Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
284 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 55 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 | #10613761 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×#2264 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10602298 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10602286 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10598319 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10598097 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10598093 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | D5×#135 | — | 2 | 1 | no | — |
| 9 | C5×#508 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C5×Q32 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #65483 | — | 2 | 1 | no | — |
| 12 | #65477 | — | 2 | 1 | no | — |
| 13 | #65207 | — | 2 | 1 | no | — |
| 14 | #65204 | — | 2 | 1 | no | — |
| 15 | #65203 | — | 2 | 1 | no | — |
| 16 | #65200 | — | 2 | 1 | no | — |
| 17 | #65193 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#3930 | C22 | 1 | 1 | yes | characteristic |
| 19 | C5×Q64 | — | 2 | 1 | no | — |
| 20 | C5×SD64 | — | 2 | 1 | no | — |
| 21 | C5×M64 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | #64784 | — | 2 | 1 | no | — |
| 23 | #64742 | — | 2 | 1 | no | — |
| 24 | #64741 | — | 2 | 1 | no | — |
| 25 | #64725 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #64719 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | #13668 | — | 5 | 1 | no | maximal, Sylow |
| 28 | D5×#41 | — | 4 | 1 | no | — |
| 29 | C5×#135 | — | 2 | 1 | no | — |
| 30 | C2×C5×Q16 | — | 2 | 1 | no | — |
| 31 | #4096 | — | 4 | 1 | no | — |
| 32 | #4092 | — | 4 | 1 | no | — |
| 33 | #4065 | — | 2 | 1 | no | — |
| 34 | D5×Q16 | — | 2 | 1 | no | — |
| 35 | #4062 | — | 4 | 1 | no | — |
| 36 | D5×SD16 | — | 4 | 1 | no | — |
| 37 | #4058 | — | 2 | 1 | no | — |
| 38 | D5×D8 | — | 2 | 1 | no | — |
| 39 | #4050 | — | 2 | 1 | no | — |
| 40 | C2×C8×D5 | D4 | 1 | 1 | yes | characteristic |
| 41 | C5×Q32 | — | 2 | 1 | no | — |
| 42 | C5×Q32 | — | 2 | 1 | no | — |
| 43 | C5×Q32 | D4 | 1 | 1 | yes | characteristic |
| 44 | C5×SD32 | — | 2 | 1 | no | — |
| 45 | C5×D16 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×C5×C16 | C23 | 1 | 1 | yes | characteristic |
| 47 | #3967 | — | 4 | 1 | no | — |
| 48 | #3961 | — | 2 | 1 | no | — |
| 49 | #3960 | — | 2 | 1 | no | — |
| 50 | #3959 | — | 2 | 1 | no | — |
| 51 | #3958 | — | 2 | 1 | no | — |
| 52 | #3950 | — | 2 | 1 | no | — |
| 53 | C2×#672 | D4 | 1 | 1 | yes | characteristic |
| 54 | Dic40 | D4 | 1 | 1 | yes | characteristic |
| 55 | #3932 | — | 2 | 1 | no | — |
| 56 | D80 | D4 | 1 | 1 | yes | characteristic |
| 57 | #3930 | — | 2 | 1 | no | — |
| 58 | #3930 | — | 2 | 1 | no | — |
| 59 | C5×C32 | — | 2 | 1 | no | — |
| 60 | #3926 | — | 2 | 1 | no | — |
| 61 | #3414 | — | 5 | 1 | no | — |
| 62 | #2264 | — | 5 | 1 | no | — |
| 63 | #2264 | D5 | 1 | 1 | yes | characteristic, complemented |
| 64 | #2219 | — | 5 | 1 | no | — |
| 65 | #1418 | — | 5 | 1 | no | — |
