← #33242722
Subgroups — #33242722
Order 18688.
696 conjugacy classes of subgroups.
The subgroup lattice has 268 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#17989019 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22621691 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22612181 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22612017 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22611972 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22611856 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C4×Dic584 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | D4×C16×C73 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#14556613 | C22 | 1 | 4 | yes | — |
| 10 | #17989258 | — | 2 | 2 | no | — |
| 11 | #17989019 | — | 2 | 8 | no | — |
| 12 | C2×Dic584 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#14556536 | C22 | 1 | 4 | yes | — |
| 14 | C2×D1168 | C22 | 1 | 2 | yes | — |
| 15 | C2×D1168 | C22 | 1 | 2 | yes | — |
| 16 | #17988964 | C22 | 1 | 2 | yes | — |
| 17 | #17988823 | C22 | 1 | 4 | yes | — |
| 18 | #17988788 | — | 2 | 2 | no | — |
| 19 | #17988734 | — | 2 | 4 | no | — |
| 20 | C4×Dic292 | C22 | 1 | 2 | yes | — |
| 21 | #17988413 | C22 | 1 | 1 | yes | characteristic |
| 22 | #17988392 | C22 | 1 | 2 | yes | — |
| 23 | #17988389 | C22 | 1 | 4 | yes | — |
| 24 | C22×C16×C73 | C22 | 1 | 2 | yes | — |
| 25 | C8×D4×C73 | C22 | 1 | 1 | yes | characteristic |
| 26 | C4⋊C16×C73 | C22 | 1 | 1 | yes | characteristic |
| 27 | C22⋊C16×C73 | C22 | 1 | 2 | yes | — |
| 28 | C4×C16×C73 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#11924028 | — | 2 | 4 | no | — |
| 30 | #14556613 | — | 2 | 8 | no | — |
| 31 | #14556613 | — | 4 | 4 | no | — |
| 32 | C2×Dic292 | C23 | 1 | 2 | yes | — |
| 33 | C2×#11924017 | — | 2 | 4 | no | — |
| 34 | C2×D584 | — | 2 | 2 | no | — |
| 35 | C2×D584 | C23 | 1 | 4 | yes | — |
| 36 | #14556588 | — | 2 | 2 | no | — |
| 37 | #14556587 | — | 4 | 2 | no | — |
| 38 | #14556565 | — | 2 | 4 | no | — |
| 39 | #14556564 | — | 4 | 4 | no | — |
| 40 | C4×Dic146 | — | 2 | 2 | no | — |
| 41 | #14556553 | — | 4 | 2 | no | — |
| 42 | Dic584 | D4 | 1 | 4 | yes | complemented |
| 43 | #14556536 | — | 2 | 8 | no | — |
| 44 | D1168 | — | 2 | 4 | no | — |
| 45 | D1168 | — | 4 | 2 | no | — |
| 46 | #14556525 | C23 | 1 | 2 | yes | — |
| 47 | #14556516 | — | 2 | 2 | no | — |
| 48 | #14556513 | — | 2 | 4 | no | — |
| 49 | C22×C8×C73 | C23 | 1 | 2 | yes | — |
| 50 | C4×D4×C73 | D4 | 1 | 1 | yes | characteristic |
| 51 | C2×C16×C73 | — | 2 | 4 | no | — |
| 52 | C2×C16×C73 | C23 | 1 | 2 | yes | — |
| 53 | C2×C16×C73 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×C16×C73 | C23 | 1 | 1 | yes | characteristic |
| 55 | C4⋊C8×C73 | D4 | 1 | 1 | yes | characteristic |
| 56 | C22⋊C8×C73 | D4 | 1 | 2 | yes | — |
| 57 | C4×C8×C73 | C23 | 1 | 1 | yes | characteristic |
| 58 | #11924028 | — | 4 | 8 | no | — |
| 59 | #11924028 | — | 8 | 4 | no | — |
