — |
| 35 | D4×Dic79 | — | 4 | 4 | no | — |
| 36 | #14579120 | — | 2 | 4 | no | — |
| 37 | #14579120 | C23 | 1 | 2 | yes | — |
| 38 | C2×C4⋊Dic79 | — | 2 | 4 | no | — |
| 39 | C2×C4⋊Dic79 | C23 | 1 | 4 | yes | — |
| 40 | #14579097 | — | 2 | 2 | no | — |
| 41 | #14579095 | — | 2 | 2 | no | — |
| 42 | #14579095 | — | 2 | 2 | no | — |
| 43 | #14579095 | — | 2 | 4 | no | — |
| 44 | #14579095 | D4 | 1 | 2 | yes | — |
| 45 | #14579093 | — | 2 | 2 | no | — |
| 46 | #14579093 | — | 2 | 2 | no | — |
| 47 | #14579093 | — | 2 | 4 | no | — |
| 48 | #14579093 | D4 | 1 | 1 | yes | characteristic |
| 49 | #14579093 | D4 | 1 | 1 | yes | characteristic |
| 50 | #14579085 | — | 2 | 4 | no | — |
| 51 | #14579082 | — | 2 | 4 | no | — |
| 52 | #14579071 | — | 2 | 2 | no | — |
| 53 | C8×Dic79 | — | 2 | 2 | no | — |
| 54 | C79⋊#95 | — | 4 | 2 | no | — |
| 55 | C2×#41×C79 | C23 | 1 | 2 | yes | — |
| 56 | #135×C79 | C2×C4 | 1 | 4 | yes | — |
| 57 | #135×C79 | Q8 | 1 | 4 | yes | — |
| 58 | C2×Q16×C79 | D4 | 1 | 1 | yes | characteristic |
| 59 | C2×SD16×C79 | D4 | 1 | 2 | yes | — |
| 60 | C2×D8×C79 | D4 | 1 | 1 | yes | characteristic |
| 61 | #131×C79 | — | 2 | 4 | no | — |
| 62 | #131×C79 | C2×C4 | 1 | 8 | yes | complemented |
| 63 | C2×M16×C79 | — | 2 | 2 | no | — |
| 64 | C2×M16×C79 | C23 | 1 | 2 | yes | — |
| 65 | C22×C8×C79 | — | 2 | 2 | no | — |
| 66 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 67 | C22×C8×C79 | C23 | 1 | 1 | yes | characteristic |
| 68 | C2×C42×C79 | C23 | 1 | 1 | yes | characteristic |
| 69 | #108×C79 | C2×C4 | 1 | 4 | yes | complemented |
| 70 | #104×C79 | C2×C4 | 1 | 8 | yes | complemented |
| 71 | C4×C8×C79 | C2×C4 | 1 | 4 | yes | complemented |
| 72 | C22×Dic79 | — | 2 | 4 | no | — |
| 73 | C22×Dic79 | — | 4 | 4 | no | — |
| 74 | #11929608 | — | 2 | 2 | no | — |
| 75 | #11929608 | — | 2 | 2 | no | — |
| 76 | C2×C79⋊C8 | — | 2 | 2 | no | — |
| 77 | C4⋊Dic79 | — | 2 | 4 | no | — |
| 78 | C4⋊Dic79 | — | 2 | 4 | no | — |
| 79 | C4⋊Dic79 | — | 4 | 4 | no | — |
| 80 | C4⋊Dic79 | C2×D4 | 1 | 2 | yes | — |
| 81 | C4⋊Dic79 | C2×D4 | 1 | 2 | yes | — |
| 82 | C22⋊Dic79 | — | 2 | 2 | no | — |
| 83 | C22⋊Dic79 | — | 4 | 4 | no | — |
| 84 | C4×Dic79 | — | 2 | 2 | no | — |
| 85 | C4×Dic79 | — | 4 | 4 | no | — |
| 86 | #41×C79 | — | 2 | 2 | no | — |
| 87 | #41×C79 | — | 2 | 2 | no | — |
| 88 | #41×C79 | C22×C4 | 1 | 4 | yes | — |
| 89 | #41×C79 | C2×Q8 | 1 | 4 | yes | — |
| 90 | C2×Q8×C79 | C2×D4 | 1 | 2 | yes | — |
| 91 | C2×D4×C79 | — | 2 | 2 | no | — |
| 92 | C2×D4×C79 | C2×D4 | 1 | 2 | yes | — |
| 93 | C22×C4×C79 | — | 2 | 2 | no | — |
