← #13013168
Subgroups — #13013168
Order 1280.
10796 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #10619442 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10619440 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#3573 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10619069 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | #10619068 | C2 | 1 | 15 | yes | maximal, complemented |
| 6 | #10619064 | C2 | 1 | 15 | yes | maximal, complemented |
| 7 | C2×#66085 | C2 | 1 | 15 | yes | maximal, complemented |
| 8 | C5×#585 | C22 | 1 | 1 | yes | characteristic |
| 9 | #66287 | — | 2 | 6 | no | — |
| 10 | D5×2−1+4 | — | 2 | 6 | no | — |
| 11 | #66285 | — | 2 | 10 | no | — |
| 12 | D5×H(5,𝔽2) | — | 2 | 10 | no | — |
| 13 | #66283 | C22 | 1 | 15 | yes | complemented |
| 14 | #66283 | C22 | 1 | 15 | yes | complemented |
| 15 | C2×#4150 | C22 | 1 | 15 | yes | complemented |
| 16 | C2×#4149 | C22 | 1 | 15 | yes | complemented |
| 17 | C5×#568 | C22 | 1 | 15 | yes | complemented |
| 18 | C2×C5×#131 | C22 | 1 | 15 | yes | complemented |
| 19 | #66087 | C22 | 1 | 60 | yes | complemented |
| 20 | #66086 | C22 | 1 | 60 | yes | complemented |
| 21 | #66085 | C22 | 1 | 60 | yes | complemented |
| 22 | #66085 | C22 | 1 | 120 | yes | complemented |
| 23 | #66085 | C22 | 1 | 20 | yes | complemented |
| 24 | #66082 | C22 | 1 | 90 | yes | complemented |
| 25 | C2×#4055 | C22 | 1 | 45 | yes | complemented |
| 26 | C2×#4054 | C22 | 1 | 45 | yes | complemented |
| 27 | C2×#4050 | C22 | 1 | 45 | yes | complemented |
| 28 | C22×#677 | C22 | 1 | 15 | yes | complemented |
| 29 | #63092 | — | 5 | 1 | no | maximal, Sylow |
| 30 | C5×2−1+4 | D4 | 1 | 6 | yes | — |
| 31 | C5×H(5,𝔽2) | D4 | 1 | 10 | yes | — |
| 32 | C2×C5×#41 | C23 | 1 | 15 | yes | — |
| 33 | #4150 | — | 2 | 60 | no | — |
| 34 | #4150 | C23 | 1 | 20 | yes | — |
| 35 | #4150 | C23 | 1 | 60 | yes | complemented |
| 36 | #4149 | — | 2 | 60 | no | — |
| 37 | #4149 | C23 | 1 | 60 | yes | — |
| 38 | #4149 | C23 | 1 | 20 | yes | complemented |
| 39 | D5×#41 | — | 2 | 60 | no | — |
| 40 | D5×#41 | — | 2 | 60 | no | — |
| 41 | #4147 | — | 2 | 15 | no | — |
| 42 | #4147 | — | 2 | 15 | no | — |
| 43 | C2×#713 | — | 2 | 15 | no | — |
| 44 | C2×Q8×D5 | — | 2 | 15 | no | — |
| 45 | #4144 | — | 2 | 45 | no | — |
| 46 | #4144 | — | 2 | 45 | no | — |
| 47 | C2×#711 | — | 2 | 45 | no | — |
| 48 | C2×D4×D5 | — | 2 | 45 | no | — |
| 49 | C2×#709 | C23 | 1 | 45 | yes | — |
