Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1312 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 1312 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10617747 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10617553 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10604357 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #10604145 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×#4149 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×#4149 | C22 | 1 | 1 | yes | characteristic |
| 7 | #66260 | — | 2 | 2 | no | — |
| 8 | C4×C5×#41 | C4 | 1 | 1 | yes | characteristic |
| 9 | #65911 | — | 2 | 2 | no | — |
| 10 | D4×D20 | — | 2 | 6 | no | — |
| 11 | #65859 | C4 | 1 | 3 | yes | — |
| 12 | #65859 | C22 | 1 | 3 | yes | — |
| 13 | #65811 | C22 | 1 | 3 | yes | — |
| 14 | C2×#4020 | C4 | 1 | 3 | yes | — |
| 15 | D5⋊#100 | — | 2 | 6 | no | — |
| 16 | #65761 | C22 | 1 | 12 | yes | complemented |
| 17 | #64856 | C22 | 1 | 12 | yes | complemented |
| 18 | #64855 | C22 | 1 | 4 | yes | complemented |
| 19 | #33373 | — | 5 | 1 | no | maximal, Sylow |
| 20 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | #4149 | — | 4 | 4 | no | — |
| 22 | #4149 | — | 4 | 4 | no | — |
| 23 | C2×#713 | — | 2 | 1 | no | — |
| 24 | C2×#713 | D4 | 1 | 2 | yes | — |
| 25 | C2×D4×D5 | — | 2 | 3 | no | — |
| 26 | C2×D4×D5 | D4 | 1 | 6 | yes | — |
| 27 | C2×#709 | C2×C4 | 1 | 3 | yes | — |
| 28 | C2×#709 | C23 | 1 | 3 | yes | — |
| 29 | C22×D20 | — | 2 | 6 | no | — |
| 30 | C22×D20 | C2×C4 | 1 | 3 | yes | — |
| 31 | C22×D20 | C2×C4 | 1 | 3 | yes | — |
| 32 | #4133 | — | 4 | 8 | no | — |
| 33 | D5⋊M16 | — | 2 | 12 | no | — |
| 34 | C4×C5×Q8 | — | 2 | 1 | no | — |
| 35 | C4×C5×D4 | — | 2 | 3 | no | — |
| 36 | C5×#117 | C2×C4 | 1 | 3 | yes | — |
| 37 | C2×C42×C5 | C2×C4 | 1 | 3 | yes | — |
| 38 | #4076 | — | 2 | 6 | no | — |
| 39 | #4041 | — | 2 | 6 | no | — |
| 40 | #4041 | — | 2 | 6 | no | — |
| 41 | #4028 | — | 2 | 6 | no | — |
| 42 | #4028 | — | 2 | 6 | no | — |
| 43 | #4021 | C2×C4 | 1 | 3 | yes | — |
| 44 | #4021 | C23 | 1 | 3 | yes | — |
| 45 | #4020 | — | 2 | 6 | no | — |
| 46 | #4020 | C2×C4 | 1 | 3 | yes | — |
| 47 | #4020 | C2×C4 | 1 | 6 | yes | — |
| 48 | C4×D20 | — | 2 | 6 | no | — |
| 49 | #4015 | C23 | 1 | 1 | yes | characteristic |
| 50 | #4013 | — | 2 | 12 | no | — |
| 51 | #4005 | C23 | 1 | 8 | yes | — |
| 52 | #3498 | — | 5 | 1 | no | — |
| 53 | #2888 | — | 5 | 2 | no | — |
| 54 | #2134 | — | 5 | 6 | no | — |
| 55 | #2132 | — | 5 | 6 | no | — |
| 56 | C23×D5 | — | 2 | 3 | no | — |
| 57 | C23×D5 | C22⋊C4 | 1 | 6 | yes | — |
| 58 | C5×#41 | — | 2 | 4 | no | — |
| 59 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C2×C5×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 61 | C22×C4×C5 | — | 2 | 3 | no | — |
| 62 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 63 | C2×C22⋊D5 | — | 2 | 3 | no | — |
| 64 | C2×C22⋊D5 | C2×D4 | 1 | 6 | yes | — |
| 65 | #713 | — | 2 | 8 | no | — |
| 66 | #713 | — | 4 | 4 | no | — |
