← #12017709
Subgroups — #12017709
Order 1280.
418 conjugacy classes of subgroups.
The subgroup lattice has 250 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#65193 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C5×#346 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#64730 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#64726 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10600588 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#4050 | C22 | 1 | 2 | yes | — |
| 7 | C22×C5×C16 | C22 | 1 | 1 | yes | characteristic |
| 8 | C22×C5×C16 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 10 | #65399 | — | 2 | 2 | no | — |
| 11 | C2×#3951 | C22 | 1 | 2 | yes | — |
| 12 | C2×#3937 | — | 2 | 2 | no | — |
| 13 | C2×Dic40 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #65193 | — | 2 | 2 | no | — |
| 15 | #65193 | — | 2 | 2 | no | — |
| 16 | C2×#3932 | C4 | 1 | 2 | yes | complemented |
| 17 | C2×D80 | C4 | 1 | 1 | yes | characteristic, complemented |
| 18 | C5×#346 | — | 2 | 2 | no | — |
| 19 | #64730 | — | 2 | 4 | no | — |
| 20 | #64726 | — | 2 | 2 | no | — |
| 21 | #13242 | — | 5 | 1 | no | maximal, Sylow |
| 22 | C2×#709 | — | 2 | 2 | no | — |
| 23 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 24 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 25 | C2×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 26 | C2×#681 | — | 2 | 2 | no | — |
| 27 | C2×Dic20 | C2×C4 | 1 | 2 | yes | — |
| 28 | #4050 | — | 2 | 2 | no | — |
| 29 | #4050 | — | 2 | 2 | no | — |
| 30 | #4050 | — | 2 | 2 | no | — |
| 31 | #4050 | — | 2 | 2 | no | — |
| 32 | C2×D40 | C2×C4 | 1 | 2 | yes | — |
| 33 | C2×#677 | — | 2 | 2 | no | — |
| 34 | C2×C5×C16 | — | 2 | 1 | no | — |
| 35 | C2×C5×C16 | — | 2 | 1 | no | — |
| 36 | C2×C5×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C2×C5×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C2×C5×C16 | D4 | 1 | 2 | yes | — |
| 39 | C2×C5×C16 | D4 | 1 | 2 | yes | — |
| 40 | C2×C5×C16 | D4 | 1 | 2 | yes | — |
| 41 | C4×C5×C8 | — | 2 | 1 | no | — |
| 42 | C4×C5×C8 | — | 2 | 1 | no | — |
| 43 | #3952 | — | 4 | 4 | no | — |
| 44 | #3951 | — | 2 | 2 | no | — |
| 45 | #3951 | — | 2 | 2 | no | — |
| 46 | #3937 | — | 4 | 4 | no | — |
| 47 | Dic40 | — | 2 | 2 | no | — |
| 48 | #3932 | — | 2 | 4 | no | — |
| 49 | D80 | — | 2 | 2 | no | — |
| 50 | C2×#508 | — | 5 | 1 | no | — |
| 51 | C2×#368 | — | 5 | 1 | no | — |
| 52 | C2×#359 | — | 5 | 2 | no | — |
| 53 | C2×#346 | D5 | 1 | 1 | yes | characteristic, complemented |
| 54 | #1974 | — | 5 | 2 | no | — |
| 55 | C22×C4×C5 | — | 2 | 1 | no | — |
| 56 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 57 | C2×C22⋊D5 | — | 4 | 2 | no | — |
