Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
3274 conjugacy classes of subgroups; only the 411 normal subgroups are shown.
This group has 3274 conjugacy classes of subgroups — too many to draw. Only its 411 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 | C22×C7×#120 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#90019 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11729885 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #11729302 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C26×C7 | C22 | 1 | 1 | yes | characteristic |
| 6 | C23×C7×D4 | C22 | 1 | 3 | yes | — |
| 7 | C2×C7×#120 | — | 2 | 8 | no | — |
| 8 | C2×C7×#120 | C4 | 1 | 8 | yes | complemented |
| 9 | C22×C7×C22⋊C4 | C22 | 1 | 3 | yes | — |
| 10 | C22×C22⋊Dic7 | C22 | 1 | 12 | yes | — |
| 11 | C22×C22⋊Dic7 | C22 | 1 | 4 | yes | complemented |
| 12 | #90019 | — | 2 | 8 | no | — |
| 13 | C2×C7⋊#95 | C22 | 1 | 12 | yes | complemented |
| 14 | #14657 | — | 7 | 1 | no | maximal, Sylow |
| 15 | C25×C7 | — | 4 | 12 | no | — |
| 16 | C25×C7 | C2×C4 | 1 | 4 | yes | — |
| 17 | C25×C7 | D4 | 1 | 8 | yes | — |
| 18 | C25×C7 | C23 | 1 | 1 | yes | characteristic |
| 19 | C25×C7 | C23 | 1 | 3 | yes | — |
| 20 | C22×C7×D4 | — | 2 | 12 | no | — |
| 21 | C22×C7×D4 | — | 4 | 12 | no | — |
| 22 | C22×C7×D4 | C2×C4 | 1 | 12 | yes | — |
| 23 | C23×C4×C7 | C23 | 1 | 3 | yes | — |
| 24 | C23×Dic7 | C23 | 1 | 8 | yes | — |
| 25 | C7×#120 | — | 2 | 32 | no | — |
| 26 | C2×C7×C22⋊C4 | — | 2 | 12 | no | — |
| 27 | C2×C7×C22⋊C4 | C2×C4 | 1 | 12 | yes | — |
| 28 | C2×C22⋊Dic7 | — | 2 | 24 | no | — |
| 29 | C2×C22⋊Dic7 | — | 4 | 24 | no | — |
| 30 | C2×C22⋊Dic7 | — | 4 | 12 | no | — |
| 31 | C7⋊#95 | — | 4 | 12 | no | — |
| 32 | C22×#120 | D7 | 1 | 1 | yes | characteristic |
| 33 | C2×#379 | — | 7 | 2 | no | — |
| 34 | #1440 | — | 7 | 6 | no | — |
| 35 | #1438 | — | 7 | 6 | no | — |
| 36 | C24×C7 | — | 2 | 24 | no | — |
| 37 | C24×C7 | — | 2 | 12 | no | — |
| 38 | C24×C7 | — | 2 | 2 | no | — |
| 39 | C24×C7 | — | 2 | 12 | no | — |
| 40 | C24×C7 | — | 4 | 24 | no | — |
| 41 | C24×C7 | — | 4 | 48 | no | — |
| 42 | C24×C7 | — | 4 | 48 | no | — |
| 43 | C24×C7 | — | 4 | 16 | no | — |
| 44 | C24×C7 | C22⋊C4 | 1 | 16 | yes | complemented |
| 45 | C24×C7 | C22×C4 | 1 | 2 | yes | — |
| 46 | C24×C7 | C22×C4 | 1 | 6 | yes | — |
| 47 | C24×C7 | C2×D4 | 1 | 12 | yes | — |