| 94 | C22×C4×C79 | — | 2 | 1 | no | — |
| 95 | C22×C4×C79 | C24 | 1 | 1 | yes | characteristic |
| 96 | Q16×C79 | C4⋊C4 | 1 | 4 | yes | — |
| 97 | SD16×C79 | C4⋊C4 | 1 | 8 | yes | — |
| 98 | D8×C79 | C4⋊C4 | 1 | 4 | yes | — |
| 99 | M16×C79 | — | 2 | 8 | no | — |
| 100 | M16×C79 | — | 2 | 2 | no | — |
| 101 | M16×C79 | C22×C4 | 1 | 4 | yes | — |
| 102 | C2×C8×C79 | — | 2 | 8 | no | — |
| 103 | C2×C8×C79 | — | 2 | 2 | no | — |
| 104 | C2×C8×C79 | C22×C4 | 1 | 2 | yes | — |
| 105 | C2×C8×C79 | C22×C4 | 1 | 2 | yes | — |
| 106 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C2×C8×C79 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C8×C79 | C2×Q8 | 1 | 2 | yes | — |
| 111 | C2×C8×C79 | #41 | 1 | 2 | yes | — |
| 112 | C42×C79 | — | 2 | 1 | no | — |
| 113 | C42×C79 | C22×C4 | 1 | 2 | yes | — |
| 114 | C23×C79 | — | 2 | 2 | no | — |
| 115 | C23×C79 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 116 | Q8×C79 | — | 2 | 2 | no | — |
| 117 | Q8×C79 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 118 | D4×C79 | — | 2 | 4 | no | — |
| 119 | D4×C79 | — | 2 | 4 | no | — |
| 120 | D4×C79 | — | 2 | 2 | no | — |
| 121 | D4×C79 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 122 | C2×C4×C79 | — | 2 | 2 | no | — |
| 123 | C2×C4×C79 | — | 2 | 4 | no | — |
| 124 | C2×C4×C79 | — | 2 | 4 | no | — |
| 125 | C2×C4×C79 | — | 2 | 2 | no | — |
| 126 | C2×C4×C79 | — | 4 | 2 | no | — |
| 127 | C2×C4×C79 | C23×C4 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C79 | C22×D4 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C79 | C22×D4 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C79 | C22×Q8 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C79 | C2×#41 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C79 | 2−1+4 | 1 | 1 | yes | characteristic |
| 133 | C2×Dic79 | — | 2 | 2 | no | — |
| 134 | C2×Dic79 | — | 2 | 2 | no | — |
| 135 | C2×Dic79 | — | 4 | 4 | no | — |
| 136 | C2×Dic79 | — | 8 | 4 | no | — |
| 137 | C2×Dic79 | — | 16 | 4 | no | — |
| 138 | C8×C79 | — | 2 | 4 | no | — |
| 139 | C8×C79 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 140 | C8×C79 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 141 | C8×C79 | C4×D4 | 1 | 2 | yes | — |
| 142 | C8×C79 | C4×D4 | 1 | 2 | yes | — |
| 143 | C79⋊C8 | — | 4 | 2 | no | — |
| 144 | C22×C79 | — | 2 | 2 | no | — |
| 145 | C22×C79 | — | 2 | 4 | no | — |
| 146 | C22×C79 | — | 4 | 4 | no | — |
| 147 | C22×C79 | #520 | 1 | 1 | yes | characteristic |
| 148 | C22×C79 | D42 | 1 | 1 | yes | characteristic |
| 149 | C22×C79 | #554 | 1 | 1 | yes | characteristic |
| 150 | C4×C79 | — | 2 | 4 | no | — |