| 66 | #1418 | — | 5 | 1 | no | — |
| 67 | #1400 | — | 5 | 1 | no | — |
| 68 | C5×#41 | — | 4 | 1 | no | — |
| 69 | C2×C5×Q8 | — | 4 | 1 | no | — |
| 70 | #713 | — | 4 | 1 | no | — |
| 71 | Q8×D5 | — | 4 | 1 | no | — |
| 72 | #711 | — | 4 | 1 | no | — |
| 73 | #711 | — | 8 | 1 | no | — |
| 74 | D4×D5 | — | 4 | 1 | no | — |
| 75 | D4×D5 | — | 8 | 1 | no | — |
| 76 | #709 | — | 4 | 1 | no | — |
| 77 | #709 | — | 8 | 1 | no | — |
| 78 | C2×C4×D5 | — | 8 | 1 | no | — |
| 79 | C2×C4×D5 | D8 | 1 | 1 | yes | characteristic |
| 80 | C5×Q16 | — | 2 | 1 | no | — |
| 81 | C5×Q16 | — | 4 | 1 | no | — |
| 82 | C5×Q16 | — | 4 | 1 | no | — |
| 83 | C5×SD16 | — | 4 | 1 | no | — |
| 84 | C5×D8 | — | 2 | 1 | no | — |
| 85 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | #689 | — | 4 | 1 | no | — |
| 87 | #688 | — | 4 | 1 | no | — |
| 88 | #687 | — | 4 | 1 | no | — |
| 89 | #686 | — | 4 | 1 | no | — |
| 90 | #685 | — | 8 | 1 | no | — |
| 91 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 92 | #683 | — | 8 | 1 | no | — |
| 93 | C2×#168 | D8 | 1 | 1 | yes | characteristic |
| 94 | Dic20 | — | 2 | 1 | no | — |
| 95 | D40 | — | 2 | 1 | no | — |
| 96 | #677 | — | 4 | 1 | no | — |
| 97 | C8×D5 | — | 2 | 1 | no | — |
| 98 | C8×D5 | — | 2 | 1 | no | — |
| 99 | C5×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C5×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | #672 | — | 2 | 1 | no | — |
| 102 | C2×#135 | — | 10 | 1 | no | — |
| 103 | #510 | — | 10 | 1 | no | — |
| 104 | #510 | — | 10 | 1 | no | — |
| 105 | #509 | — | 10 | 1 | no | — |
| 106 | #509 | — | 10 | 1 | no | — |
| 107 | #508 | — | 5 | 1 | no | — |
| 108 | #508 | — | 10 | 1 | no | — |
| 109 | #508 | C2×D5 | 1 | 1 | yes | characteristic |
| 110 | C2×Q32 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 111 | C2×M32 | — | 5 | 1 | no | — |
| 112 | #467 | — | 10 | 1 | no | — |
| 113 | Q64 | — | 2 | 1 | no | — |
| 114 | Q64 | — | 10 | 1 | no | — |
| 115 | SD64 | — | 2 | 1 | no | — |
| 116 | SD64 | — | 10 | 1 | no | — |
| 117 | M64 | — | 5 | 1 | no | — |
| 118 | M64 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 119 | #367 | — | 5 | 1 | no | — |
| 120 | #358 | — | 10 | 1 | no | — |
| 121 | C22×D5 | — | 8 | 1 | no | — |
| 122 | C22×D5 | D16 | 1 | 1 | yes | characteristic |
| 123 | C5×Q8 | — | 4 | 1 | no | — |
| 124 | C5×Q8 | — | 8 | 1 | no | — |
| 125 | C5×Q8 | — | 8 | 1 | no | — |
| 126 | C5×D4 | — | 4 | 1 | no | — |
| 127 | C5×D4 | — | 8 | 1 | no | — |
| 128 | C2×C4×C5 | — | 8 | 1 | no | — |
| 129 | C2×C4×C5 | — | 8 | 1 | no | — |
| 130 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 131 | C22⋊D5 | — | 8 | 1 | no | — |
| 132 | C22⋊D5 | — | 8 | 1 | no | — |
| 133 | C22⋊D5 | — | 8 | 1 | no | — |
| 134 | C2×Dic5 | — | 8 | 1 | no | — |
| 135 | C2×Dic5 | — | 8 | 1 | no | — |
| 136 | C2×Dic5 | D16 | 1 | 1 | yes | characteristic |
| 137 | D20 | — | 4 | 1 | no | — |
| 138 | D20 | — | 8 | 1 | no | — |
| 139 | C4×D5 | — | 2 | 1 | no | — |
| 140 | C4×D5 | — | 2 | 1 | no | — |
| 141 | C4×D5 | — | 8 | 1 | no | — |
| 142 | C4×D5 | — | 8 | 1 | no | — |
