Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
400 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 89 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 | #11739205 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | D7×#364 | 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 | #11725376 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#6479 | — | 2 | 2 | no | — |
| 7 | #90416 | C4 | 1 | 1 | yes | characteristic |
| 8 | #90407 | C4 | 1 | 1 | yes | characteristic |
| 9 | C7×#492 | C22 | 1 | 1 | yes | characteristic |
| 10 | #89940 | — | 2 | 2 | no | — |
| 11 | #89864 | C4 | 1 | 1 | yes | characteristic |
| 12 | #89845 | C4 | 1 | 1 | yes | characteristic |
| 13 | D7×M32 | — | 2 | 2 | no | — |
| 14 | #89465 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×C8×D7 | C22 | 1 | 1 | yes | characteristic |
| 16 | C7×#364 | C22 | 1 | 2 | yes | — |
| 17 | #89306 | — | 2 | 2 | no | — |
| 18 | #89258 | C22 | 1 | 2 | yes | — |
| 19 | #11922 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C2×#1161 | — | 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 | #6510 | — | 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×#1141 | — | 2 | 2 | no | — |
| 31 | #6479 | — | 4 | 2 | no | — |
| 32 | #6479 | — | 4 | 2 | no | — |
| 33 | #6479 | — | 4 | 2 | no | — |
| 34 | #6479 | — | 4 | 2 | no | — |
| 35 | C2×#1132 | — | 2 | 2 | no | — |
| 36 | C2×C8×D7 | D4 | 1 | 2 | yes | — |
| 37 | #6458 | — | 4 | 1 | no | — |
| 38 | #6456 | — | 4 | 1 | no | — |
| 39 | #6437 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C42×D7 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | #6432 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C7×M32 | — | 2 | 2 | no | — |
| 43 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C8×Dic7 | D4 | 1 | 2 | yes | — |
| 45 | #6386 | — | 2 | 2 | no | — |
| 46 | C4×C7⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | #6372 | — | 4 | 2 | no | — |
| 48 | D7×C16 | — | 4 | 2 | no | — |
| 49 | #3147 | — | 7 | 1 | no | — |
| 50 | C2×#364 | — | 7 | 2 | no | — |
| 51 | #1359 | — | 7 | 1 | no | — |
| 52 | #1359 | — | 7 | 2 | no | — |
| 53 | #1359 | D7 | 1 | 1 | yes | characteristic, complemented |
| 54 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 55 | C2×C7×D4 | — | 2 | 1 | no | — |
| 56 | C2×C7×D4 | — | 4 | 1 | no | — |
| 57 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 58 | C22×Dic7 | — | 4 | 1 | no | — |
| 59 | #1161 | — | 4 | 2 | no | — |
| 60 | #1161 | — | 4 | 2 | no | — |
| 61 | #1159 | — | 4 | 2 | no | — |
| 62 | #1159 | — | 4 | 2 | no | — |
| 63 | C2×D28 | — | 2 | 1 | no | — |
| 64 | C2×D28 | — | 4 | 1 | no | — |
| 65 | C2×D28 | — | 4 | 1 | no | — |
| 66 | C2×C4×D7 | — | 2 | 1 | no | — |
| 67 | C2×C4×D7 | — | 4 | 1 | no | — |
