← #10619183
Subgroups — #10619183
Order 640.
1748 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C5×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#4149 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #66168 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #66166 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #66164 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #41×Dic5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #66139 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×#4086 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | D4×C22⋊D5 | C2 | 1 | 12 | yes | maximal, complemented |
| 10 | C5×H(5,𝔽2) | — | 2 | 8 | no | — |
| 11 | C2×C5×#41 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C5×#41 | C22 | 1 | 3 | yes | complemented |
| 13 | C2×C5×#41 | C22 | 1 | 2 | yes | complemented |
| 14 | C22×C5×D4 | C22 | 1 | 6 | yes | complemented |
| 15 | C22×C5×D4 | C22 | 1 | 3 | yes | complemented |
| 16 | C22×C22⋊D5 | C22 | 1 | 12 | yes | complemented |
| 17 | #4149 | — | 2 | 8 | no | — |
| 18 | C2×#713 | C22 | 1 | 2 | yes | complemented |
| 19 | C2×D4×D5 | C22 | 1 | 6 | yes | complemented |
| 20 | C2×#709 | C22 | 1 | 3 | yes | complemented |
| 21 | C22×D20 | C22 | 1 | 3 | yes | complemented |
| 22 | #4099 | C22 | 1 | 12 | yes | complemented |
| 23 | #4093 | C22 | 1 | 3 | yes | complemented |
| 24 | Q8×Dic5 | C22 | 1 | 2 | yes | complemented |
| 25 | #4090 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #4086 | C22 | 1 | 12 | yes | complemented |
| 27 | #4086 | C22 | 1 | 3 | yes | complemented |
| 28 | #4086 | C22 | 1 | 6 | yes | complemented |
| 29 | #4085 | C22 | 1 | 12 | yes | complemented |
| 30 | #4084 | C22 | 1 | 12 | yes | complemented |
| 31 | #4083 | C22 | 1 | 12 | yes | complemented |
| 32 | #4082 | C22 | 1 | 3 | yes | complemented |
| 33 | D4×Dic5 | C22 | 1 | 6 | yes | complemented |
| 34 | #4076 | C22 | 1 | 12 | yes | complemented |
| 35 | C4×C22⋊D5 | C22 | 1 | 12 | yes | complemented |
| 36 | #4072 | C22 | 1 | 3 | yes | complemented |
| 37 | C2×C4×Dic5 | C22 | 1 | 3 | yes | complemented |
| 38 | #3498 | — | 5 | 1 | no | maximal, Sylow |
| 39 | C24×C5 | C23 | 1 | 6 | yes | — |
| 40 | C23×D5 | C23 | 1 | 6 | yes | — |
| 41 | C5×#41 | — | 2 | 8 | no | — |
| 42 | C5×#41 | — | 2 | 12 | no | — |
| 43 | C5×#41 | D4 | 1 | 8 | yes | complemented |
| 44 | C2×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C5×D4 | — | 2 | 24 | no | — |
| 47 | C2×C5×D4 | — | 2 | 12 | no | — |
| 48 | C2×C5×D4 | C23 | 1 | 3 | yes | — |
| 49 | C2×C5×D4 | C23 | 1 | 6 | yes | — |
| 50 | C2×C5×D4 | C23 | 1 | 6 | yes | — |
| 51 | C2×C5×D4 | C23 | 1 | 3 | yes | — |
| 52 | C22×C4×C5 | C23 | 1 | 6 | yes | — |