| 58 | #709 | — | 4 | 2 | no | — |
| 59 | #709 | — | 4 | 2 | no | — |
| 60 | #709 | — | 4 | 2 | no | — |
| 61 | #709 | — | 4 | 2 | no | — |
| 62 | C2×D20 | — | 2 | 2 | no | — |
| 63 | C2×C4×D5 | — | 4 | 2 | no | — |
| 64 | C2×Dic10 | — | 2 | 2 | no | — |
| 65 | C2×C5×C8 | — | 2 | 1 | no | — |
| 66 | C2×C5×C8 | — | 2 | 1 | no | — |
| 67 | C2×C5×C8 | D8 | 1 | 2 | yes | — |
| 68 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 69 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C2×C5×C8 | #41 | 1 | 1 | yes | characteristic |
| 73 | C2×C5×C8 | #41 | 1 | 1 | yes | characteristic |
| 74 | C42×C5 | — | 2 | 1 | no | — |
| 75 | C42×C5 | D8 | 1 | 2 | yes | — |
| 76 | #681 | — | 4 | 2 | no | — |
| 77 | #681 | — | 4 | 2 | no | — |
| 78 | C2×#168 | — | 4 | 2 | no | — |
| 79 | Dic20 | — | 2 | 2 | no | — |
| 80 | Dic20 | — | 2 | 2 | no | — |
| 81 | D40 | — | 2 | 2 | no | — |
| 82 | D40 | — | 2 | 2 | no | — |
| 83 | #677 | — | 4 | 2 | no | — |
| 84 | #677 | — | 4 | 2 | no | — |
| 85 | C5×C16 | — | 2 | 2 | no | — |
| 86 | C5×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 87 | C5×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 88 | C2×#135 | — | 5 | 2 | no | — |
| 89 | #508 | — | 10 | 2 | no | — |
| 90 | #508 | — | 10 | 2 | no | — |
| 91 | C2×Q32 | — | 5 | 1 | no | — |
| 92 | C2×SD32 | — | 5 | 2 | no | — |
| 93 | C2×D16 | — | 5 | 1 | no | — |
| 94 | C22×C16 | C2×D5 | 1 | 1 | yes | characteristic |
| 95 | C22×C16 | C2×D5 | 1 | 1 | yes | characteristic |
| 96 | C2×#108 | — | 5 | 2 | no | — |
| 97 | C2×#104 | — | 10 | 2 | no | — |
| 98 | #408 | — | 10 | 2 | no | — |
| 99 | C2×C4×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 100 | #368 | — | 10 | 2 | no | — |
| 101 | #359 | — | 10 | 4 | no | — |
| 102 | #346 | — | 2 | 2 | no | — |
| 103 | C23×C5 | #135 | 1 | 1 | yes | characteristic |
| 104 | C22×D5 | — | 4 | 2 | no | — |
| 105 | C2×C4×C5 | — | 2 | 2 | no | — |
| 106 | C2×C4×C5 | — | 4 | 2 | no | — |
| 107 | C2×C4×C5 | C4×D4 | 1 | 1 | yes | characteristic |
| 108 | C2×C4×C5 | #121 | 1 | 1 | yes | characteristic |
| 109 | C2×C4×C5 | #127 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C5 | #135 | 1 | 1 | yes | characteristic |
| 113 | C22⋊D5 | — | 8 | 2 | no | — |
| 114 | C22⋊D5 | — | 8 | 2 | no | — |
| 115 | C2×Dic5 | — | 4 | 2 | no | — |
| 116 | D20 | — | 4 | 2 | no | — |
| 117 | D20 | — | 4 | 2 | no | — |
| 118 | C4×D5 | — | 8 | 2 | no | — |
| 119 | C4×D5 | — | 8 | 2 | no | — |
| 120 | Dic10 | — | 4 | 2 | no | — |
| 121 | Dic10 | — | 4 | 2 | no | — |
| 122 | C5×C8 | — | 2 | 2 | no | — |
| 123 | C5×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 124 | C5×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 125 | C5×C8 | #121 | 1 | 1 | yes | characteristic |