| 68 | C2×C4×D7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C2×Dic14 | — | 2 | 1 | no | — |
| 70 | C7×SD16 | — | 4 | 2 | no | — |
| 71 | C7×SD16 | — | 4 | 2 | no | — |
| 72 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 73 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 74 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C22⋊Dic7 | — | 4 | 2 | no | — |
| 76 | #1144 | — | 4 | 2 | no | — |
| 77 | #1144 | — | 4 | 2 | no | — |
| 78 | #1141 | — | 4 | 2 | no | — |
| 79 | #1141 | — | 4 | 2 | no | — |
| 80 | (C2×D7)⋊C4 | — | 4 | 2 | no | — |
| 81 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 82 | C4×Dic7 | — | 2 | 1 | no | — |
| 83 | C4×Dic7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C2×C7⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 85 | #1132 | — | 4 | 2 | no | — |
| 86 | #1132 | — | 4 | 2 | no | — |
| 87 | C8×D7 | D8 | 1 | 2 | yes | — |
| 88 | C8×D7 | D8 | 1 | 2 | yes | — |
| 89 | C8×D7 | SD16 | 1 | 2 | yes | — |
| 90 | C8×D7 | SD16 | 1 | 2 | yes | — |
| 91 | C7×C16 | — | 4 | 2 | no | — |
| 92 | C7⋊C16 | — | 4 | 2 | no | — |
| 93 | C2×#135 | — | 14 | 2 | no | — |
| 94 | #532 | — | 7 | 1 | no | — |
| 95 | #532 | — | 7 | 1 | no | — |
| 96 | C2×M32 | — | 14 | 2 | no | — |
| 97 | #495 | — | 14 | 2 | no | — |
| 98 | #494 | — | 7 | 1 | no | — |
| 99 | #493 | — | 7 | 1 | no | — |
| 100 | #492 | — | 7 | 1 | no | — |
| 101 | #492 | C2×D7 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C8 | — | 7 | 1 | no | — |
| 103 | #364 | — | 7 | 2 | no | — |
| 104 | #364 | — | 14 | 2 | no | — |
| 105 | #364 | C2×D7 | 1 | 2 | yes | — |
| 106 | C23×C7 | — | 4 | 1 | no | — |
| 107 | C22×D7 | — | 4 | 1 | no | — |
| 108 | C22×D7 | #100 | 1 | 1 | yes | characteristic |
| 109 | C7×Q8 | — | 4 | 2 | no | — |
| 110 | C7×D4 | — | 4 | 2 | no | — |
| 111 | C7×D4 | — | 8 | 2 | no | — |
| 112 | C2×C4×C7 | — | 2 | 1 | no | — |
| 113 | C2×C4×C7 | — | 4 | 1 | no | — |
| 114 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C22⋊D7 | — | 8 | 2 | no | — |
| 116 | C2×Dic7 | — | 2 | 1 | no | — |
| 117 | C2×Dic7 | — | 4 | 1 | no | — |
| 118 | C2×Dic7 | — | 8 | 1 | no | — |
| 119 | C2×Dic7 | — | 8 | 1 | no | — |
| 120 | C2×Dic7 | #100 | 1 | 1 | yes | characteristic |
| 121 | D28 | — | 4 | 2 | no | — |
| 122 | D28 | — | 8 | 2 | no | — |
| 123 | D28 | — | 8 | 2 | no | — |
| 124 | C4×D7 | — | 4 | 2 | no | — |
| 125 | C4×D7 | — | 8 | 1 | no | — |
| 126 | C4×D7 | — | 8 | 1 | no | — |
| 127 | C4×D7 | #102 | 1 | 2 | yes | — |
| 128 | C4×D7 | #102 | 1 | 2 | yes | — |
| 129 | Dic14 | — | 4 | 2 | no | — |
| 130 | C7×C8 | C2×D8 | 1 | 2 | yes | — |
| 131 | C7×C8 | C2×SD16 | 1 | 2 | yes | — |
| 132 | C7⋊C8 | #102 | 1 | 2 | yes | — |
| 133 | C7⋊C8 | #102 | 1 | 2 | yes | — |
| 134 | C2×#41 | — | 14 | 1 | no | — |
| 135 | C2×#41 | — | 14 | 1 | no | — |