| 53 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 54 | C2×C22⋊D5 | — | 4 | 24 | no | — |
| 55 | C2×C22⋊D5 | C23 | 1 | 6 | yes | — |
| 56 | C2×C22⋊D5 | C23 | 1 | 12 | yes | — |
| 57 | C2×C22⋊D5 | C23 | 1 | 24 | yes | — |
| 58 | C22×Dic5 | C23 | 1 | 6 | yes | — |
| 59 | #713 | — | 2 | 8 | no | — |
| 60 | D4×D5 | — | 2 | 24 | no | — |
| 61 | #709 | — | 2 | 12 | no | — |
| 62 | C2×D20 | — | 2 | 12 | no | — |
| 63 | C2×D20 | C23 | 1 | 3 | yes | — |
| 64 | C2×D20 | C23 | 1 | 6 | yes | — |
| 65 | C2×C4×D5 | C23 | 1 | 6 | yes | — |
| 66 | C2×Dic10 | C23 | 1 | 1 | yes | characteristic |
| 67 | C22⋊Dic5 | C23 | 1 | 6 | yes | — |
| 68 | C22⋊Dic5 | C23 | 1 | 12 | yes | — |
| 69 | #685 | C23 | 1 | 12 | yes | — |
| 70 | C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 71 | #683 | C23 | 1 | 4 | yes | — |
| 72 | C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 73 | C4×Dic5 | C23 | 1 | 3 | yes | — |
| 74 | C4×Dic5 | C23 | 1 | 6 | yes | — |
| 75 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 76 | C2×H(5,𝔽2) | D5 | 1 | 1 | yes | characteristic, complemented |
| 77 | #550 | — | 5 | 4 | no | — |
| 78 | D42 | — | 5 | 12 | no | — |
| 79 | #535 | — | 5 | 3 | no | — |
| 80 | #535 | — | 5 | 3 | no | — |
| 81 | #532 | — | 5 | 3 | no | — |
| 82 | C2×#127 | — | 5 | 3 | no | — |
| 83 | C4×#41 | — | 5 | 1 | no | — |
| 84 | C23×C5 | — | 2 | 12 | no | — |
| 85 | C23×C5 | — | 4 | 12 | no | — |
| 86 | C23×C5 | C24 | 1 | 6 | yes | — |
| 87 | C23×C5 | C24 | 1 | 3 | yes | — |
| 88 | C22×D5 | — | 2 | 12 | no | — |
| 89 | C22×D5 | — | 4 | 12 | no | — |
| 90 | C22×D5 | C24 | 1 | 6 | yes | — |
| 91 | C5×Q8 | — | 2 | 2 | no | — |
| 92 | C5×Q8 | C2×D4 | 1 | 4 | yes | complemented |
| 93 | C5×D4 | — | 2 | 12 | no | — |
| 94 | C5×D4 | — | 2 | 6 | no | — |
| 95 | C5×D4 | — | 2 | 12 | no | — |
| 96 | C5×D4 | C2×D4 | 1 | 12 | yes | complemented |
| 97 | C2×C4×C5 | — | 2 | 12 | no | — |
| 98 | C2×C4×C5 | C2×D4 | 1 | 12 | yes | complemented |
| 99 | C2×C4×C5 | C24 | 1 | 2 | yes | — |
| 100 | C2×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 101 | C2×C4×C5 | C24 | 1 | 3 | yes | — |
| 102 | C22⋊D5 | — | 2 | 12 | no | — |
| 103 | C22⋊D5 | — | 2 | 24 | no | — |
| 104 | C22⋊D5 | — | 4 | 24 | no | — |
| 105 | C2×Dic5 | — | 2 | 12 | no | — |
| 106 | C2×Dic5 | C24 | 1 | 6 | yes | — |
| 107 | C2×Dic5 | C24 | 1 | 2 | yes | — |
| 108 | C2×Dic5 | C24 | 1 | 2 | yes | — |
| 109 | D20 | — | 2 | 6 | no | — |
| 110 | D20 | — | 2 | 12 | no | — |
| 111 | C4×D5 | — | 2 | 12 | no | — |
| 112 | Dic10 | — | 2 | 2 | no | — |
| 113 | H(5,𝔽2) | — | 2 | 8 | no | — |
| 114 | H(5,𝔽2) | — | 10 | 8 | no | — |
| 115 | C2×#41 | — | 5 | 2 | no | — |
