← #17978807
Subgroups — #17978807
Order 4352.
532 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #14545198 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #14545169 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #14545168 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #14545143 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #14545142 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #14545118 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #14545117 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C8×#63254 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C2×#10587530 | C22 | 1 | 2 | yes | — |
| 10 | C2×#10587538 | C22 | 1 | 4 | yes | — |
| 11 | #11763730 | C22 | 1 | 4 | yes | — |
| 12 | #11763730 | C22 | 1 | 2 | yes | — |
| 13 | #11763730 | C22 | 1 | 1 | yes | characteristic |
| 14 | #11763730 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4⋊#63216 | C22 | 1 | 2 | yes | — |
| 16 | #11763704 | C22 | 1 | 2 | yes | — |
| 17 | C22⋊#63216 | C22 | 1 | 4 | yes | — |
| 18 | #11763695 | C22 | 1 | 4 | yes | — |
| 19 | C4×#63216 | C22 | 1 | 2 | yes | — |
| 20 | C4×#63254 | C4 | 1 | 1 | yes | characteristic |
| 21 | C8×Dic34 | C4 | 1 | 1 | yes | characteristic |
| 22 | #11762933 | C22 | 1 | 1 | yes | characteristic |
| 23 | D4×#3622 | C22 | 1 | 1 | yes | characteristic |
| 24 | C8×C22⋊D17 | C4 | 1 | 2 | yes | — |
| 25 | #11762872 | C22 | 1 | 2 | yes | — |
| 26 | #11762861 | C4 | 1 | 1 | yes | characteristic |
| 27 | #11762859 | C4 | 1 | 2 | yes | — |
| 28 | C2×C8×Dic17 | C22 | 1 | 2 | yes | — |
| 29 | C4×C8×D17 | C22 | 1 | 1 | yes | characteristic |
| 30 | C8×D4×C17 | C4 | 1 | 1 | yes | characteristic |
| 31 | C2×#63254 | C8 | 1 | 1 | yes | characteristic |
| 32 | C22×C8×C17 | C2×C4 | 1 | 2 | yes | — |
| 33 | C4×D4×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | D4×Dic17 | C8 | 1 | 1 | yes | characteristic |
| 35 | C4×C22⋊D17 | C2×C4 | 1 | 2 | yes | — |
| 36 | C2×C4×Dic17 | C2×C4 | 1 | 2 | yes | — |
| 37 | C22×#3622 | C23 | 1 | 2 | yes | — |
| 38 | C2×C8×D17 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C2×C8×D17 | C23 | 1 | 1 | yes | characteristic |
| 40 | #10587578 | C8 | 1 | 1 | yes | characteristic |
| 41 | #10587573 | C8 | 1 | 1 | yes | characteristic |
| 42 | #10587567 | C8 | 1 | 2 | yes | — |
| 43 | #10587563 | C8 | 1 | 2 | yes | — |
| 44 | C42×D17 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C4×Dic34 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | #10587538 | — | 4 | 4 | no | — |
| 47 | C2×#63216 | C23 | 1 | 4 | yes | — |
| 48 | C2×#63216 | C23 | 1 | 2 | yes | — |
| 49 | C2×#63216 | C23 | 1 | 2 | yes | — |
| 50 | #10587530 | — | 4 | 2 | no | — |
| 51 | C17×C4⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C17×C22⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 53 | C4×C8×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C22⋊#3622 | C2×C4 | 1 | 2 | yes | — |