| 50 | C2×#709 | C23 | 1 | 45 | yes | complemented |
| 51 | C22×D20 | C23 | 1 | 15 | yes | — |
| 52 | C22×Dic10 | C23 | 1 | 15 | yes | complemented |
| 53 | C5×#131 | C23 | 1 | 20 | yes | complemented |
| 54 | C5×#131 | C23 | 1 | 60 | yes | complemented |
| 55 | C2×C5×M16 | C23 | 1 | 45 | yes | complemented |
| 56 | C22×C5×C8 | C23 | 1 | 15 | yes | complemented |
| 57 | #4055 | C23 | 1 | 180 | yes | complemented |
| 58 | #4055 | C23 | 1 | 60 | yes | complemented |
| 59 | #4054 | C23 | 1 | 180 | yes | complemented |
| 60 | #4054 | C23 | 1 | 60 | yes | complemented |
| 61 | C2×Dic20 | C23 | 1 | 45 | yes | complemented |
| 62 | #4050 | C23 | 1 | 60 | yes | complemented |
| 63 | #4050 | C23 | 1 | 180 | yes | complemented |
| 64 | C2×D40 | C23 | 1 | 45 | yes | complemented |
| 65 | C2×#677 | C23 | 1 | 45 | yes | complemented |
| 66 | C2×#677 | C23 | 1 | 90 | yes | complemented |
| 67 | C2×#677 | C23 | 1 | 15 | yes | complemented |
| 68 | 2−1+6 | — | 5 | 1 | no | — |
| 69 | H(7,𝔽2) | — | 5 | 1 | no | — |
| 70 | #3586 | — | 5 | 15 | no | — |
| 71 | #3585 | — | 5 | 15 | no | — |
| 72 | #3584 | — | 5 | 15 | no | — |
| 73 | C2×#577 | — | 5 | 15 | no | — |
| 74 | #3573 | D5 | 1 | 1 | yes | characteristic, complemented |
| 75 | C23×D5 | — | 2 | 15 | no | — |
| 76 | C5×#41 | C2×D4 | 1 | 60 | yes | — |
| 77 | C5×#41 | C24 | 1 | 20 | yes | — |
| 78 | C2×C5×Q8 | C2×D4 | 1 | 15 | yes | — |
| 79 | C2×C5×D4 | C2×D4 | 1 | 45 | yes | — |
| 80 | C22×C4×C5 | C24 | 1 | 15 | yes | — |
| 81 | C2×C22⋊D5 | — | 2 | 45 | no | — |
| 82 | C2×C22⋊D5 | — | 2 | 45 | no | — |
| 83 | C22×Dic5 | — | 2 | 15 | no | — |
| 84 | #713 | — | 2 | 60 | no | — |
| 85 | #713 | — | 2 | 20 | no | — |
| 86 | Q8×D5 | — | 2 | 60 | no | — |
| 87 | Q8×D5 | — | 2 | 20 | no | — |
| 88 | #711 | — | 2 | 180 | no | — |
| 89 | #711 | — | 2 | 60 | no | — |
| 90 | D4×D5 | — | 2 | 180 | no | — |
| 91 | D4×D5 | — | 2 | 60 | no | — |
| 92 | #709 | — | 2 | 90 | no | — |
| 93 | #709 | — | 2 | 90 | no | — |
| 94 | #709 | C24 | 1 | 60 | yes | — |
| 95 | #709 | C24 | 1 | 60 | yes | — |
| 96 | C2×D20 | — | 2 | 45 | no | — |
| 97 | C2×D20 | C24 | 1 | 45 | yes | — |
| 98 | C2×D20 | C24 | 1 | 15 | yes | — |
| 99 | C2×C4×D5 | — | 2 | 45 | no | — |
| 100 | C2×C4×D5 | — | 2 | 45 | no | — |
| 101 | C2×Dic10 | — | 2 | 45 | no | — |
| 102 | C2×Dic10 | C24 | 1 | 45 | yes | — |
| 103 | C2×Dic10 | C24 | 1 | 15 | yes | — |
| 104 | C5×M16 | C24 | 1 | 60 | yes | — |
| 105 | C2×C5×C8 | C24 | 1 | 15 | yes | — |