| 48 | C24×C7 | C24 | 1 | 1 | yes | characteristic |
| 49 | C2×C7×D4 | — | 2 | 24 | no | — |
| 50 | C2×C7×D4 | — | 4 | 24 | no | — |
| 51 | C2×C7×D4 | — | 4 | 48 | no | — |
| 52 | C22×C4×C7 | — | 2 | 6 | no | — |
| 53 | C22×C4×C7 | — | 2 | 12 | no | — |
| 54 | C22×C4×C7 | C22×C4 | 1 | 6 | yes | — |
| 55 | C22×Dic7 | — | 2 | 8 | no | — |
| 56 | C22×Dic7 | — | 2 | 24 | no | — |
| 57 | C22×Dic7 | — | 2 | 24 | no | — |
| 58 | C7×C22⋊C4 | — | 2 | 24 | no | — |
| 59 | C22⋊Dic7 | — | 4 | 48 | no | — |
| 60 | C22⋊Dic7 | — | 4 | 16 | no | — |
| 61 | C26 | C2×D7 | 1 | 1 | yes | characteristic |
| 62 | C23×D4 | C2×D7 | 1 | 3 | yes | — |
| 63 | C2×#120 | — | 2 | 8 | no | — |
| 64 | C2×#120 | Dic7 | 1 | 8 | yes | complemented |
| 65 | C22×C22⋊C4 | — | 7 | 12 | no | — |
| 66 | C22×C22⋊C4 | — | 7 | 4 | no | — |
| 67 | C22×C22⋊C4 | C2×D7 | 1 | 3 | yes | — |
| 68 | #379 | — | 14 | 8 | no | — |
| 69 | C2×#95 | — | 7 | 12 | no | — |
| 70 | C23×C7 | — | 2 | 12 | no | — |
| 71 | C23×C7 | — | 2 | 24 | no | — |
| 72 | C23×C7 | — | 2 | 24 | no | — |
| 73 | C23×C7 | — | 2 | 24 | no | — |
| 74 | C23×C7 | — | 2 | 24 | no | — |
| 75 | C23×C7 | — | 2 | 2 | no | — |
| 76 | C23×C7 | — | 2 | 12 | no | — |
| 77 | C23×C7 | — | 4 | 48 | no | — |
| 78 | C23×C7 | — | 4 | 64 | no | — |
| 79 | C23×C7 | — | 4 | 128 | no | — |
| 80 | C23×C7 | — | 4 | 36 | no | — |
| 81 | C23×C7 | — | 4 | 12 | no | — |
| 82 | C23×C7 | — | 4 | 12 | no | — |
| 83 | C23×C7 | — | 4 | 12 | no | — |
| 84 | C23×C7 | — | 4 | 4 | no | — |
| 85 | C23×C7 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 86 | C23×C7 | C23×C4 | 1 | 1 | yes | characteristic |
| 87 | C23×C7 | C22×D4 | 1 | 2 | yes | — |
| 88 | C23×C7 | H(5,𝔽2) | 1 | 12 | yes | — |
| 89 | C7×D4 | — | 4 | 48 | no | — |
| 90 | C2×C4×C7 | — | 2 | 6 | no | — |
| 91 | C2×C4×C7 | — | 2 | 12 | no | — |
| 92 | C2×C4×C7 | — | 4 | 12 | no | — |
| 93 | C2×Dic7 | — | 4 | 24 | no | — |
| 94 | C2×Dic7 | — | 4 | 24 | no | — |
| 95 | C2×Dic7 | — | 4 | 8 | no | — |
| 96 | C25 | — | 4 | 12 | no | — |
| 97 | C25 | C2×Dic7 | 1 | 4 | yes | — |
| 98 | C25 | C22⋊D7 | 1 | 8 | yes | — |
| 99 | C25 | C22×D7 | 1 | 1 | yes | characteristic |
| 100 | C25 | C22×D7 | 1 | 3 | yes | — |
| 101 | C22×D4 | — | 2 | 12 | no | — |
| 102 | C22×D4 | — | 4 | 12 | no | — |
| 103 | C22×D4 | C2×Dic7 | 1 | 12 | yes | — |
| 104 | C23×C4 | — | 7 | 8 | no | — |
| 105 | C23×C4 | C22×D7 | 1 | 3 | yes | — |
