← #13107815
Subgroups — #13107815
Order 1600.
1784 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#87900 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C5×#709 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11717495 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #11717473 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11717472 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #11717471 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | D5×#709 | C2 | 1 | 8 | yes | maximal, complemented |
| 8 | #11717463 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #11717462 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | C22×C4×C52 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C22⋊(C5⋊D5) | C22 | 1 | 2 | yes | complemented |
| 12 | #87900 | C22 | 1 | 4 | yes | complemented |
| 13 | #87900 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×C4⋊(C5⋊D5) | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×C4×C5⋊D5 | C22 | 1 | 2 | yes | complemented |
| 16 | C2×#6045 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×C5×C22⋊D5 | C22 | 1 | 4 | yes | complemented |
| 18 | C5×#709 | C22 | 1 | 4 | yes | complemented |
| 19 | C5×#709 | C22 | 1 | 4 | yes | complemented |
| 20 | C5×#709 | C22 | 1 | 4 | yes | complemented |
| 21 | C5×#709 | C22 | 1 | 4 | yes | complemented |
| 22 | C2×C5×D20 | C22 | 1 | 2 | yes | complemented |
| 23 | C2×C4×C5×D5 | C22 | 1 | 4 | yes | complemented |
| 24 | C2×C5×Dic10 | C22 | 1 | 2 | yes | complemented |
| 25 | #87886 | C22 | 1 | 8 | yes | complemented |
| 26 | D5×C22⋊D5 | C22 | 1 | 16 | yes | complemented |
| 27 | #87880 | C22 | 1 | 8 | yes | complemented |
| 28 | #87879 | C22 | 1 | 16 | yes | complemented |
| 29 | #87877 | C22 | 1 | 4 | yes | complemented |
| 30 | D5×D20 | C22 | 1 | 8 | yes | complemented |
| 31 | C4×D52 | C22 | 1 | 8 | yes | complemented |
| 32 | #87874 | C22 | 1 | 8 | yes | complemented |
| 33 | #87873 | C22 | 1 | 8 | yes | complemented |
| 34 | #87872 | C22 | 1 | 4 | yes | complemented |
| 35 | #87871 | C22 | 1 | 8 | yes | complemented |
| 36 | #87870 | C22 | 1 | 8 | yes | complemented |
| 37 | D5×Dic10 | C22 | 1 | 8 | yes | complemented |
| 38 | #66283 | — | 5 | 2 | no | maximal |
| 39 | C23×C52 | C23 | 1 | 1 | yes | characteristic |
| 40 | C22×C5⋊D5 | C23 | 1 | 2 | yes | — |
| 41 | C22×C5×D5 | C23 | 1 | 4 | yes | — |
| 42 | C2×D52 | C23 | 1 | 8 | yes | — |
| 43 | C2×C4×C52 | C23 | 1 | 2 | yes | — |
| 44 | C2×C4×C52 | C23 | 1 | 2 | yes | — |
| 45 | C2×C4×C52 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C4×C52 | C23 | 1 | 1 | yes | characteristic |
| 47 | C22⋊(C5⋊D5) | C23 | 1 | 8 | yes | — |
| 48 | C2×#1039 | C23 | 1 | 2 | yes | — |
| 49 | C4⋊(C5⋊D5) | C23 | 1 | 4 | yes | — |
| 50 | C4×C5⋊D5 | C23 | 1 | 4 | yes | — |
| 51 | C4×C5⋊D5 | C23 | 1 | 4 | yes | — |
