Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
804 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 149 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 | C2×#13133192 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#13133190 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #16810547 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #16810533 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #16810532 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #16810494 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C53×#396 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×D212 | C22 | 1 | 3 | yes | — |
| 9 | C22×D212 | C22 | 1 | 3 | yes | — |
| 10 | C22×C22⋊D53 | C22 | 1 | 3 | yes | — |
| 11 | C22×C22⋊D53 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×C4×D53 | C22 | 1 | 3 | yes | — |
| 13 | #13133192 | — | 2 | 12 | no | — |
| 14 | #13133190 | — | 2 | 12 | no | — |
| 15 | C2×#11719919 | C22 | 1 | 3 | yes | — |
| 16 | C2×#11719921 | C22 | 1 | 3 | yes | — |
| 17 | C2×#11719921 | C22 | 1 | 3 | yes | — |
| 18 | C2×#11719921 | C22 | 1 | 3 | yes | — |
| 19 | #13133124 | C22 | 1 | 3 | yes | — |
| 20 | C2×C4⋊C4×C53 | C22 | 1 | 3 | yes | — |
| 21 | C2×C22⋊C4×C53 | C22 | 1 | 3 | yes | — |
| 22 | #95×C53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C24×C53 | C23 | 1 | 1 | yes | characteristic |
| 24 | C23×D53 | C23 | 1 | 1 | yes | characteristic |
| 25 | C23×D53 | C23 | 1 | 3 | yes | — |
| 26 | C22×C4×C53 | C23 | 1 | 3 | yes | — |
| 27 | C22×C4×C53 | C23 | 1 | 3 | yes | — |
| 28 | C2×C22⋊D53 | — | 2 | 6 | no | — |
| 29 | C2×C22⋊D53 | — | 2 | 6 | no | — |
| 30 | C2×C22⋊D53 | — | 4 | 3 | no | — |
| 31 | C2×C22⋊D53 | — | 4 | 3 | no | — |
| 32 | C22×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×Dic53 | C23 | 1 | 3 | yes | — |
| 34 | C2×D212 | — | 2 | 6 | no | — |
| 35 | C2×D212 | — | 2 | 6 | no | — |
| 36 | C2×D212 | — | 2 | 6 | no | — |
| 37 | C2×D212 | — | 2 | 6 | no | — |
| 38 | C2×D212 | — | 4 | 3 | no | — |
| 39 | C2×D212 | — | 4 | 3 | no | — |
| 40 | C2×C4×D53 | — | 2 | 6 | no | — |
| 41 | C2×C4×D53 | — | 2 | 6 | no | — |
| 42 | C2×C4×D53 | — | 4 | 3 | no | — |
| 43 | C4⋊C4×C53 | — | 2 | 6 | no | — |
| 44 | C22⋊C4×C53 | — | 2 | 6 | no | — |
| 45 | #11719921 | — | 2 | 6 | no | — |
| 46 | #11719921 | — | 4 | 3 | no | — |
| 47 | #11719921 | D4 | 1 | 12 | yes | complemented |
| 48 | #11719919 | — | 2 | 6 | no | — |
| 49 | C23×C53 | — | 2 | 3 | no | — |
| 50 | C23×C53 | — | 2 | 3 | no | — |
| 51 | C23×C53 | — | 2 | 1 | no | — |
| 52 | C23×C53 | C24 | 1 | 1 | yes | characteristic |
| 53 | C22×D53 | — | 2 | 3 | no | — |
| 54 | C22×D53 | — | 2 | 3 | no | — |
| 55 | C22×D53 | — | 2 | 3 | no | — |
| 56 | C22×D53 | — | 2 | 3 | no | — |
| 57 | C22×D53 | — | 2 | 3 | no | — |
| 58 | C22×D53 | — | 2 | 3 | no | — |
| 59 | C22×D53 | — | 2 | 3 | no | — |
| 60 | C22×D53 | — | 2 | 3 | no | — |
| 61 | C22×D53 | — | 2 | 1 | no | — |
| 62 | C22×D53 | C2×D4 | 1 | 6 | yes | — |
| 63 | C2×C4×C53 | — | 2 | 3 | no | — |
| 64 | C2×C4×C53 | — | 2 | 3 | no | — |
| 65 | C2×C4×C53 | — | 2 | 3 | no | — |
