Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
268 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 37 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 | #11727747 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#1320 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11725494 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #90482 | — | 2 | 1 | no | — |
| 5 | #90439 | C4 | 1 | 1 | yes | characteristic |
| 6 | C7×#407 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C22⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 8 | #89911 | — | 2 | 1 | no | — |
| 9 | C7×#349 | — | 2 | 1 | no | — |
| 10 | #89352 | C4 | 1 | 1 | yes | characteristic |
| 11 | #7397 | — | 7 | 1 | no | maximal, Sylow |
| 12 | C23×C4×C7 | D4 | 1 | 1 | yes | characteristic |
| 13 | C2×#1157 | — | 2 | 1 | no | — |
| 14 | C2×C7×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 15 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 16 | #6516 | — | 4 | 1 | no | — |
| 17 | #6493 | — | 4 | 1 | no | — |
| 18 | #6492 | — | 4 | 1 | no | — |
| 19 | C4×C22⋊D7 | — | 4 | 2 | no | — |
| 20 | #6489 | — | 2 | 1 | no | — |
| 21 | #6487 | — | 4 | 1 | no | — |
| 22 | C22×C7⋊C8 | — | 4 | 1 | no | — |
| 23 | #6472 | — | 4 | 1 | no | — |
| 24 | #6471 | — | 4 | 1 | no | — |
| 25 | #6467 | — | 4 | 1 | no | — |
| 26 | C7×M32 | — | 4 | 1 | no | — |
| 27 | C7×C22⋊C8 | — | 4 | 1 | no | — |
| 28 | C22⋊(C7⋊C8) | — | 2 | 2 | no | — |
| 29 | C22⋊(C7⋊C8) | — | 4 | 1 | no | — |
| 30 | #6399 | — | 4 | 2 | no | — |
| 31 | #2010 | — | 7 | 1 | no | — |
| 32 | #1320 | D7 | 1 | 1 | yes | characteristic, complemented |
| 33 | #1298 | — | 7 | 1 | no | — |
| 34 | C24×C7 | SD16 | 1 | 1 | yes | characteristic |
| 35 | C22×C4×C7 | — | 2 | 1 | no | — |
| 36 | C22×C4×C7 | — | 2 | 1 | no | — |
| 37 | C22×C4×C7 | — | 4 | 1 | no | — |
| 38 | C22×C4×C7 | — | 4 | 1 | no | — |
| 39 | C22×C4×C7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 40 | C22×C4×C7 | D8 | 1 | 1 | yes | characteristic |
| 41 | C2×C22⋊D7 | — | 4 | 2 | no | — |
| 42 | C2×C22⋊D7 | — | 4 | 1 | no | — |
| 43 | #1157 | — | 4 | 1 | no | — |
| 44 | #1157 | — | 4 | 1 | no | — |
| 45 | #1157 | — | 4 | 1 | no | — |
| 46 | #1157 | — | 4 | 1 | no | — |
| 47 | C2×D28 | — | 2 | 1 | no | — |
| 48 | C2×C4×D7 | — | 4 | 1 | no | — |
| 49 | C2×Dic14 | — | 2 | 1 | no | — |
| 50 | C7×M16 | — | 4 | 1 | no | — |
| 51 | C7×M16 | — | 4 | 1 | no | — |
| 52 | C2×C7×C8 | — | 2 | 1 | no | — |
| 53 | C2×C7×C8 | — | 2 | 1 | no | — |
| 54 | C22⋊Dic7 | — | 4 | 2 | no | — |
| 55 | C22⋊Dic7 | — | 4 | 1 | no | — |
| 56 | (C2×D7)⋊C4 | — | 4 | 2 | no | — |
| 57 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 58 | Dic7⋊C4 | — | 4 | 2 | no | — |
| 59 | C4×Dic7 | — | 2 | 2 | no | — |
| 60 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 61 | C2×C7⋊C8 | — | 8 | 1 | no | — |
| 62 | C2×C7⋊C8 | — | 8 | 1 | no | — |
| 63 | Dic28 | — | 4 | 1 | no | — |
| 64 | D56 | — | 4 | 1 | no | — |
| 65 | #1132 | — | 4 | 2 | no | — |
| 66 | C7×C16 | — | 8 | 1 | no | — |
| 67 | #575 | — | 7 | 1 | no | — |
| 68 | #525 | — | 14 | 1 | no | — |
| 69 | #421 | — | 14 | 1 | no | — |
| 70 | #407 | C2×D7 | 1 | 1 | yes | characteristic |