| 126 | C5×C8 | #124 | 1 | 1 | yes | characteristic |
| 127 | #168 | — | 8 | 2 | no | — |
| 128 | C2×#41 | — | 10 | 2 | no | — |
| 129 | #135 | — | 10 | 2 | no | — |
| 130 | #135 | — | 10 | 2 | no | — |
| 131 | #135 | — | 10 | 2 | no | — |
| 132 | #135 | — | 10 | 2 | no | — |
| 133 | C2×Q16 | — | 5 | 2 | no | — |
| 134 | C2×SD16 | — | 10 | 2 | no | — |
| 135 | C2×D8 | — | 5 | 2 | no | — |
| 136 | C2×M16 | — | 10 | 2 | no | — |
| 137 | C22×C8 | D20 | 1 | 1 | yes | characteristic |
| 138 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 139 | C2×C42 | D20 | 1 | 1 | yes | characteristic |
| 140 | Q32 | — | 10 | 2 | no | — |
| 141 | SD32 | — | 10 | 4 | no | — |
| 142 | D16 | — | 10 | 2 | no | — |
| 143 | C2×C16 | — | 2 | 1 | no | — |
| 144 | C2×C16 | — | 2 | 1 | no | — |
| 145 | C2×C16 | C4×D5 | 1 | 1 | yes | characteristic |
| 146 | C2×C16 | C4×D5 | 1 | 1 | yes | characteristic |
| 147 | C2×C16 | D20 | 1 | 2 | yes | — |
| 148 | C2×C16 | D20 | 1 | 2 | yes | — |
| 149 | C2×C16 | C22⋊D5 | 1 | 2 | yes | — |
| 150 | #108 | — | 10 | 2 | no | — |
| 151 | #108 | — | 10 | 2 | no | — |
| 152 | #104 | — | 20 | 4 | no | — |
| 153 | C22⋊C8 | — | 20 | 4 | no | — |
| 154 | C4×C8 | — | 2 | 1 | no | — |
| 155 | C4×C8 | — | 2 | 1 | no | — |
| 156 | C22×C5 | — | 2 | 2 | no | — |
| 157 | C22×C5 | C4×D8 | 1 | 1 | yes | characteristic |
| 158 | C22×C5 | #466 | 1 | 1 | yes | characteristic |
| 159 | C22×C5 | #495 | 1 | 1 | yes | characteristic |
| 160 | C2×D5 | — | 8 | 4 | no | — |
| 161 | C2×D5 | — | 8 | 2 | no | — |
| 162 | C4×C5 | — | 4 | 2 | no | — |
| 163 | C4×C5 | #390 | 1 | 1 | yes | characteristic |
| 164 | C4×C5 | C4×D8 | 1 | 1 | yes | characteristic |
| 165 | C4×C5 | #466 | 1 | 1 | yes | characteristic |
| 166 | C4×C5 | #493 | 1 | 1 | yes | characteristic |
| 167 | Dic5 | — | 8 | 2 | no | — |
| 168 | #41 | — | 20 | 2 | no | — |
| 169 | #41 | — | 20 | 2 | no | — |
| 170 | #41 | — | 20 | 2 | no | — |
| 171 | #41 | — | 20 | 2 | no | — |
| 172 | C2×Q8 | — | 10 | 2 | no | — |
| 173 | C2×D4 | — | 10 | 2 | no | — |
| 174 | C2×D4 | — | 20 | 2 | no | — |
| 175 | C22×C4 | — | 2 | 1 | no | — |
| 176 | C22×C4 | — | 20 | 2 | no | — |
| 177 | C22×C4 | C2×D20 | 1 | 1 | yes | characteristic |
| 178 | Q16 | — | 10 | 2 | no | — |
| 179 | Q16 | — | 10 | 2 | no | — |
| 180 | SD16 | — | 20 | 2 | no | — |
| 181 | SD16 | — | 20 | 2 | no | — |
| 182 | D8 | — | 10 | 2 | no | — |
| 183 | D8 | — | 10 | 2 | no | — |
| 184 | M16 | — | 20 | 2 | no | — |
| 185 | M16 | — | 20 | 2 | no | — |
| 186 | C2×C8 | — | 2 | 1 | no | — |
| 187 | C2×C8 | — | 2 | 1 | no | — |
| 188 | C2×C8 | — | 20 | 2 | no | — |
| 189 | C2×C8 | D40 | 1 | 2 | yes | — |
