← #28507187
Subgroups — #28507187
Order 15104.
472 conjugacy classes of subgroups.
The subgroup lattice has 320 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #21434524 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#16837223 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #21425831 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #21425823 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C59×#1772 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14273749 | — | 2 | 1 | no | — |
| 7 | C2×#14273749 | C22 | 1 | 2 | yes | — |
| 8 | #16837952 | — | 2 | 1 | no | — |
| 9 | C2×#14273667 | C22 | 1 | 2 | yes | — |
| 10 | #16837786 | — | 2 | 1 | no | — |
| 11 | #16837786 | C4 | 1 | 2 | yes | complemented |
| 12 | #16837782 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | #16837779 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #16837775 | — | 2 | 1 | no | — |
| 15 | #16837223 | — | 2 | 1 | no | — |
| 16 | #16837223 | — | 2 | 1 | no | — |
| 17 | #16837211 | — | 2 | 4 | no | — |
| 18 | #16837208 | — | 2 | 4 | no | — |
| 19 | C2×#108×C59 | C22 | 1 | 1 | yes | characteristic |
| 20 | C59×#427 | C22 | 1 | 1 | yes | characteristic |
| 21 | C59×#405 | — | 2 | 1 | no | — |
| 22 | C2×C4×C8×C59 | C22 | 1 | 1 | yes | characteristic |
| 23 | C59×#339 | — | 2 | 1 | no | — |
| 24 | C2×#11745685 | — | 2 | 2 | no | — |
| 25 | #14273749 | — | 2 | 2 | no | — |
| 26 | #14273749 | — | 2 | 2 | no | — |
| 27 | #14273749 | — | 2 | 2 | no | — |
| 28 | #14273749 | — | 2 | 2 | no | — |
| 29 | #14273749 | — | 4 | 1 | no | — |
| 30 | #14273749 | — | 4 | 1 | no | — |
| 31 | #14273749 | — | 4 | 1 | no | — |
| 32 | #14273749 | — | 4 | 1 | no | — |
| 33 | C2×Dic236 | — | 2 | 1 | no | — |
| 34 | C2×Dic236 | C2×C4 | 1 | 2 | yes | — |
| 35 | C2×#11745660 | — | 2 | 2 | no | — |
| 36 | C2×#11745660 | — | 2 | 2 | no | — |
| 37 | C2×D472 | — | 2 | 1 | no | — |
| 38 | C2×D472 | C2×C4 | 1 | 2 | yes | — |
| 39 | #14273727 | — | 2 | 1 | no | — |
| 40 | #14273724 | — | 2 | 1 | no | — |
| 41 | #14273715 | — | 2 | 2 | no | — |
| 42 | #14273701 | — | 4 | 2 | no | — |
| 43 | C4×D236 | — | 4 | 2 | no | — |
| 44 | #14273667 | — | 2 | 2 | no | — |
| 45 | #14273667 | — | 2 | 2 | no | — |
| 46 | C2×C59⋊C16 | — | 2 | 2 | no | — |
| 47 | C2×M16×C59 | — | 2 | 1 | no | — |
| 48 | C22×C8×C59 | D4 | 1 | 1 | yes | characteristic |
| 49 | C22×C8×C59 | C23 | 1 | 1 | yes | characteristic |
| 50 | #117×C59 | — | 2 | 1 | no | — |
| 51 | C2×C42×C59 | D4 | 1 | 1 | yes | characteristic |
| 52 | #108×C59 | — | 2 | 1 | no | — |
| 53 | #108×C59 | D4 | 1 | 2 | yes | complemented |
| 54 | #107×C59 | — | 2 | 1 | no | — |
| 55 | #106×C59 | D4 | 1 | 2 | yes | complemented |
| 56 | #97×C59 | — | 4 | 1 | no | — |
| 57 | C4×C8×C59 | — | 4 | 1 | no | — |
| 58 | C4×C8×C59 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C4×C8×C59 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | C4×C8×C59 | D4 | 1 | 1 | yes | characteristic |