| 106 | C2×C5×C8 | C24 | 1 | 45 | yes | — |
| 107 | Dic20 | C24 | 1 | 60 | yes | — |
| 108 | D40 | C24 | 1 | 60 | yes | — |
| 109 | #677 | C24 | 1 | 10 | yes | — |
| 110 | #677 | C24 | 1 | 6 | yes | — |
| 111 | #677 | C24 | 1 | 90 | yes | — |
| 112 | #677 | C24 | 1 | 30 | yes | — |
| 113 | #585 | — | 5 | 15 | no | — |
| 114 | #585 | — | 5 | 15 | no | — |
| 115 | #585 | — | 10 | 6 | no | — |
| 116 | #585 | — | 10 | 10 | no | — |
| 117 | #585 | C2×D5 | 1 | 1 | yes | characteristic |
| 118 | C2×2−1+4 | — | 5 | 15 | no | — |
| 119 | C2×2−1+4 | — | 10 | 6 | no | — |
| 120 | C2×H(5,𝔽2) | — | 5 | 15 | no | — |
| 121 | C2×H(5,𝔽2) | — | 10 | 10 | no | — |
| 122 | #578 | — | 5 | 60 | no | — |
| 123 | #577 | — | 5 | 20 | no | — |
| 124 | #577 | — | 5 | 60 | no | — |
| 125 | #577 | — | 5 | 120 | no | — |
| 126 | #576 | — | 5 | 60 | no | — |
| 127 | #575 | — | 5 | 90 | no | — |
| 128 | C2×#137 | — | 5 | 45 | no | — |
| 129 | C2×AGL(1,ℤ/8) | — | 5 | 45 | no | — |
| 130 | C2×#135 | — | 5 | 45 | no | — |
| 131 | C22×SD16 | — | 5 | 15 | no | — |
| 132 | #568 | C2×D5 | 1 | 15 | yes | complemented |
| 133 | C2×#131 | C2×D5 | 1 | 15 | yes | complemented |
| 134 | C23×C5 | C22×D4 | 1 | 15 | yes | — |
| 135 | C22×D5 | — | 2 | 15 | no | — |
| 136 | C22×D5 | — | 2 | 45 | no | — |
| 137 | C22×D5 | — | 16 | 15 | no | — |
| 138 | C5×Q8 | C22×D4 | 1 | 20 | yes | — |
| 139 | C5×D4 | C22×D4 | 1 | 60 | yes | — |
| 140 | C2×C4×C5 | C22×D4 | 1 | 45 | yes | — |
| 141 | C2×C4×C5 | C25 | 1 | 15 | yes | — |
| 142 | C22⋊D5 | — | 2 | 60 | no | — |
| 143 | C22⋊D5 | — | 2 | 60 | no | — |
| 144 | C2×Dic5 | — | 2 | 45 | no | — |
| 145 | C2×Dic5 | — | 2 | 15 | no | — |
| 146 | D20 | — | 2 | 45 | no | — |
| 147 | D20 | — | 2 | 15 | no | — |
| 148 | D20 | C25 | 1 | 10 | yes | — |
| 149 | D20 | C25 | 1 | 6 | yes | — |
| 150 | C4×D5 | — | 2 | 60 | no | — |
| 151 | C4×D5 | — | 2 | 60 | no | — |
| 152 | Dic10 | — | 2 | 45 | no | — |
| 153 | Dic10 | — | 2 | 15 | no | — |
| 154 | Dic10 | C25 | 1 | 10 | yes | — |
| 155 | Dic10 | C25 | 1 | 6 | yes | — |
| 156 | C5×C8 | C25 | 1 | 1 | yes | characteristic |
| 157 | C5×C8 | C25 | 1 | 15 | yes | — |
| 158 | 2−1+4 | — | 5 | 20 | no | — |
| 159 | 2−1+4 | — | 5 | 60 | no | — |
| 160 | 2−1+4 | — | 10 | 60 | no | — |
| 161 | 2−1+4 | — | 10 | 15 | no | — |
| 162 | 2−1+4 | — | 10 | 15 | no | — |
| 163 | 2−1+4 | D20 | 1 | 6 | yes | — |
| 164 | H(5,𝔽2) | — | 5 | 60 | no | — |
| 165 | H(5,𝔽2) | — | 5 | 20 | no | — |
