Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
326 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 61 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 | #11739160 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11726842 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C7×#1355 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11725589 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11725326 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | D7×#127 | C4 | 1 | 1 | yes | characteristic |
| 7 | #90379 | C4 | 1 | 1 | yes | characteristic |
| 8 | C7×#490 | C22 | 1 | 1 | yes | characteristic |
| 9 | #89843 | C4 | 1 | 1 | yes | characteristic |
| 10 | #89839 | C4 | 1 | 1 | yes | characteristic |
| 11 | #89542 | — | 2 | 2 | no | — |
| 12 | #89532 | — | 2 | 2 | no | — |
| 13 | #89485 | C22 | 1 | 1 | yes | characteristic |
| 14 | #89479 | C22 | 1 | 1 | yes | characteristic |
| 15 | C7×#347 | C22 | 1 | 2 | yes | complemented |
| 16 | #89309 | — | 2 | 2 | no | — |
| 17 | #89306 | — | 2 | 2 | no | — |
| 18 | #89261 | C22 | 1 | 2 | yes | complemented |
| 19 | #11915 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C2×D4×D7 | — | 2 | 1 | no | — |
| 21 | C2×D4×D7 | — | 4 | 1 | no | — |
| 22 | C7×#127 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C7×#125 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | #6503 | — | 2 | 1 | no | — |
| 25 | #6503 | — | 4 | 1 | no | — |
| 26 | C2×#1131 | D4 | 1 | 2 | yes | — |
| 27 | #6459 | — | 4 | 1 | no | — |
| 28 | #6455 | — | 2 | 1 | no | — |
| 29 | #6451 | — | 2 | 1 | no | — |
| 30 | #6450 | — | 4 | 1 | no | — |
| 31 | #6437 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C42×D7 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | #6433 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C7×M32 | — | 2 | 2 | no | — |
| 35 | C7×#102 | — | 2 | 2 | no | — |
| 36 | C7×#97 | C23 | 1 | 1 | yes | characteristic |
| 37 | #6406 | — | 2 | 2 | no | — |
| 38 | #6395 | — | 2 | 2 | no | — |
| 39 | C8×Dic7 | D4 | 1 | 2 | yes | — |
| 40 | #6386 | — | 2 | 2 | no | — |
| 41 | #6383 | — | 2 | 2 | no | — |
| 42 | #6377 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | #3138 | — | 7 | 1 | no | — |
| 44 | #2165 | — | 7 | 2 | no | — |
| 45 | #1355 | — | 7 | 1 | no | — |
| 46 | #1355 | — | 7 | 2 | no | — |
| 47 | #1355 | D7 | 1 | 1 | yes | characteristic, complemented |
| 48 | C23×D7 | — | 4 | 1 | no | — |
| 49 | C2×C7×D4 | — | 2 | 1 | no | — |
| 50 | C2×C7×D4 | — | 4 | 1 | no | — |
| 51 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 52 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 53 | D4×D7 | — | 4 | 4 | no | — |
| 54 | D4×D7 | — | 8 | 4 | no | — |
| 55 | C2×D28 | — | 2 | 1 | no | — |
| 56 | C2×D28 | — | 4 | 1 | no | — |
| 57 | C2×C4×D7 | — | 2 | 1 | no | — |
| 58 | C2×C4×D7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C2×Dic14 | — | 4 | 1 | no | — |
| 60 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 61 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 62 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 63 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 64 | (C2×D7)⋊C4 | — | 4 | 1 | no | — |
| 65 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 66 | Dic7⋊C4 | — | 4 | 1 | no | — |