| 61 | C4×C8×C59 | D4 | 1 | 1 | yes | characteristic |
| 62 | C22×C4×C59 | — | 2 | 1 | no | — |
| 63 | C22×C4×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C2×C22⋊D59 | — | 4 | 2 | no | — |
| 65 | #11745685 | — | 4 | 2 | no | — |
| 66 | #11745685 | — | 4 | 2 | no | — |
| 67 | #11745685 | — | 4 | 2 | no | — |
| 68 | #11745685 | — | 4 | 2 | no | — |
| 69 | C2×D236 | — | 2 | 2 | no | — |
| 70 | C2×D236 | — | 4 | 2 | no | — |
| 71 | C2×C4×D59 | — | 4 | 2 | no | — |
| 72 | C2×Dic118 | — | 2 | 2 | no | — |
| 73 | M16×C59 | — | 2 | 2 | no | — |
| 74 | M16×C59 | — | 4 | 1 | no | — |
| 75 | C2×C8×C59 | — | 4 | 1 | no | — |
| 76 | C2×C8×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 78 | C2×C8×C59 | D8 | 1 | 1 | yes | characteristic |
| 79 | C2×C8×C59 | D8 | 1 | 1 | yes | characteristic |
| 80 | C2×C8×C59 | SD16 | 1 | 1 | yes | characteristic |
| 81 | C2×C8×C59 | SD16 | 1 | 1 | yes | characteristic |
| 82 | C2×C8×C59 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C8×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C59 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C59 | #41 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C59 | #41 | 1 | 1 | yes | characteristic |
| 88 | C4⋊C4×C59 | — | 2 | 2 | no | — |
| 89 | C22⋊C4×C59 | — | 4 | 1 | no | — |
| 90 | C42×C59 | — | 2 | 1 | no | — |
| 91 | C42×C59 | — | 4 | 1 | no | — |
| 92 | C42×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 93 | C42×C59 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 94 | (C2×D59)⋊C4 | — | 4 | 4 | no | — |
| 95 | C4⋊Dic59 | — | 4 | 2 | no | — |
| 96 | Dic236 | — | 2 | 2 | no | — |
| 97 | Dic236 | — | 2 | 2 | no | — |
| 98 | Dic236 | — | 4 | 1 | no | — |
| 99 | Dic236 | — | 4 | 1 | no | — |
| 100 | D472 | — | 2 | 2 | no | — |
| 101 | D472 | — | 2 | 2 | no | — |
| 102 | D472 | — | 4 | 1 | no | — |
| 103 | D472 | — | 4 | 1 | no | — |
| 104 | #11745660 | — | 4 | 2 | no | — |
| 105 | #11745660 | — | 4 | 2 | no | — |
| 106 | #11745660 | — | 4 | 2 | no | — |
| 107 | #11745660 | — | 4 | 2 | no | — |
| 108 | C59⋊C16 | — | 4 | 2 | no | — |
| 109 | C23×C59 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 110 | C22×D59 | — | 4 | 2 | no | — |
| 111 | C2×C4×C59 | — | 2 | 1 | no | — |
| 112 | C2×C4×C59 | — | 2 | 1 | no | — |
| 113 | C2×C4×C59 | — | 4 | 2 | no | — |
| 114 | C2×C4×C59 | — | 4 | 2 | no | — |
| 115 | C2×C4×C59 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C59 | #121 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C59 | #121 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C59 | C2×D8 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C59 | C2×SD16 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C59 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 121 | C22⋊D59 | — | 8 | 2 | no | — |
| 122 | C22⋊D59 | — | 8 | 2 | no | — |
| 123 | C2×Dic59 | — | 4 | 2 | no | — |
| 124 | D236 | — | 4 | 2 | no | — |
