← #520×D109 #28376224
Subgroups — #520×D109 #28376224
Order 13952.
1640 conjugacy classes of subgroups.
The subgroup lattice has 290 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#41×D109 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20295509 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20295508 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #20295507 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #20295506 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #20295505 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C4×Q8×D109 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C4×D4×D109 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | #117×D109 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | C2×C4⋊C4×D109 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #520×C109 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#13182030 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#13182029 | C22 | 1 | 3 | yes | — |
| 14 | C2×#13182028 | C22 | 1 | 3 | yes | — |
| 15 | #41×D109 | C4 | 1 | 16 | yes | complemented |
| 16 | C2×Q8×D109 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×D4×D109 | C22 | 1 | 3 | yes | — |
| 18 | C22×C4×D109 | C22 | 1 | 6 | yes | — |
| 19 | C22×C4×D109 | C22 | 1 | 3 | yes | — |
| 20 | Q8×Dic109 | C22 | 1 | 2 | yes | complemented |
| 21 | C4×Dic218 | C22 | 1 | 6 | yes | complemented |
| 22 | #16812664 | C22 | 1 | 6 | yes | complemented |
| 23 | C4×D436 | C22 | 1 | 6 | yes | complemented |
| 24 | D4×Dic109 | C22 | 1 | 6 | yes | complemented |
| 25 | #16812661 | C22 | 1 | 6 | yes | complemented |
| 26 | C4×C22⋊D109 | C22 | 1 | 12 | yes | complemented |
| 27 | #16812659 | C22 | 1 | 12 | yes | complemented |
| 28 | #16812658 | C22 | 1 | 3 | yes | complemented |
| 29 | #16812657 | C22 | 1 | 3 | yes | complemented |
| 30 | #16812657 | C22 | 1 | 3 | yes | complemented |
| 31 | #16812656 | C22 | 1 | 6 | yes | complemented |
| 32 | #16812655 | C22 | 1 | 6 | yes | complemented |
| 33 | C2×C4⋊Dic109 | C22 | 1 | 3 | yes | complemented |
| 34 | C2×#13182008 | C22 | 1 | 6 | yes | complemented |
| 35 | C4⋊C4×D109 | C22 | 1 | 3 | yes | — |
| 36 | C4⋊C4×D109 | C22 | 1 | 6 | yes | — |
| 37 | C4⋊C4×D109 | C22 | 1 | 1 | yes | characteristic |
| 38 | C4⋊C4×D109 | C22 | 1 | 6 | yes | complemented |
| 39 | C4⋊C4×D109 | C22 | 1 | 6 | yes | complemented |
| 40 | C22⋊C4×D109 | C22 | 1 | 6 | yes | — |
| 41 | C42×D109 | C22 | 1 | 6 | yes | — |
| 42 | C2×#41×C109 | C22 | 1 | 1 | yes | characteristic |
| 43 | C4×Q8×C109 | C22 | 1 | 2 | yes | complemented |
| 44 | C4×D4×C109 | C22 | 1 | 6 | yes | complemented |
| 45 | #117×C109 | C22 | 1 | 3 | yes | complemented |
