← #10618674
Subgroups — #10618674
Order 640.
1122 conjugacy classes of subgroups.
The subgroup lattice has 258 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#550 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #66168 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #66161 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #66052 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #66050 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #65953 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #65952 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | C4×#713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #65904 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #65891 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | #65889 | C2 | 1 | 3 | yes | maximal, complemented |
| 12 | C2×C5×#41 | C22 | 1 | 2 | yes | — |
| 13 | C2×#713 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#711 | C22 | 1 | 6 | yes | — |
| 15 | C5×#127 | C22 | 1 | 3 | yes | — |
| 16 | C5×#121 | C22 | 1 | 6 | yes | — |
| 17 | C4×C5×Q8 | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×C5×D4 | C22 | 1 | 3 | yes | — |
| 19 | Q8×Dic5 | C22 | 1 | 2 | yes | — |
| 20 | Q8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 21 | #4086 | C22 | 1 | 6 | yes | — |
| 22 | #4085 | C22 | 1 | 12 | yes | — |
| 23 | #4084 | C22 | 1 | 6 | yes | — |
| 24 | #4084 | C22 | 1 | 6 | yes | — |
| 25 | #4084 | C22 | 1 | 3 | yes | — |
| 26 | #4082 | C22 | 1 | 3 | yes | — |
| 27 | D4×Dic5 | C22 | 1 | 6 | yes | — |
| 28 | D4×Dic5 | C22 | 1 | 6 | yes | — |
| 29 | D4×Dic5 | C22 | 1 | 6 | yes | — |
| 30 | #4076 | C22 | 1 | 6 | yes | — |
| 31 | C4×C22⋊D5 | C22 | 1 | 6 | yes | — |
| 32 | #4072 | C22 | 1 | 6 | yes | — |
| 33 | C2×C4⋊Dic5 | C22 | 1 | 6 | yes | — |
| 34 | #4043 | C22 | 1 | 3 | yes | — |
| 35 | #4039 | C22 | 1 | 3 | yes | — |
| 36 | #4038 | C22 | 1 | 3 | yes | — |
| 37 | #4036 | C22 | 1 | 1 | yes | characteristic |
| 38 | #4036 | C22 | 1 | 6 | yes | — |
| 39 | #4032 | C22 | 1 | 12 | yes | — |
| 40 | #4029 | C22 | 1 | 6 | yes | — |
| 41 | #4025 | C22 | 1 | 6 | yes | — |
| 42 | C4×D20 | C22 | 1 | 3 | yes | — |
| 43 | C42×D5 | C22 | 1 | 3 | yes | — |
| 44 | #4015 | C22 | 1 | 3 | yes | — |
| 45 | C4×Dic10 | C22 | 1 | 3 | yes | — |
| 46 | #3517 | — | 5 | 1 | no | maximal, Sylow |
| 47 | C5×#41 | — | 2 | 8 | no | — |
| 48 | C2×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C5×D4 | C23 | 1 | 3 | yes | — |
| 50 | C2×C5×D4 | C23 | 1 | 6 | yes | — |
| 51 | C2×C5×D4 | C23 | 1 | 6 | yes | — |
| 52 | C22×C4×C5 | C23 | 1 | 6 | yes | — |
| 53 | C2×C22⋊D5 | C23 | 1 | 12 | yes | — |
| 54 | C2×C22⋊D5 | C23 | 1 | 6 | yes | — |
| 55 | C22×Dic5 | C23 | 1 | 12 | yes | — |
| 56 | #713 | — | 2 | 4 | no | — |