| 106 | #120 | — | 2 | 32 | no | — |
| 107 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 108 | C2×C22⋊C4 | — | 14 | 24 | no | — |
| 109 | C2×C22⋊C4 | — | 28 | 24 | no | — |
| 110 | C2×C22⋊C4 | — | 28 | 12 | no | — |
| 111 | C2×C22⋊C4 | C2×Dic7 | 1 | 12 | yes | — |
| 112 | #95 | — | 28 | 12 | no | — |
| 113 | C22×C7 | — | 2 | 12 | no | — |
| 114 | C22×C7 | — | 2 | 24 | no | — |
| 115 | C22×C7 | — | 4 | 64 | no | — |
| 116 | C22×C7 | — | 4 | 24 | no | — |
| 117 | C22×C7 | — | 4 | 48 | no | — |
| 118 | C22×C7 | — | 4 | 6 | no | — |
| 119 | C22×C7 | — | 4 | 8 | no | — |
| 120 | C22×C7 | — | 8 | 8 | no | — |
| 121 | C22×C7 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 122 | C22×C7 | #518 | 1 | 6 | yes | — |
| 123 | C22×C7 | #534 | 1 | 12 | yes | — |
| 124 | C22×C7 | #560 | 1 | 12 | yes | — |
| 125 | C22×C7 | H(3,𝔽4) | 1 | 4 | yes | — |
| 126 | C4×C7 | — | 4 | 6 | no | — |
| 127 | Dic7 | — | 8 | 8 | no | — |
| 128 | C24 | — | 2 | 24 | no | — |
| 129 | C24 | — | 2 | 12 | no | — |
| 130 | C24 | — | 2 | 12 | no | — |
| 131 | C24 | — | 2 | 2 | no | — |
| 132 | C24 | — | 4 | 24 | no | — |
| 133 | C24 | — | 4 | 48 | no | — |
| 134 | C24 | — | 4 | 16 | no | — |
| 135 | C24 | — | 4 | 48 | no | — |
| 136 | C24 | C22⋊Dic7 | 1 | 16 | yes | complemented |
| 137 | C24 | C22×Dic7 | 1 | 2 | yes | — |
| 138 | C24 | C22×Dic7 | 1 | 6 | yes | — |
| 139 | C24 | C2×C22⋊D7 | 1 | 12 | yes | — |
| 140 | C24 | C23×D7 | 1 | 1 | yes | characteristic |
| 141 | C2×D4 | — | 2 | 24 | no | — |
| 142 | C2×D4 | — | 4 | 24 | no | — |
| 143 | C2×D4 | — | 4 | 48 | no | — |
| 144 | C22×C4 | — | 2 | 12 | no | — |
| 145 | C22×C4 | — | 2 | 6 | no | — |
| 146 | C22×C4 | — | 14 | 24 | no | — |
| 147 | C22×C4 | — | 14 | 24 | no | — |
| 148 | C22×C4 | — | 14 | 8 | no | — |
| 149 | C22×C4 | C22×Dic7 | 1 | 6 | yes | — |
| 150 | C22⋊C4 | — | 2 | 24 | no | — |
| 151 | C22⋊C4 | — | 28 | 48 | no | — |
| 152 | C22⋊C4 | — | 28 | 16 | no | — |
| 153 | C2×C7 | — | 4 | 12 | no | — |
| 154 | C2×C7 | — | 8 | 2 | no | — |
| 155 | C2×C7 | #2308 | 1 | 3 | yes | — |
| 156 | C2×C7 | #2375 | 1 | 4 | yes | — |
| 157 | C2×C7 | #2812 | 1 | 8 | yes | — |
| 158 | C23 | — | 2 | 12 | no | — |
| 159 | C23 | — | 2 | 24 | no | — |
| 160 | C23 | — | 2 | 24 | no | — |
| 161 | C23 | — | 2 | 24 | no | — |
| 162 | C23 | — | 2 | 24 | no | — |
| 163 | C23 | — | 2 | 12 | no | — |
| 164 | C23 | — | 2 | 2 | no | — |
