Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
1114 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×#562 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #66034 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #66022 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #66018 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #66002 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #65997 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #65989 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #65970 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #65963 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #65933 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #65931 | C2 | 1 | 4 | yes | maximal, complemented |
| 12 | C2×#713 | C22 | 1 | 2 | yes | — |
| 13 | C2×#711 | C22 | 1 | 2 | yes | — |
| 14 | C2×#711 | C22 | 1 | 4 | yes | — |
| 15 | C2×#709 | C22 | 1 | 4 | yes | — |
| 16 | C5×#125 | C22 | 1 | 1 | yes | characteristic |
| 17 | C5×#124 | C22 | 1 | 2 | yes | — |
| 18 | C5×#123 | C22 | 1 | 4 | yes | — |
| 19 | C5×C22⋊Q8 | C22 | 1 | 4 | yes | — |
| 20 | C5×#121 | C22 | 1 | 4 | yes | — |
| 21 | #4092 | C22 | 1 | 4 | yes | — |
| 22 | #4090 | C22 | 1 | 2 | yes | — |
| 23 | #4086 | C22 | 1 | 4 | yes | — |
| 24 | #4086 | C22 | 1 | 2 | yes | — |
| 25 | #4085 | C22 | 1 | 4 | yes | — |
| 26 | #4085 | C22 | 1 | 4 | yes | — |
| 27 | #4085 | C22 | 1 | 4 | yes | — |
| 28 | #4076 | C22 | 1 | 4 | yes | — |
| 29 | C4×C22⋊D5 | C22 | 1 | 4 | yes | — |
| 30 | C2×#683 | C22 | 1 | 4 | yes | — |
| 31 | #4042 | C22 | 1 | 4 | yes | — |
| 32 | #4042 | C22 | 1 | 4 | yes | — |
| 33 | #4042 | C22 | 1 | 4 | yes | — |
| 34 | #4041 | C22 | 1 | 4 | yes | — |
| 35 | #4041 | C22 | 1 | 4 | yes | — |
| 36 | #4040 | C22 | 1 | 4 | yes | — |
| 37 | #4038 | C22 | 1 | 2 | yes | — |
| 38 | D5×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 39 | #4035 | C22 | 1 | 4 | yes | — |
| 40 | #4034 | C22 | 1 | 2 | yes | — |
| 41 | #4033 | C22 | 1 | 4 | yes | — |
| 42 | #4032 | C22 | 1 | 4 | yes | — |
| 43 | #4030 | C22 | 1 | 4 | yes | — |
| 44 | #4030 | C22 | 1 | 4 | yes | — |
| 45 | #4030 | C22 | 1 | 4 | yes | — |
| 46 | #4030 | C22 | 1 | 4 | yes | — |
| 47 | #4030 | C22 | 1 | 4 | yes | — |
| 48 | #4029 | C22 | 1 | 4 | yes | — |
| 49 | #4027 | C22 | 1 | 4 | yes | — |
| 50 | #4027 | C22 | 1 | 4 | yes | — |
| 51 | #4027 | C22 | 1 | 4 | yes | — |
| 52 | #4024 | C22 | 1 | 4 | yes | — |
| 53 | #4024 | C22 | 1 | 4 | yes | — |
| 54 | #4021 | C22 | 1 | 2 | yes | — |
| 55 | C42×D5 | C22 | 1 | 1 | yes | characteristic |
| 56 | #4016 | C22 | 1 | 1 | yes | characteristic |
| 57 | #3548 | — | 5 | 1 | no | maximal, Sylow |
| 58 | C2×C5×Q8 | C23 | 1 | 2 | yes | — |
| 59 | C2×C5×D4 | C23 | 1 | 4 | yes | — |
| 60 | C2×C5×D4 | C23 | 1 | 2 | yes | — |