| 60 | C2×Dic146 | — | 2 | 2 | no | — |
| 61 | C2×D292 | — | 2 | 4 | no | — |
| 62 | C2×D292 | — | 4 | 2 | no | — |
| 63 | C2×D292 | — | 4 | 2 | no | — |
| 64 | C2×C22⋊D73 | — | 4 | 4 | no | — |
| 65 | C2×C22⋊D73 | — | 4 | 4 | no | — |
| 66 | C2×C4×D73 | — | 4 | 4 | no | — |
| 67 | Dic292 | — | 2 | 2 | no | — |
| 68 | Dic292 | C2×D4 | 1 | 4 | yes | — |
| 69 | #11924017 | — | 4 | 8 | no | — |
| 70 | D584 | — | 2 | 4 | no | — |
| 71 | D584 | — | 2 | 4 | no | — |
| 72 | D584 | — | 8 | 2 | no | — |
| 73 | C4⋊Dic73 | — | 2 | 2 | no | — |
| 74 | #11924007 | — | 4 | 2 | no | — |
| 75 | #11924005 | — | 4 | 4 | no | — |
| 76 | C4×Dic73 | — | 4 | 2 | no | — |
| 77 | C2×D4×C73 | D8 | 1 | 1 | yes | characteristic |
| 78 | C22×C4×C73 | C2×D4 | 1 | 2 | yes | — |
| 79 | C2×C8×C73 | — | 2 | 2 | no | — |
| 80 | C2×C8×C73 | — | 2 | 2 | no | — |
| 81 | C2×C8×C73 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×C8×C73 | C2×D4 | 1 | 2 | yes | — |
| 83 | C2×C8×C73 | C24 | 1 | 1 | yes | characteristic |
| 84 | C4⋊C4×C73 | D8 | 1 | 1 | yes | characteristic |
| 85 | C22⋊C4×C73 | D8 | 1 | 2 | yes | — |
| 86 | C42×C73 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C16×C73 | — | 2 | 2 | no | — |
| 88 | C16×C73 | — | 2 | 1 | no | — |
| 89 | C16×C73 | C2×D4 | 1 | 2 | yes | — |
| 90 | C23×C73 | C2×D8 | 1 | 2 | yes | — |
| 91 | C22×D73 | — | 4 | 4 | no | — |
| 92 | D4×C73 | — | 2 | 2 | no | — |
| 93 | C2×C4×C73 | — | 2 | 2 | no | — |
| 94 | C2×C4×C73 | — | 2 | 2 | no | — |
| 95 | C2×C4×C73 | C2×D8 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C73 | C2×D8 | 1 | 2 | yes | — |
| 97 | C2×C4×C73 | C2×D8 | 1 | 1 | yes | characteristic |
| 98 | C2×C4×C73 | C22×D4 | 1 | 1 | yes | characteristic |
| 99 | C22⋊D73 | — | 8 | 8 | no | — |
| 100 | C22⋊D73 | — | 16 | 4 | no | — |
| 101 | C2×Dic73 | — | 4 | 2 | no | — |
| 102 | C2×Dic73 | — | 4 | 2 | no | — |
| 103 | D292 | — | 4 | 4 | no | — |
| 104 | D292 | — | 4 | 4 | no | — |
| 105 | D292 | — | 8 | 4 | no | — |
| 106 | D292 | — | 16 | 2 | no | — |
| 107 | C4×D73 | — | 8 | 8 | no | — |
| 108 | Dic146 | — | 2 | 4 | no | — |
| 109 | Dic146 | — | 4 | 2 | no | — |
| 110 | C8×C73 | — | 2 | 2 | no | — |
| 111 | C8×C73 | — | 2 | 1 | no | — |
| 112 | C8×C73 | C22×D4 | 1 | 1 | yes | characteristic |
| 113 | C8×C73 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 114 | C22×C73 | — | 2 | 2 | no | — |
| 115 | C22×C73 | — | 2 | 2 | no | — |
| 116 | C22×C73 | — | 2 | 2 | no | — |
| 117 | C22×C73 | C22×D8 | 1 | 1 | yes | characteristic |
| 118 | C2×D73 | — | 8 | 4 | no | — |
| 119 | C2×D73 | — | 8 | 4 | no | — |
| 120 | C2×D73 | — | 8 | 4 | no | — |
| 121 | C4×C73 | — | 2 | 1 | no | — |
| 122 | C4×C73 | — | 2 | 2 | no | — |
| 123 | C4×C73 | #508 | 1 | 2 | yes | complemented |
| 124 | C4×C73 | D42 | 1 | 1 | yes | characteristic |
| 125 | C4×C73 | #576 | 1 | 1 | yes | characteristic |
| 126 | Dic73 | — | 4 | 4 | no | — |
| 127 | Dic73 | — | 8 | 2 | no | — |