| 52 | #6045 | C23 | 1 | 4 | yes | — |
| 53 | C5×C22⋊D5 | C23 | 1 | 8 | yes | — |
| 54 | C5×C22⋊D5 | C23 | 1 | 8 | yes | — |
| 55 | C2×C5×Dic5 | C23 | 1 | 4 | yes | — |
| 56 | C5×D20 | C23 | 1 | 4 | yes | — |
| 57 | C5×D20 | C23 | 1 | 4 | yes | — |
| 58 | C4×C5×D5 | C23 | 1 | 8 | yes | — |
| 59 | C4×C5×D5 | C23 | 1 | 8 | yes | — |
| 60 | C5×Dic10 | C23 | 1 | 4 | yes | — |
| 61 | C5×Dic10 | C23 | 1 | 4 | yes | — |
| 62 | #6039 | C23 | 1 | 8 | yes | — |
| 63 | #6038 | C23 | 1 | 16 | yes | — |
| 64 | #6037 | C23 | 1 | 8 | yes | — |
| 65 | #6036 | C23 | 1 | 8 | yes | — |
| 66 | D5×Dic5 | C23 | 1 | 16 | yes | — |
| 67 | C2×C5×#41 | — | 5 | 2 | no | — |
| 68 | #4150 | — | 5 | 4 | no | — |
| 69 | #4150 | — | 5 | 4 | no | — |
| 70 | #4149 | — | 5 | 4 | no | — |
| 71 | #4149 | — | 5 | 4 | no | — |
| 72 | D5×#41 | — | 5 | 8 | no | — |
| 73 | D5×#41 | — | 5 | 8 | no | — |
| 74 | #4147 | — | 5 | 4 | no | — |
| 75 | #4144 | — | 5 | 8 | no | — |
| 76 | #4144 | — | 5 | 4 | no | — |
| 77 | C2×#709 | — | 5 | 2 | no | — |
| 78 | C2×#709 | — | 5 | 8 | no | — |
| 79 | C2×#709 | — | 10 | 2 | no | — |
| 80 | C2×#709 | D5 | 1 | 2 | yes | complemented |
| 81 | C22×C52 | — | 4 | 1 | no | — |
| 82 | C22×C52 | C24 | 1 | 2 | yes | — |
| 83 | C22×C52 | C24 | 1 | 1 | yes | characteristic |
| 84 | C2×C5⋊D5 | — | 4 | 2 | no | — |
| 85 | C2×C5⋊D5 | C24 | 1 | 4 | yes | — |
| 86 | C2×C5×D5 | — | 4 | 4 | no | — |
| 87 | C2×C5×D5 | C24 | 1 | 8 | yes | — |
| 88 | D52 | — | 4 | 8 | no | — |
| 89 | C4×C52 | C24 | 1 | 1 | yes | characteristic |
| 90 | C4×C52 | C24 | 1 | 2 | yes | — |
| 91 | C4×C52 | C24 | 1 | 1 | yes | characteristic |
| 92 | #1039 | C24 | 1 | 4 | yes | — |
| 93 | C5×Dic5 | C24 | 1 | 8 | yes | — |
| 94 | C5×#41 | — | 5 | 4 | no | — |
| 95 | C5×#41 | — | 5 | 4 | no | — |
| 96 | C5×#41 | — | 5 | 4 | no | — |
| 97 | C5×#41 | — | 5 | 4 | no | — |
| 98 | C2×C5×Q8 | — | 5 | 2 | no | — |
| 99 | C2×C5×D4 | — | 5 | 4 | no | — |
| 100 | C2×C5×D4 | — | 5 | 2 | no | — |
| 101 | C22×C4×C5 | — | 2 | 2 | no | — |
| 102 | C22×C4×C5 | — | 5 | 4 | no | — |
| 103 | C22×C4×C5 | C2×D5 | 1 | 2 | yes | complemented |
| 104 | C2×C22⋊D5 | — | 5 | 4 | no | — |
| 105 | C2×C22⋊D5 | — | 5 | 16 | no | — |
| 106 | C2×C22⋊D5 | — | 10 | 4 | no | — |
| 107 | C2×C22⋊D5 | C2×D5 | 1 | 4 | yes | complemented |
| 108 | #713 | — | 5 | 8 | no | — |
| 109 | #713 | — | 5 | 8 | no | — |
| 110 | Q8×D5 | — | 5 | 8 | no | — |
| 111 | Q8×D5 | — | 5 | 8 | no | — |
| 112 | #711 | — | 5 | 16 | no | — |
| 113 | #711 | — | 5 | 16 | no | — |
| 114 | #711 | — | 5 | 8 | no | — |
| 115 | #711 | — | 5 | 8 | no | — |
| 116 | D4×D5 | — | 5 | 16 | no | — |
| 117 | D4×D5 | — | 5 | 16 | no | — |
| 118 | D4×D5 | — | 5 | 8 | no | — |
| 119 | D4×D5 | — | 5 | 8 | no | — |
| 120 | #709 | — | 5 | 4 | no | — |
| 121 | #709 | — | 5 | 4 | no | — |