| 61 | C22×C4×C5 | C23 | 1 | 4 | yes | — |
| 62 | C2×C22⋊D5 | C23 | 1 | 4 | yes | — |
| 63 | C2×C22⋊D5 | C23 | 1 | 4 | yes | — |
| 64 | C2×C22⋊D5 | C23 | 1 | 4 | yes | — |
| 65 | C2×C22⋊D5 | C23 | 1 | 4 | yes | — |
| 66 | C2×C22⋊D5 | C23 | 1 | 4 | yes | — |
| 67 | C22×Dic5 | C23 | 1 | 4 | yes | — |
| 68 | C22×Dic5 | C23 | 1 | 4 | yes | — |
| 69 | #713 | — | 4 | 4 | no | — |
| 70 | #711 | — | 4 | 8 | no | — |
| 71 | #711 | — | 4 | 4 | no | — |
| 72 | #709 | — | 4 | 8 | no | — |
| 73 | C2×D20 | C23 | 1 | 4 | yes | — |
| 74 | C2×D20 | C23 | 1 | 2 | yes | — |
| 75 | C2×D20 | C23 | 1 | 4 | yes | — |
| 76 | C2×C4×D5 | C23 | 1 | 4 | yes | — |
| 77 | C2×C4×D5 | C23 | 1 | 2 | yes | — |
| 78 | C2×C4×D5 | C23 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×D5 | C23 | 1 | 4 | yes | — |
| 80 | C2×C4×D5 | C23 | 1 | 4 | yes | — |
| 81 | C2×Dic10 | C23 | 1 | 4 | yes | — |
| 82 | C2×Dic10 | C23 | 1 | 2 | yes | — |
| 83 | C5×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | C5×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 85 | C5×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 86 | C5×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 87 | C5×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 88 | C5×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 89 | C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 90 | C22⋊Dic5 | C23 | 1 | 4 | yes | — |
| 91 | #685 | C23 | 1 | 4 | yes | — |
| 92 | #685 | C23 | 1 | 4 | yes | — |
| 93 | #685 | C23 | 1 | 4 | yes | — |
| 94 | #685 | C23 | 1 | 4 | yes | — |
| 95 | #685 | C23 | 1 | 4 | yes | — |
| 96 | C4⋊Dic5 | C23 | 1 | 2 | yes | — |
| 97 | C4⋊Dic5 | C23 | 1 | 4 | yes | — |
| 98 | #683 | C23 | 1 | 4 | yes | — |
| 99 | #683 | C23 | 1 | 4 | yes | — |
| 100 | #683 | C23 | 1 | 4 | yes | — |
| 101 | #683 | C23 | 1 | 4 | yes | — |
| 102 | #683 | C23 | 1 | 4 | yes | — |
| 103 | #683 | C23 | 1 | 4 | yes | — |
| 104 | C4×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 105 | C4×Dic5 | C23 | 1 | 2 | yes | — |
| 106 | C4×Dic5 | C23 | 1 | 4 | yes | — |
| 107 | #562 | — | 5 | 1 | no | — |
| 108 | #562 | — | 5 | 2 | no | — |
| 109 | #562 | D5 | 1 | 1 | yes | characteristic, complemented |
| 110 | #554 | — | 5 | 4 | no | — |
| 111 | #550 | — | 5 | 4 | no | — |
| 112 | #547 | — | 5 | 4 | no | — |
| 113 | #540 | — | 5 | 4 | no | — |
| 114 | #537 | — | 5 | 4 | no | — |
| 115 | #537 | — | 5 | 4 | no | — |
| 116 | #533 | — | 5 | 1 | no | — |
| 117 | #532 | — | 5 | 2 | no | — |
| 118 | C23×C5 | C24 | 1 | 4 | yes | — |
| 119 | C22×D5 | C24 | 1 | 4 | yes | — |
| 120 | C22×D5 | C24 | 1 | 1 | yes | characteristic |
| 121 | C5×Q8 | — | 4 | 2 | no | — |
| 122 | C5×D4 | — | 4 | 4 | no | — |
| 123 | C5×D4 | — | 4 | 2 | no | — |
| 124 | C2×C4×C5 | — | 4 | 4 | no | — |
| 125 | C2×C4×C5 | C24 | 1 | 2 | yes | — |
