← #13051007
Subgroups — #13051007
Order 1344.
228 conjugacy classes of subgroups.
The subgroup lattice has 196 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C3⋊(C7⋊C8) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C7×C4⋊Dic3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C3×C4⋊Dic7 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #89324 | — | 3 | 1 | no | maximal, Hall |
| 5 | C22×C3×C4×C7 | C22 | 1 | 1 | yes | characteristic |
| 6 | C22×C7×Dic3 | — | 2 | 1 | no | — |
| 7 | C22×C3×Dic7 | — | 2 | 1 | no | — |
| 8 | C2×C3⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 9 | C2×C3⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 10 | C2×C3⋊(C7⋊C8) | C4 | 1 | 2 | yes | complemented |
| 11 | C7×C4⋊Dic3 | — | 2 | 1 | no | — |
| 12 | C7×C4⋊Dic3 | C4 | 1 | 2 | yes | complemented |
| 13 | C3×C4⋊Dic7 | — | 2 | 1 | no | — |
| 14 | C3×C4⋊Dic7 | C4 | 1 | 2 | yes | complemented |
| 15 | C2×C7×C4⋊C4 | — | 3 | 1 | no | — |
| 16 | C2×C4⋊Dic7 | D3 | 1 | 1 | yes | characteristic |
| 17 | C22×C7⋊C8 | — | 3 | 1 | no | — |
| 18 | #4529 | — | 7 | 1 | no | maximal, Hall |
| 19 | C23×C3×C7 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×C3×C4×C7 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C2×C3×C4×C7 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C3×C4×C7 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C2×C3×C4×C7 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×C3×C4×C7 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×C3×C4×C7 | Q8 | 1 | 1 | yes | characteristic |
| 26 | C2×C7×Dic3 | — | 2 | 2 | no | — |
| 27 | C2×C7×Dic3 | — | 4 | 1 | no | — |
| 28 | C2×C7×Dic3 | — | 4 | 1 | no | — |
| 29 | C2×C3×Dic7 | — | 2 | 2 | no | — |
| 30 | C2×C3×Dic7 | — | 4 | 1 | no | — |
| 31 | C2×C3×Dic7 | — | 4 | 1 | no | — |
| 32 | C3⋊(C7⋊C8) | — | 2 | 2 | no | — |
| 33 | C22×C4×C7 | — | 6 | 1 | no | — |
| 34 | C22×C4×C7 | C2×D3 | 1 | 1 | yes | characteristic |
| 35 | C22×Dic7 | — | 2 | 1 | no | — |
| 36 | C7×C4⋊C4 | — | 3 | 2 | no | — |
| 37 | C7×C4⋊C4 | — | 6 | 1 | no | — |
| 38 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 39 | C4⋊Dic7 | Dic3 | 1 | 2 | yes | complemented |
| 40 | C2×C7⋊C8 | — | 3 | 2 | no | — |
| 41 | C2×C7⋊C8 | — | 6 | 1 | no | — |
| 42 | C2×C7⋊C8 | — | 6 | 1 | no | — |
| 43 | C2×C3×C4⋊C4 | — | 7 | 1 | no | — |
| 44 | C2×C4⋊Dic3 | D7 | 1 | 1 | yes | characteristic |
| 45 | C22×C3⋊C8 | — | 7 | 1 | no | — |
| 46 | C22×C3×C7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 47 | C22×C3×C7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 48 | C22×C3×C7 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 49 | C22×C3×C7 | D8 | 1 | 1 | yes | characteristic |
| 50 | C22×C3×C7 | SD16 | 1 | 1 | yes | characteristic |
| 51 | C22×C3×C7 | SD16 | 1 | 1 | yes | characteristic |
| 52 | C22×C3×C7 | Q16 | 1 | 1 | yes | characteristic |
| 53 | C3×C4×C7 | C42 | 1 | 1 | yes | characteristic |
| 54 | C3×C4×C7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 55 | C3×C4×C7 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 56 | C3×C4×C7 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 57 | C3×Dic7 | — | 4 | 2 | no | — |
| 58 | C7×Dic3 | — | 4 | 2 | no | — |
| 59 | #340 | — | 21 | 1 | no | Sylow |
| 60 | C23×C7 | #67 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C7 | — | 6 | 2 | no | — |
| 62 | C2×C4×C7 | — | 12 | 1 | no | — |
| 63 | C2×C4×C7 | — | 12 | 1 | no | — |