| 67 | Dic7⋊C4 | — | 4 | 1 | no | — |
| 68 | C4×Dic7 | — | 2 | 1 | no | — |
| 69 | C4×Dic7 | — | 4 | 1 | no | — |
| 70 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×C7⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 72 | #1131 | — | 4 | 2 | no | — |
| 73 | C7×C16 | — | 4 | 2 | no | — |
| 74 | C7⋊C16 | — | 4 | 2 | no | — |
| 75 | #533 | — | 7 | 1 | no | — |
| 76 | C2×#127 | — | 7 | 1 | no | — |
| 77 | #490 | — | 7 | 1 | no | — |
| 78 | #490 | — | 7 | 1 | no | — |
| 79 | #490 | — | 7 | 1 | no | — |
| 80 | #490 | C2×D7 | 1 | 1 | yes | characteristic |
| 81 | #486 | — | 14 | 2 | no | — |
| 82 | #418 | — | 14 | 2 | no | — |
| 83 | C4×M16 | — | 7 | 1 | no | — |
| 84 | #364 | — | 14 | 2 | no | — |
| 85 | #349 | — | 14 | 2 | no | — |
| 86 | #347 | — | 7 | 2 | no | — |
| 87 | #347 | C2×D7 | 1 | 2 | yes | complemented |
| 88 | C23×C7 | — | 4 | 1 | no | — |
| 89 | C22×D7 | — | 4 | 1 | no | — |
| 90 | C22×D7 | — | 8 | 2 | no | — |
| 91 | C22×D7 | — | 8 | 4 | no | — |
| 92 | C22×D7 | #100 | 1 | 1 | yes | characteristic |
| 93 | C7×D4 | — | 4 | 2 | no | — |
| 94 | C7×D4 | — | 8 | 2 | no | — |
| 95 | C2×C4×C7 | — | 2 | 1 | no | — |
| 96 | C2×C4×C7 | — | 4 | 1 | no | — |
| 97 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C22⋊D7 | — | 8 | 2 | no | — |
| 99 | C22⋊D7 | — | 8 | 4 | no | — |
| 100 | C2×Dic7 | — | 2 | 1 | no | — |
| 101 | C2×Dic7 | — | 4 | 1 | no | — |
| 102 | C2×Dic7 | #100 | 1 | 1 | yes | characteristic |
| 103 | D28 | — | 4 | 2 | no | — |
| 104 | D28 | — | 8 | 2 | no | — |
| 105 | C4×D7 | — | 2 | 2 | no | — |
| 106 | C4×D7 | — | 4 | 2 | no | — |
| 107 | Dic14 | — | 8 | 2 | no | — |
| 108 | C7×C8 | — | 2 | 2 | no | — |
| 109 | C7⋊C8 | — | 2 | 2 | no | — |
| 110 | C22×D4 | — | 14 | 1 | no | — |
| 111 | C22×D4 | — | 28 | 1 | no | — |
| 112 | C2×M16 | — | 7 | 2 | no | — |
| 113 | C4⋊Q8 | — | 14 | 1 | no | — |
| 114 | #127 | — | 7 | 1 | no | — |
| 115 | #127 | — | 14 | 1 | no | — |
| 116 | #127 | — | 28 | 1 | no | — |
| 117 | #127 | C4×D7 | 1 | 1 | yes | characteristic |
| 118 | #125 | — | 7 | 1 | no | — |
| 119 | #125 | C4×D7 | 1 | 1 | yes | characteristic |
| 120 | C22⋊Q8 | — | 28 | 1 | no | — |
| 121 | C4×Q8 | — | 28 | 1 | no | — |
| 122 | #117 | — | 14 | 1 | no | — |
| 123 | C2×C42 | — | 7 | 1 | no | — |
| 124 | M32 | — | 2 | 2 | no | — |
| 125 | M32 | — | 14 | 2 | no | — |
| 126 | #102 | — | 2 | 2 | no | — |
| 127 | #102 | — | 14 | 2 | no | — |
| 128 | #102 | — | 14 | 2 | no | — |
| 129 | #102 | — | 14 | 2 | no | — |
| 130 | #97 | — | 7 | 1 | no | — |
| 131 | #97 | C22×D7 | 1 | 1 | yes | characteristic |
| 132 | C4×C8 | — | 7 | 2 | no | — |
| 133 | C22×C7 | — | 8 | 2 | no | — |
| 134 | C22×C7 | — | 8 | 1 | no | — |
| 135 | C22×C7 | C2×#100 | 1 | 1 | yes | characteristic |
| 136 | C2×D7 | — | 2 | 1 | no | — |
| 137 | C2×D7 | — | 4 | 1 | no | — |
| 138 | C2×D7 | — | 8 | 2 | no | — |
| 139 | C2×D7 | — | 8 | 1 | no | — |
| 140 | C2×D7 | — | 16 | 4 | no | — |
| 141 | C4×C7 | — | 4 | 1 | no | — |
| 142 | C4×C7 | — | 8 | 1 | no | — |
| 143 | C4×C7 | #421 | 1 | 2 | yes | — |
| 144 | Dic7 | — | 2 | 2 | no | — |
| 145 | Dic7 | — | 2 | 1 | no | — |
| 146 | Dic7 | — | 8 | 1 | no | — |
| 147 | C24 | — | 28 | 1 | no | — |
| 148 | C2×Q8 | — | 28 | 1 | no | — |