| 71 | C2×C22⋊C8 | — | 14 | 1 | no | — |
| 72 | #349 | — | 2 | 1 | no | — |
| 73 | H(3,ℤ/4) | — | 7 | 1 | no | — |
| 74 | C23×C7 | — | 2 | 1 | no | — |
| 75 | C23×C7 | — | 4 | 1 | no | — |
| 76 | C23×C7 | — | 4 | 2 | no | — |
| 77 | C23×C7 | #102 | 1 | 1 | yes | characteristic |
| 78 | C22×D7 | — | 4 | 1 | no | — |
| 79 | C2×C4×C7 | — | 2 | 1 | no | — |
| 80 | C2×C4×C7 | — | 2 | 1 | no | — |
| 81 | C2×C4×C7 | — | 2 | 1 | no | — |
| 82 | C2×C4×C7 | — | 4 | 1 | no | — |
| 83 | C2×C4×C7 | — | 4 | 1 | no | — |
| 84 | C2×C4×C7 | — | 4 | 1 | no | — |
| 85 | C2×C4×C7 | — | 8 | 1 | no | — |
| 86 | C2×C4×C7 | #99 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C7 | #104 | 1 | 1 | yes | characteristic |
| 88 | C22⋊D7 | — | 8 | 1 | no | — |
| 89 | C22⋊D7 | — | 8 | 4 | no | — |
| 90 | C22⋊D7 | — | 8 | 1 | no | — |
| 91 | C2×Dic7 | — | 4 | 2 | no | — |
| 92 | C2×Dic7 | — | 4 | 1 | no | — |
| 93 | D28 | — | 4 | 1 | no | — |
| 94 | D28 | — | 4 | 1 | no | — |
| 95 | C4×D7 | — | 8 | 1 | no | — |
| 96 | C4×D7 | — | 8 | 1 | no | — |
| 97 | Dic14 | — | 4 | 1 | no | — |
| 98 | Dic14 | — | 4 | 1 | no | — |
| 99 | C7×C8 | — | 4 | 1 | no | — |
| 100 | C7×C8 | — | 4 | 1 | no | — |
| 101 | C7⋊C8 | — | 8 | 2 | no | — |
| 102 | C2×#41 | — | 14 | 1 | no | — |
| 103 | C23×C4 | D28 | 1 | 1 | yes | characteristic |
| 104 | #137 | — | 28 | 1 | no | — |
| 105 | AGL(1,ℤ/8) | — | 28 | 1 | no | — |
| 106 | #135 | — | 28 | 1 | no | — |
| 107 | C2×M16 | C4×D7 | 1 | 1 | yes | characteristic |
| 108 | C2×M16 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 109 | C22×C8 | — | 28 | 1 | no | — |
| 110 | #123 | — | 28 | 1 | no | — |
| 111 | C22⋊Q8 | — | 28 | 1 | no | — |
| 112 | #121 | — | 28 | 1 | no | — |
| 113 | #120 | — | 28 | 1 | no | — |
| 114 | C4×D4 | — | 28 | 2 | no | — |
| 115 | #117 | — | 14 | 1 | no | — |
| 116 | M32 | — | 4 | 1 | no | — |
| 117 | #104 | — | 28 | 2 | no | — |
| 118 | C22⋊C8 | — | 4 | 1 | no | — |
| 119 | C22⋊C8 | — | 14 | 2 | no | — |
| 120 | C22⋊C8 | — | 28 | 1 | no | — |
| 121 | C22×C7 | — | 2 | 2 | no | — |
| 122 | C22×C7 | — | 2 | 1 | no | — |
| 123 | C22×C7 | — | 4 | 1 | no | — |
| 124 | C22×C7 | — | 4 | 1 | no | — |
| 125 | C22×C7 | — | 4 | 2 | no | — |
| 126 | C22×C7 | — | 4 | 1 | no | — |
| 127 | C22×C7 | — | 8 | 1 | no | — |
| 128 | C22×C7 | #327 | 1 | 1 | yes | characteristic |
| 129 | C2×D7 | — | 8 | 2 | no | — |
| 130 | C2×D7 | — | 8 | 1 | no | — |
| 131 | C4×C7 | — | 2 | 1 | no | — |
| 132 | C4×C7 | — | 4 | 1 | no | — |
| 133 | C4×C7 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 134 | C4×C7 | #353 | 1 | 1 | yes | characteristic |
| 135 | Dic7 | — | 8 | 2 | no | — |
| 136 | Dic7 | — | 8 | 1 | no | — |
| 137 | C24 | #1132 | 1 | 1 | yes | characteristic |
| 138 | #41 | — | 28 | 1 | no | — |
| 139 | #41 | — | 28 | 1 | no | — |
| 140 | #41 | — | 28 | 1 | no | — |
| 141 | #41 | — | 28 | 1 | no | — |
| 142 | C2×Q8 | — | 14 | 1 | no | — |
| 143 | C2×D4 | — | 14 | 1 | no | — |
| 144 | C2×D4 | — | 28 | 2 | no | — |
| 145 | C2×D4 | — | 28 | 1 | no | — |
| 146 | C22×C4 | — | 2 | 1 | no | — |
| 147 | C22×C4 | — | 2 | 1 | no | — |
| 148 | C22×C4 | — | 4 | 1 | no | — |
| 149 | C22×C4 | — | 4 | 1 | no | — |