| 151 | C4×C79 | — | 4 | 2 | no | — |
| 152 | C4×C79 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 153 | C4×C79 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 154 | C4×C79 | #547 | 1 | 1 | yes | characteristic |
| 155 | C4×C79 | D4×Q8 | 1 | 1 | yes | characteristic |
| 156 | Dic79 | — | 4 | 2 | no | — |
| 157 | Dic79 | — | 4 | 2 | no | — |
| 158 | Dic79 | — | 8 | 4 | no | — |
| 159 | #22690 | — | 79 | 1 | no | maximal, Sylow |
| 160 | C2×C79 | — | 2 | 1 | no | — |
| 161 | C2×C79 | — | 2 | 1 | no | — |
| 162 | C2×C79 | — | 4 | 4 | no | — |
| 163 | C2×C79 | D4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 164 | C2×C79 | #3290 | 1 | 2 | yes | — |
| 165 | C2×#443 | D79 | 1 | 1 | yes | characteristic |
| 166 | #2951 | — | 79 | 2 | no | — |
| 167 | #2912 | — | 79 | 2 | no | — |
| 168 | #1952 | — | 79 | 2 | no | — |
| 169 | #1860 | — | 79 | 4 | no | — |
| 170 | #1846 | — | 79 | 1 | no | — |
| 171 | #1815 | — | 79 | 2 | no | — |
| 172 | #1772 | — | 79 | 1 | no | — |
| 173 | C79 | #22690 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 174 | C2×#135 | C2×D79 | 1 | 1 | yes | characteristic |
| 175 | C2×#131 | C2×D79 | 1 | 2 | yes | — |
| 176 | #520 | — | 79 | 4 | no | — |
| 177 | #443 | Dic79 | 1 | 8 | yes | complemented |
| 178 | #440 | — | 158 | 4 | no | — |
| 179 | C4×Q16 | — | 158 | 2 | no | — |
| 180 | C4×D8 | — | 158 | 2 | no | — |
| 181 | C2×#108 | — | 79 | 1 | no | — |
| 182 | C2×#108 | C2×D79 | 1 | 1 | yes | characteristic |
| 183 | #428 | — | 79 | 4 | no | — |
| 184 | #428 | — | 158 | 4 | no | — |
| 185 | #427 | — | 79 | 1 | no | — |
| 186 | #427 | — | 158 | 2 | no | — |
| 187 | C2×#107 | — | 79 | 2 | no | — |
| 188 | C2×#106 | — | 79 | 2 | no | — |
| 189 | C2×#106 | — | 158 | 2 | no | — |
| 190 | C2×#104 | C2×D79 | 1 | 2 | yes | — |
| 191 | #417 | — | 79 | 4 | no | — |
| 192 | #405 | — | 79 | 2 | no | — |
| 193 | C2×C4×C8 | C2×D79 | 1 | 1 | yes | characteristic |
| 194 | #382 | — | 79 | 2 | no | — |
| 195 | #343 | — | 79 | 4 | no | — |
| 196 | #339 | — | 79 | 2 | no | — |
| 197 | C2×#41 | C22×D79 | 1 | 2 | yes | — |
| 198 | #135 | Dic158 | 1 | 4 | yes | — |
| 199 | #135 | C2×Dic79 | 1 | 4 | yes | — |
| 200 | C2×Q16 | D316 | 1 | 1 | yes | characteristic |
| 201 | C2×SD16 | D316 | 1 | 2 | yes | — |
| 202 | C2×D8 | D316 | 1 | 1 | yes | characteristic |
| 203 | #131 | — | 2 | 4 | no | — |
| 204 | #131 | C2×Dic79 | 1 | 8 | yes | complemented |
| 205 | C2×M16 | — | 2 | 2 | no | — |
| 206 | C2×M16 | — | 79 | 2 | no | — |
| 207 | C2×M16 | C22×D79 | 1 | 2 | yes | — |
| 208 | C22×C8 | — | 2 | 2 | no | — |