| 190 | C2×C8 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 191 | C2×C8 | C2×D20 | 1 | 1 | yes | characteristic |
| 192 | C2×C8 | C2×D20 | 1 | 1 | yes | characteristic |
| 193 | C2×C8 | #709 | 1 | 1 | yes | characteristic |
| 194 | C2×C8 | #709 | 1 | 1 | yes | characteristic |
| 195 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 196 | C42 | — | 2 | 1 | no | — |
| 197 | C42 | D40 | 1 | 2 | yes | — |
| 198 | C16 | — | 2 | 2 | no | — |
| 199 | C16 | #685 | 1 | 2 | yes | — |
| 200 | C16 | #685 | 1 | 2 | yes | — |
| 201 | C2×C5 | — | 2 | 1 | no | — |
| 202 | C2×C5 | — | 2 | 1 | no | — |
| 203 | C2×C5 | #1970 | 1 | 1 | yes | characteristic |
| 204 | C2×C5 | #2179 | 1 | 2 | yes | — |
| 205 | D5 | — | 16 | 2 | no | — |
| 206 | C23 | — | 20 | 2 | no | — |
| 207 | C23 | #4050 | 1 | 1 | yes | characteristic |
| 208 | Q8 | — | 20 | 2 | no | — |
| 209 | Q8 | — | 20 | 2 | no | — |
| 210 | D4 | — | 20 | 2 | no | — |
| 211 | D4 | — | 20 | 2 | no | — |
| 212 | D4 | — | 40 | 2 | no | — |
| 213 | D4 | — | 40 | 2 | no | — |
| 214 | C2×C4 | — | 2 | 2 | no | — |
| 215 | C2×C4 | — | 4 | 2 | no | — |
| 216 | C2×C4 | — | 20 | 2 | no | — |
| 217 | C2×C4 | — | 40 | 2 | no | — |
| 218 | C2×C4 | — | 40 | 2 | no | — |
| 219 | C2×C4 | C4×D20 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | #4020 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | C2×D40 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | C2×D40 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #4050 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #4076 | 1 | 1 | yes | characteristic |
| 225 | C8 | — | 2 | 2 | no | — |
| 226 | C8 | — | 40 | 2 | no | — |
| 227 | C8 | C4×D20 | 1 | 1 | yes | characteristic |
| 228 | C8 | #4021 | 1 | 1 | yes | characteristic |
| 229 | C8 | C2×#685 | 1 | 1 | yes | characteristic |
| 230 | C8 | #4076 | 1 | 1 | yes | characteristic |
| 231 | C5 | #13242 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 232 | C22 | — | 2 | 2 | no | — |
| 233 | C22 | — | 40 | 2 | no | — |
| 234 | C22 | — | 40 | 4 | no | — |
| 235 | C22 | C4×D40 | 1 | 1 | yes | characteristic |
| 236 | C22 | #64985 | 1 | 1 | yes | characteristic |
| 237 | C22 | #65404 | 1 | 1 | yes | characteristic |
| 238 | C4 | — | 4 | 2 | no | — |
| 239 | C4 | — | 40 | 2 | no | — |
| 240 | C4 | C4×D40 | 1 | 1 | yes | characteristic |
| 241 | C4 | #64984 | 1 | 1 | yes | characteristic |
| 242 | C4 | #65228 | 1 | 1 | yes | characteristic |
| 243 | C4 | #65404 | 1 | 1 | yes | characteristic |
| 244 | C2 | — | 2 | 1 | no | — |
| 245 | C2 | — | 2 | 1 | no | — |
| 246 | C2 | — | 80 | 2 | no | — |
| 247 | C2 | #10599878 | 1 | 2 | yes | — |
| 248 | C2 | #10600569 | 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