| 143 | C4×D5 | — | 8 | 1 | no | — |
| 144 | Dic10 | — | 4 | 1 | no | — |
| 145 | Dic10 | — | 8 | 1 | no | — |
| 146 | C5×C8 | #120 | 1 | 1 | yes | characteristic |
| 147 | C5×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 148 | #168 | — | 2 | 1 | no | — |
| 149 | C2×#41 | — | 20 | 1 | no | — |
| 150 | #135 | — | 2 | 1 | no | — |
| 151 | #135 | — | 10 | 1 | no | — |
| 152 | #135 | — | 10 | 1 | no | — |
| 153 | #135 | — | 10 | 1 | no | — |
| 154 | #135 | — | 20 | 1 | no | — |
| 155 | #135 | — | 20 | 1 | no | — |
| 156 | C2×Q16 | — | 2 | 1 | no | — |
| 157 | C2×Q16 | — | 10 | 1 | no | — |
| 158 | C2×SD16 | — | 20 | 1 | no | — |
| 159 | C2×D8 | — | 10 | 1 | no | — |
| 160 | C22×C8 | — | 5 | 1 | no | — |
| 161 | C22⋊Q8 | — | 20 | 1 | no | — |
| 162 | Q32 | — | 2 | 1 | no | — |
| 163 | Q32 | — | 2 | 1 | no | — |
| 164 | Q32 | — | 5 | 1 | no | — |
| 165 | Q32 | — | 10 | 1 | no | — |
| 166 | Q32 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 167 | SD32 | — | 2 | 1 | no | — |
| 168 | SD32 | — | 10 | 1 | no | — |
| 169 | SD32 | — | 10 | 1 | no | — |
| 170 | SD32 | — | 10 | 1 | no | — |
| 171 | D16 | — | 5 | 1 | no | — |
| 172 | D16 | — | 10 | 1 | no | — |
| 173 | D16 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 174 | M32 | — | 10 | 1 | no | — |
| 175 | M32 | — | 10 | 1 | no | — |
| 176 | C2×C16 | — | 5 | 1 | no | — |
| 177 | C2×C16 | C22×D5 | 1 | 1 | yes | characteristic |
| 178 | #107 | — | 10 | 1 | no | — |
| 179 | #103 | — | 20 | 1 | no | — |
| 180 | C32 | — | 2 | 1 | no | — |
| 181 | C32 | — | 10 | 1 | no | — |
| 182 | C22×C5 | — | 8 | 1 | no | — |
| 183 | C22×C5 | C2×D16 | 1 | 1 | yes | characteristic |
| 184 | C2×D5 | — | 2 | 1 | no | — |
| 185 | C2×D5 | — | 4 | 1 | no | — |
| 186 | C2×D5 | — | 8 | 1 | no | — |
| 187 | C2×D5 | — | 16 | 2 | no | — |
| 188 | C4×C5 | — | 8 | 1 | no | — |
| 189 | C4×C5 | — | 16 | 1 | no | — |
| 190 | C4×C5 | #447 | 1 | 1 | yes | characteristic |
| 191 | C4×C5 | #509 | 1 | 1 | yes | characteristic |
| 192 | Dic5 | — | 2 | 1 | no | — |
| 193 | Dic5 | — | 8 | 1 | no | — |
| 194 | Dic5 | — | 16 | 1 | no | — |
| 195 | #41 | — | 4 | 1 | no | — |
| 196 | #41 | — | 20 | 1 | no | — |
| 197 | #41 | — | 20 | 1 | no | — |
| 198 | #41 | — | 20 | 1 | no | — |
| 199 | #41 | — | 40 | 1 | no | — |
| 200 | #41 | — | 40 | 1 | no | — |
| 201 | C2×Q8 | — | 4 | 1 | no | — |
| 202 | C2×Q8 | — | 20 | 1 | no | — |
| 203 | C2×D4 | — | 20 | 1 | no | — |
| 204 | C2×D4 | — | 40 | 1 | no | — |
| 205 | C22×C4 | — | 5 | 1 | no | — |
| 206 | C22×C4 | — | 40 | 1 | no | — |
| 207 | Q16 | — | 2 | 1 | no | — |
| 208 | Q16 | — | 4 | 1 | no | — |
| 209 | Q16 | — | 4 | 1 | no | — |
| 210 | Q16 | — | 10 | 1 | no | — |
| 211 | Q16 | — | 20 | 1 | no | — |
| 212 | SD16 | — | 4 | 1 | no | — |
| 213 | SD16 | — | 20 | 1 | no | — |
| 214 | SD16 | — | 20 | 1 | no | — |
| 215 | SD16 | — | 20 | 1 | no | — |
| 216 | D8 | — | 2 | 1 | no | — |
| 217 | D8 | — | 10 | 1 | no | — |
| 218 | D8 | — | 20 | 1 | no | — |
| 219 | C2×C8 | — | 5 | 1 | no | — |
| 220 | C2×C8 | — | 10 | 1 | no | — |
| 221 | C2×C8 | — | 10 | 1 | no | — |