| 136 | #135 | — | 28 | 2 | no | — |
| 137 | #135 | — | 28 | 2 | no | — |
| 138 | #135 | — | 28 | 2 | no | — |
| 139 | #135 | — | 28 | 2 | no | — |
| 140 | C2×Q16 | — | 14 | 2 | no | — |
| 141 | C2×SD16 | — | 2 | 2 | no | — |
| 142 | C2×SD16 | — | 14 | 2 | no | — |
| 143 | C2×D8 | — | 14 | 2 | no | — |
| 144 | C22×C8 | — | 7 | 2 | no | — |
| 145 | C4⋊Q8 | — | 7 | 1 | no | — |
| 146 | C4⋊Q8 | C4×D7 | 1 | 1 | yes | characteristic |
| 147 | #127 | — | 7 | 1 | no | — |
| 148 | #127 | C4×D7 | 1 | 1 | yes | characteristic |
| 149 | #124 | — | 14 | 1 | no | — |
| 150 | #124 | — | 14 | 1 | no | — |
| 151 | #121 | — | 28 | 1 | no | — |
| 152 | #121 | — | 28 | 1 | no | — |
| 153 | C4×D4 | — | 28 | 1 | no | — |
| 154 | C4×D4 | — | 28 | 1 | no | — |
| 155 | C2×C42 | — | 7 | 1 | no | — |
| 156 | M32 | — | 2 | 2 | no | — |
| 157 | M32 | — | 14 | 2 | no | — |
| 158 | M32 | — | 28 | 2 | no | — |
| 159 | C2×C16 | — | 28 | 2 | no | — |
| 160 | C4×C8 | — | 7 | 2 | no | — |
| 161 | C4×C8 | — | 7 | 1 | no | — |
| 162 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 163 | C22×C7 | — | 8 | 2 | no | — |
| 164 | C22×C7 | — | 8 | 1 | no | — |
| 165 | C22×C7 | C2×#100 | 1 | 1 | yes | characteristic |
| 166 | C2×D7 | — | 4 | 1 | no | — |
| 167 | C2×D7 | — | 8 | 2 | no | — |
| 168 | C2×D7 | — | 8 | 1 | no | — |
| 169 | C2×D7 | #331 | 1 | 2 | yes | — |
| 170 | C4×C7 | — | 2 | 2 | no | — |
| 171 | C4×C7 | — | 8 | 1 | no | — |
| 172 | C4×C7 | C2×#102 | 1 | 2 | yes | — |
| 173 | Dic7 | — | 4 | 1 | no | — |
| 174 | Dic7 | — | 8 | 1 | no | — |
| 175 | Dic7 | #331 | 1 | 1 | yes | characteristic |
| 176 | Dic7 | #331 | 1 | 1 | yes | characteristic |
| 177 | #41 | — | 28 | 2 | no | — |
| 178 | #41 | — | 28 | 2 | no | — |
| 179 | #41 | — | 28 | 2 | no | — |
| 180 | #41 | — | 28 | 2 | no | — |
| 181 | C2×Q8 | — | 2 | 1 | no | — |
| 182 | C2×Q8 | — | 14 | 1 | no | — |
| 183 | C2×D4 | — | 2 | 1 | no | — |
| 184 | C2×D4 | — | 4 | 1 | no | — |
| 185 | C2×D4 | — | 14 | 1 | no | — |
| 186 | C2×D4 | — | 28 | 1 | no | — |
| 187 | C2×D4 | — | 28 | 1 | no | — |
| 188 | C2×D4 | — | 28 | 1 | no | — |
| 189 | C22×C4 | — | 7 | 1 | no | — |
| 190 | C22×C4 | — | 14 | 1 | no | — |
| 191 | C22×C4 | — | 28 | 1 | no | — |
| 192 | C22×C4 | — | 28 | 1 | no | — |
| 193 | Q16 | — | 28 | 2 | no | — |
| 194 | Q16 | — | 28 | 2 | no | — |
| 195 | SD16 | — | 4 | 2 | no | — |
| 196 | SD16 | — | 4 | 2 | no | — |
| 197 | SD16 | — | 28 | 2 | no | — |
| 198 | SD16 | — | 28 | 2 | no | — |
| 199 | D8 | — | 28 | 2 | no | — |
| 200 | D8 | — | 28 | 2 | no | — |
| 201 | C2×C8 | — | 7 | 2 | no | — |
| 202 | C2×C8 | — | 7 | 2 | no | — |
| 203 | C2×C8 | — | 7 | 2 | no | — |
| 204 | C2×C8 | — | 7 | 2 | no | — |
| 205 | C2×C8 | — | 7 | 2 | no | — |
