Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1220 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 1220 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 | #10619184 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10617539 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10604996 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #10604918 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C2×C5×H(5,𝔽2) | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#4150 | C22 | 1 | 1 | yes | characteristic |
| 7 | #66262 | — | 2 | 2 | no | — |
| 8 | #66166 | C22 | 1 | 3 | yes | — |
| 9 | #66165 | — | 2 | 2 | no | — |
| 10 | #41×Dic5 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #66159 | C4 | 1 | 3 | yes | complemented |
| 12 | #66140 | — | 2 | 6 | no | — |
| 13 | #66139 | C22 | 1 | 3 | yes | — |
| 14 | C2×#4082 | C4 | 1 | 3 | yes | complemented |
| 15 | #65771 | — | 2 | 6 | no | — |
| 16 | #65745 | C22 | 1 | 12 | yes | complemented |
| 17 | #64928 | C22 | 1 | 4 | yes | complemented |
| 18 | #64922 | C22 | 1 | 12 | yes | complemented |
| 19 | #33362 | — | 5 | 1 | no | maximal, Sylow |
| 20 | C5×H(5,𝔽2) | — | 4 | 4 | no | — |
| 21 | C2×C5×#41 | — | 2 | 1 | no | — |
| 22 | C2×C5×#41 | C2×C4 | 1 | 3 | yes | — |
| 23 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C22×C5×D4 | — | 2 | 3 | no | — |
| 25 | C22×C5×D4 | C2×C4 | 1 | 3 | yes | — |
| 26 | #4150 | — | 4 | 4 | no | — |
| 27 | C2×Q8×D5 | D4 | 1 | 2 | yes | — |
| 28 | C2×#711 | D4 | 1 | 6 | yes | — |
| 29 | C2×#709 | C23 | 1 | 3 | yes | — |
| 30 | C22×Dic10 | C2×C4 | 1 | 3 | yes | — |
| 31 | Q8×AGL(1,𝔽5) | — | 4 | 8 | no | — |
| 32 | #4130 | — | 2 | 12 | no | — |
| 33 | #4099 | — | 2 | 6 | no | — |
| 34 | C2×C22⋊Dic5 | — | 2 | 6 | no | — |
| 35 | #4093 | C23 | 1 | 1 | yes | characteristic |
| 36 | Q8×Dic5 | — | 2 | 1 | no | — |
| 37 | #4090 | C2×C4 | 1 | 3 | yes | — |
| 38 | #4086 | C23 | 1 | 3 | yes | — |
| 39 | #4085 | — | 2 | 6 | no | — |
| 40 | #4084 | — | 2 | 6 | no | — |
| 41 | #4082 | — | 2 | 6 | no | — |
| 42 | #4082 | C2×C4 | 1 | 3 | yes | — |
| 43 | #4082 | C2×C4 | 1 | 6 | yes | — |
| 44 | #4081 | — | 2 | 6 | no | — |
| 45 | D4×Dic5 | — | 2 | 3 | no | — |
| 46 | C4×C22⋊D5 | — | 2 | 6 | no | — |
| 47 | #4072 | C2×C4 | 1 | 3 | yes | — |
| 48 | #4070 | — | 2 | 6 | no | — |
| 49 | C2×C4×Dic5 | C2×C4 | 1 | 3 | yes | — |
| 50 | #4011 | — | 2 | 12 | no | — |
| 51 | #3999 | C23 | 1 | 8 | yes | — |
| 52 | #3500 | — | 5 | 1 | no | — |
| 53 | #2884 | — | 5 | 2 | no | — |
| 54 | #2128 | — | 5 | 6 | no | — |
| 55 | #2126 | — | 5 | 6 | no | — |
| 56 | C24×C5 | — | 2 | 3 | no | — |
| 57 | C5×#41 | — | 2 | 4 | no | — |
| 58 | C5×#41 | — | 4 | 6 | no | — |
| 59 | C5×#41 | — | 4 | 4 | no | — |
| 60 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C2×C5×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 62 | C2×C5×D4 | — | 2 | 3 | no | — |
| 63 | C2×C5×D4 | — | 2 | 3 | no | — |
| 64 | C2×C5×D4 | — | 4 | 6 | no | — |
| 65 | C2×C5×D4 | — | 4 | 12 | no | — |
