Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
772 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #16811458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #16811188 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #16810811 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #16810810 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #16810809 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #16810595 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C53×B(4,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13133264 | — | 2 | 3 | no | — |
| 9 | #13133262 | — | 2 | 3 | no | — |
| 10 | #13133262 | C22 | 1 | 3 | yes | — |
| 11 | C2×#11719947 | — | 2 | 3 | no | — |
| 12 | C2×#11719947 | C22 | 1 | 3 | yes | — |
| 13 | #41×D53 | — | 2 | 3 | no | — |
| 14 | C22×C22⋊D53 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #13133213 | — | 2 | 3 | no | — |
| 16 | #13133203 | — | 2 | 3 | no | — |
| 17 | #13133190 | — | 2 | 3 | no | — |
| 18 | #13133189 | — | 2 | 3 | no | — |
| 19 | #13133187 | C22 | 1 | 3 | yes | complemented |
| 20 | #13133186 | C22 | 1 | 3 | yes | complemented |
| 21 | #13133185 | C22 | 1 | 3 | yes | complemented |
| 22 | #13133173 | C22 | 1 | 3 | yes | complemented |
| 23 | #13133132 | C22 | 1 | 3 | yes | complemented |
| 24 | #13133131 | C22 | 1 | 6 | yes | complemented |
| 25 | H(5,𝔽2)×C53 | C22 | 1 | 1 | yes | characteristic |
| 26 | #120×C53 | C22 | 1 | 3 | yes | complemented |
| 27 | #99×C53 | C22 | 1 | 3 | yes | complemented |
| 28 | C24×C53 | C23 | 1 | 1 | yes | characteristic |
| 29 | C23×D53 | C23 | 1 | 1 | yes | characteristic |
| 30 | #41×C53 | — | 2 | 3 | no | — |
| 31 | C2×D4×C53 | — | 2 | 3 | no | — |
| 32 | C2×D4×C53 | — | 2 | 3 | no | — |
| 33 | C2×D4×C53 | C23 | 1 | 3 | yes | — |
| 34 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 35 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 36 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 37 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 38 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 39 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 40 | C2×C22⋊D53 | D4 | 1 | 3 | yes | complemented |
| 41 | C2×C22⋊D53 | C23 | 1 | 3 | yes | — |
| 42 | C22×Dic53 | — | 2 | 1 | no | — |
| 43 | C22×Dic53 | D4 | 1 | 3 | yes | complemented |
| 44 | C22×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 45 | #11719949 | — | 2 | 1 | no | — |
| 46 | Q8×D53 | — | 2 | 3 | no | — |
| 47 | #11719947 | — | 2 | 3 | no | — |
| 48 | #11719947 | — | 2 | 6 | no | — |
| 49 | #11719947 | — | 2 | 6 | no | — |
| 50 | #11719947 | — | 2 | 6 | no | — |
| 51 | #11719947 | — | 2 | 6 | no | — |
| 52 | D4×D53 | — | 2 | 3 | no | — |
| 53 | D4×D53 | — | 2 | 6 | no | — |
| 54 | #11719945 | — | 2 | 6 | no | — |
| 55 | #11719945 | — | 2 | 3 | no | — |
| 56 | C2×D212 | — | 2 | 3 | no | — |
| 57 | C2×C4×D53 | — | 2 | 3 | no | — |
| 58 | C2×C4×D53 | D4 | 1 | 3 | yes | complemented |
| 59 | C2×Dic106 | — | 2 | 3 | no | — |
| 60 | C2×Dic106 | D4 | 1 | 3 | yes | complemented |
| 61 | C22⋊C4×C53 | — | 2 | 3 | no | — |