| 126 | C2×C4×C5 | C24 | 1 | 4 | yes | — |
| 127 | C2×C4×C5 | C24 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C5 | C24 | 1 | 4 | yes | — |
| 129 | C22⋊D5 | — | 2 | 8 | no | — |
| 130 | C22⋊D5 | — | 2 | 8 | no | — |
| 131 | C22⋊D5 | — | 4 | 4 | no | — |
| 132 | C22⋊D5 | — | 4 | 4 | no | — |
| 133 | C22⋊D5 | — | 4 | 4 | no | — |
| 134 | C2×Dic5 | — | 2 | 8 | no | — |
| 135 | C2×Dic5 | — | 4 | 4 | no | — |
| 136 | C2×Dic5 | C24 | 1 | 4 | yes | — |
| 137 | C2×Dic5 | C24 | 1 | 4 | yes | — |
| 138 | C2×Dic5 | C24 | 1 | 2 | yes | — |
| 139 | C2×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 140 | D20 | — | 4 | 4 | no | — |
| 141 | D20 | — | 4 | 2 | no | — |
| 142 | D20 | — | 4 | 4 | no | — |
| 143 | C4×D5 | — | 2 | 8 | no | — |
| 144 | C4×D5 | — | 2 | 4 | no | — |
| 145 | C4×D5 | — | 2 | 2 | no | — |
| 146 | C4×D5 | — | 4 | 4 | no | — |
| 147 | C4×D5 | — | 4 | 4 | no | — |
| 148 | Dic10 | — | 2 | 8 | no | — |
| 149 | Dic10 | — | 4 | 2 | no | — |
| 150 | C2×#41 | — | 5 | 2 | no | — |
| 151 | C2×#41 | — | 5 | 2 | no | — |
| 152 | C2×#41 | — | 5 | 4 | no | — |
| 153 | C2×#41 | — | 5 | 4 | no | — |
| 154 | C4⋊Q8 | — | 5 | 2 | no | — |
| 155 | C4⋊Q8 | — | 5 | 2 | no | — |
| 156 | #127 | — | 5 | 2 | no | — |
| 157 | #127 | — | 5 | 4 | no | — |
| 158 | #125 | — | 5 | 4 | no | — |
| 159 | #125 | — | 5 | 1 | no | — |
| 160 | #125 | C2×D5 | 1 | 1 | yes | characteristic |
| 161 | #124 | — | 5 | 2 | no | — |
| 162 | #124 | C2×D5 | 1 | 2 | yes | — |
| 163 | #123 | — | 5 | 4 | no | — |
| 164 | #123 | — | 5 | 4 | no | — |
| 165 | #123 | — | 5 | 4 | no | — |
| 166 | #123 | C2×D5 | 1 | 4 | yes | — |
| 167 | C22⋊Q8 | — | 5 | 4 | no | — |
| 168 | C22⋊Q8 | — | 5 | 4 | no | — |
| 169 | C22⋊Q8 | — | 5 | 4 | no | — |
| 170 | C22⋊Q8 | — | 5 | 4 | no | — |
| 171 | C22⋊Q8 | — | 5 | 4 | no | — |
| 172 | C22⋊Q8 | — | 5 | 4 | no | — |
| 173 | C22⋊Q8 | C2×D5 | 1 | 4 | yes | — |
| 174 | #121 | — | 5 | 4 | no | — |
| 175 | #121 | — | 5 | 4 | no | — |
| 176 | #121 | — | 5 | 4 | no | — |
| 177 | #121 | — | 5 | 4 | no | — |
| 178 | #121 | — | 5 | 4 | no | — |
| 179 | #121 | — | 5 | 4 | no | — |
| 180 | #121 | — | 5 | 4 | no | — |
| 181 | #121 | — | 5 | 4 | no | — |
| 182 | #121 | — | 5 | 4 | no | — |
| 183 | #121 | — | 5 | 4 | no | — |
| 184 | #121 | — | 5 | 4 | no | — |
| 185 | #121 | C2×D5 | 1 | 4 | yes | — |
| 186 | C4×Q8 | — | 5 | 4 | no | — |
| 187 | C4×D4 | — | 5 | 4 | no | — |
| 188 | C4×D4 | — | 5 | 4 | no | — |
| 189 | C4×D4 | — | 5 | 4 | no | — |
| 190 | C4×D4 | — | 5 | 4 | no | — |
| 191 | #117 | — | 5 | 2 | no | — |
| 192 | C2×C4⋊C4 | — | 5 | 4 | no | — |
| 193 | C2×C4⋊C4 | — | 5 | 4 | no | — |
| 194 | C2×C42 | — | 5 | 1 | no | — |
| 195 | C22×C5 | — | 2 | 4 | no | — |
| 196 | C22×C5 | — | 2 | 4 | no | — |
| 197 | C22×C5 | — | 2 | 4 | no | — |