| 57 | #711 | — | 4 | 12 | no | — |
| 58 | C2×D20 | C23 | 1 | 3 | yes | — |
| 59 | C2×C4×D5 | C23 | 1 | 3 | yes | — |
| 60 | C2×C4×D5 | C23 | 1 | 3 | yes | — |
| 61 | C2×C4×D5 | C23 | 1 | 3 | yes | — |
| 62 | C2×Dic10 | C23 | 1 | 3 | yes | — |
| 63 | C5×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 64 | C5×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 65 | C5×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 66 | C42×C5 | C23 | 1 | 3 | yes | — |
| 67 | C22⋊Dic5 | C23 | 1 | 6 | yes | — |
| 68 | C22⋊Dic5 | C23 | 1 | 6 | yes | — |
| 69 | C22⋊Dic5 | C23 | 1 | 6 | yes | — |
| 70 | #685 | C23 | 1 | 6 | yes | — |
| 71 | #685 | C23 | 1 | 6 | yes | — |
| 72 | C4⋊Dic5 | C23 | 1 | 1 | yes | characteristic |
| 73 | C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 74 | C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 75 | C4⋊Dic5 | C23 | 1 | 3 | yes | — |
| 76 | C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 77 | C4⋊Dic5 | C23 | 1 | 2 | yes | — |
| 78 | #683 | C23 | 1 | 6 | yes | — |
| 79 | #683 | C23 | 1 | 2 | yes | — |
| 80 | C4×Dic5 | C23 | 1 | 6 | yes | — |
| 81 | C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 82 | C4×Dic5 | C23 | 1 | 3 | yes | — |
| 83 | C4×Dic5 | C23 | 1 | 3 | yes | — |
| 84 | #557 | — | 5 | 1 | no | — |
| 85 | #553 | — | 5 | 6 | no | — |
| 86 | #550 | — | 5 | 2 | no | — |
| 87 | #550 | — | 5 | 3 | no | — |
| 88 | #550 | D5 | 1 | 1 | yes | characteristic, complemented |
| 89 | #547 | — | 5 | 3 | no | — |
| 90 | #540 | — | 5 | 6 | no | — |
| 91 | #532 | — | 5 | 3 | no | — |
| 92 | #529 | — | 5 | 3 | no | — |
| 93 | #520 | — | 5 | 2 | no | — |
| 94 | C4×#41 | — | 5 | 1 | no | — |
| 95 | C23×C5 | C24 | 1 | 6 | yes | — |
| 96 | C22×D5 | C24 | 1 | 3 | yes | — |
| 97 | C5×Q8 | #41 | 1 | 4 | yes | complemented |
| 98 | C5×D4 | — | 2 | 12 | no | — |
| 99 | C5×D4 | — | 2 | 6 | no | — |
| 100 | C5×D4 | — | 4 | 6 | no | — |
| 101 | C2×C4×C5 | — | 2 | 12 | no | — |
| 102 | C2×C4×C5 | C24 | 1 | 3 | yes | — |
| 103 | C2×C4×C5 | C24 | 1 | 2 | yes | — |
| 104 | C2×C4×C5 | C24 | 1 | 3 | yes | — |
| 105 | C2×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 106 | C22⋊D5 | — | 4 | 12 | no | — |
| 107 | C22⋊D5 | — | 4 | 6 | no | — |
| 108 | C2×Dic5 | — | 4 | 12 | no | — |
| 109 | C2×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 110 | C2×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 111 | C2×Dic5 | C24 | 1 | 2 | yes | — |
| 112 | C2×Dic5 | C24 | 1 | 3 | yes | — |
| 113 | C2×Dic5 | C24 | 1 | 6 | yes | — |
| 114 | D20 | — | 2 | 6 | no | — |
| 115 | C4×D5 | — | 2 | 6 | no | — |
| 116 | C4×D5 | — | 2 | 6 | no | — |
| 117 | C4×D5 | — | 4 | 3 | no | — |
| 118 | Dic10 | — | 2 | 6 | no | — |
| 119 | C2×#41 | — | 5 | 1 | no | — |