| 62 | C22⋊C4×C53 | C23 | 1 | 3 | yes | — |
| 63 | C22⋊Dic53 | — | 2 | 3 | no | — |
| 64 | C22⋊Dic53 | C23 | 1 | 3 | yes | — |
| 65 | #11719921 | — | 2 | 3 | no | — |
| 66 | #11719921 | — | 2 | 3 | no | — |
| 67 | #11719919 | — | 2 | 3 | no | — |
| 68 | C4×Dic53 | — | 2 | 3 | no | — |
| 69 | C23×C53 | — | 2 | 1 | no | — |
| 70 | C23×C53 | — | 2 | 3 | no | — |
| 71 | C23×C53 | — | 8 | 1 | no | — |
| 72 | C23×C53 | C2×D4 | 1 | 3 | yes | — |
| 73 | C23×C53 | C24 | 1 | 1 | yes | characteristic |
| 74 | C22×D53 | — | 2 | 3 | no | — |
| 75 | C22×D53 | — | 2 | 3 | no | — |
| 76 | C22×D53 | — | 8 | 1 | no | — |
| 77 | C22×D53 | C2×D4 | 1 | 3 | yes | — |
| 78 | Q8×C53 | — | 2 | 1 | no | — |
| 79 | D4×C53 | — | 2 | 3 | no | — |
| 80 | D4×C53 | — | 2 | 6 | no | — |
| 81 | D4×C53 | — | 8 | 3 | no | — |
| 82 | C2×C4×C53 | — | 2 | 3 | no | — |
| 83 | C2×C4×C53 | — | 2 | 3 | no | — |
| 84 | C2×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 85 | C22⋊D53 | — | 2 | 6 | no | — |
| 86 | C22⋊D53 | — | 2 | 6 | no | — |
| 87 | C22⋊D53 | — | 2 | 6 | no | — |
| 88 | C22⋊D53 | — | 4 | 3 | no | — |
| 89 | C22⋊D53 | — | 4 | 1 | no | — |
| 90 | C22⋊D53 | — | 8 | 3 | no | — |
| 91 | C22⋊D53 | — | 8 | 3 | no | — |
| 92 | C2×Dic53 | — | 2 | 3 | no | — |
| 93 | C2×Dic53 | — | 2 | 3 | no | — |
| 94 | C2×Dic53 | — | 2 | 3 | no | — |
| 95 | C2×Dic53 | — | 2 | 3 | no | — |
| 96 | C2×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 97 | C2×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 98 | C2×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 99 | D212 | — | 2 | 3 | no | — |
| 100 | D212 | — | 8 | 3 | no | — |
| 101 | C4×D53 | — | 2 | 6 | no | — |
| 102 | C4×D53 | — | 2 | 3 | no | — |
| 103 | C4×D53 | — | 2 | 3 | no | — |
| 104 | Dic106 | — | 2 | 3 | no | — |
| 105 | Dic106 | — | 2 | 6 | no | — |
| 106 | C22×C53 | — | 2 | 3 | no | — |
| 107 | C22×C53 | — | 4 | 1 | no | — |
| 108 | C22×C53 | — | 4 | 3 | no | — |
| 109 | C22×C53 | — | 4 | 3 | no | — |
| 110 | C22×C53 | — | 4 | 3 | no | — |
| 111 | C22×C53 | — | 4 | 1 | no | — |
| 112 | C22×C53 | — | 8 | 1 | no | — |
| 113 | C22×C53 | C22×D4 | 1 | 3 | yes | — |
| 114 | C2×D53 | — | 2 | 3 | no | — |
| 115 | C2×D53 | — | 4 | 3 | no | — |
| 116 | C2×D53 | — | 4 | 3 | no | — |
| 117 | C2×D53 | — | 4 | 3 | no | — |
| 118 | C2×D53 | — | 4 | 1 | no | — |
| 119 | C2×D53 | — | 8 | 3 | no | — |
| 120 | C4×C53 | — | 2 | 3 | no | — |
| 121 | C4×C53 | — | 4 | 3 | no | — |
| 122 | Dic53 | — | 2 | 3 | no | — |
| 123 | Dic53 | — | 4 | 3 | no | — |
| 124 | Dic53 | #120 | 1 | 3 | yes | — |
| 125 | Dic53 | #120 | 1 | 1 | yes | characteristic |
| 126 | #3027 | — | 53 | 1 | no | maximal, Sylow |
| 127 | C2×C53 | — | 2 | 3 | no | — |
| 128 | C2×C53 | — | 4 | 3 | no | — |
| 129 | C2×C53 | — | 8 | 1 | no | — |
| 130 | C2×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 131 | D53 | — | 4 | 3 | no | — |
| 132 | D53 | — | 8 | 1 | no | — |
| 133 | #585 | — | 53 | 1 | no | — |
| 134 | #535 | — | 53 | 3 | no | — |
| 135 | B(4,𝔽2) | — | 53 | 1 | no | — |
| 136 | B(4,𝔽2) | — | 53 | 3 | no | — |