| 122 | #709 | — | 5 | 4 | no | — |
| 123 | #709 | — | 5 | 4 | no | — |
| 124 | #709 | — | 5 | 16 | no | — |
| 125 | #709 | — | 5 | 8 | no | — |
| 126 | #709 | — | 5 | 16 | no | — |
| 127 | #709 | — | 5 | 8 | no | — |
| 128 | #709 | — | 10 | 8 | no | — |
| 129 | #709 | — | 10 | 8 | no | — |
| 130 | #709 | C2×D5 | 1 | 4 | yes | complemented |
| 131 | #709 | C2×D5 | 1 | 4 | yes | complemented |
| 132 | #709 | C2×D5 | 1 | 4 | yes | complemented |
| 133 | #709 | C2×D5 | 1 | 4 | yes | complemented |
| 134 | C2×D20 | — | 5 | 2 | no | — |
| 135 | C2×D20 | — | 5 | 8 | no | — |
| 136 | C2×D20 | — | 10 | 2 | no | — |
| 137 | C2×D20 | C2×D5 | 1 | 2 | yes | complemented |
| 138 | C2×C4×D5 | — | 5 | 4 | no | — |
| 139 | C2×C4×D5 | — | 5 | 16 | no | — |
| 140 | C2×C4×D5 | — | 10 | 4 | no | — |
| 141 | C2×C4×D5 | C2×D5 | 1 | 4 | yes | complemented |
| 142 | C2×Dic10 | — | 5 | 2 | no | — |
| 143 | C2×Dic10 | — | 5 | 8 | no | — |
| 144 | C2×Dic10 | — | 10 | 2 | no | — |
| 145 | C2×Dic10 | C2×D5 | 1 | 2 | yes | complemented |
| 146 | #585 | — | 25 | 1 | no | Sylow |
| 147 | C2×C52 | — | 2 | 1 | no | — |
| 148 | C2×C52 | — | 2 | 2 | no | — |
| 149 | C2×C52 | C25 | 1 | 1 | yes | characteristic |
| 150 | C5⋊D5 | — | 2 | 4 | no | — |
| 151 | C5×D5 | — | 2 | 8 | no | — |
| 152 | C23×C5 | — | 2 | 2 | no | — |
| 153 | C23×C5 | — | 5 | 4 | no | — |
| 154 | C23×C5 | C22×D5 | 1 | 2 | yes | — |
| 155 | C22×D5 | — | 5 | 4 | no | — |
| 156 | C22×D5 | — | 5 | 16 | no | — |
| 157 | C22×D5 | — | 10 | 4 | no | — |
| 158 | C22×D5 | C22×D5 | 1 | 4 | yes | — |
| 159 | C5×Q8 | — | 5 | 4 | no | — |
| 160 | C5×Q8 | — | 5 | 4 | no | — |
| 161 | C5×D4 | — | 5 | 8 | no | — |
| 162 | C5×D4 | — | 5 | 8 | no | — |
| 163 | C5×D4 | — | 5 | 4 | no | — |
| 164 | C5×D4 | — | 5 | 4 | no | — |
| 165 | C2×C4×C5 | — | 2 | 2 | no | — |
| 166 | C2×C4×C5 | — | 2 | 2 | no | — |
| 167 | C2×C4×C5 | — | 2 | 4 | no | — |
| 168 | C2×C4×C5 | — | 2 | 4 | no | — |
| 169 | C2×C4×C5 | — | 5 | 4 | no | — |
| 170 | C2×C4×C5 | — | 5 | 8 | no | — |
| 171 | C2×C4×C5 | — | 5 | 8 | no | — |
| 172 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 173 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 174 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 175 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 176 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 177 | C2×C4×C5 | C22×D5 | 1 | 2 | yes | — |
| 178 | C22⋊D5 | — | 5 | 8 | no | — |
| 179 | C22⋊D5 | — | 5 | 8 | no | — |
| 180 | C22⋊D5 | — | 5 | 16 | no | — |
| 181 | C22⋊D5 | — | 5 | 16 | no | — |
| 182 | C22⋊D5 | — | 10 | 16 | no | — |
| 183 | C22⋊D5 | C22×D5 | 1 | 8 | yes | — |
| 184 | C22⋊D5 | C22×D5 | 1 | 8 | yes | — |
| 185 | C2×Dic5 | — | 5 | 4 | no | — |
| 186 | C2×Dic5 | — | 5 | 16 | no | — |
| 187 | C2×Dic5 | — | 10 | 4 | no | — |
| 188 | C2×Dic5 | C22×D5 | 1 | 4 | yes | — |