| 150 | C22×C4 | — | 28 | 1 | no | — |
| 151 | C22×C4 | D56 | 1 | 1 | yes | characteristic |
| 152 | C22×C4 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 153 | Q16 | — | 28 | 1 | no | — |
| 154 | SD16 | — | 28 | 2 | no | — |
| 155 | D8 | — | 28 | 1 | no | — |
| 156 | M16 | — | 4 | 1 | no | — |
| 157 | M16 | — | 4 | 1 | no | — |
| 158 | C2×C8 | — | 2 | 1 | no | — |
| 159 | C2×C8 | — | 2 | 1 | no | — |
| 160 | C2×C8 | — | 28 | 2 | no | — |
| 161 | C2×C8 | — | 56 | 1 | no | — |
| 162 | C2×C8 | — | 56 | 1 | no | — |
| 163 | C4⋊C4 | — | 28 | 1 | no | — |
| 164 | C4⋊C4 | — | 28 | 2 | no | — |
| 165 | C22⋊C4 | — | 28 | 2 | no | — |
| 166 | C22⋊C4 | — | 28 | 2 | no | — |
| 167 | C22⋊C4 | — | 28 | 1 | no | — |
| 168 | C42 | — | 14 | 2 | no | — |
| 169 | C16 | — | 8 | 1 | no | — |
| 170 | C2×C7 | — | 2 | 1 | no | — |
| 171 | C2×C7 | — | 4 | 1 | no | — |
| 172 | C2×C7 | — | 4 | 2 | no | — |
| 173 | C2×C7 | #1343 | 1 | 1 | yes | characteristic |
| 174 | D7 | — | 16 | 1 | no | — |
| 175 | C23 | — | 2 | 1 | no | — |
| 176 | C23 | — | 4 | 1 | no | — |
| 177 | C23 | — | 4 | 2 | no | — |
| 178 | C23 | — | 28 | 1 | no | — |
| 179 | C23 | #6395 | 1 | 1 | yes | characteristic |
| 180 | Q8 | — | 28 | 1 | no | — |
| 181 | Q8 | — | 28 | 1 | no | — |
| 182 | D4 | — | 28 | 1 | no | — |
| 183 | D4 | — | 28 | 1 | no | — |
| 184 | D4 | — | 56 | 1 | no | — |
| 185 | D4 | — | 56 | 4 | no | — |
| 186 | D4 | — | 56 | 1 | no | — |
| 187 | C2×C4 | — | 2 | 1 | no | — |
| 188 | C2×C4 | — | 2 | 1 | no | — |
| 189 | C2×C4 | — | 2 | 1 | no | — |
| 190 | C2×C4 | — | 4 | 1 | no | — |
| 191 | C2×C4 | — | 4 | 1 | no | — |
| 192 | C2×C4 | — | 4 | 1 | no | — |
| 193 | C2×C4 | — | 8 | 1 | no | — |
| 194 | C2×C4 | — | 28 | 2 | no | — |
| 195 | C2×C4 | — | 28 | 1 | no | — |
| 196 | C2×C4 | — | 56 | 1 | no | — |
| 197 | C2×C4 | — | 56 | 1 | no | — |
| 198 | C2×C4 | #6380 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | #6399 | 1 | 1 | yes | characteristic |
| 200 | C8 | — | 4 | 1 | no | — |
| 201 | C8 | — | 4 | 1 | no | — |
| 202 | C8 | — | 56 | 2 | no | — |
| 203 | C7 | #7397 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 204 | C22 | — | 2 | 2 | no | — |
| 205 | C22 | — | 2 | 1 | no | — |
| 206 | C22 | — | 4 | 1 | no | — |
| 207 | C22 | — | 4 | 1 | no | — |
| 208 | C22 | — | 4 | 2 | no | — |
| 209 | C22 | — | 4 | 1 | no | — |
| 210 | C22 | — | 8 | 1 | no | — |
| 211 | C22 | — | 56 | 2 | no | — |
| 212 | C22 | — | 56 | 1 | no | — |
| 213 | C22 | #89265 | 1 | 1 | yes | characteristic |
| 214 | C4 | — | 2 | 1 | no | — |
| 215 | C4 | — | 4 | 1 | no | — |
| 216 | C4 | — | 56 | 2 | no | — |
| 217 | C4 | — | 56 | 1 | no | — |
| 218 | C4 | #89269 | 1 | 1 | yes | characteristic |
| 219 | C4 | #89270 | 1 | 1 | yes | characteristic |
| 220 | C2 | — | 2 | 1 | no | — |
| 221 | C2 | — | 4 | 1 | no | — |
| 222 | C2 | — | 4 | 2 | no | — |
| 223 | C2 | — | 112 | 1 | no | — |
| 224 | C2 | #11725254 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1320 | C2 | semidirect |
| 3. #11725494 | C2 | semidirect |
| 32. #1320 | D7 | semidirect |
| 203. C7 | #7397 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11727747 | 2 |
| C7×#1320 | 2 |
| #11725494 | 2 |
| #7397 | 7 |