| 125 | D236 | — | 4 | 2 | no | — |
| 126 | D236 | — | 8 | 4 | no | — |
| 127 | C4×D59 | — | 8 | 2 | no | — |
| 128 | C4×D59 | — | 8 | 2 | no | — |
| 129 | Dic118 | — | 4 | 2 | no | — |
| 130 | Dic118 | — | 4 | 2 | no | — |
| 131 | C8×C59 | — | 4 | 2 | no | — |
| 132 | C8×C59 | #102 | 1 | 1 | yes | characteristic |
| 133 | C8×C59 | #102 | 1 | 1 | yes | characteristic |
| 134 | C8×C59 | #102 | 1 | 1 | yes | characteristic |
| 135 | C8×C59 | #102 | 1 | 1 | yes | characteristic |
| 136 | C8×C59 | C4×D4 | 1 | 1 | yes | characteristic |
| 137 | C8×C59 | C4×D4 | 1 | 1 | yes | characteristic |
| 138 | C8×C59 | #124 | 1 | 1 | yes | characteristic |
| 139 | C8×C59 | #127 | 1 | 1 | yes | characteristic |
| 140 | #13259 | — | 59 | 1 | no | maximal, Sylow |
| 141 | C22×C59 | — | 4 | 1 | no | — |
| 142 | C22×C59 | #418 | 1 | 1 | yes | characteristic |
| 143 | C22×C59 | #459 | 1 | 1 | yes | characteristic |
| 144 | C22×C59 | #460 | 1 | 1 | yes | characteristic |
| 145 | C2×D59 | — | 8 | 4 | no | — |
| 146 | C2×D59 | — | 8 | 2 | no | — |
| 147 | C4×C59 | — | 2 | 2 | no | — |
| 148 | C4×C59 | — | 4 | 1 | no | — |
| 149 | C4×C59 | — | 8 | 2 | no | — |
| 150 | C4×C59 | #390 | 1 | 1 | yes | characteristic |
| 151 | C4×C59 | C2×#102 | 1 | 1 | yes | characteristic |
| 152 | C4×C59 | #459 | 1 | 1 | yes | characteristic |
| 153 | C4×C59 | #460 | 1 | 1 | yes | characteristic |
| 154 | Dic59 | — | 8 | 2 | no | — |
| 155 | #3147 | — | 59 | 1 | no | — |
| 156 | C2×#364 | — | 59 | 1 | no | — |
| 157 | #2145 | — | 59 | 2 | no | — |
| 158 | #2143 | — | 59 | 2 | no | — |
| 159 | #1772 | D59 | 1 | 1 | yes | characteristic, complemented |
| 160 | C2×C59 | — | 2 | 1 | no | — |
| 161 | C2×C59 | — | 2 | 1 | no | — |
| 162 | C2×C59 | — | 2 | 1 | no | — |
| 163 | C2×C59 | #1982 | 1 | 1 | yes | characteristic |
| 164 | D59 | — | 16 | 2 | no | — |
| 165 | C2×#135 | — | 59 | 2 | no | — |
| 166 | C2×#135 | — | 118 | 1 | no | — |
| 167 | #532 | — | 118 | 1 | no | — |
| 168 | C2×M32 | — | 59 | 2 | no | — |
| 169 | #495 | — | 59 | 2 | no | — |
| 170 | #495 | — | 118 | 1 | no | — |
| 171 | #494 | — | 59 | 1 | no | — |
| 172 | #493 | — | 59 | 1 | no | — |
| 173 | #492 | — | 118 | 1 | no | — |
| 174 | C2×#108 | C2×D59 | 1 | 1 | yes | characteristic |
| 175 | #427 | C2×D59 | 1 | 1 | yes | characteristic |
| 176 | #405 | — | 2 | 1 | no | — |
| 177 | C2×C4×C8 | C2×D59 | 1 | 1 | yes | characteristic |
| 178 | #364 | — | 118 | 1 | no | — |
| 179 | #364 | — | 118 | 1 | no | — |
| 180 | #360 | — | 118 | 4 | no | — |
| 181 | #359 | — | 118 | 4 | no | — |
| 182 | #339 | — | 2 | 1 | no | — |
| 183 | C59 | #13259 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C2×#41 | — | 118 | 2 | no | — |
| 185 | #135 | — | 118 | 2 | no | — |
| 186 | #135 | — | 118 | 2 | no | — |
| 187 | #135 | — | 118 | 2 | no | — |
| 188 | #135 | — | 118 | 2 | no | — |
| 189 | #135 | — | 236 | 1 | no | — |
| 190 | #135 | — | 236 | 1 | no | — |