| 189 | D20 | — | 5 | 16 | no | — |
| 190 | D20 | — | 5 | 4 | no | — |
| 191 | D20 | — | 5 | 4 | no | — |
| 192 | D20 | — | 10 | 8 | no | — |
| 193 | D20 | C22×D5 | 1 | 4 | yes | — |
| 194 | D20 | C22×D5 | 1 | 4 | yes | — |
| 195 | C4×D5 | — | 5 | 16 | no | — |
| 196 | C4×D5 | — | 5 | 8 | no | — |
| 197 | C4×D5 | — | 5 | 8 | no | — |
| 198 | C4×D5 | — | 5 | 16 | no | — |
| 199 | C4×D5 | — | 10 | 8 | no | — |
| 200 | C4×D5 | — | 10 | 8 | no | — |
| 201 | C4×D5 | C22×D5 | 1 | 8 | yes | — |
| 202 | C4×D5 | C22×D5 | 1 | 8 | yes | — |
| 203 | Dic10 | — | 5 | 4 | no | — |
| 204 | Dic10 | — | 5 | 4 | no | — |
| 205 | Dic10 | — | 5 | 16 | no | — |
| 206 | Dic10 | — | 10 | 8 | no | — |
| 207 | Dic10 | C22×D5 | 1 | 4 | yes | — |
| 208 | Dic10 | C22×D5 | 1 | 4 | yes | — |
| 209 | 2−1+4 | — | 25 | 4 | no | — |
| 210 | 2−1+4 | — | 25 | 2 | no | — |
| 211 | H(5,𝔽2) | — | 25 | 4 | no | — |
| 212 | H(5,𝔽2) | — | 25 | 4 | no | — |
| 213 | H(5,𝔽2) | — | 25 | 2 | no | — |
| 214 | C2×#41 | — | 5 | 2 | no | — |
| 215 | C2×#41 | — | 25 | 8 | no | — |
| 216 | C2×#41 | — | 25 | 4 | no | — |
| 217 | C2×#41 | — | 25 | 1 | no | — |
| 218 | C52 | #585 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 219 | C22×C5 | — | 2 | 2 | no | — |
| 220 | C22×C5 | — | 2 | 4 | no | — |
| 221 | C22×C5 | — | 4 | 2 | no | — |
| 222 | C22×C5 | — | 4 | 4 | no | — |
| 223 | C22×C5 | — | 5 | 8 | no | — |
| 224 | C22×C5 | — | 20 | 4 | no | — |
| 225 | C22×C5 | C23×D5 | 1 | 2 | yes | — |
| 226 | C22×C5 | C23×D5 | 1 | 2 | yes | — |
| 227 | C22×C5 | C23×D5 | 1 | 2 | yes | — |
| 228 | C2×D5 | — | 4 | 4 | no | — |
| 229 | C2×D5 | — | 5 | 8 | no | — |
| 230 | C2×D5 | — | 10 | 8 | no | — |
| 231 | C2×D5 | — | 20 | 16 | no | — |
| 232 | C2×D5 | — | 20 | 4 | no | — |
| 233 | C2×D5 | — | 20 | 8 | no | — |
| 234 | C2×D5 | C23×D5 | 1 | 8 | yes | — |
| 235 | C4×C5 | — | 2 | 4 | no | — |
| 236 | C4×C5 | — | 2 | 2 | no | — |
| 237 | C4×C5 | — | 2 | 2 | no | — |
| 238 | C4×C5 | — | 5 | 8 | no | — |
| 239 | C4×C5 | C23×D5 | 1 | 2 | yes | — |
| 240 | C4×C5 | C23×D5 | 1 | 2 | yes | — |
| 241 | C4×C5 | C23×D5 | 1 | 2 | yes | — |
| 242 | C4×C5 | C23×D5 | 1 | 2 | yes | — |
| 243 | Dic5 | — | 5 | 8 | no | — |
| 244 | Dic5 | — | 10 | 8 | no | — |
| 245 | Dic5 | C23×D5 | 1 | 8 | yes | — |
| 246 | #41 | — | 5 | 4 | no | — |
| 247 | #41 | — | 5 | 4 | no | — |
| 248 | #41 | — | 5 | 4 | no | — |
| 249 | #41 | — | 5 | 4 | no | — |
| 250 | #41 | — | 25 | 8 | no | — |
| 251 | #41 | — | 25 | 16 | no | — |
| 252 | #41 | — | 25 | 8 | no | — |
| 253 | #41 | — | 25 | 8 | no | — |
| 254 | #41 | — | 25 | 8 | no | — |
| 255 | #41 | — | 25 | 8 | no | — |
| 256 | #41 | — | 25 | 4 | no | — |
| 257 | #41 | — | 25 | 4 | no | — |
| 258 | C2×Q8 | — | 5 | 2 | no | — |
| 259 | C2×Q8 | — | 25 | 8 | no | — |
| 260 | C2×Q8 | — | 25 | 4 | no | — |