| 55 | #10587491 | C2×C4 | 1 | 2 | yes | — |
| 56 | #10587485 | C2×C4 | 1 | 2 | yes | — |
| 57 | #10587485 | C2×C4 | 1 | 2 | yes | — |
| 58 | C8×Dic17 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C8×Dic17 | C2×C4 | 1 | 2 | yes | — |
| 60 | C8×Dic17 | C2×C4 | 1 | 2 | yes | — |
| 61 | C8×Dic17 | C23 | 1 | 1 | yes | characteristic |
| 62 | C8×Dic17 | C23 | 1 | 2 | yes | — |
| 63 | C4⋊#3622 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | C4×#3622 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×D4×C17 | C2×C8 | 1 | 1 | yes | characteristic |
| 66 | C22×C4×C17 | C22×C4 | 1 | 2 | yes | — |
| 67 | C2×C22⋊D17 | C2×C8 | 1 | 2 | yes | — |
| 68 | C22×Dic17 | C2×C8 | 1 | 2 | yes | — |
| 69 | C22×Dic17 | C2×C8 | 1 | 2 | yes | — |
| 70 | #63254 | — | 2 | 4 | no | — |
| 71 | C2×C4×D17 | C2×C8 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×D17 | C2×C8 | 1 | 1 | yes | characteristic |
| 73 | C2×C4×D17 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C2×Dic34 | C2×C8 | 1 | 1 | yes | characteristic |
| 75 | C2×C8×C17 | — | 4 | 2 | no | — |
| 76 | C2×C8×C17 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C17 | C22×C4 | 1 | 2 | yes | — |
| 78 | C2×C8×C17 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C4⋊C4×C17 | C2×C8 | 1 | 1 | yes | characteristic |
| 80 | C22⋊C4×C17 | C2×C8 | 1 | 2 | yes | — |
| 81 | C42×C17 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C22⋊Dic17 | C2×C8 | 1 | 2 | yes | — |
| 83 | #63227 | C2×C8 | 1 | 2 | yes | — |
| 84 | C4⋊Dic17 | C2×C8 | 1 | 1 | yes | characteristic |
| 85 | #63225 | C2×C8 | 1 | 2 | yes | — |
| 86 | #63225 | C2×C8 | 1 | 2 | yes | — |
| 87 | C4×Dic17 | C2×C8 | 1 | 1 | yes | characteristic |
| 88 | C4×Dic17 | C2×C8 | 1 | 1 | yes | characteristic |
| 89 | C4×Dic17 | C2×C8 | 1 | 2 | yes | — |
| 90 | C4×Dic17 | C2×C8 | 1 | 2 | yes | — |
| 91 | C4×Dic17 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C4×Dic17 | C22×C4 | 1 | 2 | yes | — |
| 93 | C2×#3622 | — | 2 | 2 | no | — |
| 94 | C2×#3622 | — | 2 | 2 | no | — |
| 95 | C2×#3622 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×#3622 | C22×C4 | 1 | 2 | yes | — |
| 97 | C2×#3622 | C24 | 1 | 1 | yes | characteristic |
| 98 | C8×D17 | — | 2 | 1 | no | — |
| 99 | C8×D17 | — | 2 | 1 | no | — |
| 100 | C8×D17 | — | 4 | 1 | no | — |
| 101 | #63216 | — | 2 | 4 | no | — |
| 102 | #63216 | — | 2 | 2 | no | — |
| 103 | #63216 | — | 2 | 2 | no | — |
| 104 | #33505 | — | 17 | 1 | no | maximal, Sylow |
| 105 | C23×C17 | C22×C8 | 1 | 2 | yes | — |
| 106 | C22×D17 | C22×C8 | 1 | 1 | yes | characteristic |
| 107 | D4×C17 | — | 2 | 2 | no | — |
| 108 | C2×C4×C17 | — | 2 | 2 | no | — |
| 109 | C2×C4×C17 | — | 2 | 2 | no | — |