| 46 | C2×C4⋊C4×C109 | C22 | 1 | 3 | yes | complemented |
| 47 | C23×D109 | C23 | 1 | 3 | yes | — |
| 48 | #13182030 | C2×C4 | 1 | 8 | yes | complemented |
| 49 | #13182029 | C2×C4 | 1 | 24 | yes | complemented |
| 50 | #13182028 | C2×C4 | 1 | 24 | yes | complemented |
| 51 | C2×Dic218 | C23 | 1 | 3 | yes | — |
| 52 | Q8×D109 | C2×C4 | 1 | 8 | yes | complemented |
| 53 | C2×D436 | C23 | 1 | 3 | yes | — |
| 54 | C2×C22⋊D109 | C23 | 1 | 6 | yes | — |
| 55 | D4×D109 | C2×C4 | 1 | 24 | yes | complemented |
| 56 | C22×Dic109 | C23 | 1 | 6 | yes | — |
| 57 | C22×Dic109 | C23 | 1 | 3 | yes | — |
| 58 | C2×C4×D109 | — | 4 | 12 | no | — |
| 59 | C2×C4×D109 | C2×C4 | 1 | 24 | yes | complemented |
| 60 | C2×C4×D109 | C23 | 1 | 2 | yes | — |
| 61 | C2×C4×D109 | C23 | 1 | 6 | yes | — |
| 62 | C2×C4×D109 | C23 | 1 | 12 | yes | — |
| 63 | C2×C4×D109 | C23 | 1 | 3 | yes | — |
| 64 | C2×C4×D109 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×D109 | C23 | 1 | 3 | yes | — |
| 66 | C2×C4×D109 | C23 | 1 | 3 | yes | — |
| 67 | C4⋊Dic109 | C23 | 1 | 3 | yes | — |
| 68 | C4⋊Dic109 | C23 | 1 | 6 | yes | — |
| 69 | C4⋊Dic109 | C23 | 1 | 1 | yes | characteristic |
| 70 | #13182008 | C23 | 1 | 6 | yes | — |
| 71 | #13182008 | C23 | 1 | 12 | yes | — |
| 72 | #13182008 | C23 | 1 | 2 | yes | — |
| 73 | C22⋊Dic109 | C23 | 1 | 6 | yes | — |
| 74 | #13182006 | C23 | 1 | 6 | yes | — |
| 75 | #13182006 | C23 | 1 | 6 | yes | — |
| 76 | C4×Dic109 | C23 | 1 | 6 | yes | — |
| 77 | C4×Dic109 | C23 | 1 | 6 | yes | — |
| 78 | C4×Dic109 | C23 | 1 | 6 | yes | — |
| 79 | #41×C109 | C2×C4 | 1 | 8 | yes | complemented |
| 80 | C2×Q8×C109 | C23 | 1 | 1 | yes | characteristic |
| 81 | C2×D4×C109 | C23 | 1 | 3 | yes | — |
| 82 | C22×C4×C109 | C23 | 1 | 6 | yes | — |
| 83 | C22×C4×C109 | C23 | 1 | 3 | yes | — |
| 84 | C4⋊C4×C109 | C23 | 1 | 3 | yes | — |
| 85 | C4⋊C4×C109 | C23 | 1 | 6 | yes | — |
| 86 | C4⋊C4×C109 | C23 | 1 | 1 | yes | characteristic |
| 87 | C22⋊C4×C109 | C23 | 1 | 6 | yes | — |
| 88 | C42×C109 | C23 | 1 | 6 | yes | — |
| 89 | C23×C109 | C24 | 1 | 3 | yes | — |
| 90 | C22×D109 | — | 2 | 6 | no | — |
| 91 | C22×D109 | — | 2 | 6 | no | — |
| 92 | C22×D109 | C22×C4 | 1 | 12 | yes | — |
| 93 | C22×D109 | C24 | 1 | 3 | yes | — |
| 94 | C22×D109 | C24 | 1 | 1 | yes | characteristic |
| 95 | Q8×C109 | C22×C4 | 1 | 4 | yes | complemented |
| 96 | D4×C109 | C22×C4 | 1 | 12 | yes | complemented |
| 97 | C2×C4×C109 | — | 4 | 6 | no | — |
| 98 | C2×C4×C109 | C22×C4 | 1 | 12 | yes | complemented |
| 99 | C2×C4×C109 | C24 | 1 | 2 | yes | — |
| 100 | C2×C4×C109 | C24 | 1 | 6 | yes | — |
| 101 | C2×C4×C109 | C24 | 1 | 3 | yes | — |
| 102 | C2×C4×C109 | C24 | 1 | 1 | yes | characteristic |