| 198 | C22×C5 | C25 | 1 | 1 | yes | characteristic |
| 199 | C2×D5 | — | 2 | 4 | no | — |
| 200 | C2×D5 | — | 2 | 2 | no | — |
| 201 | C2×D5 | — | 2 | 4 | no | — |
| 202 | C2×D5 | — | 2 | 1 | no | — |
| 203 | C2×D5 | — | 2 | 4 | no | — |
| 204 | C4×C5 | — | 2 | 4 | no | — |
| 205 | C4×C5 | — | 2 | 2 | no | — |
| 206 | C4×C5 | — | 2 | 4 | no | — |
| 207 | C4×C5 | — | 2 | 1 | no | — |
| 208 | Dic5 | — | 2 | 4 | no | — |
| 209 | Dic5 | — | 2 | 4 | no | — |
| 210 | Dic5 | H(5,𝔽2) | 1 | 4 | yes | — |
| 211 | Dic5 | 2−1+4 | 1 | 2 | yes | — |
| 212 | #41 | — | 20 | 8 | no | — |
| 213 | #41 | — | 20 | 4 | no | — |
| 214 | #41 | — | 20 | 4 | no | — |
| 215 | #41 | — | 20 | 8 | no | — |
| 216 | C2×Q8 | — | 5 | 4 | no | — |
| 217 | C2×Q8 | — | 5 | 2 | no | — |
| 218 | C2×Q8 | C22×D5 | 1 | 2 | yes | — |
| 219 | C2×D4 | — | 5 | 4 | no | — |
| 220 | C2×D4 | — | 5 | 4 | no | — |
| 221 | C2×D4 | — | 5 | 4 | no | — |
| 222 | C2×D4 | — | 5 | 4 | no | — |
| 223 | C2×D4 | — | 5 | 4 | no | — |
| 224 | C2×D4 | — | 5 | 4 | no | — |
| 225 | C2×D4 | — | 5 | 4 | no | — |
| 226 | C2×D4 | — | 5 | 2 | no | — |
| 227 | C2×D4 | C22×D5 | 1 | 4 | yes | — |
| 228 | C2×D4 | C22×D5 | 1 | 2 | yes | — |
| 229 | C22×C4 | — | 5 | 4 | no | — |
| 230 | C22×C4 | — | 5 | 4 | no | — |
| 231 | C22×C4 | — | 5 | 1 | no | — |
| 232 | C22×C4 | — | 5 | 2 | no | — |
| 233 | C22×C4 | — | 5 | 4 | no | — |
| 234 | C22×C4 | — | 5 | 4 | no | — |
| 235 | C22×C4 | — | 5 | 4 | no | — |
| 236 | C22×C4 | C22×D5 | 1 | 4 | yes | — |
| 237 | C4⋊C4 | — | 5 | 4 | no | — |
| 238 | C4⋊C4 | — | 5 | 4 | no | — |
| 239 | C4⋊C4 | — | 5 | 4 | no | — |
| 240 | C4⋊C4 | — | 5 | 4 | no | — |
| 241 | C4⋊C4 | — | 5 | 4 | no | — |
| 242 | C4⋊C4 | — | 5 | 2 | no | — |
| 243 | C4⋊C4 | — | 5 | 4 | no | — |
| 244 | C4⋊C4 | — | 5 | 4 | no | — |
| 245 | C4⋊C4 | C22×D5 | 1 | 4 | yes | — |
| 246 | C4⋊C4 | C22×D5 | 1 | 2 | yes | — |
| 247 | C4⋊C4 | C22×D5 | 1 | 4 | yes | — |
| 248 | C22⋊C4 | — | 5 | 4 | no | — |
| 249 | C22⋊C4 | — | 5 | 4 | no | — |
| 250 | C22⋊C4 | — | 5 | 4 | no | — |
| 251 | C22⋊C4 | — | 5 | 4 | no | — |
| 252 | C22⋊C4 | — | 5 | 4 | no | — |
| 253 | C22⋊C4 | — | 5 | 4 | no | — |
| 254 | C22⋊C4 | C22×D5 | 1 | 4 | yes | — |
| 255 | C22⋊C4 | C22×D5 | 1 | 4 | yes | — |
| 256 | C22⋊C4 | C22×D5 | 1 | 4 | yes | — |
| 257 | C42 | — | 5 | 1 | no | — |
| 258 | C42 | — | 5 | 2 | no | — |
| 259 | C42 | — | 5 | 4 | no | — |
| 260 | C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 261 | C2×C5 | — | 4 | 4 | no | — |
| 262 | C2×C5 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 263 | C2×C5 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 264 | D5 | — | 4 | 4 | no | — |
| 265 | D5 | — | 4 | 1 | no | — |
| 266 | C23 | — | 5 | 1 | no | — |
| 267 | C23 | — | 5 | 4 | no | — |
| 268 | C23 | C23×D5 | 1 | 4 | yes | — |