| 67 | D4×D5 | — | 4 | 12 | no | — |
| 68 | D4×D5 | — | 4 | 12 | no | — |
| 69 | #709 | — | 4 | 6 | no | — |
| 70 | #709 | — | 4 | 6 | no | — |
| 71 | C2×D20 | — | 2 | 6 | no | — |
| 72 | C2×D20 | — | 2 | 3 | no | — |
| 73 | C2×D20 | — | 2 | 12 | no | — |
| 74 | C2×D20 | — | 4 | 6 | no | — |
| 75 | C2×D20 | — | 4 | 6 | no | — |
| 76 | C2×D20 | — | 8 | 12 | no | — |
| 77 | C2×D20 | C22⋊C4 | 1 | 6 | yes | — |
| 78 | C2×D20 | C22×C4 | 1 | 3 | yes | — |
| 79 | C2×D20 | C22×C4 | 1 | 3 | yes | — |
| 80 | C2×C4×D5 | — | 2 | 3 | no | — |
| 81 | C2×C4×D5 | C2×D4 | 1 | 6 | yes | — |
| 82 | C2×Dic10 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C2×Dic10 | C24 | 1 | 1 | yes | characteristic |
| 84 | #704 | — | 2 | 8 | no | — |
| 85 | #700 | — | 4 | 12 | no | — |
| 86 | D5⋊C8 | — | 4 | 12 | no | — |
| 87 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 88 | C5×C22⋊C4 | — | 2 | 3 | no | — |
| 89 | C42×C5 | — | 2 | 3 | no | — |
| 90 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C42×C5 | C22×C4 | 1 | 3 | yes | — |
| 92 | #685 | — | 2 | 6 | no | — |
| 93 | #685 | — | 2 | 6 | no | — |
| 94 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 95 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 96 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 97 | #568 | — | 10 | 2 | no | — |
| 98 | #550 | — | 10 | 2 | no | — |
| 99 | D42 | — | 10 | 6 | no | — |
| 100 | #535 | — | 5 | 3 | no | — |
| 101 | #535 | — | 5 | 3 | no | — |
| 102 | #532 | — | 5 | 3 | no | — |
| 103 | C2×#127 | — | 5 | 3 | no | — |
| 104 | C4×#41 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 105 | #413 | — | 5 | 12 | no | — |
| 106 | C2×#100 | — | 10 | 6 | no | — |
| 107 | #356 | — | 5 | 4 | no | — |
| 108 | #355 | — | 5 | 12 | no | — |
| 109 | C23×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 110 | C22×D5 | — | 2 | 3 | no | — |
| 111 | C22×D5 | — | 4 | 6 | no | — |
| 112 | C22×D5 | — | 4 | 6 | no | — |
| 113 | C22×D5 | — | 8 | 6 | no | — |
| 114 | C22×D5 | — | 8 | 6 | no | — |
| 115 | C22×D5 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 116 | C5×Q8 | #99 | 1 | 4 | yes | — |
| 117 | C5×D4 | — | 2 | 6 | no | — |
| 118 | C2×C4×C5 | — | 2 | 1 | no | — |
| 119 | C2×C4×C5 | — | 2 | 6 | no | — |
| 120 | C2×C4×C5 | — | 2 | 3 | no | — |
| 121 | C2×C4×C5 | — | 4 | 6 | no | — |
| 122 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 123 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 124 | C22⋊D5 | — | 4 | 6 | no | — |
| 125 | C22⋊D5 | — | 4 | 6 | no | — |
| 126 | C2×Dic5 | — | 2 | 1 | no | — |
| 127 | C2×Dic5 | C22×D4 | 1 | 2 | yes | — |
| 128 | D20 | — | 2 | 12 | no | — |
| 129 | D20 | — | 4 | 3 | no | — |
| 130 | D20 | — | 4 | 3 | no | — |
| 131 | D20 | — | 4 | 12 | no | — |
| 132 | D20 | — | 4 | 6 | no | — |
| 133 | D20 | — | 8 | 12 | no | — |
| 134 | C4×D5 | — | 2 | 12 | no | — |
| 135 | C4×D5 | — | 4 | 6 | no | — |
| 136 | Dic10 | — | 4 | 1 | no | — |
| 137 | Dic10 | — | 4 | 1 | no | — |
| 138 | #170 | — | 4 | 8 | no | — |