| 103 | C22⋊D109 | C22×C4 | 1 | 24 | yes | complemented |
| 104 | C2×Dic109 | — | 4 | 6 | no | — |
| 105 | C2×Dic109 | C22×C4 | 1 | 12 | yes | complemented |
| 106 | C2×Dic109 | C24 | 1 | 2 | yes | — |
| 107 | C2×Dic109 | C24 | 1 | 6 | yes | — |
| 108 | C2×Dic109 | C24 | 1 | 3 | yes | — |
| 109 | C2×Dic109 | C24 | 1 | 1 | yes | characteristic |
| 110 | D436 | C22×C4 | 1 | 12 | yes | complemented |
| 111 | C4×D109 | — | 2 | 12 | no | — |
| 112 | C4×D109 | — | 2 | 4 | no | — |
| 113 | C4×D109 | — | 4 | 12 | no | — |
| 114 | C4×D109 | C22×C4 | 1 | 4 | yes | — |
| 115 | C4×D109 | C22×C4 | 1 | 12 | yes | — |
| 116 | C4×D109 | C22×C4 | 1 | 12 | yes | complemented |
| 117 | C4×D109 | C22×C4 | 1 | 12 | yes | complemented |
| 118 | Dic218 | C22×C4 | 1 | 12 | yes | complemented |
| 119 | C22×C109 | — | 2 | 3 | no | — |
| 120 | C22×C109 | — | 2 | 3 | no | — |
| 121 | C22×C109 | C23×C4 | 1 | 6 | yes | — |
| 122 | C22×C109 | C25 | 1 | 1 | yes | characteristic |
| 123 | C2×D109 | — | 2 | 3 | no | — |
| 124 | C2×D109 | — | 2 | 3 | no | — |
| 125 | C2×D109 | — | 2 | 24 | no | — |
| 126 | C2×D109 | C23×C4 | 1 | 6 | yes | — |
| 127 | C2×D109 | C23×C4 | 1 | 2 | yes | — |
| 128 | C2×D109 | H(5,𝔽2) | 1 | 2 | yes | — |
| 129 | C2×D109 | 2−1+4 | 1 | 2 | yes | — |
| 130 | C4×C109 | — | 2 | 6 | no | — |
| 131 | C4×C109 | — | 2 | 2 | no | — |
| 132 | C4×C109 | C23×C4 | 1 | 6 | yes | — |
| 133 | C4×C109 | C23×C4 | 1 | 2 | yes | — |
| 134 | Dic109 | — | 2 | 6 | no | — |
| 135 | Dic109 | — | 2 | 2 | no | — |
| 136 | Dic109 | C23×C4 | 1 | 6 | yes | — |
| 137 | Dic109 | C23×C4 | 1 | 2 | yes | — |
| 138 | C2×C109 | — | 2 | 6 | no | — |
| 139 | C2×C109 | C24×C4 | 1 | 1 | yes | characteristic |
| 140 | C2×C109 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 141 | C2×C109 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 142 | D109 | — | 2 | 6 | no | — |
| 143 | D109 | #520 | 1 | 4 | yes | direct product factor |
| 144 | C2×#520 | — | 109 | 1 | no | maximal, Sylow |
| 145 | C109 | C2×#520 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 146 | C22×#41 | — | 109 | 1 | no | — |
| 147 | #520 | — | 109 | 1 | no | — |
| 148 | #520 | — | 109 | 2 | no | — |
| 149 | #520 | — | 109 | 6 | no | — |
| 150 | #520 | — | 109 | 3 | no | — |
| 151 | #520 | — | 109 | 3 | no | — |
| 152 | #520 | D109 | 1 | 1 | yes | characteristic, direct product factor |
| 153 | C2×C4×Q8 | — | 109 | 2 | no | — |
| 154 | C2×C4×D4 | — | 109 | 6 | no | — |
| 155 | C2×#117 | — | 109 | 3 | no | — |
| 156 | C22×C4⋊C4 | — | 109 | 3 | no | — |
| 157 | C2×#41 | — | 109 | 16 | no | — |
| 158 | C2×#41 | — | 109 | 1 | no | — |
| 159 | C2×#41 | — | 109 | 3 | no | — |
| 160 | C2×#41 | — | 109 | 3 | no | — |