| 269 | Q8 | — | 4 | 2 | no | — |
| 270 | Q8 | — | 10 | 8 | no | — |
| 271 | Q8 | — | 20 | 2 | no | — |
| 272 | D4 | — | 4 | 4 | no | — |
| 273 | D4 | — | 4 | 2 | no | — |
| 274 | D4 | — | 10 | 8 | no | — |
| 275 | D4 | — | 10 | 8 | no | — |
| 276 | D4 | — | 20 | 4 | no | — |
| 277 | D4 | — | 20 | 4 | no | — |
| 278 | D4 | — | 20 | 2 | no | — |
| 279 | D4 | — | 20 | 4 | no | — |
| 280 | D4 | — | 20 | 4 | no | — |
| 281 | D4 | — | 20 | 4 | no | — |
| 282 | C2×C4 | — | 4 | 4 | no | — |
| 283 | C2×C4 | — | 5 | 1 | no | — |
| 284 | C2×C4 | — | 5 | 2 | no | — |
| 285 | C2×C4 | — | 5 | 4 | no | — |
| 286 | C2×C4 | — | 5 | 4 | no | — |
| 287 | C2×C4 | — | 10 | 8 | no | — |
| 288 | C2×C4 | — | 10 | 4 | no | — |
| 289 | C2×C4 | — | 10 | 2 | no | — |
| 290 | C2×C4 | — | 10 | 8 | no | — |
| 291 | C2×C4 | — | 20 | 4 | no | — |
| 292 | C2×C4 | — | 20 | 4 | no | — |
| 293 | C2×C4 | — | 20 | 4 | no | — |
| 294 | C2×C4 | C23×D5 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | C23×D5 | 1 | 2 | yes | — |
| 296 | C2×C4 | C23×D5 | 1 | 4 | yes | — |
| 297 | C2×C4 | C23×D5 | 1 | 4 | yes | — |
| 298 | C5 | #3548 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 299 | C22 | — | 2 | 4 | no | — |
| 300 | C22 | — | 2 | 4 | no | — |
| 301 | C22 | — | 2 | 4 | no | — |
| 302 | C22 | — | 10 | 4 | no | — |
| 303 | C22 | — | 10 | 2 | no | — |
| 304 | C22 | — | 10 | 4 | no | — |
| 305 | C22 | — | 10 | 1 | no | — |
| 306 | C22 | — | 10 | 4 | no | — |
| 307 | C22 | C24×D5 | 1 | 1 | yes | characteristic |
| 308 | C4 | — | 2 | 4 | no | — |
| 309 | C4 | — | 2 | 4 | no | — |
| 310 | C4 | — | 2 | 2 | no | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 5 | 4 | no | — |
| 313 | C4 | — | 5 | 2 | no | — |
| 314 | C4 | — | 10 | 4 | no | — |
| 315 | C4 | — | 10 | 4 | no | — |
| 316 | C2 | — | 4 | 4 | no | — |
| 317 | C2 | — | 20 | 1 | no | — |
| 318 | C2 | — | 20 | 4 | no | — |
| 319 | C2 | D5×H(5,𝔽2) | 1 | 2 | yes | — |
| 320 | C2 | D5×2−1+4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×#562 | C2 |
| hypercenter | #562 | D5 |
| hypocenter | C5 | #3548 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #562 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#562 | C2 | semidirect |
| 2. #66034 | C2 | semidirect |
| 3. #66022 | C2 | semidirect |
| 4. #66018 | C2 | semidirect |
| 5. #66002 | C2 | semidirect |
| 6. #65997 | C2 | semidirect |
| 7. #65989 | C2 | semidirect |
| 8. #65970 | C2 | semidirect |
| 9. #65963 | C2 | semidirect |
| 10. #65933 | C2 | semidirect |
| 11. #65931 | C2 | semidirect |
| 109. #562 | D5 | semidirect |
| 298. C5 | #3548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#562 | 2 |
| #66034 | 2 |
| #66022 | 2 |
| #66018 | 2 |
| #66002 | 2 |
| #65997 | 2 |
| #65989 | 2 |
| #65970 | 2 |
| #65963 | 2 |
| #65933 | 2 |
| #65931 | 2 |
| #3548 | 5 |