| 66 | C2×C4×C53 | — | 2 | 3 | no | — |
| 67 | C2×C4×C53 | — | 2 | 3 | no | — |
| 68 | C2×C4×C53 | C2×D4 | 1 | 6 | yes | — |
| 69 | C22⋊D53 | — | 4 | 12 | no | — |
| 70 | C22⋊D53 | — | 4 | 4 | no | — |
| 71 | C2×Dic53 | — | 2 | 3 | no | — |
| 72 | C2×Dic53 | — | 2 | 3 | no | — |
| 73 | C2×Dic53 | — | 2 | 3 | no | — |
| 74 | C2×Dic53 | C2×D4 | 1 | 6 | yes | — |
| 75 | D212 | — | 4 | 12 | no | — |
| 76 | D212 | — | 4 | 12 | no | — |
| 77 | C4×D53 | — | 4 | 12 | no | — |
| 78 | C22×C53 | — | 4 | 3 | no | — |
| 79 | C22×C53 | — | 4 | 3 | no | — |
| 80 | C22×C53 | — | 4 | 1 | no | — |
| 81 | C22×C53 | C22×D4 | 1 | 3 | yes | — |
| 82 | C22×C53 | H(5,𝔽2) | 1 | 3 | yes | — |
| 83 | C22×C53 | 2−1+4 | 1 | 1 | yes | characteristic |
| 84 | C2×D53 | — | 2 | 6 | no | — |
| 85 | C2×D53 | — | 2 | 6 | no | — |
| 86 | C2×D53 | — | 2 | 6 | no | — |
| 87 | C2×D53 | — | 4 | 3 | no | — |
| 88 | C2×D53 | — | 4 | 3 | no | — |
| 89 | C2×D53 | — | 4 | 3 | no | — |
| 90 | C2×D53 | — | 4 | 3 | no | — |
| 91 | C2×D53 | — | 4 | 3 | no | — |
| 92 | C2×D53 | — | 4 | 3 | no | — |
| 93 | C2×D53 | — | 4 | 1 | no | — |
| 94 | C4×C53 | — | 2 | 6 | no | — |
| 95 | C4×C53 | — | 4 | 3 | no | — |
| 96 | Dic53 | — | 2 | 6 | no | — |
| 97 | Dic53 | — | 4 | 1 | no | — |
| 98 | #2675 | — | 53 | 1 | no | maximal, Sylow |
| 99 | C2×C53 | — | 8 | 1 | no | — |
| 100 | C2×C53 | #537 | 1 | 3 | yes | — |
| 101 | C2×C53 | D42 | 1 | 3 | yes | — |
| 102 | C2×C53 | #560 | 1 | 1 | yes | characteristic |
| 103 | D53 | — | 4 | 6 | no | — |
| 104 | D53 | — | 8 | 1 | no | — |
| 105 | C2×#121 | — | 53 | 3 | no | — |
| 106 | C2×#121 | — | 53 | 3 | no | — |
| 107 | #396 | — | 53 | 1 | no | — |
| 108 | #396 | D53 | 1 | 1 | yes | characteristic, complemented |
| 109 | #392 | — | 53 | 3 | no | — |
| 110 | #392 | — | 53 | 1 | no | — |
| 111 | #385 | — | 53 | 3 | no | — |
| 112 | C53 | #2675 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 113 | C22×D4 | — | 53 | 3 | no | — |
| 114 | C22×D4 | — | 53 | 1 | no | — |
| 115 | C22×D4 | — | 53 | 3 | no | — |
| 116 | C22×D4 | — | 53 | 3 | no | — |
| 117 | C23×C4 | — | 53 | 3 | no | — |
| 118 | #121 | — | 106 | 12 | no | — |
| 119 | #121 | — | 106 | 12 | no | — |
| 120 | C2×C4⋊C4 | — | 53 | 3 | no | — |
| 121 | C2×C4⋊C4 | C2×D53 | 1 | 3 | yes | — |
| 122 | C2×C22⋊C4 | — | 53 | 3 | no | — |
| 123 | C2×C22⋊C4 | — | 53 | 3 | no | — |
| 124 | C2×C22⋊C4 | — | 53 | 3 | no | — |
| 125 | C2×C22⋊C4 | C2×D53 | 1 | 3 | yes | — |
| 126 | #95 | — | 53 | 3 | no | — |
| 127 | #95 | C2×D53 | 1 | 1 | yes | characteristic |
| 128 | C24 | — | 53 | 3 | no | — |
| 129 | C24 | — | 53 | 1 | no | — |
| 130 | C24 | C22×D53 | 1 | 1 | yes | characteristic |
| 131 | C2×D4 | — | 106 | 6 | no | — |
| 132 | C2×D4 | — | 106 | 6 | no | — |
| 133 | C2×D4 | — | 106 | 6 | no | — |
| 134 | C2×D4 | — | 106 | 6 | no | — |
| 135 | C2×D4 | — | 106 | 6 | no | — |
| 136 | C2×D4 | — | 106 | 6 | no | — |
| 137 | C2×D4 | — | 212 | 3 | no | — |
| 138 | C2×D4 | — | 212 | 3 | no | — |
| 139 | C2×D4 | — | 212 | 3 | no | — |
| 140 | C2×D4 | — | 212 | 3 | no | — |
| 141 | C22×C4 | — | 53 | 3 | no | — |