| 137 | B(4,𝔽2) | — | 53 | 3 | no | — |
| 138 | B(4,𝔽2) | D53 | 1 | 1 | yes | characteristic, complemented |
| 139 | #410 | — | 53 | 3 | no | — |
| 140 | C53 | #3027 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 141 | 2−1+4 | — | 106 | 3 | no | — |
| 142 | H(5,𝔽2) | — | 53 | 3 | no | — |
| 143 | H(5,𝔽2) | — | 106 | 3 | no | — |
| 144 | H(5,𝔽2) | C2×D53 | 1 | 1 | yes | characteristic |
| 145 | C2×#41 | — | 53 | 3 | no | — |
| 146 | C2×#41 | — | 106 | 3 | no | — |
| 147 | C2×#41 | — | 106 | 3 | no | — |
| 148 | C22×D4 | — | 53 | 1 | no | — |
| 149 | #127 | — | 106 | 3 | no | — |
| 150 | #124 | — | 106 | 3 | no | — |
| 151 | #121 | — | 106 | 3 | no | — |
| 152 | #121 | — | 106 | 3 | no | — |
| 153 | #120 | — | 53 | 3 | no | — |
| 154 | #120 | — | 53 | 3 | no | — |
| 155 | #120 | — | 53 | 3 | no | — |
| 156 | #120 | C2×D53 | 1 | 3 | yes | complemented |
| 157 | #117 | — | 53 | 3 | no | — |
| 158 | #99 | — | 53 | 3 | no | — |
| 159 | #99 | — | 53 | 6 | no | — |
| 160 | #99 | C2×D53 | 1 | 3 | yes | complemented |
| 161 | C24 | — | 53 | 1 | no | — |
| 162 | C24 | C22×D53 | 1 | 1 | yes | characteristic |
| 163 | #41 | — | 2 | 3 | no | — |
| 164 | #41 | — | 106 | 6 | no | — |
| 165 | #41 | — | 106 | 6 | no | — |
| 166 | #41 | — | 106 | 6 | no | — |
| 167 | #41 | — | 106 | 6 | no | — |
| 168 | #41 | — | 106 | 3 | no | — |
| 169 | #41 | — | 106 | 6 | no | — |
| 170 | #41 | — | 106 | 1 | no | — |
| 171 | #41 | — | 106 | 3 | no | — |
| 172 | C2×Q8 | — | 53 | 3 | no | — |
| 173 | C2×Q8 | — | 106 | 3 | no | — |
| 174 | C2×Q8 | — | 106 | 3 | no | — |
| 175 | C2×D4 | — | 2 | 3 | no | — |
| 176 | C2×D4 | — | 2 | 3 | no | — |
| 177 | C2×D4 | — | 53 | 3 | no | — |
| 178 | C2×D4 | — | 53 | 3 | no | — |
| 179 | C2×D4 | — | 106 | 3 | no | — |
| 180 | C2×D4 | — | 106 | 3 | no | — |
| 181 | C2×D4 | — | 106 | 3 | no | — |
| 182 | C2×D4 | — | 106 | 6 | no | — |
| 183 | C2×D4 | — | 106 | 3 | no | — |
| 184 | C2×D4 | — | 106 | 3 | no | — |
| 185 | C2×D4 | — | 106 | 3 | no | — |
| 186 | C2×D4 | — | 106 | 3 | no | — |
| 187 | C2×D4 | — | 106 | 3 | no | — |
| 188 | C2×D4 | C22×D53 | 1 | 3 | yes | — |
| 189 | C22×C4 | — | 53 | 1 | no | — |
| 190 | C22×C4 | — | 53 | 3 | no | — |
| 191 | C22×C4 | — | 53 | 3 | no | — |
| 192 | C22×C4 | — | 106 | 3 | no | — |
| 193 | C22×C4 | — | 106 | 1 | no | — |
| 194 | C4⋊C4 | — | 106 | 3 | no | — |
| 195 | C22⋊C4 | — | 2 | 3 | no | — |
| 196 | C22⋊C4 | — | 53 | 3 | no | — |
| 197 | C22⋊C4 | — | 106 | 3 | no | — |
| 198 | C22⋊C4 | — | 106 | 3 | no | — |
| 199 | C22⋊C4 | — | 106 | 3 | no | — |
| 200 | C22⋊C4 | C22×D53 | 1 | 3 | yes | — |
| 201 | C42 | — | 106 | 3 | no | — |
| 202 | C23 | — | 2 | 1 | no | — |
| 203 | C23 | — | 2 | 3 | no | — |
| 204 | C23 | — | 8 | 1 | no | — |
| 205 | C23 | — | 53 | 3 | no | — |
| 206 | C23 | — | 106 | 3 | no | — |
| 207 | C23 | — | 106 | 3 | no | — |
| 208 | C23 | — | 424 | 1 | no | — |
| 209 | C23 | D4×D53 | 1 | 3 | yes | — |
| 210 | C23 | C23×D53 | 1 | 1 | yes | characteristic |
| 211 | Q8 | — | 2 | 1 | no | — |
| 212 | Q8 | — | 106 | 6 | no | — |
| 213 | Q8 | — | 106 | 3 | no | — |