| 191 | #135 | — | 236 | 1 | no | — |
| 192 | #135 | — | 236 | 1 | no | — |
| 193 | C2×Q16 | — | 59 | 2 | no | — |
| 194 | C2×Q16 | — | 118 | 1 | no | — |
| 195 | C2×SD16 | — | 118 | 2 | no | — |
| 196 | C2×SD16 | — | 118 | 2 | no | — |
| 197 | C2×D8 | — | 59 | 2 | no | — |
| 198 | C2×D8 | — | 118 | 1 | no | — |
| 199 | C2×M16 | — | 2 | 1 | no | — |
| 200 | C22×C8 | C22⋊D59 | 1 | 1 | yes | characteristic |
| 201 | C22×C8 | C22×D59 | 1 | 1 | yes | characteristic |
| 202 | C4⋊Q8 | — | 118 | 1 | no | — |
| 203 | #127 | — | 118 | 1 | no | — |
| 204 | #124 | — | 118 | 2 | no | — |
| 205 | #121 | — | 236 | 2 | no | — |
| 206 | C4×D4 | — | 236 | 2 | no | — |
| 207 | #117 | — | 2 | 1 | no | — |
| 208 | C2×C42 | C22⋊D59 | 1 | 1 | yes | characteristic |
| 209 | M32 | — | 118 | 2 | no | — |
| 210 | M32 | — | 118 | 2 | no | — |
| 211 | C2×C16 | — | 118 | 2 | no | — |
| 212 | #108 | — | 2 | 1 | no | — |
| 213 | #108 | D236 | 1 | 2 | yes | complemented |
| 214 | #107 | — | 2 | 1 | no | — |
| 215 | #106 | D236 | 1 | 2 | yes | complemented |
| 216 | #97 | — | 4 | 1 | no | — |
| 217 | C4×C8 | — | 4 | 1 | no | — |
| 218 | C4×C8 | C4×D59 | 1 | 1 | yes | characteristic |
| 219 | C4×C8 | C4×D59 | 1 | 1 | yes | characteristic |
| 220 | C4×C8 | D236 | 1 | 1 | yes | characteristic |
| 221 | C4×C8 | D236 | 1 | 1 | yes | characteristic |
| 222 | #41 | — | 236 | 2 | no | — |
| 223 | #41 | — | 236 | 2 | no | — |
| 224 | #41 | — | 236 | 2 | no | — |
| 225 | #41 | — | 236 | 2 | no | — |
| 226 | C2×Q8 | — | 118 | 2 | no | — |
| 227 | C2×D4 | — | 118 | 2 | no | — |
| 228 | C2×D4 | — | 236 | 2 | no | — |
| 229 | C2×D4 | — | 236 | 2 | no | — |
| 230 | C22×C4 | — | 2 | 1 | no | — |
| 231 | C22×C4 | — | 236 | 2 | no | — |
| 232 | C22×C4 | C2×C22⋊D59 | 1 | 1 | yes | characteristic |
| 233 | Q16 | — | 118 | 2 | no | — |
| 234 | Q16 | — | 118 | 2 | no | — |
| 235 | Q16 | — | 236 | 1 | no | — |
| 236 | Q16 | — | 236 | 1 | no | — |
| 237 | SD16 | — | 236 | 2 | no | — |
| 238 | SD16 | — | 236 | 2 | no | — |
| 239 | SD16 | — | 236 | 2 | no | — |
| 240 | SD16 | — | 236 | 2 | no | — |
| 241 | D8 | — | 118 | 2 | no | — |
| 242 | D8 | — | 118 | 2 | no | — |
| 243 | D8 | — | 236 | 1 | no | — |
| 244 | D8 | — | 236 | 1 | no | — |
| 245 | M16 | — | 2 | 2 | no | — |
| 246 | M16 | — | 4 | 1 | no | — |
| 247 | C2×C8 | — | 4 | 1 | no | — |
| 248 | C2×C8 | (C2×D59)⋊C4 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | (C2×D59)⋊C4 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | #11745669 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | #11745669 | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | #11745671 | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | #11745671 | 1 | 1 | yes | characteristic |
| 254 | C2×C8 | C2×C4×D59 | 1 | 1 | yes | characteristic |
| 255 | C2×C8 | C2×D236 | 1 | 1 | yes | characteristic |