| 261 | C2×Q8 | — | 25 | 1 | no | — |
| 262 | C2×D4 | — | 5 | 4 | no | — |
| 263 | C2×D4 | — | 5 | 2 | no | — |
| 264 | C2×D4 | — | 25 | 8 | no | — |
| 265 | C2×D4 | — | 25 | 8 | no | — |
| 266 | C2×D4 | — | 25 | 4 | no | — |
| 267 | C2×D4 | — | 25 | 16 | no | — |
| 268 | C2×D4 | — | 25 | 2 | no | — |
| 269 | C2×D4 | — | 25 | 1 | no | — |
| 270 | C22×C4 | — | 5 | 4 | no | — |
| 271 | C22×C4 | — | 25 | 8 | no | — |
| 272 | C22×C4 | — | 25 | 2 | no | — |
| 273 | C22×C4 | D52 | 1 | 1 | yes | characteristic, complemented |
| 274 | C2×C5 | — | 2 | 4 | no | — |
| 275 | C2×C5 | — | 2 | 2 | no | — |
| 276 | C2×C5 | — | 2 | 2 | no | — |
| 277 | C2×C5 | — | 2 | 2 | no | — |
| 278 | C2×C5 | — | 2 | 2 | no | — |
| 279 | C2×C5 | — | 4 | 4 | no | — |
| 280 | C2×C5 | — | 10 | 8 | no | — |
| 281 | C2×C5 | C24×D5 | 1 | 2 | yes | — |
| 282 | D5 | — | 2 | 8 | no | — |
| 283 | D5 | — | 10 | 8 | no | — |
| 284 | D5 | — | 20 | 8 | no | — |
| 285 | C23 | — | 5 | 4 | no | — |
| 286 | C23 | — | 25 | 8 | no | — |
| 287 | C23 | — | 25 | 2 | no | — |
| 288 | C23 | C2×D52 | 1 | 1 | yes | characteristic |
| 289 | Q8 | — | 5 | 4 | no | — |
| 290 | Q8 | — | 5 | 4 | no | — |
| 291 | Q8 | — | 25 | 8 | no | — |
| 292 | Q8 | — | 25 | 4 | no | — |
| 293 | D4 | — | 5 | 8 | no | — |
| 294 | D4 | — | 5 | 8 | no | — |
| 295 | D4 | — | 5 | 4 | no | — |
| 296 | D4 | — | 5 | 4 | no | — |
| 297 | D4 | — | 25 | 16 | no | — |
| 298 | D4 | — | 25 | 8 | no | — |
| 299 | D4 | — | 25 | 8 | no | — |
| 300 | D4 | — | 25 | 4 | no | — |
| 301 | C2×C4 | — | 5 | 4 | no | — |
| 302 | C2×C4 | — | 5 | 8 | no | — |
| 303 | C2×C4 | — | 5 | 8 | no | — |
| 304 | C2×C4 | — | 25 | 8 | no | — |
| 305 | C2×C4 | — | 25 | 16 | no | — |
| 306 | C2×C4 | — | 25 | 2 | no | — |
| 307 | C2×C4 | — | 25 | 4 | no | — |
| 308 | C2×C4 | — | 25 | 4 | no | — |
| 309 | C2×C4 | C2×D52 | 1 | 2 | yes | — |
| 310 | C2×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | C2×D52 | 1 | 2 | yes | — |
| 312 | C2×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 313 | C5 | — | 2 | 2 | no | — |
| 314 | C5 | #66283 | 1 | 2 | yes | complemented |
| 315 | C22 | — | 4 | 1 | no | — |
| 316 | C22 | — | 5 | 8 | no | — |
| 317 | C22 | — | 20 | 4 | no | — |
| 318 | C22 | — | 25 | 4 | no | — |
| 319 | C22 | — | 100 | 2 | no | — |
| 320 | C22 | — | 100 | 8 | no | — |
| 321 | C22 | C22×D52 | 1 | 2 | yes | — |
| 322 | C22 | C22×D52 | 1 | 1 | yes | characteristic |
| 323 | C4 | — | 5 | 8 | no | — |
| 324 | C4 | — | 25 | 4 | no | — |
| 325 | C4 | C22×D52 | 1 | 2 | yes | — |
| 326 | C4 | C22×D52 | 1 | 1 | yes | characteristic |
| 327 | C4 | C22×D52 | 1 | 1 | yes | characteristic |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 10 | 8 | no | — |
| 331 | C2 | — | 50 | 4 | no | — |
| 332 | C2 | C23×D52 | 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