| 222 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 223 | C4⋊C4 | — | 20 | 1 | no | — |
| 224 | C4⋊C4 | — | 40 | 1 | no | — |
| 225 | C22⋊C4 | — | 40 | 1 | no | — |
| 226 | C16 | — | 10 | 1 | no | — |
| 227 | C16 | D4×D5 | 1 | 1 | yes | characteristic |
| 228 | C16 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 229 | C2×C5 | — | 2 | 1 | no | — |
| 230 | C2×C5 | — | 16 | 1 | no | — |
| 231 | C2×C5 | #2184 | 1 | 1 | yes | characteristic |
| 232 | D5 | — | 4 | 1 | no | — |
| 233 | D5 | — | 16 | 1 | no | — |
| 234 | C23 | — | 5 | 1 | no | — |
| 235 | C23 | — | 40 | 1 | no | — |
| 236 | Q8 | — | 4 | 1 | no | — |
| 237 | Q8 | — | 8 | 1 | no | — |
| 238 | Q8 | — | 8 | 1 | no | — |
| 239 | Q8 | — | 20 | 1 | no | — |
| 240 | Q8 | — | 40 | 1 | no | — |
| 241 | D4 | — | 4 | 1 | no | — |
| 242 | D4 | — | 8 | 1 | no | — |
| 243 | D4 | — | 20 | 1 | no | — |
| 244 | D4 | — | 40 | 1 | no | — |
| 245 | D4 | — | 40 | 1 | no | — |
| 246 | D4 | — | 40 | 1 | no | — |
| 247 | D4 | — | 40 | 1 | no | — |
| 248 | C2×C4 | — | 5 | 1 | no | — |
| 249 | C2×C4 | — | 8 | 1 | no | — |
| 250 | C2×C4 | — | 8 | 1 | no | — |
| 251 | C2×C4 | — | 10 | 1 | no | — |
| 252 | C2×C4 | — | 10 | 1 | no | — |
| 253 | C2×C4 | — | 40 | 1 | no | — |
| 254 | C2×C4 | — | 40 | 1 | no | — |
| 255 | C2×C4 | — | 40 | 1 | no | — |
| 256 | C2×C4 | — | 40 | 1 | no | — |
| 257 | C2×C4 | — | 40 | 1 | no | — |
| 258 | C2×C4 | D5×D8 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 10 | 1 | no | — |
| 260 | C8 | #4057 | 1 | 1 | yes | characteristic |
| 261 | C8 | #4083 | 1 | 1 | yes | characteristic |
| 262 | C5 | #13668 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 263 | C22 | — | 8 | 1 | no | — |
| 264 | C22 | — | 10 | 1 | no | — |
| 265 | C22 | — | 20 | 1 | no | — |
| 266 | C22 | — | 40 | 1 | no | — |
| 267 | C22 | — | 80 | 2 | no | — |
| 268 | C22 | D5×D16 | 1 | 1 | yes | characteristic |
| 269 | C4 | — | 8 | 1 | no | — |
| 270 | C4 | — | 10 | 1 | no | — |
| 271 | C4 | — | 16 | 1 | no | — |
| 272 | C4 | — | 40 | 1 | no | — |
| 273 | C4 | — | 80 | 1 | no | — |
| 274 | C4 | #65200 | 1 | 1 | yes | characteristic |
| 275 | C4 | #65445 | 1 | 1 | yes | characteristic |
| 276 | C2 | — | 2 | 1 | no | — |
| 277 | C2 | — | 16 | 1 | no | — |
| 278 | C2 | — | 20 | 1 | no | — |
| 279 | C2 | — | 80 | 1 | no | — |
| 280 | C2 | #10602260 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C5×#2264 | C2 | semidirect |
| 3. #10602298 | C2 | semidirect |
| 4. #10602286 | C2 | semidirect |
| 5. #10598319 | C2 | semidirect |
| 6. #10598097 | C2 | semidirect |
| 7. #10598093 | C2 | semidirect |
| 10. C2×C5×Q32 | C22 | semidirect |
| 21. C5×M64 | C22 | semidirect |
| 25. #64725 | C22 | semidirect |
| 26. #64719 | C22 | semidirect |
| 63. #2264 | D5 | semidirect |
| 110. C2×Q32 | C2×D5 | semidirect |
| 118. M64 | C2×D5 | semidirect |
| 262. C5 | #13668 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10613761 | 2 |
| C5×#2264 | 2 |
| #10602298 | 2 |
| #10602286 | 2 |
| #10598319 | 2 |
| #10598097 | 2 |
| #10598093 | 2 |
| #13668 | 5 |