Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
360 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 69 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 | #11739207 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11727052 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C7×#1359 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11725593 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11725378 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#6478 | — | 2 | 2 | no | — |
| 7 | D7×C4⋊Q8 | C4 | 1 | 1 | yes | characteristic |
| 8 | #90407 | C4 | 1 | 1 | yes | characteristic |
| 9 | C7×#492 | C22 | 1 | 1 | yes | characteristic |
| 10 | #89939 | — | 2 | 2 | no | — |
| 11 | #89864 | C4 | 1 | 1 | yes | characteristic |
| 12 | #89848 | C4 | 1 | 1 | yes | characteristic |
| 13 | #89471 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×#1131 | C22 | 1 | 1 | yes | characteristic |
| 15 | C7×#364 | C22 | 1 | 2 | yes | — |
| 16 | #89309 | — | 2 | 2 | no | — |
| 17 | #89307 | — | 2 | 2 | no | — |
| 18 | #89258 | C22 | 1 | 2 | yes | — |
| 19 | #11993 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C2×Q8×D7 | — | 2 | 1 | no | — |
| 21 | C2×#1159 | — | 2 | 1 | no | — |
| 22 | C2×C7×SD16 | — | 2 | 2 | no | — |
| 23 | C7×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C7×#127 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | #6507 | — | 2 | 1 | no | — |
| 26 | C2×#1144 | — | 2 | 2 | no | — |
| 27 | #6501 | — | 4 | 1 | no | — |
| 28 | #6499 | — | 2 | 1 | no | — |
| 29 | D4×Dic7 | — | 4 | 1 | no | — |
| 30 | C2×#1142 | — | 2 | 2 | no | — |
| 31 | #6478 | — | 4 | 4 | no | — |
| 32 | #6478 | — | 4 | 4 | no | — |
| 33 | C2×Dic28 | — | 2 | 2 | no | — |
| 34 | C2×#1131 | D4 | 1 | 2 | yes | — |
| 35 | D7×C4⋊C4 | — | 4 | 1 | no | — |
| 36 | #6451 | — | 4 | 1 | no | — |
| 37 | C42×D7 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | #6432 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | #6432 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C7×M32 | — | 2 | 2 | no | — |
| 41 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 42 | #6389 | D4 | 1 | 2 | yes | — |
| 43 | #6386 | — | 2 | 2 | no | — |
| 44 | C4×C7⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | #3152 | — | 7 | 1 | no | — |
| 46 | #2165 | — | 7 | 2 | no | — |
| 47 | #1359 | — | 7 | 1 | no | — |
| 48 | #1359 | D7 | 1 | 1 | yes | characteristic, complemented |
| 49 | #1358 | — | 7 | 2 | no | — |
| 50 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 51 | C2×C7×D4 | — | 2 | 1 | no | — |
| 52 | C2×C7×D4 | — | 4 | 1 | no | — |
| 53 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 54 | C22×Dic7 | — | 4 | 1 | no | — |
| 55 | Q8×D7 | — | 4 | 4 | no | — |
| 56 | #1159 | — | 4 | 4 | no | — |
| 57 | C2×C4×D7 | — | 2 | 1 | no | — |
| 58 | C2×C4×D7 | — | 4 | 1 | no | — |
| 59 | C2×C4×D7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C2×Dic14 | — | 2 | 1 | no | — |
| 61 | C2×Dic14 | — | 2 | 1 | no | — |
| 62 | C2×Dic14 | — | 4 | 1 | no | — |
| 63 | C7×SD16 | — | 4 | 2 | no | — |
| 64 | C7×SD16 | — | 4 | 2 | no | — |
| 65 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 66 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 67 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C22⋊Dic7 | — | 4 | 2 | no | — |
| 69 | #1144 | — | 4 | 4 | no | — |
| 70 | #1142 | — | 4 | 4 | no | — |