| 142 | C22×C4 | — | 53 | 1 | no | — |
| 143 | C22×C4 | — | 106 | 6 | no | — |
| 144 | C22×C4 | — | 106 | 6 | no | — |
| 145 | C22×C4 | — | 212 | 3 | no | — |
| 146 | C22×C4 | C22×D53 | 1 | 3 | yes | — |
| 147 | C22×C4 | C22×D53 | 1 | 3 | yes | — |
| 148 | C4⋊C4 | — | 2 | 6 | no | — |
| 149 | C4⋊C4 | — | 106 | 6 | no | — |
| 150 | C22⋊C4 | — | 2 | 6 | no | — |
| 151 | C22⋊C4 | — | 53 | 12 | no | — |
| 152 | C22⋊C4 | — | 106 | 6 | no | — |
| 153 | C22⋊C4 | — | 212 | 3 | no | — |
| 154 | C23 | — | 2 | 3 | no | — |
| 155 | C23 | — | 2 | 3 | no | — |
| 156 | C23 | — | 2 | 1 | no | — |
| 157 | C23 | — | 53 | 6 | no | — |
| 158 | C23 | — | 106 | 3 | no | — |
| 159 | C23 | — | 106 | 3 | no | — |
| 160 | C23 | — | 106 | 3 | no | — |
| 161 | C23 | — | 106 | 3 | no | — |
| 162 | C23 | — | 106 | 3 | no | — |
| 163 | C23 | — | 106 | 3 | no | — |
| 164 | C23 | — | 106 | 3 | no | — |
| 165 | C23 | — | 106 | 3 | no | — |
| 166 | C23 | — | 106 | 1 | no | — |
| 167 | C23 | C23×D53 | 1 | 1 | yes | characteristic |
| 168 | D4 | — | 212 | 12 | no | — |
| 169 | D4 | — | 212 | 12 | no | — |
| 170 | D4 | — | 212 | 12 | no | — |
| 171 | D4 | — | 212 | 4 | no | — |
| 172 | C2×C4 | — | 2 | 3 | no | — |
| 173 | C2×C4 | — | 2 | 3 | no | — |
| 174 | C2×C4 | — | 2 | 3 | no | — |
| 175 | C2×C4 | — | 2 | 3 | no | — |
| 176 | C2×C4 | — | 2 | 3 | no | — |
| 177 | C2×C4 | — | 53 | 6 | no | — |
| 178 | C2×C4 | — | 106 | 3 | no | — |
| 179 | C2×C4 | — | 106 | 3 | no | — |
| 180 | C2×C4 | — | 106 | 3 | no | — |
| 181 | C2×C4 | — | 212 | 12 | no | — |
| 182 | C2×C4 | D4×D53 | 1 | 6 | yes | — |
| 183 | C22 | — | 4 | 1 | no | — |
| 184 | C22 | — | 4 | 3 | no | — |
| 185 | C22 | — | 4 | 3 | no | — |
| 186 | C22 | — | 106 | 6 | no | — |
| 187 | C22 | — | 106 | 6 | no | — |
| 188 | C22 | — | 106 | 6 | no | — |
| 189 | C22 | — | 212 | 1 | no | — |
| 190 | C22 | — | 212 | 3 | no | — |
| 191 | C22 | — | 212 | 3 | no | — |
| 192 | C22 | — | 212 | 3 | no | — |
| 193 | C22 | — | 212 | 3 | no | — |
| 194 | C22 | — | 212 | 3 | no | — |
| 195 | C22 | — | 212 | 3 | no | — |
| 196 | C22 | C2×D4×D53 | 1 | 3 | yes | — |
| 197 | C22 | #13133263 | 1 | 3 | yes | — |
| 198 | C22 | #13133265 | 1 | 1 | yes | characteristic |
| 199 | C4 | — | 2 | 6 | no | — |
| 200 | C4 | — | 4 | 3 | no | — |
| 201 | C4 | — | 106 | 6 | no | — |
| 202 | C4 | — | 212 | 1 | no | — |
| 203 | C2 | — | 8 | 1 | no | — |
| 204 | C2 | — | 424 | 1 | no | — |
| 205 | C2 | — | 212 | 6 | no | — |
| 206 | C2 | #16811203 | 1 | 3 | yes | — |
| 207 | C2 | #16811257 | 1 | 3 | yes | — |
| 208 | C2 | #16811364 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#396 | C2 |
| hypercenter | #396 | D53 |
| hypocenter | C53 | #2675 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#13133192 | C2 | semidirect |
| 2. C2×#13133190 | C2 | semidirect |
| 3. #16810547 | C2 | semidirect |
| 4. #16810533 | C2 | semidirect |
| 5. #16810532 | C2 | semidirect |
| 6. #16810494 | C2 | semidirect |
| 7. C53×#396 | C2 | semidirect |
| 47. #11719921 | D4 | semidirect |
| 108. #396 | D53 | semidirect |
| 112. C53 | #2675 | semidirect |