| 110 | C2×C4×C17 | C22×C8 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C17 | C22×C8 | 1 | 2 | yes | — |
| 112 | C2×C4×C17 | C22×C8 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C17 | C23×C4 | 1 | 1 | yes | characteristic |
| 114 | C22⋊D17 | — | 2 | 4 | no | — |
| 115 | C2×Dic17 | — | 2 | 4 | no | — |
| 116 | C2×Dic17 | — | 2 | 4 | no | — |
| 117 | C2×Dic17 | C22×C8 | 1 | 1 | yes | characteristic |
| 118 | C2×Dic17 | C22×C8 | 1 | 2 | yes | — |
| 119 | C2×Dic17 | C22×C8 | 1 | 1 | yes | characteristic |
| 120 | C2×Dic17 | C22×C8 | 1 | 2 | yes | — |
| 121 | C2×Dic17 | C22×C8 | 1 | 1 | yes | characteristic |
| 122 | C4×D17 | — | 2 | 1 | no | — |
| 123 | C4×D17 | — | 2 | 1 | no | — |
| 124 | C4×D17 | — | 2 | 2 | no | — |
| 125 | C4×D17 | — | 2 | 2 | no | — |
| 126 | Dic34 | M32 | 1 | 4 | yes | complemented |
| 127 | C8×C17 | — | 2 | 1 | no | — |
| 128 | C8×C17 | — | 2 | 1 | no | — |
| 129 | C8×C17 | — | 2 | 2 | no | — |
| 130 | #3622 | — | 2 | 2 | no | — |
| 131 | #3622 | — | 2 | 1 | no | — |
| 132 | #3622 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 133 | #3622 | 2−1+4 | 1 | 1 | yes | characteristic |
| 134 | C8×#41 | — | 17 | 1 | no | — |
| 135 | Q8×C16 | — | 17 | 2 | no | — |
| 136 | #2169 | — | 17 | 2 | no | — |
| 137 | #2169 | — | 17 | 4 | no | — |
| 138 | #2162 | — | 17 | 2 | no | — |
| 139 | #2162 | — | 17 | 1 | no | — |
| 140 | #2150 | — | 17 | 1 | no | — |
| 141 | #2150 | — | 17 | 2 | no | — |
| 142 | C22×C17 | — | 2 | 2 | no | — |
| 143 | C22×C17 | — | 2 | 2 | no | — |
| 144 | C22×C17 | — | 2 | 2 | no | — |
| 145 | C22×C17 | C23×C8 | 1 | 1 | yes | characteristic |
| 146 | C2×D17 | — | 2 | 1 | no | — |
| 147 | C2×D17 | — | 2 | 1 | no | — |
| 148 | C2×D17 | — | 2 | 1 | no | — |
| 149 | C4×C17 | — | 2 | 1 | no | — |
| 150 | C4×C17 | — | 2 | 2 | no | — |
| 151 | C4×C17 | C2×M32 | 1 | 2 | yes | — |
| 152 | C4×C17 | #520 | 1 | 1 | yes | characteristic |
| 153 | C4×C17 | #568 | 1 | 1 | yes | characteristic |
| 154 | Dic17 | — | 2 | 1 | no | — |
| 155 | Dic17 | — | 2 | 2 | no | — |
| 156 | Dic17 | — | 2 | 1 | no | — |
| 157 | Dic17 | C2×M32 | 1 | 4 | yes | — |
| 158 | Dic17 | #504 | 1 | 2 | yes | — |
| 159 | C4×#41 | — | 17 | 1 | no | — |
| 160 | C2×M32 | — | 17 | 4 | no | — |
| 161 | C2×M32 | — | 17 | 2 | no | — |
| 162 | C8×Q8 | — | 17 | 1 | no | — |
| 163 | C8×Q8 | — | 17 | 1 | no | — |
| 164 | C8×D4 | — | 17 | 2 | no | — |
| 165 | C8×D4 | — | 17 | 1 | no | — |
| 166 | C8×D4 | — | 17 | 2 | no | — |
| 167 | C8×D4 | #595 | 1 | 1 | yes | characteristic |
| 168 | #431 | — | 17 | 1 | no | — |
| 169 | #431 | — | 17 | 2 | no | — |
| 170 | C2×C4×C8 | — | 17 | 2 | no | — |
| 171 | C2×C4×C8 | — | 17 | 1 | no | — |
| 172 | C4⋊C16 | — | 17 | 2 | no | — |
| 173 | C4⋊C16 | — | 17 | 2 | no | — |
| 174 | C4⋊C16 | — | 17 | 1 | no | — |