| 71 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 72 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 73 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 74 | C4×Dic7 | — | 2 | 1 | no | — |
| 75 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C2×C7⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 77 | Dic28 | — | 4 | 2 | no | — |
| 78 | Dic28 | — | 4 | 2 | no | — |
| 79 | #1131 | — | 2 | 2 | no | — |
| 80 | #1131 | — | 2 | 2 | no | — |
| 81 | C7×C16 | — | 4 | 2 | no | — |
| 82 | C7⋊C16 | — | 4 | 2 | no | — |
| 83 | C2×#137 | — | 14 | 2 | no | — |
| 84 | #532 | — | 7 | 1 | no | — |
| 85 | C2×C4⋊Q8 | — | 7 | 1 | no | — |
| 86 | #497 | — | 14 | 2 | no | — |
| 87 | #494 | — | 7 | 1 | no | — |
| 88 | #494 | — | 7 | 1 | no | — |
| 89 | #492 | — | 7 | 1 | no | — |
| 90 | #492 | C2×D7 | 1 | 1 | yes | characteristic |
| 91 | C4×M16 | — | 7 | 1 | no | — |
| 92 | #364 | — | 7 | 2 | no | — |
| 93 | #364 | C2×D7 | 1 | 2 | yes | — |
| 94 | #349 | — | 14 | 2 | no | — |
| 95 | #347 | — | 14 | 2 | no | — |
| 96 | C23×C7 | — | 4 | 1 | no | — |
| 97 | C22×D7 | #100 | 1 | 1 | yes | characteristic |
| 98 | C7×Q8 | — | 4 | 2 | no | — |
| 99 | C7×D4 | — | 4 | 2 | no | — |
| 100 | C7×D4 | — | 8 | 2 | no | — |
| 101 | C2×C4×C7 | — | 2 | 1 | no | — |
| 102 | C2×C4×C7 | — | 4 | 1 | no | — |
| 103 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C22⋊D7 | — | 8 | 2 | no | — |
| 105 | C2×Dic7 | — | 2 | 1 | no | — |
| 106 | C2×Dic7 | — | 4 | 1 | no | — |
| 107 | C2×Dic7 | — | 4 | 1 | no | — |
| 108 | C2×Dic7 | — | 8 | 2 | no | — |
| 109 | C2×Dic7 | #100 | 1 | 1 | yes | characteristic |
| 110 | C4×D7 | — | 2 | 2 | no | — |
| 111 | C4×D7 | — | 4 | 2 | no | — |
| 112 | C4×D7 | — | 8 | 2 | no | — |
| 113 | Dic14 | — | 4 | 2 | no | — |
| 114 | Dic14 | — | 4 | 2 | no | — |
| 115 | Dic14 | — | 8 | 2 | no | — |
| 116 | C7×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 117 | C7×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 118 | C7⋊C8 | — | 2 | 2 | no | — |
| 119 | C2×#41 | — | 14 | 1 | no | — |
| 120 | C22×Q8 | — | 14 | 1 | no | — |
| 121 | #137 | — | 28 | 4 | no | — |
| 122 | #137 | — | 28 | 4 | no | — |
| 123 | C2×Q16 | — | 14 | 2 | no | — |
| 124 | C2×Q16 | — | 14 | 2 | no | — |
| 125 | C2×SD16 | — | 2 | 2 | no | — |
| 126 | C2×SD16 | — | 14 | 2 | no | — |
| 127 | C2×M16 | — | 7 | 2 | no | — |
| 128 | C4⋊Q8 | — | 7 | 1 | no | — |
| 129 | C4⋊Q8 | — | 7 | 1 | no | — |
| 130 | C4⋊Q8 | — | 14 | 1 | no | — |
| 131 | C4⋊Q8 | — | 28 | 1 | no | — |
| 132 | C4⋊Q8 | C4×D7 | 1 | 1 | yes | characteristic |
| 133 | #127 | C4×D7 | 1 | 1 | yes | characteristic |
| 134 | #124 | — | 14 | 1 | no | — |
| 135 | #121 | — | 28 | 1 | no | — |
| 136 | C4×D4 | — | 28 | 1 | no | — |
| 137 | C2×C4⋊C4 | — | 28 | 1 | no | — |
| 138 | C2×C42 | — | 7 | 1 | no | — |
| 139 | M32 | — | 2 | 2 | no | — |
| 140 | M32 | — | 14 | 2 | no | — |
| 141 | #97 | — | 7 | 2 | no | — |
| 142 | C4×C8 | — | 7 | 1 | no | — |
| 143 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 144 | C22×C7 | — | 8 | 2 | no | — |
| 145 | C22×C7 | — | 8 | 1 | no | — |
| 146 | C22×C7 | C2×#100 | 1 | 1 | yes | characteristic |
| 147 | C2×D7 | — | 2 | 1 | no | — |
| 148 | C2×D7 | — | 4 | 1 | no | — |
| 149 | C4×C7 | — | 2 | 2 | no | — |
| 150 | C4×C7 | — | 8 | 1 | no | — |
| 151 | C4×C7 | #418 | 1 | 2 | yes | — |
| 152 | Dic7 | — | 2 | 1 | no | — |
| 153 | Dic7 | — | 4 | 1 | no | — |