| 206 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 207 | C4⋊C4 | — | 4 | 1 | no | — |
| 208 | C4⋊C4 | — | 28 | 1 | no | — |
| 209 | C22⋊C4 | — | 28 | 2 | no | — |
| 210 | C22⋊C4 | — | 28 | 2 | no | — |
| 211 | C42 | — | 7 | 1 | no | — |
| 212 | C42 | — | 14 | 1 | no | — |
| 213 | C42 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 214 | C16 | — | 4 | 2 | no | — |
| 215 | C16 | — | 28 | 2 | no | — |
| 216 | C2×C7 | — | 2 | 1 | no | — |
| 217 | C2×C7 | — | 16 | 1 | no | — |
| 218 | C2×C7 | C2×#331 | 1 | 1 | yes | characteristic |
| 219 | D7 | — | 2 | 2 | no | — |
| 220 | D7 | — | 16 | 1 | no | — |
| 221 | C23 | — | 4 | 1 | no | — |
| 222 | C23 | — | 7 | 1 | no | — |
| 223 | C23 | — | 28 | 1 | no | — |
| 224 | Q8 | — | 4 | 2 | no | — |
| 225 | Q8 | — | 28 | 2 | no | — |
| 226 | D4 | — | 4 | 2 | no | — |
| 227 | D4 | — | 8 | 2 | no | — |
| 228 | D4 | — | 28 | 2 | no | — |
| 229 | D4 | — | 56 | 2 | no | — |
| 230 | D4 | — | 56 | 2 | no | — |
| 231 | D4 | — | 56 | 2 | no | — |
| 232 | C2×C4 | — | 2 | 1 | no | — |
| 233 | C2×C4 | — | 4 | 1 | no | — |
| 234 | C2×C4 | — | 7 | 1 | no | — |
| 235 | C2×C4 | — | 7 | 2 | no | — |
| 236 | C2×C4 | — | 7 | 2 | no | — |
| 237 | C2×C4 | — | 14 | 1 | no | — |
| 238 | C2×C4 | — | 28 | 2 | no | — |
| 239 | C2×C4 | — | 28 | 1 | no | — |
| 240 | C2×C4 | — | 56 | 1 | no | — |
| 241 | C2×C4 | — | 56 | 1 | no | — |
| 242 | C2×C4 | — | 56 | 1 | no | — |
| 243 | C2×C4 | — | 56 | 1 | no | — |
| 244 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 245 | C8 | — | 7 | 2 | no | — |
| 246 | C8 | — | 7 | 2 | no | — |
| 247 | C8 | D7×D8 | 1 | 2 | yes | — |
| 248 | C8 | D7×SD16 | 1 | 2 | yes | — |
| 249 | C7 | #11922 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 250 | C22 | — | 7 | 2 | no | — |
| 251 | C22 | — | 8 | 2 | no | — |
| 252 | C22 | — | 8 | 1 | no | — |
| 253 | C22 | — | 28 | 1 | no | — |
| 254 | C22 | — | 56 | 2 | no | — |
| 255 | C22 | — | 56 | 1 | no | — |
| 256 | C22 | D7×#100 | 1 | 1 | yes | characteristic |
| 257 | C4 | — | 2 | 2 | no | — |
| 258 | C4 | — | 7 | 1 | no | — |
| 259 | C4 | — | 7 | 1 | no | — |
| 260 | C4 | — | 8 | 1 | no | — |
| 261 | C4 | — | 28 | 1 | no | — |
| 262 | C4 | — | 56 | 1 | no | — |
| 263 | C4 | D7×#102 | 1 | 2 | yes | — |
| 264 | C2 | — | 2 | 1 | no | — |
| 265 | C2 | — | 14 | 2 | no | — |
| 266 | C2 | — | 16 | 1 | no | — |
| 267 | C2 | — | 112 | 1 | no | — |
| 268 | C2 | D7×#331 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. D7×#364 | C2 | semidirect |
| 3. C7×#1359 | C2 | semidirect |
| 4. #11725593 | C2 | semidirect |
| 5. #11725376 | C2 | semidirect |
| 53. #1359 | D7 | semidirect |
| 249. C7 | #11922 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11739205 | 2 |
| D7×#364 | 2 |
| C7×#1359 | 2 |
| #11725593 | 2 |
| #11725376 | 2 |
| #11922 | 7 |