| 166 | H(5,𝔽2) | — | 10 | 60 | no | — |
| 167 | H(5,𝔽2) | — | 10 | 45 | no | — |
| 168 | H(5,𝔽2) | — | 10 | 45 | no | — |
| 169 | H(5,𝔽2) | D20 | 1 | 10 | yes | — |
| 170 | C2×#41 | — | 5 | 45 | no | — |
| 171 | C2×#41 | — | 5 | 45 | no | — |
| 172 | C2×#41 | — | 10 | 60 | no | — |
| 173 | C2×#41 | — | 10 | 60 | no | — |
| 174 | C2×#41 | — | 10 | 15 | no | — |
| 175 | C2×#41 | — | 10 | 45 | no | — |
| 176 | C2×#41 | C22×D5 | 1 | 15 | yes | — |
| 177 | C22×Q8 | — | 5 | 15 | no | — |
| 178 | C22×Q8 | — | 10 | 15 | no | — |
| 179 | C22×D4 | — | 5 | 15 | no | — |
| 180 | C22×D4 | — | 10 | 45 | no | — |
| 181 | #137 | — | 5 | 180 | no | — |
| 182 | #137 | — | 5 | 60 | no | — |
| 183 | AGL(1,ℤ/8) | — | 5 | 180 | no | — |
| 184 | AGL(1,ℤ/8) | — | 5 | 60 | no | — |
| 185 | #135 | — | 5 | 60 | no | — |
| 186 | #135 | — | 5 | 180 | no | — |
| 187 | C2×Q16 | — | 5 | 45 | no | — |
| 188 | C2×SD16 | — | 5 | 45 | no | — |
| 189 | C2×SD16 | — | 5 | 90 | no | — |
| 190 | C2×SD16 | — | 5 | 15 | no | — |
| 191 | C2×D8 | — | 5 | 45 | no | — |
| 192 | #131 | C22×D5 | 1 | 20 | yes | complemented |
| 193 | #131 | C22×D5 | 1 | 60 | yes | complemented |
| 194 | C2×M16 | C22×D5 | 1 | 45 | yes | complemented |
| 195 | C22×C8 | C22×D5 | 1 | 15 | yes | complemented |
| 196 | C22×C5 | — | 4 | 15 | no | — |
| 197 | C22×C5 | C23×D4 | 1 | 15 | yes | — |
| 198 | C2×D5 | — | 2 | 6 | no | — |
| 199 | C2×D5 | — | 2 | 10 | no | — |
| 200 | C2×D5 | — | 8 | 15 | no | — |
| 201 | C2×D5 | — | 8 | 45 | no | — |
| 202 | C4×C5 | C23×D4 | 1 | 15 | yes | — |
| 203 | C4×C5 | C26 | 1 | 1 | yes | characteristic |
| 204 | Dic5 | — | 2 | 10 | no | — |
| 205 | Dic5 | — | 2 | 6 | no | — |
| 206 | C24 | — | 10 | 15 | no | — |
| 207 | #41 | — | 5 | 60 | no | — |
| 208 | #41 | — | 5 | 60 | no | — |
| 209 | #41 | — | 10 | 60 | no | — |
| 210 | #41 | — | 10 | 20 | no | — |
| 211 | #41 | — | 10 | 60 | no | — |
| 212 | #41 | — | 10 | 180 | no | — |
| 213 | #41 | — | 10 | 90 | no | — |
| 214 | #41 | — | 10 | 90 | no | — |
| 215 | #41 | C2×D20 | 1 | 60 | yes | — |
| 216 | #41 | C23×D5 | 1 | 20 | yes | — |
| 217 | C2×Q8 | — | 5 | 45 | no | — |
| 218 | C2×Q8 | — | 5 | 15 | no | — |
| 219 | C2×Q8 | — | 10 | 60 | no | — |
| 220 | C2×Q8 | — | 10 | 20 | no | — |
| 221 | C2×Q8 | — | 10 | 45 | no | — |
| 222 | C2×Q8 | C2×D20 | 1 | 15 | yes | — |
| 223 | C2×D4 | — | 5 | 15 | no | — |
| 224 | C2×D4 | — | 5 | 45 | no | — |
| 225 | C2×D4 | — | 10 | 45 | no | — |