| 149 | C2×D4 | — | 2 | 1 | no | — |
| 150 | C2×D4 | — | 4 | 1 | no | — |
| 151 | C2×D4 | — | 14 | 1 | no | — |
| 152 | C2×D4 | — | 28 | 1 | no | — |
| 153 | C2×D4 | — | 28 | 1 | no | — |
| 154 | C2×D4 | — | 28 | 4 | no | — |
| 155 | C2×D4 | — | 28 | 2 | no | — |
| 156 | C2×D4 | — | 56 | 4 | no | — |
| 157 | C22×C4 | — | 7 | 1 | no | — |
| 158 | C22×C4 | — | 14 | 1 | no | — |
| 159 | M16 | — | 28 | 2 | no | — |
| 160 | C2×C8 | — | 7 | 2 | no | — |
| 161 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 162 | C4⋊C4 | — | 2 | 1 | no | — |
| 163 | C4⋊C4 | — | 4 | 1 | no | — |
| 164 | C4⋊C4 | — | 14 | 1 | no | — |
| 165 | C4⋊C4 | — | 28 | 1 | no | — |
| 166 | C4⋊C4 | — | 28 | 1 | no | — |
| 167 | C22⋊C4 | — | 28 | 1 | no | — |
| 168 | C42 | — | 7 | 1 | no | — |
| 169 | C42 | — | 14 | 1 | no | — |
| 170 | C42 | — | 28 | 1 | no | — |
| 171 | C42 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 172 | C16 | — | 4 | 2 | no | — |
| 173 | C16 | — | 28 | 2 | no | — |
| 174 | C2×C7 | — | 2 | 1 | no | — |
| 175 | C2×C7 | — | 16 | 1 | no | — |
| 176 | C2×C7 | #1533 | 1 | 1 | yes | characteristic |
| 177 | D7 | — | 4 | 1 | no | — |
| 178 | D7 | — | 16 | 1 | no | — |
| 179 | C23 | — | 4 | 1 | no | — |
| 180 | C23 | — | 7 | 1 | no | — |
| 181 | C23 | — | 28 | 1 | no | — |
| 182 | C23 | — | 56 | 2 | no | — |
| 183 | C23 | — | 56 | 4 | no | — |
| 184 | Q8 | — | 56 | 2 | no | — |
| 185 | D4 | — | 4 | 2 | no | — |
| 186 | D4 | — | 8 | 2 | no | — |
| 187 | D4 | — | 28 | 2 | no | — |
| 188 | D4 | — | 56 | 2 | no | — |
| 189 | D4 | — | 56 | 2 | no | — |
| 190 | D4 | — | 56 | 4 | no | — |
| 191 | C2×C4 | — | 2 | 1 | no | — |
| 192 | C2×C4 | — | 4 | 1 | no | — |
| 193 | C2×C4 | — | 7 | 1 | no | — |
| 194 | C2×C4 | — | 14 | 1 | no | — |
| 195 | C2×C4 | — | 14 | 2 | no | — |
| 196 | C2×C4 | — | 28 | 2 | no | — |
| 197 | C2×C4 | — | 28 | 1 | no | — |
| 198 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 199 | C8 | — | 2 | 2 | no | — |
| 200 | C8 | — | 14 | 2 | no | — |
| 201 | C7 | #11915 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 202 | C22 | — | 8 | 2 | no | — |
| 203 | C22 | — | 8 | 1 | no | — |
| 204 | C22 | — | 14 | 1 | no | — |
| 205 | C22 | — | 28 | 1 | no | — |
| 206 | C22 | — | 56 | 2 | no | — |
| 207 | C22 | — | 56 | 1 | no | — |
| 208 | C22 | — | 112 | 4 | no | — |
| 209 | C22 | D7×#100 | 1 | 1 | yes | characteristic |
| 210 | C4 | — | 4 | 1 | no | — |
| 211 | C4 | — | 8 | 1 | no | — |
| 212 | C4 | — | 14 | 2 | no | — |
| 213 | C4 | — | 14 | 1 | no | — |
| 214 | C4 | — | 56 | 1 | no | — |
| 215 | C4 | #89593 | 1 | 2 | yes | — |
| 216 | C2 | — | 2 | 1 | no | — |
| 217 | C2 | — | 16 | 1 | no | — |
| 218 | C2 | — | 28 | 1 | no | — |
| 219 | C2 | — | 112 | 1 | no | — |
| 220 | C2 | #11726101 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11726842 | C2 | semidirect |
| 3. C7×#1355 | C2 | semidirect |
| 4. #11725589 | C2 | semidirect |
| 5. #11725326 | C2 | semidirect |
| 15. C7×#347 | C22 | semidirect |
| 18. #89261 | C22 | semidirect |
| 47. #1355 | D7 | semidirect |
| 87. #347 | C2×D7 | semidirect |
| 201. C7 | #11915 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11739160 | 2 |
| #11726842 | 2 |
| C7×#1355 | 2 |
| #11725589 | 2 |
| #11725326 | 2 |
| #11915 | 7 |