| 139 | H(5,𝔽2) | — | 20 | 4 | no | — |
| 140 | H(5,𝔽2) | — | 20 | 4 | no | — |
| 141 | C2×#41 | — | 5 | 2 | no | — |
| 142 | C2×#41 | — | 5 | 3 | no | — |
| 143 | C2×#41 | — | 5 | 3 | no | — |
| 144 | C2×#41 | — | 10 | 1 | no | — |
| 145 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 146 | C22×D4 | — | 5 | 6 | no | — |
| 147 | C22×D4 | — | 5 | 3 | no | — |
| 148 | C22×D4 | — | 5 | 3 | no | — |
| 149 | C22×D4 | — | 10 | 6 | no | — |
| 150 | C22×D4 | — | 10 | 3 | no | — |
| 151 | #131 | — | 20 | 8 | no | — |
| 152 | C2×M16 | — | 10 | 12 | no | — |
| 153 | C4⋊Q8 | — | 5 | 1 | no | — |
| 154 | #127 | — | 5 | 3 | no | — |
| 155 | #127 | — | 5 | 6 | no | — |
| 156 | #127 | — | 10 | 6 | no | — |
| 157 | #124 | — | 5 | 3 | no | — |
| 158 | #124 | — | 5 | 3 | no | — |
| 159 | #121 | — | 10 | 6 | no | — |
| 160 | #121 | — | 10 | 6 | no | — |
| 161 | #121 | — | 10 | 6 | no | — |
| 162 | #120 | — | 10 | 6 | no | — |
| 163 | #120 | — | 10 | 6 | no | — |
| 164 | C4×Q8 | — | 2 | 1 | no | — |
| 165 | C4×D4 | — | 2 | 3 | no | — |
| 166 | C4×D4 | — | 10 | 6 | no | — |
| 167 | #117 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 168 | C2×C42 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 169 | #101 | — | 5 | 8 | no | — |
| 170 | #100 | — | 10 | 12 | no | — |
| 171 | C22×C5 | — | 2 | 3 | no | — |
| 172 | C22×C5 | — | 4 | 3 | no | — |
| 173 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 174 | C2×D5 | — | 2 | 6 | no | — |
| 175 | C2×D5 | — | 4 | 6 | no | — |
| 176 | C2×D5 | — | 4 | 6 | no | — |
| 177 | C2×D5 | — | 4 | 3 | no | — |
| 178 | C2×D5 | — | 16 | 6 | no | — |
| 179 | C2×D5 | — | 16 | 6 | no | — |
| 180 | C4×C5 | — | 2 | 2 | no | — |
| 181 | C4×C5 | — | 2 | 1 | no | — |
| 182 | C4×C5 | — | 4 | 3 | no | — |
| 183 | C4×C5 | C2×#99 | 1 | 6 | yes | — |
| 184 | Dic5 | — | 2 | 2 | no | — |
| 185 | Dic5 | — | 4 | 1 | no | — |
| 186 | C24 | — | 5 | 6 | no | — |
| 187 | C24 | — | 10 | 3 | no | — |
| 188 | #41 | — | 2 | 4 | no | — |
| 189 | #41 | — | 10 | 8 | no | — |
| 190 | #41 | — | 20 | 6 | no | — |
| 191 | #41 | — | 20 | 6 | no | — |
| 192 | #41 | — | 20 | 4 | no | — |
| 193 | C2×Q8 | — | 5 | 1 | no | — |
| 194 | C2×Q8 | — | 5 | 1 | no | — |
| 195 | C2×Q8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 196 | C2×D4 | — | 5 | 6 | no | — |
| 197 | C2×D4 | — | 5 | 6 | no | — |
| 198 | C2×D4 | — | 5 | 3 | no | — |
| 199 | C2×D4 | — | 5 | 3 | no | — |
| 200 | C2×D4 | — | 10 | 6 | no | — |
| 201 | C2×D4 | — | 10 | 12 | no | — |
| 202 | C2×D4 | — | 10 | 3 | no | — |
| 203 | C2×D4 | — | 10 | 3 | no | — |
| 204 | C2×D4 | — | 20 | 6 | no | — |
| 205 | C2×D4 | — | 20 | 6 | no | — |
| 206 | C2×D4 | — | 20 | 12 | no | — |
| 207 | C2×D4 | — | 20 | 12 | no | — |
| 208 | C2×D4 | — | 40 | 12 | no | — |
| 209 | C2×D4 | C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 210 | C22×C4 | — | 2 | 3 | no | — |
| 211 | C22×C4 | — | 5 | 6 | no | — |
| 212 | C22×C4 | — | 10 | 3 | no | — |
| 213 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 214 | M16 | — | 10 | 8 | no | — |