| 256 | C2×C8 | C2×D236 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | C2×D236 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | #11745685 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | #11745685 | 1 | 1 | yes | characteristic |
| 260 | C4⋊C4 | — | 2 | 2 | no | — |
| 261 | C4⋊C4 | — | 236 | 2 | no | — |
| 262 | C22⋊C4 | — | 4 | 1 | no | — |
| 263 | C22⋊C4 | — | 236 | 4 | no | — |
| 264 | C42 | — | 2 | 1 | no | — |
| 265 | C42 | — | 4 | 1 | no | — |
| 266 | C42 | (C2×D59)⋊C4 | 1 | 1 | yes | characteristic |
| 267 | C42 | (C2×D59)⋊C4 | 1 | 1 | yes | characteristic |
| 268 | C16 | — | 236 | 2 | no | — |
| 269 | C23 | — | 236 | 2 | no | — |
| 270 | C23 | #14273752 | 1 | 1 | yes | characteristic |
| 271 | Q8 | — | 236 | 2 | no | — |
| 272 | Q8 | — | 236 | 2 | no | — |
| 273 | D4 | — | 472 | 4 | no | — |
| 274 | D4 | — | 472 | 2 | no | — |
| 275 | D4 | — | 472 | 2 | no | — |
| 276 | D4 | — | 236 | 2 | no | — |
| 277 | D4 | — | 236 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 1 | no | — |
| 279 | C2×C4 | — | 2 | 1 | no | — |
| 280 | C2×C4 | — | 4 | 2 | no | — |
| 281 | C2×C4 | — | 4 | 2 | no | — |
| 282 | C2×C4 | — | 472 | 2 | no | — |
| 283 | C2×C4 | — | 472 | 2 | no | — |
| 284 | C2×C4 | — | 236 | 2 | no | — |
| 285 | C2×C4 | C2×(C2×D59)⋊C4 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #14273701 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #14273701 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C2×#11745669 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | C2×#11745671 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #14273752 | 1 | 1 | yes | characteristic |
| 291 | C8 | — | 4 | 2 | no | — |
| 292 | C8 | #14273647 | 1 | 1 | yes | characteristic |
| 293 | C8 | #14273647 | 1 | 1 | yes | characteristic |
| 294 | C8 | #14273647 | 1 | 1 | yes | characteristic |
| 295 | C8 | #14273647 | 1 | 1 | yes | characteristic |
| 296 | C8 | C4×D236 | 1 | 1 | yes | characteristic |
| 297 | C8 | C4×D236 | 1 | 1 | yes | characteristic |
| 298 | C8 | #14273715 | 1 | 1 | yes | characteristic |
| 299 | C8 | #14273724 | 1 | 1 | yes | characteristic |
| 300 | C22 | — | 4 | 1 | no | — |
| 301 | C22 | — | 472 | 4 | no | — |
| 302 | C22 | — | 472 | 2 | no | — |
| 303 | C22 | #16837406 | 1 | 1 | yes | characteristic |
| 304 | C22 | #16837597 | 1 | 1 | yes | characteristic |
| 305 | C22 | #16837602 | 1 | 1 | yes | characteristic |
| 306 | C4 | — | 2 | 2 | no | — |
| 307 | C4 | — | 4 | 1 | no | — |
| 308 | C4 | — | 8 | 2 | no | — |
| 309 | C4 | — | 472 | 2 | no | — |
| 310 | C4 | #16837304 | 1 | 1 | yes | characteristic |
| 311 | C4 | C2×#14273647 | 1 | 1 | yes | characteristic |
| 312 | C4 | #16837597 | 1 | 1 | yes | characteristic |
| 313 | C4 | #16837602 | 1 | 1 | yes | characteristic |
| 314 | C2 | — | 2 | 1 | no | — |
| 315 | C2 | — | 2 | 1 | no | — |
| 316 | C2 | — | 2 | 1 | no | — |
| 317 | C2 | — | 944 | 2 | no | — |
| 318 | C2 | #21425023 | 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