| 165 | C23 | — | 4 | 48 | no | — |
| 166 | C23 | — | 4 | 64 | no | — |
| 167 | C23 | — | 4 | 128 | no | — |
| 168 | C23 | — | 4 | 12 | no | — |
| 169 | C23 | — | 4 | 36 | no | — |
| 170 | C23 | — | 4 | 12 | no | — |
| 171 | C23 | — | 4 | 4 | no | — |
| 172 | C23 | — | 4 | 12 | no | — |
| 173 | C23 | C2×C22⋊Dic7 | 1 | 12 | yes | — |
| 174 | C23 | #6548 | 1 | 12 | yes | — |
| 175 | C23 | C23×Dic7 | 1 | 1 | yes | characteristic |
| 176 | C23 | C22×C22⋊D7 | 1 | 2 | yes | — |
| 177 | D4 | — | 4 | 48 | no | — |
| 178 | C2×C4 | — | 2 | 12 | no | — |
| 179 | C2×C4 | — | 2 | 6 | no | — |
| 180 | C2×C4 | — | 4 | 12 | no | — |
| 181 | C2×C4 | — | 28 | 24 | no | — |
| 182 | C2×C4 | — | 28 | 8 | no | — |
| 183 | C2×C4 | — | 28 | 24 | no | — |
| 184 | C7 | #14657 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 185 | C22 | — | 2 | 12 | no | — |
| 186 | C22 | — | 2 | 24 | no | — |
| 187 | C22 | — | 4 | 64 | no | — |
| 188 | C22 | — | 4 | 24 | no | — |
| 189 | C22 | — | 4 | 48 | no | — |
| 190 | C22 | — | 4 | 6 | no | — |
| 191 | C22 | — | 4 | 8 | no | — |
| 192 | C22 | — | 8 | 8 | no | — |
| 193 | C22 | #90285 | 1 | 4 | yes | — |
| 194 | C22 | #90286 | 1 | 12 | yes | — |
| 195 | C22 | #90489 | 1 | 6 | yes | — |
| 196 | C22 | #90495 | 1 | 12 | yes | — |
| 197 | C22 | C22×C22⋊Dic7 | 1 | 1 | yes | characteristic |
| 198 | C4 | — | 4 | 6 | no | — |
| 199 | C4 | — | 56 | 8 | no | — |
| 200 | C2 | — | 4 | 12 | no | — |
| 201 | C2 | — | 8 | 2 | no | — |
| 202 | C2 | #11740025 | 1 | 4 | yes | — |
| 203 | C2 | #11740056 | 1 | 8 | yes | — |
| 204 | C2 | #11741488 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D7 |
| commutator | C23×C7 | C23×C4 |
| Fitting, prime radical | C22×C7×#120 | C2 |
| socle, cosocle | C24×C7 | C24 |
| hypercenter | C22×#120 | D7 |
| hypocenter | C7 | #14657 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C24 ◁ C22×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#90019 | C2 | semidirect |
| 3. #11729885 | C2 | semidirect |
| 4. #11729302 | C2 | semidirect |
| 8. C2×C7×#120 | C4 | semidirect |
| 11. C22×C22⋊Dic7 | C22 | semidirect |
| 13. C2×C7⋊#95 | C22 | semidirect |
| 44. C24×C7 | C22⋊C4 | semidirect |
| 64. C2×#120 | Dic7 | semidirect |
| 136. C24 | C22⋊Dic7 | semidirect |
| 184. C7 | #14657 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×#120 | 2 |
| C2×#90019 | 2 |
| #11729885 | 2 |
| #11729302 | 2 |
| #14657 | 7 |