| 161 | C2×#41 | C2×D109 | 1 | 1 | yes | characteristic |
| 162 | C22×Q8 | — | 109 | 1 | no | — |
| 163 | C22×D4 | — | 109 | 3 | no | — |
| 164 | C23×C4 | — | 109 | 3 | no | — |
| 165 | C23×C4 | — | 109 | 6 | no | — |
| 166 | C4×Q8 | — | 109 | 2 | no | — |
| 167 | C4×Q8 | — | 109 | 6 | no | — |
| 168 | C4×Q8 | — | 109 | 6 | no | — |
| 169 | C4×Q8 | C2×D109 | 1 | 2 | yes | complemented |
| 170 | C4×D4 | — | 109 | 6 | no | — |
| 171 | C4×D4 | — | 109 | 12 | no | — |
| 172 | C4×D4 | — | 109 | 6 | no | — |
| 173 | C4×D4 | — | 109 | 12 | no | — |
| 174 | C4×D4 | — | 109 | 6 | no | — |
| 175 | C4×D4 | C2×D109 | 1 | 6 | yes | complemented |
| 176 | #117 | — | 109 | 3 | no | — |
| 177 | #117 | — | 109 | 3 | no | — |
| 178 | #117 | — | 109 | 3 | no | — |
| 179 | #117 | — | 109 | 6 | no | — |
| 180 | #117 | — | 109 | 6 | no | — |
| 181 | #117 | C2×D109 | 1 | 3 | yes | complemented |
| 182 | C2×C4⋊C4 | — | 109 | 6 | no | — |
| 183 | C2×C4⋊C4 | — | 109 | 3 | no | — |
| 184 | C2×C4⋊C4 | — | 109 | 3 | no | — |
| 185 | C2×C4⋊C4 | — | 109 | 6 | no | — |
| 186 | C2×C4⋊C4 | — | 109 | 6 | no | — |
| 187 | C2×C4⋊C4 | — | 109 | 6 | no | — |
| 188 | C2×C4⋊C4 | — | 109 | 1 | no | — |
| 189 | C2×C4⋊C4 | C2×D109 | 1 | 3 | yes | complemented |
| 190 | C2×C22⋊C4 | — | 109 | 6 | no | — |
| 191 | C2×C42 | — | 109 | 6 | no | — |
| 192 | C24 | — | 109 | 3 | no | — |
| 193 | #41 | — | 109 | 8 | no | — |
| 194 | #41 | — | 109 | 24 | no | — |
| 195 | #41 | — | 109 | 24 | no | — |
| 196 | #41 | C4×D109 | 1 | 8 | yes | complemented |
| 197 | C2×Q8 | — | 109 | 8 | no | — |
| 198 | C2×Q8 | — | 109 | 3 | no | — |
| 199 | C2×Q8 | C22×D109 | 1 | 1 | yes | characteristic |
| 200 | C2×D4 | — | 109 | 6 | no | — |
| 201 | C2×D4 | — | 109 | 24 | no | — |
| 202 | C2×D4 | — | 109 | 3 | no | — |
| 203 | C2×D4 | C22×D109 | 1 | 3 | yes | — |
| 204 | C22×C4 | — | 109 | 3 | no | — |
| 205 | C22×C4 | — | 109 | 24 | no | — |
| 206 | C22×C4 | — | 109 | 3 | no | — |
| 207 | C22×C4 | — | 109 | 1 | no | — |
| 208 | C22×C4 | — | 109 | 3 | no | — |
| 209 | C22×C4 | — | 109 | 3 | no | — |
| 210 | C22×C4 | — | 109 | 6 | no | — |
| 211 | C22×C4 | — | 109 | 12 | no | — |
| 212 | C22×C4 | — | 109 | 2 | no | — |
| 213 | C22×C4 | — | 109 | 6 | no | — |
| 214 | C22×C4 | — | 436 | 12 | no | — |
| 215 | C22×C4 | C22×D109 | 1 | 6 | yes | — |
| 216 | C22×C4 | C22×D109 | 1 | 3 | yes | — |
| 217 | C4⋊C4 | — | 109 | 6 | no | — |
| 218 | C4⋊C4 | — | 109 | 12 | no | — |
| 219 | C4⋊C4 | — | 109 | 2 | no | — |
| 220 | C4⋊C4 | — | 109 | 3 | no | — |
| 221 | C4⋊C4 | — | 109 | 6 | no | — |
| 222 | C4⋊C4 | — | 109 | 1 | no | — |
| 223 | C4⋊C4 | C22×D109 | 1 | 3 | yes | — |
| 224 | C4⋊C4 | C22×D109 | 1 | 6 | yes | — |