| 66 | C2×C5×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 67 | C2×C5×D4 | C22×C4 | 1 | 3 | yes | — |
| 68 | C22×C4×C5 | — | 2 | 3 | no | — |
| 69 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 70 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 71 | C2×C22⋊D5 | C2×D4 | 1 | 6 | yes | — |
| 72 | C22×Dic5 | — | 2 | 3 | no | — |
| 73 | C22×Dic5 | C22⋊C4 | 1 | 6 | yes | — |
| 74 | Q8×D5 | — | 2 | 8 | no | — |
| 75 | #711 | — | 4 | 12 | no | — |
| 76 | #709 | — | 4 | 6 | no | — |
| 77 | C2×D20 | C24 | 1 | 1 | yes | characteristic |
| 78 | C2×C4×D5 | C2×D4 | 1 | 6 | yes | — |
| 79 | C2×Dic10 | — | 4 | 6 | no | — |
| 80 | C2×Dic10 | C22⋊C4 | 1 | 6 | yes | — |
| 81 | C2×Dic10 | C22×C4 | 1 | 3 | yes | — |
| 82 | C22⋊AGL(1,𝔽5) | — | 2 | 8 | no | — |
| 83 | C4⋊AGL(1,𝔽5) | — | 4 | 12 | no | — |
| 84 | C4×AGL(1,𝔽5) | — | 4 | 12 | no | — |
| 85 | C22⋊Dic5 | — | 2 | 3 | no | — |
| 86 | C22⋊Dic5 | — | 2 | 6 | no | — |
| 87 | C22⋊Dic5 | — | 2 | 12 | no | — |
| 88 | #685 | — | 2 | 2 | no | — |
| 89 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 90 | #683 | — | 2 | 6 | no | — |
| 91 | C4×Dic5 | — | 2 | 3 | no | — |
| 92 | C4×Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C4×Dic5 | C22×C4 | 1 | 3 | yes | — |
| 94 | C2×2−1+4 | — | 5 | 1 | no | — |
| 95 | C2×H(5,𝔽2) | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 96 | #548 | — | 10 | 2 | no | — |
| 97 | #546 | — | 10 | 6 | no | — |
| 98 | #536 | — | 5 | 3 | no | — |
| 99 | #535 | — | 5 | 3 | no | — |
| 100 | #532 | — | 5 | 3 | no | — |
| 101 | C2×#124 | — | 5 | 3 | no | — |
| 102 | #519 | — | 10 | 2 | no | — |
| 103 | C4×#41 | — | 5 | 1 | no | — |
| 104 | #410 | — | 5 | 12 | no | — |
| 105 | #410 | — | 10 | 6 | no | — |
| 106 | #354 | — | 5 | 4 | no | — |
| 107 | #353 | — | 5 | 12 | no | — |
| 108 | C23×C5 | — | 2 | 3 | no | — |
| 109 | C23×C5 | — | 4 | 6 | no | — |
| 110 | C23×C5 | — | 8 | 6 | no | — |
| 111 | C23×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 112 | C22×D5 | C22×D4 | 1 | 2 | yes | — |
| 113 | C2×AGL(1,𝔽5) | — | 4 | 8 | no | — |
| 114 | C5×Q8 | — | 4 | 1 | no | — |
| 115 | C5×Q8 | #99 | 1 | 4 | yes | — |
| 116 | C5×D4 | — | 2 | 6 | no | — |
| 117 | C5×D4 | — | 4 | 3 | no | — |
| 118 | C5×D4 | — | 4 | 6 | no | — |
| 119 | C5×D4 | — | 4 | 6 | no | — |
| 120 | C2×C4×C5 | — | 2 | 6 | no | — |
| 121 | C2×C4×C5 | — | 2 | 1 | no | — |
| 122 | C2×C4×C5 | — | 4 | 6 | no | — |
| 123 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 124 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 125 | C22⋊D5 | — | 4 | 6 | no | — |
| 126 | C22⋊D5 | — | 8 | 6 | no | — |
| 127 | C2×Dic5 | — | 2 | 1 | no | — |
| 128 | C2×Dic5 | — | 2 | 3 | no | — |
| 129 | C2×Dic5 | — | 4 | 6 | no | — |
| 130 | C2×Dic5 | — | 8 | 3 | no | — |
| 131 | C2×Dic5 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 132 | D20 | — | 4 | 1 | no | — |
| 133 | C4×D5 | — | 2 | 12 | no | — |
| 134 | Dic10 | — | 2 | 12 | no | — |
| 135 | Dic10 | — | 4 | 3 | no | — |
| 136 | 2−1+4 | — | 20 | 4 | no | — |