| 128 | #33853 | — | 73 | 1 | no | maximal, Sylow |
| 129 | C2×C73 | — | 4 | 2 | no | — |
| 130 | C2×C73 | D4×D8 | 1 | 1 | yes | characteristic |
| 131 | C2×C73 | C2×#508 | 1 | 1 | yes | characteristic |
| 132 | C2×C73 | #3415 | 1 | 1 | yes | characteristic |
| 133 | D73 | — | 16 | 4 | no | — |
| 134 | C2×#508 | — | 73 | 2 | no | — |
| 135 | #3287 | — | 73 | 2 | no | — |
| 136 | #2246 | — | 73 | 1 | no | — |
| 137 | #2215 | — | 73 | 2 | no | — |
| 138 | #2207 | — | 73 | 2 | no | — |
| 139 | #2186 | — | 73 | 4 | no | — |
| 140 | C4×Q32 | — | 73 | 1 | no | — |
| 141 | D4×C16 | D73 | 1 | 1 | yes | characteristic, complemented |
| 142 | C73 | #33853 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 143 | C2×#135 | — | 73 | 4 | no | — |
| 144 | #550 | — | 146 | 2 | no | — |
| 145 | #508 | — | 146 | 8 | no | — |
| 146 | C2×Q32 | — | 73 | 1 | no | — |
| 147 | C2×SD32 | — | 73 | 4 | no | — |
| 148 | C2×D16 | — | 73 | 2 | no | — |
| 149 | C2×D16 | — | 73 | 2 | no | — |
| 150 | C22×C16 | C2×D73 | 1 | 2 | yes | — |
| 151 | #493 | — | 73 | 2 | no | — |
| 152 | #466 | — | 73 | 4 | no | — |
| 153 | #460 | — | 146 | 2 | no | — |
| 154 | #449 | — | 146 | 4 | no | — |
| 155 | C4×Q16 | — | 73 | 2 | no | — |
| 156 | C8×D4 | C2×D73 | 1 | 1 | yes | characteristic |
| 157 | #366 | — | 73 | 1 | no | — |
| 158 | C4⋊C16 | C2×D73 | 1 | 1 | yes | characteristic |
| 159 | #358 | — | 73 | 2 | no | — |
| 160 | #357 | — | 73 | 4 | no | — |
| 161 | C22⋊C16 | C2×D73 | 1 | 2 | yes | — |
| 162 | C4×C16 | C2×D73 | 1 | 1 | yes | characteristic |
| 163 | C2×#41 | — | 146 | 4 | no | — |
| 164 | #135 | — | 292 | 4 | no | — |
| 165 | #135 | — | 146 | 8 | no | — |
| 166 | C2×Q16 | — | 73 | 2 | no | — |
| 167 | C2×SD16 | — | 146 | 4 | no | — |
| 168 | C2×D8 | — | 73 | 4 | no | — |
| 169 | C2×D8 | — | 146 | 2 | no | — |
| 170 | C22×C8 | C22×D73 | 1 | 2 | yes | — |
| 171 | #127 | — | 292 | 2 | no | — |
| 172 | #127 | — | 146 | 2 | no | — |
| 173 | #121 | — | 292 | 4 | no | — |
| 174 | #121 | — | 146 | 4 | no | — |
| 175 | C4×Q8 | — | 146 | 2 | no | — |
| 176 | C4×D4 | — | 292 | 2 | no | — |
| 177 | C4×D4 | D292 | 1 | 1 | yes | characteristic |
| 178 | Q32 | — | 73 | 4 | no | — |
| 179 | SD32 | — | 146 | 8 | no | — |
| 180 | D16 | — | 292 | 2 | no | — |
| 181 | D16 | — | 146 | 4 | no | — |
| 182 | C2×C16 | — | 2 | 4 | no | — |
| 183 | C2×C16 | C22×D73 | 1 | 1 | yes | characteristic |
| 184 | C2×C16 | C22×D73 | 1 | 2 | yes | — |
| 185 | C2×C16 | C22×D73 | 1 | 1 | yes | characteristic |
| 186 | #107 | — | 73 | 2 | no | — |
| 187 | C4⋊C8 | D292 | 1 | 1 | yes | characteristic |
| 188 | #103 | — | 146 | 2 | no | — |
| 189 | #102 | — | 146 | 4 | no | — |
| 190 | C22⋊C8 | D292 | 1 | 2 | yes | — |
| 191 | C4×C8 | C22×D73 | 1 | 1 | yes | characteristic |
| 192 | #41 | — | 292 | 8 | no | — |
| 193 | #41 | — | 584 | 4 | no | — |
| 194 | C2×Q8 | — | 146 | 2 | no | — |
| 195 | C2×D4 | — | 292 | 4 | no | — |
| 196 | C2×D4 | — | 292 | 4 | no | — |