| 175 | C4⋊C16 | — | 17 | 4 | no | — |
| 176 | C4⋊C16 | — | 17 | 1 | no | — |
| 177 | C22⋊C16 | — | 17 | 4 | no | — |
| 178 | C22⋊C16 | — | 17 | 2 | no | — |
| 179 | C4×C16 | — | 17 | 2 | no | — |
| 180 | C4×C16 | — | 17 | 4 | no | — |
| 181 | C2×C17 | — | 4 | 2 | no | — |
| 182 | C2×C17 | #2988 | 1 | 1 | yes | characteristic |
| 183 | C2×C17 | C22×M32 | 1 | 1 | yes | characteristic |
| 184 | C2×C17 | C2×#504 | 1 | 1 | yes | characteristic |
| 185 | D17 | — | 4 | 1 | no | — |
| 186 | C2×#41 | — | 17 | 1 | no | — |
| 187 | C22×C8 | — | 17 | 2 | no | — |
| 188 | C22×C8 | — | 17 | 1 | no | — |
| 189 | C22×C8 | — | 17 | 1 | no | — |
| 190 | C22×C8 | C2×#595 | 1 | 2 | yes | — |
| 191 | C4×Q8 | — | 17 | 1 | no | — |
| 192 | C4×Q8 | — | 17 | 1 | no | — |
| 193 | C4×D4 | — | 17 | 1 | no | — |
| 194 | C4×D4 | — | 17 | 2 | no | — |
| 195 | C4×D4 | — | 17 | 2 | no | — |
| 196 | C4×D4 | C2×#595 | 1 | 1 | yes | characteristic |
| 197 | #117 | — | 17 | 1 | no | — |
| 198 | #117 | — | 17 | 2 | no | — |
| 199 | C2×C42 | — | 17 | 2 | no | — |
| 200 | C2×C42 | — | 17 | 1 | no | — |
| 201 | M32 | — | 68 | 2 | no | — |
| 202 | M32 | — | 68 | 4 | no | — |
| 203 | C2×C16 | — | 17 | 2 | no | — |
| 204 | C2×C16 | — | 17 | 2 | no | — |
| 205 | C2×C16 | — | 17 | 4 | no | — |
| 206 | C4⋊C8 | — | 17 | 2 | no | — |
| 207 | C4⋊C8 | — | 17 | 2 | no | — |
| 208 | C4⋊C8 | — | 17 | 1 | no | — |
| 209 | C4⋊C8 | C2×#595 | 1 | 1 | yes | characteristic |
| 210 | C22⋊C8 | — | 17 | 2 | no | — |
| 211 | C22⋊C8 | — | 17 | 2 | no | — |
| 212 | C22⋊C8 | C2×#595 | 1 | 2 | yes | — |
| 213 | C4×C8 | — | 17 | 2 | no | — |
| 214 | C4×C8 | — | 17 | 2 | no | — |
| 215 | C4×C8 | — | 17 | 1 | no | — |
| 216 | C4×C8 | — | 17 | 2 | no | — |
| 217 | C4×C8 | — | 17 | 1 | no | — |
| 218 | C4×C8 | — | 17 | 1 | no | — |
| 219 | C4×C8 | C2×#595 | 1 | 1 | yes | characteristic |
| 220 | C17 | #33505 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 221 | #41 | — | 34 | 4 | no | — |
| 222 | C2×Q8 | — | 17 | 1 | no | — |
| 223 | C2×D4 | — | 17 | 2 | no | — |
| 224 | C2×D4 | #63242 | 1 | 1 | yes | characteristic |
| 225 | C22×C4 | — | 17 | 2 | no | — |
| 226 | C22×C4 | — | 17 | 2 | no | — |
| 227 | C22×C4 | — | 17 | 1 | no | — |
| 228 | C22×C4 | — | 17 | 1 | no | — |
| 229 | C22×C4 | — | 17 | 1 | no | — |
| 230 | C22×C4 | C22×#595 | 1 | 2 | yes | — |
| 231 | C2×C8 | — | 4 | 2 | no | — |
| 232 | C2×C8 | — | 17 | 1 | no | — |
| 233 | C2×C8 | — | 17 | 2 | no | — |
| 234 | C2×C8 | — | 17 | 1 | no | — |
| 235 | C2×C8 | — | 34 | 1 | no | — |
| 236 | C2×C8 | — | 34 | 1 | no | — |
| 237 | C2×C8 | — | 34 | 2 | no | — |
| 238 | C2×C8 | — | 34 | 2 | no | — |
| 239 | C2×C8 | — | 68 | 1 | no | — |
| 240 | C2×C8 | C22×#595 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | C22×#595 | 1 | 2 | yes | — |