| 215 | M16 | — | 20 | 12 | no | — |
| 216 | C2×C8 | — | 20 | 12 | no | — |
| 217 | C4⋊C4 | — | 2 | 3 | no | — |
| 218 | C4⋊C4 | — | 10 | 2 | no | — |
| 219 | C22⋊C4 | — | 2 | 3 | no | — |
| 220 | C22⋊C4 | — | 10 | 6 | no | — |
| 221 | C22⋊C4 | — | 10 | 6 | no | — |
| 222 | C42 | — | 2 | 3 | no | — |
| 223 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 224 | C42 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 225 | C2×C5 | — | 2 | 1 | no | — |
| 226 | C2×C5 | — | 4 | 3 | no | — |
| 227 | C2×C5 | C22×#99 | 1 | 1 | yes | characteristic |
| 228 | D5 | — | 4 | 6 | no | — |
| 229 | D5 | — | 8 | 3 | no | — |
| 230 | C23 | — | 5 | 6 | no | — |
| 231 | C23 | — | 10 | 3 | no | — |
| 232 | C23 | — | 20 | 6 | no | — |
| 233 | C23 | — | 20 | 6 | no | — |
| 234 | C23 | — | 40 | 6 | no | — |
| 235 | C23 | — | 40 | 6 | no | — |
| 236 | C23 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 237 | Q8 | — | 20 | 1 | no | — |
| 238 | Q8 | — | 20 | 1 | no | — |
| 239 | Q8 | #3999 | 1 | 4 | yes | — |
| 240 | D4 | — | 2 | 6 | no | — |
| 241 | D4 | — | 10 | 12 | no | — |
| 242 | D4 | — | 20 | 3 | no | — |
| 243 | D4 | — | 20 | 3 | no | — |
| 244 | D4 | — | 20 | 12 | no | — |
| 245 | D4 | — | 20 | 6 | no | — |
| 246 | D4 | — | 20 | 6 | no | — |
| 247 | D4 | — | 20 | 6 | no | — |
| 248 | D4 | — | 40 | 12 | no | — |
| 249 | C2×C4 | — | 2 | 1 | no | — |
| 250 | C2×C4 | — | 2 | 6 | no | — |
| 251 | C2×C4 | — | 2 | 3 | no | — |
| 252 | C2×C4 | — | 4 | 6 | no | — |
| 253 | C2×C4 | — | 5 | 2 | no | — |
| 254 | C2×C4 | — | 10 | 12 | no | — |
| 255 | C2×C4 | — | 10 | 1 | no | — |
| 256 | C2×C4 | — | 20 | 6 | no | — |
| 257 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 258 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 20 | 8 | no | — |
| 260 | C5 | #33373 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 261 | C22 | — | 2 | 3 | no | — |
| 262 | C22 | — | 4 | 3 | no | — |
| 263 | C22 | — | 10 | 6 | no | — |
| 264 | C22 | — | 20 | 6 | no | — |
| 265 | C22 | — | 20 | 6 | no | — |
| 266 | C22 | — | 20 | 3 | no | — |
| 267 | C22 | — | 80 | 6 | no | — |
| 268 | C22 | — | 80 | 6 | no | — |
| 269 | C22 | C22×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 270 | C4 | — | 2 | 2 | no | — |
| 271 | C4 | — | 2 | 1 | no | — |
| 272 | C4 | — | 4 | 3 | no | — |
| 273 | C4 | — | 10 | 2 | no | — |
| 274 | C4 | — | 20 | 1 | no | — |
| 275 | C4 | C2×#3999 | 1 | 6 | yes | — |
| 276 | C2 | — | 2 | 1 | no | — |
| 277 | C2 | — | 4 | 3 | no | — |
| 278 | C2 | — | 20 | 6 | no | — |
| 279 | C2 | — | 40 | 3 | no | — |
| 280 | C2 | C22×#3999 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10617553 | C2 | semidirect |
| 3. #10604357 | C2 | semidirect |
| 4. #10604145 | C2 | semidirect |
| 16. #65761 | C22 | semidirect |
| 17. #64856 | C22 | semidirect |
| 18. #64855 | C22 | semidirect |
| 260. C5 | #33373 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617747 | 2 |
| #10617553 | 2 |
| #10604357 | 2 |
| #10604145 | 2 |
| #33373 | 5 |