| 120 | C2×#41 | — | 5 | 6 | no | — |
| 121 | C2×#41 | C2×D5 | 1 | 2 | yes | — |
| 122 | C4⋊Q8 | — | 5 | 3 | no | — |
| 123 | #127 | — | 5 | 6 | no | — |
| 124 | #127 | C2×D5 | 1 | 3 | yes | — |
| 125 | #126 | — | 5 | 6 | no | — |
| 126 | #125 | — | 5 | 1 | no | — |
| 127 | #125 | — | 5 | 6 | no | — |
| 128 | #124 | — | 5 | 3 | no | — |
| 129 | #123 | — | 5 | 6 | no | — |
| 130 | #123 | — | 5 | 12 | no | — |
| 131 | C22⋊Q8 | — | 5 | 3 | no | — |
| 132 | #121 | — | 5 | 3 | no | — |
| 133 | #121 | — | 5 | 6 | no | — |
| 134 | #121 | — | 5 | 6 | no | — |
| 135 | #121 | — | 5 | 12 | no | — |
| 136 | #121 | — | 5 | 6 | no | — |
| 137 | #121 | C2×D5 | 1 | 6 | yes | — |
| 138 | C4×Q8 | — | 5 | 1 | no | — |
| 139 | C4×Q8 | — | 5 | 2 | no | — |
| 140 | C4×Q8 | — | 5 | 3 | no | — |
| 141 | C4×Q8 | C2×D5 | 1 | 1 | yes | characteristic |
| 142 | C4×D4 | — | 5 | 6 | no | — |
| 143 | C4×D4 | — | 5 | 6 | no | — |
| 144 | C4×D4 | — | 5 | 6 | no | — |
| 145 | C4×D4 | — | 5 | 6 | no | — |
| 146 | C4×D4 | — | 5 | 3 | no | — |
| 147 | C4×D4 | — | 5 | 3 | no | — |
| 148 | C4×D4 | C2×D5 | 1 | 3 | yes | — |
| 149 | #117 | — | 5 | 6 | no | — |
| 150 | #117 | — | 5 | 3 | no | — |
| 151 | C2×C4⋊C4 | — | 5 | 6 | no | — |
| 152 | C2×C42 | — | 5 | 3 | no | — |
| 153 | C22×C5 | — | 2 | 6 | no | — |
| 154 | C22×C5 | — | 2 | 6 | no | — |
| 155 | C22×C5 | — | 2 | 6 | no | — |
| 156 | C22×C5 | C25 | 1 | 1 | yes | characteristic |
| 157 | C2×D5 | — | 2 | 3 | no | — |
| 158 | C2×D5 | — | 2 | 3 | no | — |
| 159 | C2×D5 | — | 2 | 3 | no | — |
| 160 | C4×C5 | — | 2 | 2 | no | — |
| 161 | C4×C5 | — | 2 | 3 | no | — |
| 162 | C4×C5 | C2×#41 | 1 | 6 | yes | — |
| 163 | C4×C5 | H(5,𝔽2) | 1 | 2 | yes | — |
| 164 | Dic5 | — | 2 | 1 | no | — |
| 165 | Dic5 | — | 2 | 6 | no | — |
| 166 | Dic5 | — | 2 | 3 | no | — |
| 167 | Dic5 | — | 2 | 1 | no | — |
| 168 | Dic5 | — | 2 | 2 | no | — |
| 169 | #41 | — | 2 | 8 | no | — |
| 170 | #41 | — | 10 | 4 | no | — |
| 171 | #41 | — | 20 | 12 | no | — |
| 172 | C2×Q8 | — | 5 | 3 | no | — |
| 173 | C2×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 174 | C2×D4 | — | 5 | 6 | no | — |
| 175 | C2×D4 | — | 5 | 12 | no | — |
| 176 | C2×D4 | — | 5 | 3 | no | — |
| 177 | C2×D4 | C22×D5 | 1 | 6 | yes | — |
| 178 | C2×D4 | C22×D5 | 1 | 3 | yes | — |
| 179 | C2×D4 | C22×D5 | 1 | 6 | yes | — |
| 180 | C22×C4 | — | 5 | 12 | no | — |
| 181 | C22×C4 | — | 5 | 3 | no | — |
| 182 | C22×C4 | — | 5 | 3 | no | — |
| 183 | C22×C4 | — | 5 | 3 | no | — |
| 184 | C22×C4 | C22×D5 | 1 | 6 | yes | — |
| 185 | C4⋊C4 | — | 5 | 3 | no | — |
| 186 | C4⋊C4 | — | 5 | 6 | no | — |
| 187 | C4⋊C4 | — | 5 | 6 | no | — |
| 188 | C4⋊C4 | — | 5 | 2 | no | — |
| 189 | C4⋊C4 | — | 5 | 2 | no | — |