| 209 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 210 | C22×C8 | C22×D79 | 1 | 1 | yes | characteristic |
| 211 | C4×Q8 | — | 158 | 4 | no | — |
| 212 | C4×D4 | — | 316 | 4 | no | — |
| 213 | C4×D4 | — | 158 | 4 | no | — |
| 214 | #117 | — | 79 | 2 | no | — |
| 215 | #117 | — | 158 | 4 | no | — |
| 216 | C2×C4⋊C4 | — | 79 | 4 | no | — |
| 217 | C2×C4⋊C4 | — | 158 | 4 | no | — |
| 218 | C2×C42 | C22×D79 | 1 | 1 | yes | characteristic |
| 219 | #108 | — | 158 | 2 | no | — |
| 220 | #108 | C2×Dic79 | 1 | 4 | yes | complemented |
| 221 | #107 | — | 79 | 2 | no | — |
| 222 | #107 | — | 158 | 4 | no | — |
| 223 | #107 | — | 158 | 2 | no | — |
| 224 | #107 | — | 158 | 2 | no | — |
| 225 | #106 | — | 79 | 1 | no | — |
| 226 | #106 | — | 79 | 1 | no | — |
| 227 | #106 | — | 158 | 4 | no | — |
| 228 | #106 | — | 158 | 2 | no | — |
| 229 | #106 | — | 158 | 2 | no | — |
| 230 | #104 | C2×Dic79 | 1 | 8 | yes | complemented |
| 231 | #103 | — | 158 | 4 | no | — |
| 232 | #102 | — | 158 | 4 | no | — |
| 233 | #97 | — | 158 | 2 | no | — |
| 234 | C4×C8 | — | 158 | 2 | no | — |
| 235 | C4×C8 | C2×Dic79 | 1 | 4 | yes | complemented |
| 236 | #95 | — | 316 | 2 | no | — |
| 237 | #41 | — | 2 | 2 | no | — |
| 238 | #41 | — | 2 | 2 | no | — |
| 239 | #41 | C22×Dic79 | 1 | 4 | yes | — |
| 240 | #41 | C2×Dic158 | 1 | 4 | yes | — |
| 241 | C2×Q8 | C2×D316 | 1 | 2 | yes | — |
| 242 | C2×D4 | — | 2 | 2 | no | — |
| 243 | C2×D4 | C2×D316 | 1 | 2 | yes | — |
| 244 | C22×C4 | — | 2 | 2 | no | — |
| 245 | C22×C4 | — | 2 | 1 | no | — |
| 246 | C22×C4 | — | 316 | 4 | no | — |
| 247 | C22×C4 | — | 158 | 4 | no | — |
| 248 | C22×C4 | C23×D79 | 1 | 1 | yes | characteristic |
| 249 | Q16 | C4⋊Dic79 | 1 | 4 | yes | — |
| 250 | SD16 | C4⋊Dic79 | 1 | 8 | yes | — |
| 251 | D8 | C4⋊Dic79 | 1 | 4 | yes | — |
| 252 | M16 | — | 2 | 2 | no | — |
| 253 | M16 | — | 2 | 8 | no | — |
| 254 | M16 | — | 158 | 2 | no | — |
| 255 | M16 | — | 158 | 2 | no | — |
| 256 | M16 | C22×Dic79 | 1 | 4 | yes | — |
| 257 | C2×C8 | — | 2 | 2 | no | — |
| 258 | C2×C8 | — | 2 | 8 | no | — |
| 259 | C2×C8 | — | 158 | 2 | no | — |
| 260 | C2×C8 | C22×Dic79 | 1 | 2 | yes | — |
| 261 | C2×C8 | C22×Dic79 | 1 | 2 | yes | — |
| 262 | C2×C8 | D4×D79 | 1 | 1 | yes | characteristic |
| 263 | C2×C8 | D4×D79 | 1 | 1 | yes | characteristic |
| 264 | C2×C8 | C2×D316 | 1 | 1 | yes | characteristic |
| 265 | C2×C8 | C2×D316 | 1 | 1 | yes | characteristic |
| 266 | C2×C8 | C2×Dic158 | 1 | 2 | yes | — |
| 267 | C2×C8 | #11929623 | 1 | 2 | yes | — |
| 268 | C4⋊C4 | — | 316 | 4 | no | — |
| 269 | C4⋊C4 | — | 79 | 2 | no | — |