| 116 | C2×#41 | — | 5 | 3 | no | — |
| 117 | C2×#41 | C2×D5 | 1 | 1 | yes | characteristic |
| 118 | C2×#41 | C2×D5 | 1 | 3 | yes | complemented |
| 119 | C2×#41 | C2×D5 | 1 | 2 | yes | complemented |
| 120 | C22×D4 | — | 5 | 12 | no | — |
| 121 | C22×D4 | — | 5 | 6 | no | — |
| 122 | C22×D4 | — | 5 | 3 | no | — |
| 123 | C22×D4 | C2×D5 | 1 | 3 | yes | complemented |
| 124 | C22×D4 | C2×D5 | 1 | 6 | yes | complemented |
| 125 | C4⋊Q8 | — | 5 | 1 | no | — |
| 126 | #127 | — | 5 | 3 | no | — |
| 127 | #127 | — | 5 | 6 | no | — |
| 128 | #127 | — | 5 | 12 | no | — |
| 129 | #124 | — | 5 | 3 | no | — |
| 130 | #124 | — | 5 | 3 | no | — |
| 131 | #121 | — | 5 | 12 | no | — |
| 132 | #121 | — | 5 | 12 | no | — |
| 133 | #121 | — | 5 | 12 | no | — |
| 134 | #120 | — | 5 | 12 | no | — |
| 135 | #120 | — | 5 | 12 | no | — |
| 136 | C4×Q8 | — | 5 | 2 | no | — |
| 137 | C4×D4 | — | 5 | 6 | no | — |
| 138 | C4×D4 | — | 5 | 12 | no | — |
| 139 | #117 | — | 5 | 3 | no | — |
| 140 | C2×C42 | — | 5 | 3 | no | — |
| 141 | C22×C5 | — | 2 | 6 | no | — |
| 142 | C22×C5 | — | 2 | 6 | no | — |
| 143 | C22×C5 | — | 2 | 12 | no | — |
| 144 | C22×C5 | — | 8 | 12 | no | — |
| 145 | C22×C5 | C22×D4 | 1 | 6 | yes | — |
| 146 | C22×C5 | C25 | 1 | 1 | yes | characteristic |
| 147 | C2×D5 | — | 2 | 6 | no | — |
| 148 | C2×D5 | — | 2 | 12 | no | — |
| 149 | C2×D5 | — | 8 | 12 | no | — |
| 150 | C4×C5 | — | 2 | 2 | no | — |
| 151 | C4×C5 | C22×D4 | 1 | 2 | yes | — |
| 152 | C4×C5 | C22×D4 | 1 | 6 | yes | — |
| 153 | Dic5 | — | 2 | 2 | no | — |
| 154 | Dic5 | — | 2 | 6 | no | — |
| 155 | Dic5 | H(5,𝔽2) | 1 | 4 | yes | complemented |
| 156 | C24 | — | 5 | 6 | no | — |
| 157 | C24 | C22×D5 | 1 | 6 | yes | — |
| 158 | #41 | — | 2 | 8 | no | — |
| 159 | #41 | — | 2 | 12 | no | — |
| 160 | #41 | — | 10 | 8 | no | — |
| 161 | #41 | — | 10 | 12 | no | — |
| 162 | #41 | C22⋊D5 | 1 | 8 | yes | complemented |
| 163 | C2×Q8 | — | 5 | 1 | no | — |
| 164 | C2×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 165 | C2×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 166 | C2×D4 | — | 2 | 24 | no | — |
| 167 | C2×D4 | — | 2 | 12 | no | — |
| 168 | C2×D4 | — | 5 | 12 | no | — |
| 169 | C2×D4 | — | 5 | 6 | no | — |
| 170 | C2×D4 | — | 5 | 24 | no | — |
| 171 | C2×D4 | — | 5 | 6 | no | — |
| 172 | C2×D4 | — | 5 | 3 | no | — |
| 173 | C2×D4 | — | 10 | 24 | no | — |
| 174 | C2×D4 | — | 10 | 12 | no | — |
| 175 | C2×D4 | — | 20 | 24 | no | — |
| 176 | C2×D4 | C22×D5 | 1 | 6 | yes | — |
| 177 | C2×D4 | C22×D5 | 1 | 6 | yes | — |
| 178 | C2×D4 | C22×D5 | 1 | 3 | yes | — |
| 179 | C2×D4 | C22×D5 | 1 | 3 | yes | — |
| 180 | C22×C4 | — | 5 | 6 | no | — |