| 190 | C4⋊C4 | — | 5 | 6 | no | — |
| 191 | C4⋊C4 | — | 5 | 1 | no | — |
| 192 | C4⋊C4 | — | 5 | 6 | no | — |
| 193 | C4⋊C4 | C22×D5 | 1 | 3 | yes | — |
| 194 | C4⋊C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 195 | C22⋊C4 | — | 5 | 6 | no | — |
| 196 | C22⋊C4 | — | 5 | 6 | no | — |
| 197 | C22⋊C4 | — | 5 | 6 | no | — |
| 198 | C22⋊C4 | — | 5 | 6 | no | — |
| 199 | C22⋊C4 | — | 5 | 6 | no | — |
| 200 | C22⋊C4 | C22×D5 | 1 | 6 | yes | — |
| 201 | C42 | — | 5 | 1 | no | — |
| 202 | C42 | — | 5 | 3 | no | — |
| 203 | C42 | — | 5 | 3 | no | — |
| 204 | C42 | — | 5 | 6 | no | — |
| 205 | C42 | C22×D5 | 1 | 3 | yes | — |
| 206 | C2×C5 | — | 4 | 6 | no | — |
| 207 | C2×C5 | C22×#41 | 1 | 1 | yes | characteristic |
| 208 | C2×C5 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 209 | C2×C5 | #585 | 1 | 1 | yes | characteristic |
| 210 | D5 | — | 4 | 3 | no | — |
| 211 | C23 | — | 5 | 3 | no | — |
| 212 | C23 | C23×D5 | 1 | 6 | yes | — |
| 213 | Q8 | — | 10 | 6 | no | — |
| 214 | Q8 | #711 | 1 | 4 | yes | complemented |
| 215 | D4 | — | 2 | 12 | no | — |
| 216 | D4 | — | 2 | 6 | no | — |
| 217 | D4 | — | 4 | 6 | no | — |
| 218 | D4 | — | 10 | 6 | no | — |
| 219 | D4 | — | 20 | 12 | no | — |
| 220 | D4 | — | 20 | 6 | no | — |
| 221 | C2×C4 | — | 2 | 12 | no | — |
| 222 | C2×C4 | — | 5 | 1 | no | — |
| 223 | C2×C4 | — | 5 | 1 | no | — |
| 224 | C2×C4 | — | 5 | 2 | no | — |
| 225 | C2×C4 | — | 5 | 3 | no | — |
| 226 | C2×C4 | — | 5 | 6 | no | — |
| 227 | C2×C4 | — | 10 | 6 | no | — |
| 228 | C2×C4 | — | 10 | 6 | no | — |
| 229 | C2×C4 | — | 20 | 3 | no | — |
| 230 | C2×C4 | — | 20 | 12 | no | — |
| 231 | C2×C4 | C23×D5 | 1 | 3 | yes | — |
| 232 | C2×C4 | C23×D5 | 1 | 2 | yes | — |
| 233 | C2×C4 | C23×D5 | 1 | 3 | yes | — |
| 234 | C2×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 235 | C5 | #3517 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 236 | C22 | — | 2 | 6 | no | — |
| 237 | C22 | — | 2 | 6 | no | — |
| 238 | C22 | — | 2 | 6 | no | — |
| 239 | C22 | — | 10 | 3 | no | — |
| 240 | C22 | — | 10 | 3 | no | — |
| 241 | C22 | — | 10 | 3 | no | — |
| 242 | C22 | C24×D5 | 1 | 1 | yes | characteristic |
| 243 | C4 | — | 2 | 2 | no | — |
| 244 | C4 | — | 2 | 3 | no | — |
| 245 | C4 | — | 10 | 1 | no | — |
| 246 | C4 | — | 10 | 3 | no | — |
| 247 | C4 | — | 10 | 6 | no | — |
| 248 | C4 | — | 10 | 1 | no | — |
| 249 | C4 | — | 10 | 2 | no | — |
| 250 | C4 | C2×#711 | 1 | 6 | yes | — |
| 251 | C4 | #4149 | 1 | 2 | yes | — |
| 252 | C2 | — | 4 | 6 | no | — |
| 253 | C2 | — | 20 | 3 | no | — |
| 254 | C2 | C22×#711 | 1 | 1 | yes | characteristic |
| 255 | C2 | C2×#4149 | 1 | 1 | yes | characteristic |
| 256 | C2 | #66285 | 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