| 225 | C4⋊C4 | C22×D109 | 1 | 1 | yes | characteristic |
| 226 | C22⋊C4 | — | 109 | 6 | no | — |
| 227 | C22⋊C4 | — | 109 | 6 | no | — |
| 228 | C22⋊C4 | — | 109 | 6 | no | — |
| 229 | C22⋊C4 | C22×D109 | 1 | 6 | yes | — |
| 230 | C42 | — | 109 | 6 | no | — |
| 231 | C42 | — | 109 | 6 | no | — |
| 232 | C42 | — | 109 | 6 | no | — |
| 233 | C42 | C22×D109 | 1 | 6 | yes | — |
| 234 | C23 | — | 109 | 1 | no | — |
| 235 | C23 | — | 109 | 12 | no | — |
| 236 | C23 | — | 109 | 3 | no | — |
| 237 | C23 | — | 218 | 6 | no | — |
| 238 | C23 | — | 218 | 6 | no | — |
| 239 | C23 | C23×D109 | 1 | 3 | yes | — |
| 240 | Q8 | — | 109 | 12 | no | — |
| 241 | Q8 | C2×C4×D109 | 1 | 4 | yes | complemented |
| 242 | D4 | — | 109 | 24 | no | — |
| 243 | D4 | — | 109 | 12 | no | — |
| 244 | D4 | C2×C4×D109 | 1 | 12 | yes | complemented |
| 245 | C2×C4 | — | 4 | 6 | no | — |
| 246 | C2×C4 | — | 109 | 12 | no | — |
| 247 | C2×C4 | — | 109 | 3 | no | — |
| 248 | C2×C4 | — | 109 | 1 | no | — |
| 249 | C2×C4 | — | 109 | 4 | no | — |
| 250 | C2×C4 | — | 109 | 12 | no | — |
| 251 | C2×C4 | — | 109 | 12 | no | — |
| 252 | C2×C4 | — | 109 | 12 | no | — |
| 253 | C2×C4 | — | 109 | 2 | no | — |
| 254 | C2×C4 | — | 109 | 6 | no | — |
| 255 | C2×C4 | — | 436 | 12 | no | — |
| 256 | C2×C4 | — | 436 | 6 | no | — |
| 257 | C2×C4 | — | 218 | 12 | no | — |
| 258 | C2×C4 | — | 218 | 4 | no | — |
| 259 | C2×C4 | C2×C4×D109 | 1 | 12 | yes | complemented |
| 260 | C2×C4 | C23×D109 | 1 | 2 | yes | — |
| 261 | C2×C4 | C23×D109 | 1 | 6 | yes | — |
| 262 | C2×C4 | C23×D109 | 1 | 3 | yes | — |
| 263 | C2×C4 | C23×D109 | 1 | 1 | yes | characteristic |
| 264 | C22 | — | 2 | 3 | no | — |
| 265 | C22 | — | 2 | 3 | no | — |
| 266 | C22 | — | 109 | 2 | no | — |
| 267 | C22 | — | 109 | 2 | no | — |
| 268 | C22 | — | 109 | 2 | no | — |
| 269 | C22 | — | 109 | 6 | no | — |
| 270 | C22 | — | 218 | 24 | no | — |
| 271 | C22 | — | 218 | 3 | no | — |
| 272 | C22 | — | 218 | 3 | no | — |
| 273 | C22 | C22×C4×D109 | 1 | 6 | yes | — |
| 274 | C22 | C24×D109 | 1 | 1 | yes | characteristic |
| 275 | C4 | — | 2 | 6 | no | — |
| 276 | C4 | — | 2 | 2 | no | — |
| 277 | C4 | — | 109 | 2 | no | — |
| 278 | C4 | — | 109 | 6 | no | — |
| 279 | C4 | — | 218 | 6 | no | — |
| 280 | C4 | — | 218 | 2 | no | — |
| 281 | C4 | C22×C4×D109 | 1 | 2 | yes | — |
| 282 | C4 | C22×C4×D109 | 1 | 6 | yes | — |
| 283 | C2 | — | 2 | 6 | no | — |
| 284 | C2 | — | 109 | 4 | no | — |
| 285 | C2 | — | 218 | 6 | no | — |
| 286 | C2 | C23×C4×D109 | 1 | 1 | yes | characteristic |
| 287 | C2 | H(5,𝔽2)×D109 | 1 | 1 | yes | characteristic |
| 288 | C2 | 2−1+4×D109 | 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