| 154 | Dic7 | — | 8 | 1 | no | — |
| 155 | Dic7 | — | 8 | 1 | no | — |
| 156 | #41 | — | 28 | 4 | no | — |
| 157 | C2×Q8 | — | 2 | 1 | no | — |
| 158 | C2×Q8 | — | 14 | 1 | no | — |
| 159 | C2×Q8 | — | 14 | 1 | no | — |
| 160 | C2×Q8 | — | 28 | 1 | no | — |
| 161 | C2×Q8 | — | 28 | 4 | no | — |
| 162 | C2×D4 | — | 2 | 1 | no | — |
| 163 | C2×D4 | — | 4 | 1 | no | — |
| 164 | C2×D4 | — | 28 | 1 | no | — |
| 165 | C22×C4 | — | 7 | 1 | no | — |
| 166 | C22×C4 | — | 14 | 1 | no | — |
| 167 | C22×C4 | — | 28 | 1 | no | — |
| 168 | C22×C4 | — | 28 | 1 | no | — |
| 169 | Q16 | — | 28 | 4 | no | — |
| 170 | Q16 | — | 28 | 2 | no | — |
| 171 | Q16 | — | 28 | 2 | no | — |
| 172 | SD16 | — | 4 | 2 | no | — |
| 173 | SD16 | — | 4 | 2 | no | — |
| 174 | SD16 | — | 28 | 4 | no | — |
| 175 | M16 | — | 14 | 2 | no | — |
| 176 | M16 | — | 14 | 2 | no | — |
| 177 | C2×C8 | — | 7 | 2 | no | — |
| 178 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 179 | C4⋊C4 | — | 4 | 1 | no | — |
| 180 | C4⋊C4 | — | 28 | 1 | no | — |
| 181 | C4⋊C4 | — | 28 | 1 | no | — |
| 182 | C4⋊C4 | — | 28 | 2 | no | — |
| 183 | C22⋊C4 | — | 28 | 2 | no | — |
| 184 | C42 | — | 7 | 1 | no | — |
| 185 | C42 | — | 14 | 1 | no | — |
| 186 | C42 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 187 | C16 | — | 4 | 2 | no | — |
| 188 | C16 | — | 28 | 2 | no | — |
| 189 | C2×C7 | — | 2 | 1 | no | — |
| 190 | C2×C7 | — | 16 | 1 | no | — |
| 191 | C2×C7 | #1555 | 1 | 1 | yes | characteristic |
| 192 | D7 | — | 4 | 1 | no | — |
| 193 | C23 | — | 4 | 1 | no | — |
| 194 | C23 | — | 7 | 1 | no | — |
| 195 | Q8 | — | 4 | 2 | no | — |
| 196 | Q8 | — | 28 | 2 | no | — |
| 197 | Q8 | — | 28 | 2 | no | — |
| 198 | Q8 | — | 56 | 2 | no | — |
| 199 | D4 | — | 4 | 2 | no | — |
| 200 | D4 | — | 8 | 2 | no | — |
| 201 | D4 | — | 56 | 2 | no | — |
| 202 | C2×C4 | — | 2 | 1 | no | — |
| 203 | C2×C4 | — | 4 | 1 | no | — |
| 204 | C2×C4 | — | 7 | 1 | no | — |
| 205 | C2×C4 | — | 14 | 2 | no | — |
| 206 | C2×C4 | — | 14 | 1 | no | — |
| 207 | C2×C4 | — | 28 | 2 | no | — |
| 208 | C2×C4 | — | 28 | 1 | no | — |
| 209 | C2×C4 | — | 28 | 1 | no | — |
| 210 | C2×C4 | — | 56 | 2 | no | — |
| 211 | C2×C4 | — | 56 | 2 | no | — |
| 212 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 213 | C8 | — | 14 | 2 | no | — |
| 214 | C8 | #6474 | 1 | 2 | yes | — |
| 215 | C8 | #6477 | 1 | 2 | yes | — |
| 216 | C7 | #11993 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 217 | C22 | — | 8 | 2 | no | — |
| 218 | C22 | — | 8 | 1 | no | — |
| 219 | C22 | — | 14 | 1 | no | — |
| 220 | C22 | — | 28 | 1 | no | — |
| 221 | C22 | D7×#100 | 1 | 1 | yes | characteristic |
| 222 | C4 | — | 2 | 2 | no | — |
| 223 | C4 | — | 8 | 1 | no | — |
| 224 | C4 | — | 14 | 1 | no | — |
| 225 | C4 | — | 28 | 1 | no | — |
| 226 | C4 | — | 56 | 1 | no | — |
| 227 | C4 | — | 56 | 1 | no | — |
| 228 | C4 | #89542 | 1 | 2 | yes | — |
| 229 | C2 | — | 2 | 1 | no | — |
| 230 | C2 | — | 16 | 1 | no | — |
| 231 | C2 | — | 28 | 1 | no | — |
| 232 | C2 | #11726141 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11727052 | C2 | semidirect |
| 3. C7×#1359 | C2 | semidirect |
| 4. #11725593 | C2 | semidirect |
| 5. #11725378 | C2 | semidirect |
| 48. #1359 | D7 | semidirect |
| 216. C7 | #11993 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11739207 | 2 |
| #11727052 | 2 |
| C7×#1359 | 2 |
| #11725593 | 2 |
| #11725378 | 2 |
| #11993 | 7 |