| 181 | C22×C4 | — | 5 | 6 | no | — |
| 182 | C22×C4 | C22×D5 | 1 | 6 | yes | — |
| 183 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 184 | C4⋊C4 | — | 5 | 6 | no | — |
| 185 | C4⋊C4 | — | 5 | 4 | no | — |
| 186 | C22⋊C4 | — | 5 | 6 | no | — |
| 187 | C22⋊C4 | — | 5 | 12 | no | — |
| 188 | C22⋊C4 | — | 5 | 12 | no | — |
| 189 | C42 | — | 5 | 1 | no | — |
| 190 | C42 | — | 5 | 3 | no | — |
| 191 | C42 | — | 5 | 6 | no | — |
| 192 | C2×C5 | — | 2 | 6 | no | — |
| 193 | C2×C5 | — | 4 | 6 | no | — |
| 194 | C2×C5 | C23×D4 | 1 | 1 | yes | characteristic |
| 195 | C2×C5 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 196 | D5 | — | 4 | 6 | no | — |
| 197 | C23 | — | 2 | 12 | no | — |
| 198 | C23 | — | 4 | 12 | no | — |
| 199 | C23 | — | 5 | 6 | no | — |
| 200 | C23 | — | 10 | 12 | no | — |
| 201 | C23 | — | 20 | 12 | no | — |
| 202 | C23 | C23×D5 | 1 | 6 | yes | — |
| 203 | C23 | C23×D5 | 1 | 3 | yes | — |
| 204 | Q8 | — | 2 | 2 | no | — |
| 205 | Q8 | — | 10 | 2 | no | — |
| 206 | Q8 | C2×C22⋊D5 | 1 | 4 | yes | complemented |
| 207 | D4 | — | 2 | 12 | no | — |
| 208 | D4 | — | 2 | 12 | no | — |
| 209 | D4 | — | 2 | 6 | no | — |
| 210 | D4 | — | 10 | 24 | no | — |
| 211 | D4 | — | 10 | 12 | no | — |
| 212 | D4 | — | 10 | 12 | no | — |
| 213 | D4 | — | 10 | 6 | no | — |
| 214 | D4 | — | 20 | 24 | no | — |
| 215 | D4 | C2×C22⋊D5 | 1 | 12 | yes | complemented |
| 216 | C2×C4 | — | 2 | 12 | no | — |
| 217 | C2×C4 | — | 5 | 2 | no | — |
| 218 | C2×C4 | — | 5 | 6 | no | — |
| 219 | C2×C4 | — | 5 | 2 | no | — |
| 220 | C2×C4 | — | 10 | 12 | no | — |
| 221 | C2×C4 | — | 10 | 12 | no | — |
| 222 | C2×C4 | C2×C22⋊D5 | 1 | 12 | yes | complemented |
| 223 | C2×C4 | C23×D5 | 1 | 2 | yes | — |
| 224 | C2×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | C23×D5 | 1 | 3 | yes | — |
| 226 | C5 | #3498 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 227 | C22 | — | 2 | 6 | no | — |
| 228 | C22 | — | 2 | 12 | no | — |
| 229 | C22 | — | 2 | 6 | no | — |
| 230 | C22 | — | 8 | 12 | no | — |
| 231 | C22 | — | 10 | 12 | no | — |
| 232 | C22 | — | 10 | 6 | no | — |
| 233 | C22 | — | 40 | 12 | no | — |
| 234 | C22 | C22×C22⋊D5 | 1 | 6 | yes | — |
| 235 | C22 | C24×D5 | 1 | 1 | yes | characteristic |
| 236 | C4 | — | 2 | 2 | no | — |
| 237 | C4 | — | 5 | 4 | no | — |
| 238 | C4 | — | 10 | 6 | no | — |
| 239 | C4 | — | 10 | 2 | no | — |
| 240 | C4 | C22×C22⋊D5 | 1 | 6 | yes | — |
| 241 | C4 | C22×C22⋊D5 | 1 | 2 | yes | — |
| 242 | C2 | — | 2 | 6 | no | — |
| 243 | C2 | — | 4 | 6 | no | — |
| 244 | C2 | — | 20 | 6 | no | — |
| 245 | C2 | D5×H(5,𝔽2) | 1 | 2 | yes | — |
| 246 | C2 | C23×C22⋊D5 | 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