| 197 | C2×D4 | — | 292 | 2 | no | — |
| 198 | C2×D4 | — | 292 | 2 | no | — |
| 199 | C2×D4 | — | 146 | 4 | no | — |
| 200 | C2×D4 | D584 | 1 | 1 | yes | characteristic |
| 201 | C22×C4 | — | 292 | 4 | no | — |
| 202 | C22×C4 | C2×D292 | 1 | 2 | yes | — |
| 203 | Q16 | — | 73 | 4 | no | — |
| 204 | Q16 | — | 146 | 2 | no | — |
| 205 | SD16 | — | 292 | 8 | no | — |
| 206 | D8 | — | 584 | 2 | no | — |
| 207 | D8 | — | 146 | 4 | no | — |
| 208 | D8 | — | 146 | 4 | no | — |
| 209 | C2×C8 | — | 2 | 2 | no | — |
| 210 | C2×C8 | — | 2 | 2 | no | — |
| 211 | C2×C8 | C2×D292 | 1 | 1 | yes | characteristic |
| 212 | C2×C8 | C2×D292 | 1 | 2 | yes | — |
| 213 | C2×C8 | C23×D73 | 1 | 1 | yes | characteristic |
| 214 | C4⋊C4 | — | 292 | 2 | no | — |
| 215 | C4⋊C4 | — | 146 | 2 | no | — |
| 216 | C4⋊C4 | D584 | 1 | 1 | yes | characteristic |
| 217 | C22⋊C4 | — | 292 | 4 | no | — |
| 218 | C22⋊C4 | D584 | 1 | 2 | yes | — |
| 219 | C42 | — | 292 | 2 | no | — |
| 220 | C42 | C2×D292 | 1 | 1 | yes | characteristic |
| 221 | C16 | — | 2 | 2 | no | — |
| 222 | C16 | — | 2 | 1 | no | — |
| 223 | C16 | D4×D73 | 1 | 2 | yes | — |
| 224 | C23 | — | 292 | 4 | no | — |
| 225 | C23 | C2×D584 | 1 | 2 | yes | — |
| 226 | Q8 | — | 292 | 2 | no | — |
| 227 | Q8 | — | 146 | 4 | no | — |
| 228 | D4 | — | 2 | 2 | no | — |
| 229 | D4 | — | 292 | 4 | no | — |
| 230 | D4 | — | 292 | 4 | no | — |
| 231 | D4 | — | 584 | 8 | no | — |
| 232 | D4 | — | 584 | 4 | no | — |
| 233 | D4 | — | 1168 | 2 | no | — |
| 234 | D4 | — | 1168 | 4 | no | — |
| 235 | C2×C4 | — | 2 | 2 | no | — |
| 236 | C2×C4 | — | 2 | 2 | no | — |
| 237 | C2×C4 | — | 292 | 2 | no | — |
| 238 | C2×C4 | — | 292 | 2 | no | — |
| 239 | C2×C4 | — | 584 | 8 | no | — |
| 240 | C2×C4 | C2×D584 | 1 | 1 | yes | characteristic |
| 241 | C2×C4 | C2×D584 | 1 | 2 | yes | — |
| 242 | C2×C4 | C2×D584 | 1 | 1 | yes | characteristic |
| 243 | C2×C4 | C22×D292 | 1 | 1 | yes | characteristic |
| 244 | C8 | — | 2 | 1 | no | — |
| 245 | C8 | — | 2 | 2 | no | — |
| 246 | C8 | C2×D4×D73 | 1 | 1 | yes | characteristic |
| 247 | C8 | #14556637 | 1 | 1 | yes | characteristic |
| 248 | C22 | — | 2 | 2 | no | — |
| 249 | C22 | — | 2 | 2 | no | — |
| 250 | C22 | — | 2 | 2 | no | — |
| 251 | C22 | — | 584 | 4 | no | — |
| 252 | C22 | — | 584 | 4 | no | — |
| 253 | C22 | — | 584 | 4 | no | — |
| 254 | C22 | C22×D584 | 1 | 1 | yes | characteristic |
| 255 | C4 | — | 2 | 1 | no | — |
| 256 | C4 | — | 2 | 2 | no | — |
| 257 | C4 | — | 292 | 4 | no | — |
| 258 | C4 | — | 584 | 2 | no | — |
| 259 | C4 | #17989019 | 1 | 2 | yes | complemented |
| 260 | C4 | D4×D292 | 1 | 1 | yes | characteristic |
| 261 | C4 | #17989392 | 1 | 1 | yes | characteristic |
| 262 | C2 | — | 4 | 2 | no | — |
| 263 | C2 | — | 1168 | 4 | no | — |
| 264 | C2 | D4×D584 | 1 | 1 | yes | characteristic |
| 265 | C2 | C2×#17989019 | 1 | 1 | yes | characteristic |
| 266 | C2 | #22623072 | 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