| 270 | C4⋊C4 | — | 79 | 2 | no | — |
| 271 | C4⋊C4 | — | 158 | 4 | no | — |
| 272 | C4⋊C4 | — | 158 | 4 | no | — |
| 273 | C22⋊C4 | — | 316 | 4 | no | — |
| 274 | C22⋊C4 | — | 158 | 2 | no | — |
| 275 | C42 | — | 2 | 1 | no | — |
| 276 | C42 | — | 316 | 4 | no | — |
| 277 | C42 | — | 158 | 2 | no | — |
| 278 | C42 | C22×Dic79 | 1 | 2 | yes | — |
| 279 | C23 | — | 2 | 2 | no | — |
| 280 | C23 | #14579207 | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 2 | 2 | no | — |
| 282 | Q8 | C2×C4⋊Dic79 | 1 | 4 | yes | — |
| 283 | D4 | — | 2 | 4 | no | — |
| 284 | D4 | — | 2 | 4 | no | — |
| 285 | D4 | — | 2 | 2 | no | — |
| 286 | D4 | C2×C4⋊Dic79 | 1 | 4 | yes | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 2 | 4 | no | — |
| 289 | C2×C4 | — | 2 | 4 | no | — |
| 290 | C2×C4 | — | 2 | 2 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 316 | 4 | no | — |
| 293 | C2×C4 | — | 632 | 4 | no | — |
| 294 | C2×C4 | — | 158 | 2 | no | — |
| 295 | C2×C4 | — | 158 | 2 | no | — |
| 296 | C2×C4 | — | 1264 | 4 | no | — |
| 297 | C2×C4 | C23×Dic79 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | C2×D4×D79 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | C22×D316 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | C22×Dic158 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | C2×#11929623 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | #14579208 | 1 | 1 | yes | characteristic |
| 303 | C8 | — | 2 | 4 | no | — |
| 304 | C8 | — | 316 | 2 | no | — |
| 305 | C8 | C2×C4⋊Dic79 | 1 | 2 | yes | — |
| 306 | C8 | C2×C4⋊Dic79 | 1 | 2 | yes | — |
| 307 | C8 | D4×Dic79 | 1 | 2 | yes | — |
| 308 | C8 | D4×Dic79 | 1 | 2 | yes | — |
| 309 | C22 | — | 2 | 2 | no | — |
| 310 | C22 | — | 2 | 4 | no | — |
| 311 | C22 | — | 4 | 4 | no | — |
| 312 | C22 | #19099046 | 1 | 1 | yes | characteristic |
| 313 | C22 | D4×D316 | 1 | 1 | yes | characteristic |
| 314 | C22 | #19099270 | 1 | 1 | yes | characteristic |
| 315 | C4 | — | 2 | 4 | no | — |
| 316 | C4 | — | 4 | 2 | no | — |
| 317 | C4 | — | 316 | 2 | no | — |
| 318 | C4 | — | 316 | 2 | no | — |
| 319 | C4 | — | 632 | 4 | no | — |
| 320 | C4 | C22×C4⋊Dic79 | 1 | 1 | yes | characteristic |
| 321 | C4 | C2×D4×Dic79 | 1 | 1 | yes | characteristic |
| 322 | C4 | #19099217 | 1 | 1 | yes | characteristic |
| 323 | C4 | D4×Dic158 | 1 | 1 | yes | characteristic |
| 324 | C2 | — | 2 | 1 | no | — |
| 325 | C2 | — | 2 | 1 | no | — |
| 326 | C2 | — | 4 | 4 | no | — |
| 327 | C2 | D4×C4⋊Dic79 | 1 | 1 | yes | characteristic |
| 328 | C2 | #23775188 | 1 | 2 | yes | — |
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