| 64 | C2×C4×C7 | Dic6 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×C7 | C4×D3 | 1 | 1 | yes | characteristic |
| 66 | C2×C4×C7 | C4×D3 | 1 | 1 | yes | characteristic |
| 67 | C2×C4×C7 | D12 | 1 | 1 | yes | characteristic |
| 68 | C2×C4×C7 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 69 | C2×C4×C7 | #67 | 1 | 1 | yes | characteristic |
| 70 | C2×Dic7 | — | 2 | 2 | no | — |
| 71 | C2×Dic7 | — | 4 | 1 | no | — |
| 72 | C2×Dic7 | — | 4 | 1 | no | — |
| 73 | C7⋊C8 | — | 6 | 2 | no | — |
| 74 | C22×C3×C4 | — | 14 | 1 | no | — |
| 75 | C22×C3×C4 | C2×D7 | 1 | 1 | yes | characteristic |
| 76 | C22×Dic3 | — | 2 | 1 | no | — |
| 77 | C3×C4⋊C4 | — | 7 | 2 | no | — |
| 78 | C3×C4⋊C4 | — | 14 | 1 | no | — |
| 79 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 80 | C4⋊Dic3 | Dic7 | 1 | 2 | yes | complemented |
| 81 | C2×C3⋊C8 | — | 7 | 2 | no | — |
| 82 | C2×C3⋊C8 | — | 14 | 1 | no | — |
| 83 | C2×C3⋊C8 | — | 14 | 1 | no | — |
| 84 | C2×C3×C7 | #95 | 1 | 1 | yes | characteristic |
| 85 | C2×C3×C7 | #102 | 1 | 1 | yes | characteristic |
| 86 | C2×C3×C7 | #102 | 1 | 1 | yes | characteristic |
| 87 | C2×C3×C7 | #103 | 1 | 1 | yes | characteristic |
| 88 | C2×C3×C7 | #103 | 1 | 1 | yes | characteristic |
| 89 | C2×C3×C7 | #106 | 1 | 1 | yes | characteristic |
| 90 | C2×C3×C7 | #107 | 1 | 1 | yes | characteristic |
| 91 | C22×C8 | — | 21 | 1 | no | — |
| 92 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 93 | C2×C4⋊C4 | — | 7 | 1 | no | — |
| 94 | C22×C7 | Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 95 | C22×C7 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C22×C7 | #213 | 1 | 1 | yes | characteristic |
| 97 | C22×C7 | #214 | 1 | 1 | yes | characteristic |
| 98 | C22×C7 | #215 | 1 | 1 | yes | characteristic |
| 99 | C22×C7 | #216 | 1 | 1 | yes | characteristic |
| 100 | C22×C7 | #217 | 1 | 1 | yes | characteristic |
| 101 | C4×C7 | — | 12 | 2 | no | — |
| 102 | C4×C7 | C4×Dic3 | 1 | 1 | yes | characteristic |
| 103 | C4×C7 | Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C4×C7 | C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 105 | C4×C7 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 106 | Dic7 | — | 4 | 2 | no | — |
| 107 | C23×C3 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 108 | C2×C3×C4 | — | 14 | 2 | no | — |
| 109 | C2×C3×C4 | — | 28 | 1 | no | — |
| 110 | C2×C3×C4 | — | 28 | 1 | no | — |
| 111 | C2×C3×C4 | Dic14 | 1 | 1 | yes | characteristic |
| 112 | C2×C3×C4 | C4×D7 | 1 | 1 | yes | characteristic |
| 113 | C2×C3×C4 | C4×D7 | 1 | 1 | yes | characteristic |
| 114 | C2×C3×C4 | D28 | 1 | 1 | yes | characteristic |
| 115 | C2×C3×C4 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 116 | C2×C3×C4 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 117 | C2×Dic3 | — | 2 | 2 | no | — |
| 118 | C2×Dic3 | — | 4 | 1 | no | — |
| 119 | C2×Dic3 | — | 4 | 1 | no | — |
| 120 | C3⋊C8 | — | 14 | 2 | no | — |
| 121 | C3×C7 | #340 | 1 | 1 | yes | characteristic, complemented, Hall |
| 122 | C22×C4 | — | 6 | 1 | no | — |
| 123 | C22×C4 | — | 14 | 1 | no | — |
| 124 | C22×C4 | D3×D7 | 1 | 1 | yes | characteristic |
| 125 | C2×C8 | — | 21 | 2 | no | — |
| 126 | C2×C8 | — | 42 | 1 | no | — |
| 127 | C2×C8 | — | 42 | 1 | no | — |
| 128 | C4⋊C4 | — | 3 | 2 | no | — |
| 129 | C4⋊C4 | — | 6 | 1 | no | — |
| 130 | C4⋊C4 | — | 7 | 2 | no | — |
| 131 | C4⋊C4 | — | 14 | 1 | no | — |