| 137 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 138 | C2×#41 | — | 2 | 1 | no | — |
| 139 | C2×#41 | — | 5 | 6 | no | — |
| 140 | C2×#41 | — | 5 | 3 | no | — |
| 141 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 142 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 143 | C22×Q8 | — | 5 | 2 | no | — |
| 144 | C22×Q8 | — | 5 | 3 | no | — |
| 145 | C22×D4 | — | 2 | 3 | no | — |
| 146 | C22×D4 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 147 | C4⋊Q8 | — | 5 | 3 | no | — |
| 148 | #127 | — | 5 | 3 | no | — |
| 149 | #124 | — | 5 | 1 | no | — |
| 150 | #124 | — | 5 | 3 | no | — |
| 151 | #124 | — | 5 | 6 | no | — |
| 152 | #124 | — | 10 | 6 | no | — |
| 153 | #123 | — | 10 | 6 | no | — |
| 154 | C22⋊Q8 | — | 10 | 6 | no | — |
| 155 | #121 | — | 10 | 6 | no | — |
| 156 | #121 | — | 10 | 6 | no | — |
| 157 | #120 | — | 10 | 6 | no | — |
| 158 | C4×Q8 | — | 10 | 1 | no | — |
| 159 | C4×Q8 | — | 20 | 8 | no | — |
| 160 | C4×D4 | — | 10 | 6 | no | — |
| 161 | C4×D4 | — | 10 | 3 | no | — |
| 162 | #117 | — | 5 | 3 | no | — |
| 163 | #117 | — | 10 | 12 | no | — |
| 164 | C2×C22⋊C4 | — | 10 | 6 | no | — |
| 165 | C2×C42 | — | 5 | 3 | no | — |
| 166 | #99 | — | 5 | 8 | no | — |
| 167 | #99 | — | 10 | 12 | no | — |
| 168 | C22×C5 | — | 2 | 3 | no | — |
| 169 | C22×C5 | — | 4 | 3 | no | — |
| 170 | C22×C5 | — | 4 | 6 | no | — |
| 171 | C22×C5 | — | 4 | 3 | no | — |
| 172 | C22×C5 | — | 16 | 6 | no | — |
| 173 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 174 | C2×D5 | — | 2 | 2 | no | — |
| 175 | C2×D5 | — | 4 | 2 | no | — |
| 176 | AGL(1,𝔽5) | — | 8 | 8 | no | — |
| 177 | C4×C5 | — | 2 | 1 | no | — |
| 178 | C4×C5 | — | 4 | 1 | no | — |
| 179 | C4×C5 | C2×#99 | 1 | 6 | yes | — |
| 180 | Dic5 | — | 2 | 6 | no | — |
| 181 | Dic5 | — | 4 | 1 | no | — |
| 182 | Dic5 | — | 4 | 3 | no | — |
| 183 | C24 | — | 2 | 3 | no | — |
| 184 | #41 | — | 2 | 4 | no | — |
| 185 | #41 | — | 4 | 6 | no | — |
| 186 | #41 | — | 4 | 4 | no | — |
| 187 | #41 | — | 20 | 6 | no | — |
| 188 | #41 | — | 20 | 12 | no | — |
| 189 | C2×Q8 | — | 5 | 6 | no | — |
| 190 | C2×Q8 | — | 5 | 3 | no | — |
| 191 | C2×Q8 | — | 10 | 8 | no | — |
| 192 | C2×Q8 | — | 20 | 6 | no | — |
| 193 | C2×Q8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 194 | C2×Q8 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 195 | C2×D4 | — | 2 | 3 | no | — |
| 196 | C2×D4 | — | 2 | 3 | no | — |
| 197 | C2×D4 | — | 4 | 6 | no | — |
| 198 | C2×D4 | — | 4 | 12 | no | — |
| 199 | C2×D4 | — | 5 | 6 | no | — |
| 200 | C2×D4 | — | 5 | 1 | no | — |
| 201 | C2×D4 | — | 10 | 6 | no | — |
| 202 | C2×D4 | C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 203 | C2×D4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 204 | C22×C4 | — | 2 | 3 | no | — |
| 205 | C22×C4 | — | 5 | 6 | no | — |
| 206 | C22×C4 | — | 5 | 6 | no | — |
| 207 | C22×C4 | — | 10 | 3 | no | — |
| 208 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 209 | C4⋊C4 | — | 10 | 6 | no | — |
| 210 | C4⋊C4 | — | 10 | 3 | no | — |
| 211 | C4⋊C4 | — | 20 | 12 | no | — |