| 242 | C2×C8 | C22×#595 | 1 | 1 | yes | characteristic |
| 243 | C4⋊C4 | — | 17 | 2 | no | — |
| 244 | C4⋊C4 | — | 17 | 2 | no | — |
| 245 | C4⋊C4 | — | 17 | 1 | no | — |
| 246 | C4⋊C4 | #63242 | 1 | 1 | yes | characteristic |
| 247 | C22⋊C4 | — | 17 | 2 | no | — |
| 248 | C22⋊C4 | — | 17 | 2 | no | — |
| 249 | C22⋊C4 | #63242 | 1 | 2 | yes | — |
| 250 | C42 | — | 17 | 2 | no | — |
| 251 | C42 | — | 17 | 1 | no | — |
| 252 | C42 | — | 17 | 2 | no | — |
| 253 | C42 | — | 17 | 1 | no | — |
| 254 | C42 | — | 17 | 1 | no | — |
| 255 | C42 | — | 17 | 2 | no | — |
| 256 | C42 | C22×#595 | 1 | 1 | yes | characteristic |
| 257 | C16 | — | 34 | 4 | no | — |
| 258 | C16 | — | 34 | 2 | no | — |
| 259 | C16 | — | 34 | 2 | no | — |
| 260 | C23 | — | 17 | 1 | no | — |
| 261 | C23 | C2×#63242 | 1 | 2 | yes | — |
| 262 | Q8 | — | 17 | 4 | no | — |
| 263 | D4 | — | 2 | 2 | no | — |
| 264 | D4 | — | 34 | 4 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 2 | 2 | no | — |
| 267 | C2×C4 | — | 17 | 1 | no | — |
| 268 | C2×C4 | — | 17 | 2 | no | — |
| 269 | C2×C4 | — | 17 | 1 | no | — |
| 270 | C2×C4 | — | 17 | 1 | no | — |
| 271 | C2×C4 | — | 17 | 2 | no | — |
| 272 | C2×C4 | — | 34 | 2 | no | — |
| 273 | C2×C4 | — | 34 | 4 | no | — |
| 274 | C2×C4 | — | 34 | 4 | no | — |
| 275 | C2×C4 | — | 34 | 2 | no | — |
| 276 | C2×C4 | — | 34 | 1 | no | — |
| 277 | C2×C4 | — | 34 | 1 | no | — |
| 278 | C2×C4 | C2×#63242 | 1 | 2 | yes | — |
| 279 | C2×C4 | C2×#63242 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | C2×#63242 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | C23×#595 | 1 | 1 | yes | characteristic |
| 282 | C8 | — | 2 | 1 | no | — |
| 283 | C8 | — | 2 | 1 | no | — |
| 284 | C8 | — | 2 | 2 | no | — |
| 285 | C8 | — | 17 | 1 | no | — |
| 286 | C8 | — | 17 | 1 | no | — |
| 287 | C8 | — | 34 | 2 | no | — |
| 288 | C8 | — | 34 | 1 | no | — |
| 289 | C22 | — | 2 | 2 | no | — |
| 290 | C22 | — | 2 | 2 | no | — |
| 291 | C22 | — | 2 | 2 | no | — |
| 292 | C22 | — | 34 | 1 | no | — |
| 293 | C22 | — | 34 | 1 | no | — |
| 294 | C22 | — | 34 | 1 | no | — |
| 295 | C22 | C22×#63242 | 1 | 1 | yes | characteristic |
| 296 | C4 | — | 2 | 2 | no | — |
| 297 | C4 | — | 2 | 1 | no | — |
| 298 | C4 | — | 17 | 4 | no | — |
| 299 | C4 | — | 17 | 2 | no | — |
| 300 | C4 | — | 34 | 1 | no | — |
| 301 | C4 | — | 34 | 1 | no | — |
| 302 | C4 | — | 34 | 2 | no | — |
| 303 | C4 | #11763865 | 1 | 2 | yes | — |
| 304 | C4 | #11763878 | 1 | 1 | yes | characteristic |
| 305 | C4 | #11763887 | 1 | 1 | yes | characteristic |
| 306 | C2 | — | 4 | 2 | no | — |
| 307 | C2 | — | 68 | 1 | no | — |
| 308 | C2 | #14545700 | 1 | 1 | yes | characteristic |
| 309 | C2 | C2×#11763865 | 1 | 1 | yes | characteristic |
| 310 | C2 | #14545749 | 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