| 226 | C2×D4 | — | 10 | 45 | no | — |
| 227 | C2×D4 | — | 10 | 180 | no | — |
| 228 | C2×D4 | — | 10 | 60 | no | — |
| 229 | C2×D4 | — | 10 | 45 | no | — |
| 230 | C2×D4 | C2×D20 | 1 | 45 | yes | — |
| 231 | C22×C4 | — | 10 | 15 | no | — |
| 232 | C22×C4 | — | 10 | 45 | no | — |
| 233 | C22×C4 | — | 10 | 45 | no | — |
| 234 | C22×C4 | C23×D5 | 1 | 15 | yes | — |
| 235 | Q16 | — | 5 | 60 | no | — |
| 236 | SD16 | — | 5 | 10 | no | — |
| 237 | SD16 | — | 5 | 6 | no | — |
| 238 | SD16 | — | 5 | 90 | no | — |
| 239 | SD16 | — | 5 | 30 | no | — |
| 240 | D8 | — | 5 | 60 | no | — |
| 241 | M16 | C23×D5 | 1 | 60 | yes | — |
| 242 | C2×C8 | C23×D5 | 1 | 15 | yes | — |
| 243 | C2×C8 | C23×D5 | 1 | 45 | yes | — |
| 244 | C2×C5 | — | 2 | 15 | no | — |
| 245 | C2×C5 | C24×D4 | 1 | 1 | yes | characteristic |
| 246 | D5 | — | 4 | 6 | no | — |
| 247 | D5 | — | 4 | 10 | no | — |
| 248 | C23 | — | 10 | 15 | no | — |
| 249 | C23 | — | 10 | 45 | no | — |
| 250 | C23 | — | 80 | 15 | no | — |
| 251 | C23 | C22×D20 | 1 | 15 | yes | — |
| 252 | Q8 | — | 5 | 10 | no | — |
| 253 | Q8 | — | 5 | 6 | no | — |
| 254 | Q8 | — | 10 | 45 | no | — |
| 255 | Q8 | — | 10 | 15 | no | — |
| 256 | Q8 | C22×D20 | 1 | 20 | yes | — |
| 257 | D4 | — | 5 | 6 | no | — |
| 258 | D4 | — | 5 | 10 | no | — |
| 259 | D4 | — | 10 | 60 | no | — |
| 260 | D4 | — | 10 | 60 | no | — |
| 261 | D4 | — | 10 | 45 | no | — |
| 262 | D4 | — | 10 | 15 | no | — |
| 263 | D4 | C22×D20 | 1 | 60 | yes | — |
| 264 | C2×C4 | — | 10 | 15 | no | — |
| 265 | C2×C4 | — | 10 | 45 | no | — |
| 266 | C2×C4 | — | 10 | 60 | no | — |
| 267 | C2×C4 | — | 10 | 60 | no | — |
| 268 | C2×C4 | C22×D20 | 1 | 45 | yes | — |
| 269 | C2×C4 | C24×D5 | 1 | 15 | yes | — |
| 270 | C8 | C24×D5 | 1 | 1 | yes | characteristic |
| 271 | C8 | C24×D5 | 1 | 15 | yes | — |
| 272 | C5 | #63092 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 273 | C22 | — | 4 | 15 | no | — |
| 274 | C22 | — | 10 | 6 | no | — |
| 275 | C22 | — | 10 | 10 | no | — |
| 276 | C22 | — | 40 | 45 | no | — |
| 277 | C22 | — | 40 | 15 | no | — |
| 278 | C22 | C23×D20 | 1 | 15 | yes | — |
| 279 | C4 | — | 10 | 6 | no | — |
| 280 | C4 | — | 10 | 10 | no | — |
| 281 | C4 | C23×D20 | 1 | 15 | yes | — |
| 282 | C4 | C25×D5 | 1 | 1 | yes | characteristic |
| 283 | C2 | — | 2 | 15 | no | — |
| 284 | C2 | — | 20 | 10 | no | — |
| 285 | C2 | — | 20 | 6 | no | — |
| 286 | C2 | C24×D20 | 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