| 212 | C22⋊C4 | — | 10 | 2 | no | — |
| 213 | C22⋊C4 | — | 10 | 6 | no | — |
| 214 | C22⋊C4 | — | 10 | 12 | no | — |
| 215 | C22⋊C4 | — | 10 | 3 | no | — |
| 216 | C22⋊C4 | — | 10 | 8 | no | — |
| 217 | C42 | — | 5 | 1 | no | — |
| 218 | C42 | — | 5 | 3 | no | — |
| 219 | C42 | — | 10 | 3 | no | — |
| 220 | C42 | — | 20 | 12 | no | — |
| 221 | C2×C5 | — | 2 | 1 | no | — |
| 222 | C2×C5 | — | 4 | 3 | no | — |
| 223 | C2×C5 | — | 8 | 3 | no | — |
| 224 | C2×C5 | C22×#99 | 1 | 1 | yes | characteristic |
| 225 | D5 | — | 4 | 2 | no | — |
| 226 | C23 | — | 2 | 3 | no | — |
| 227 | C23 | — | 4 | 6 | no | — |
| 228 | C23 | — | 5 | 2 | no | — |
| 229 | C23 | — | 8 | 6 | no | — |
| 230 | C23 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 231 | Q8 | — | 4 | 1 | no | — |
| 232 | Q8 | — | 10 | 12 | no | — |
| 233 | Q8 | — | 20 | 3 | no | — |
| 234 | Q8 | #4011 | 1 | 4 | yes | — |
| 235 | D4 | — | 2 | 6 | no | — |
| 236 | D4 | — | 4 | 3 | no | — |
| 237 | D4 | — | 4 | 6 | no | — |
| 238 | D4 | — | 4 | 6 | no | — |
| 239 | D4 | — | 20 | 1 | no | — |
| 240 | D4 | — | 20 | 6 | no | — |
| 241 | D4 | — | 40 | 6 | no | — |
| 242 | C2×C4 | — | 2 | 6 | no | — |
| 243 | C2×C4 | — | 2 | 1 | no | — |
| 244 | C2×C4 | — | 4 | 6 | no | — |
| 245 | C2×C4 | — | 5 | 6 | no | — |
| 246 | C2×C4 | — | 10 | 12 | no | — |
| 247 | C2×C4 | — | 10 | 1 | no | — |
| 248 | C2×C4 | — | 10 | 3 | no | — |
| 249 | C2×C4 | — | 20 | 6 | no | — |
| 250 | C2×C4 | — | 20 | 8 | no | — |
| 251 | C2×C4 | — | 40 | 3 | no | — |
| 252 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 253 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 254 | C5 | #33362 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 255 | C22 | — | 2 | 3 | no | — |
| 256 | C22 | — | 4 | 3 | no | — |
| 257 | C22 | — | 4 | 6 | no | — |
| 258 | C22 | — | 4 | 3 | no | — |
| 259 | C22 | — | 10 | 2 | no | — |
| 260 | C22 | — | 16 | 6 | no | — |
| 261 | C22 | — | 20 | 2 | no | — |
| 262 | C22 | C22×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 263 | C4 | — | 2 | 1 | no | — |
| 264 | C4 | — | 4 | 1 | no | — |
| 265 | C4 | — | 10 | 6 | no | — |
| 266 | C4 | — | 20 | 1 | no | — |
| 267 | C4 | — | 20 | 3 | no | — |
| 268 | C4 | — | 40 | 8 | no | — |
| 269 | C4 | C2×#4011 | 1 | 6 | yes | — |
| 270 | C2 | — | 2 | 1 | no | — |
| 271 | C2 | — | 4 | 3 | no | — |
| 272 | C2 | — | 8 | 3 | no | — |
| 273 | C2 | — | 20 | 2 | no | — |
| 274 | C2 | C22×#4011 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10617539 | C2 | semidirect |
| 3. #10604996 | C2 | semidirect |
| 4. #10604918 | C2 | semidirect |
| 5. C2×C5×H(5,𝔽2) | C4 | semidirect |
| 10. #41×Dic5 | C4 | semidirect |
| 11. #66159 | C4 | semidirect |
| 14. C2×#4082 | C4 | semidirect |
| 16. #65745 | C22 | semidirect |
| 17. #64928 | C22 | semidirect |
| 18. #64922 | C22 | semidirect |
| 95. C2×H(5,𝔽2) | AGL(1,𝔽5) | semidirect |
| 254. C5 | #33362 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10619184 | 2 |
| #10617539 | 2 |
| #10604996 | 2 |
| #10604918 | 2 |
| #33362 | 5 |