| 214 | D4 | — | 2 | 3 | no | — |
| 215 | D4 | — | 2 | 6 | no | — |
| 216 | D4 | — | 8 | 3 | no | — |
| 217 | D4 | — | 106 | 3 | no | — |
| 218 | D4 | — | 106 | 6 | no | — |
| 219 | D4 | — | 106 | 6 | no | — |
| 220 | D4 | — | 106 | 6 | no | — |
| 221 | D4 | — | 424 | 3 | no | — |
| 222 | D4 | — | 424 | 3 | no | — |
| 223 | D4 | — | 424 | 3 | no | — |
| 224 | D4 | — | 212 | 1 | no | — |
| 225 | D4 | — | 212 | 3 | no | — |
| 226 | C2×C4 | — | 2 | 3 | no | — |
| 227 | C2×C4 | — | 2 | 3 | no | — |
| 228 | C2×C4 | — | 53 | 3 | no | — |
| 229 | C2×C4 | — | 53 | 3 | no | — |
| 230 | C2×C4 | — | 53 | 3 | no | — |
| 231 | C2×C4 | — | 106 | 3 | no | — |
| 232 | C2×C4 | — | 106 | 3 | no | — |
| 233 | C2×C4 | — | 106 | 6 | no | — |
| 234 | C2×C4 | — | 106 | 3 | no | — |
| 235 | C2×C4 | — | 106 | 3 | no | — |
| 236 | C2×C4 | — | 106 | 3 | no | — |
| 237 | C2×C4 | — | 106 | 3 | no | — |
| 238 | C2×C4 | D4×D53 | 1 | 3 | yes | — |
| 239 | C22 | — | 2 | 3 | no | — |
| 240 | C22 | — | 4 | 1 | no | — |
| 241 | C22 | — | 4 | 3 | no | — |
| 242 | C22 | — | 4 | 3 | no | — |
| 243 | C22 | — | 4 | 3 | no | — |
| 244 | C22 | — | 4 | 1 | no | — |
| 245 | C22 | — | 8 | 1 | no | — |
| 246 | C22 | — | 106 | 3 | no | — |
| 247 | C22 | — | 424 | 3 | no | — |
| 248 | C22 | — | 212 | 3 | no | — |
| 249 | C22 | — | 212 | 3 | no | — |
| 250 | C22 | — | 212 | 3 | no | — |
| 251 | C22 | — | 212 | 1 | no | — |
| 252 | C22 | C2×D4×D53 | 1 | 3 | yes | — |
| 253 | C4 | — | 2 | 3 | no | — |
| 254 | C4 | — | 4 | 3 | no | — |
| 255 | C4 | — | 53 | 1 | no | — |
| 256 | C4 | — | 53 | 3 | no | — |
| 257 | C4 | — | 106 | 3 | no | — |
| 258 | C4 | — | 212 | 3 | no | — |
| 259 | C2 | — | 2 | 3 | no | — |
| 260 | C2 | — | 4 | 3 | no | — |
| 261 | C2 | — | 8 | 1 | no | — |
| 262 | C2 | — | 424 | 1 | no | — |
| 263 | C2 | — | 212 | 3 | no | — |
| 264 | C2 | #120×D53 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #120×D53 |
| commutator | C23×C53 | C24 |
| Frattini | C23 | C23×D53 |
| Fitting | C53×B(4,𝔽2) | C2 |
| socle | C2×C53 | C2×#120 |
| hypercenter | B(4,𝔽2) | D53 |
| hypocenter | C53 | #3027 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ B(4,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811458 | C2 | semidirect |
| 2. #16811188 | C2 | semidirect |
| 3. #16810811 | C2 | semidirect |
| 4. #16810810 | C2 | semidirect |
| 5. #16810809 | C2 | semidirect |
| 6. #16810595 | C2 | semidirect |
| 7. C53×B(4,𝔽2) | C2 | semidirect |
| 14. C22×C22⋊D53 | C22 | semidirect |
| 19. #13133187 | C22 | semidirect |
| 20. #13133186 | C22 | semidirect |
| 21. #13133185 | C22 | semidirect |
| 22. #13133173 | C22 | semidirect |
| 23. #13133132 | C22 | semidirect |
| 24. #13133131 | C22 | semidirect |
| 26. #120×C53 | C22 | semidirect |
| 27. #99×C53 | C22 | semidirect |
| 40. C2×C22⋊D53 | D4 | semidirect |
| 43. C22×Dic53 | D4 | semidirect |
| 58. C2×C4×D53 | D4 | semidirect |
| 60. C2×Dic106 | D4 | semidirect |
| 138. B(4,𝔽2) | D53 | semidirect |
| 140. C53 | #3027 | semidirect |
| 156. #120 | C2×D53 | semidirect |
| 160. #99 | C2×D53 | semidirect |