| 132 | C2×C7 | #807 | 1 | 1 | yes | characteristic |
| 133 | C2×C7 | #808 | 1 | 1 | yes | characteristic |
| 134 | C2×C7 | #809 | 1 | 1 | yes | characteristic |
| 135 | C2×C7 | #810 | 1 | 1 | yes | characteristic |
| 136 | C2×C7 | C3⋊#95 | 1 | 1 | yes | characteristic |
| 137 | C2×C7 | #832 | 1 | 1 | yes | characteristic |
| 138 | C2×C7 | #835 | 1 | 1 | yes | characteristic |
| 139 | C22×C3 | Dic7⋊C4 | 1 | 1 | yes | characteristic |
| 140 | C22×C3 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 141 | C22×C3 | #1141 | 1 | 1 | yes | characteristic |
| 142 | C22×C3 | #1142 | 1 | 1 | yes | characteristic |
| 143 | C22×C3 | #1143 | 1 | 1 | yes | characteristic |
| 144 | C22×C3 | #1144 | 1 | 1 | yes | characteristic |
| 145 | C22×C3 | C22⋊Dic7 | 1 | 1 | yes | characteristic |
| 146 | C3×C4 | — | 28 | 2 | no | — |
| 147 | C3×C4 | C4×Dic7 | 1 | 1 | yes | characteristic |
| 148 | C3×C4 | Dic7⋊C4 | 1 | 1 | yes | characteristic |
| 149 | C3×C4 | C4⋊Dic7 | 1 | 1 | yes | characteristic |
| 150 | C3×C4 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 151 | Dic3 | — | 4 | 2 | no | — |
| 152 | C23 | #4245 | 1 | 1 | yes | characteristic |
| 153 | C2×C4 | — | 6 | 2 | no | — |
| 154 | C2×C4 | — | 12 | 1 | no | — |
| 155 | C2×C4 | — | 12 | 1 | no | — |
| 156 | C2×C4 | — | 14 | 2 | no | — |
| 157 | C2×C4 | — | 28 | 1 | no | — |
| 158 | C2×C4 | — | 28 | 1 | no | — |
| 159 | C2×C4 | Dic3×D7 | 1 | 1 | yes | characteristic |
| 160 | C2×C4 | D3×Dic7 | 1 | 1 | yes | characteristic |
| 161 | C2×C4 | D21⋊C4 | 1 | 1 | yes | characteristic |
| 162 | C2×C4 | #4246 | 1 | 1 | yes | characteristic |
| 163 | C2×C4 | #4247 | 1 | 1 | yes | characteristic |
| 164 | C2×C4 | #4248 | 1 | 1 | yes | characteristic |
| 165 | C8 | — | 42 | 2 | no | — |
| 166 | C7 | #4529 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 167 | C2×C3 | #6381 | 1 | 1 | yes | characteristic |
| 168 | C2×C3 | #6382 | 1 | 1 | yes | characteristic |
| 169 | C2×C3 | #6383 | 1 | 1 | yes | characteristic |
| 170 | C2×C3 | #6384 | 1 | 1 | yes | characteristic |
| 171 | C2×C3 | C7⋊#95 | 1 | 1 | yes | characteristic |
| 172 | C2×C3 | #6406 | 1 | 1 | yes | characteristic |
| 173 | C2×C3 | #6409 | 1 | 1 | yes | characteristic |
| 174 | C22 | #66560 | 1 | 1 | yes | characteristic |
| 175 | C22 | #66563 | 1 | 1 | yes | characteristic |
| 176 | C22 | #66564 | 1 | 1 | yes | characteristic |
| 177 | C22 | #66569 | 1 | 1 | yes | characteristic |
| 178 | C22 | (C2×D7)⋊Dic3 | 1 | 1 | yes | characteristic |
| 179 | C22 | (C2×D3)⋊Dic7 | 1 | 1 | yes | characteristic |
| 180 | C22 | #66577 | 1 | 1 | yes | characteristic |
| 181 | C4 | — | 12 | 2 | no | — |
| 182 | C4 | — | 28 | 2 | no | — |
| 183 | C4 | Dic3×Dic7 | 1 | 1 | yes | characteristic |
| 184 | C4 | #66575 | 1 | 1 | yes | characteristic |
| 185 | C4 | Dic7⋊Dic3 | 1 | 1 | yes | characteristic |
| 186 | C4 | Dic3⋊Dic7 | 1 | 1 | yes | characteristic |
| 187 | C3 | #89324 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 188 | C2 | #10620499 | 1 | 1 | yes | characteristic |
| 189 | C2 | #10620501 | 1 | 1 | yes | characteristic |
| 190 | C2 | #10620505 | 1 | 1 | yes | characteristic |
| 191 | C2 | #10620507 | 1 | 1 | yes | characteristic |
| 192 | C2 | #10620521 | 1 | 1 | yes | characteristic |
| 193 | C2 | #10620524 | 1 | 1 | yes | characteristic |
| 194 | C2 | #10620529 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta