Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
350 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 63 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 | C7×#3261 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11728839 | C2 | 1 | 2 | yes | maximal |
| 3 | #11728294 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11725583 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11725524 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C7×#541 | — | 2 | 1 | no | — |
| 7 | C22×D56 | C22 | 1 | 1 | yes | characteristic |
| 8 | C7×#479 | — | 2 | 2 | no | — |
| 9 | C7×#470 | — | 2 | 1 | no | — |
| 10 | C7×#467 | C22 | 1 | 2 | yes | — |
| 11 | C7×#462 | — | 2 | 2 | no | — |
| 12 | C7×#424 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#6402 | — | 2 | 2 | no | — |
| 14 | C2×#6397 | — | 2 | 1 | no | — |
| 15 | #89773 | — | 2 | 1 | no | — |
| 16 | #89347 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22⋊(C7⋊C16) | C22 | 1 | 2 | yes | — |
| 18 | #89286 | — | 2 | 2 | no | — |
| 19 | #12165 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C22×D28 | — | 2 | 1 | no | — |
| 21 | C2×C7×Q16 | — | 2 | 1 | no | — |
| 22 | C2×C7×Q16 | D4 | 1 | 2 | yes | — |
| 23 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 24 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 25 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 26 | C7×#126 | — | 4 | 2 | no | — |
| 27 | C7×#125 | — | 2 | 1 | no | — |
| 28 | C7×#125 | — | 4 | 2 | no | — |
| 29 | C7×C22⋊Q8 | — | 2 | 2 | no | — |
| 30 | C4×C7×Q8 | — | 2 | 2 | no | — |
| 31 | C7×#117 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×#1136 | — | 2 | 1 | no | — |
| 33 | C2×D56 | — | 2 | 2 | no | — |
| 34 | C2×D56 | — | 4 | 2 | no | — |
| 35 | C7×#107 | D4 | 1 | 2 | yes | — |
| 36 | C7×#106 | — | 2 | 1 | no | — |
| 37 | C7×C4⋊C8 | — | 2 | 2 | no | — |
| 38 | C7×#103 | — | 2 | 2 | no | — |
| 39 | C7×#103 | — | 2 | 2 | no | — |
| 40 | #6402 | — | 4 | 4 | no | — |
| 41 | #6397 | — | 4 | 2 | no | — |
| 42 | C2×C7⋊C16 | — | 2 | 2 | no | — |
| 43 | #6383 | — | 4 | 2 | no | — |
| 44 | #3261 | D7 | 1 | 1 | yes | characteristic, complemented |
| 45 | #2187 | — | 7 | 2 | no | — |
| 46 | #2045 | — | 7 | 1 | no | — |
| 47 | #1350 | — | 7 | 2 | no | — |
| 48 | #1323 | — | 7 | 1 | no | — |
| 49 | C23×D7 | — | 4 | 1 | no | — |
| 50 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 51 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C2×D28 | — | 2 | 2 | no | — |
| 53 | C2×D28 | — | 4 | 1 | no | — |
| 54 | C2×D28 | — | 4 | 1 | no | — |
| 55 | C2×D28 | — | 4 | 1 | no | — |
| 56 | C2×D28 | — | 4 | 1 | no | — |
| 57 | C2×D28 | — | 4 | 1 | no | — |
| 58 | C7×Q16 | — | 4 | 2 | no | — |
| 59 | C7×Q16 | — | 8 | 1 | no | — |
| 60 | C7×M16 | — | 4 | 1 | no | — |
| 61 | C2×C7×C8 | — | 2 | 1 | no | — |
| 62 | C2×C7×C8 | — | 4 | 1 | no | — |
| 63 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 64 | C7×C4⋊C4 | — | 2 | 2 | no | — |
| 65 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 66 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 67 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 68 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 69 | C7×C4⋊C4 | D8 | 1 | 2 | yes | — |
| 70 | C7×C22⋊C4 | — | 2 | 1 | no | — |
| 71 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 72 | C42×C7 | — | 2 | 1 | no | — |
| 73 | C42×C7 | — | 4 | 2 | no | — |
| 74 | #1136 | — | 4 | 1 | no | — |
| 75 | #1136 | — | 4 | 1 | no | — |
| 76 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 77 | D56 | — | 2 | 4 | no | — |
| 78 | D56 | — | 4 | 2 | no | — |
| 79 | C7⋊C16 | — | 4 | 2 | no | — |
| 80 | C22×D8 | — | 7 | 1 | no | — |
| 81 | #541 | — | 2 | 1 | no | — |
| 82 | C2×SD32 | — | 14 | 2 | no | — |
| 83 | #479 | — | 2 | 2 | no | — |
| 84 | #470 | — | 2 | 1 | no | — |
| 85 | #467 | C2×D7 | 1 | 2 | yes | — |
| 86 | #462 | — | 2 | 2 | no | — |
| 87 | #424 | C2×D7 | 1 | 1 | yes | characteristic |
| 88 | #418 | — | 14 | 1 | no | — |
| 89 | C2×#100 | — | 14 | 1 | no | — |
| 90 | #357 | — | 14 | 2 | no | — |
| 91 | C22⋊C16 | — | 7 | 2 | no | — |
| 92 | #341 | — | 7 | 1 | no | — |
| 93 | C23×C7 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 94 | C22×D7 | — | 4 | 2 | no | — |
| 95 | C22×D7 | — | 8 | 1 | no | — |
| 96 | C22×D7 | — | 8 | 1 | no | — |
| 97 | C22×D7 | — | 16 | 2 | no | — |
| 98 | C7×Q8 | — | 4 | 2 | no | — |
| 99 | C7×Q8 | — | 4 | 2 | no | — |
| 100 | C2×C4×C7 | — | 2 | 1 | no | — |
| 101 | C2×C4×C7 | — | 2 | 1 | no | — |
| 102 | C2×C4×C7 | — | 2 | 2 | no | — |
| 103 | C2×C4×C7 | — | 4 | 2 | no | — |
| 104 | C2×C4×C7 | — | 4 | 2 | no | — |
| 105 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 107 | D28 | — | 4 | 2 | no | — |
| 108 | D28 | — | 4 | 2 | no | — |
| 109 | D28 | — | 8 | 1 | no | — |
| 110 | D28 | — | 8 | 1 | no | — |
| 111 | C7×C8 | — | 2 | 2 | no | — |
| 112 | C7×C8 | — | 4 | 1 | no | — |
| 113 | C7⋊C8 | — | 8 | 1 | no | — |
| 114 | C22×D4 | — | 14 | 1 | no | — |
| 115 | C2×Q16 | — | 2 | 1 | no | — |
| 116 | C2×Q16 | C22⋊D7 | 1 | 2 | yes | — |
| 117 | C2×D8 | — | 14 | 2 | no | — |
| 118 | C2×D8 | — | 28 | 2 | no | — |
| 119 | C2×M16 | — | 14 | 1 | no | — |
| 120 | C2×M16 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 121 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 122 | C4⋊Q8 | — | 2 | 1 | no | — |
| 123 | #126 | — | 4 | 2 | no | — |
| 124 | #125 | — | 2 | 1 | no | — |
| 125 | #125 | — | 4 | 2 | no | — |
| 126 | C22⋊Q8 | — | 2 | 2 | no | — |
| 127 | C4×Q8 | — | 2 | 2 | no | — |
| 128 | #117 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 129 | SD32 | — | 28 | 4 | no | — |
| 130 | C2×C16 | — | 14 | 2 | no | — |
| 131 | #107 | C22⋊D7 | 1 | 2 | yes | — |
| 132 | #106 | — | 2 | 1 | no | — |
| 133 | C4⋊C8 | — | 2 | 2 | no | — |
| 134 | #103 | — | 2 | 2 | no | — |
| 135 | #103 | — | 2 | 2 | no | — |
| 136 | #102 | — | 28 | 2 | no | — |
| 137 | #100 | — | 28 | 2 | no | — |
| 138 | C22×C7 | — | 2 | 1 | no | — |
| 139 | C22×C7 | — | 2 | 2 | no | — |
| 140 | C22×C7 | #447 | 1 | 1 | yes | characteristic |
| 141 | C2×D7 | — | 8 | 2 | no | — |
| 142 | C2×D7 | — | 16 | 2 | no | — |
| 143 | C2×D7 | — | 16 | 2 | no | — |
| 144 | C2×D7 | — | 16 | 2 | no | — |
| 145 | C4×C7 | — | 2 | 1 | no | — |
| 146 | C4×C7 | — | 4 | 2 | no | — |
| 147 | C4×C7 | — | 8 | 2 | no | — |
| 148 | C4×C7 | — | 8 | 2 | no | — |
| 149 | C4×C7 | #454 | 1 | 1 | yes | characteristic |
| 150 | C4×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 151 | C24 | — | 28 | 1 | no | — |
| 152 | C2×Q8 | — | 2 | 2 | no | — |
| 153 | C2×D4 | — | 14 | 2 | no | — |
| 154 | C2×D4 | — | 28 | 1 | no | — |
| 155 | C2×D4 | — | 28 | 1 | no | — |
| 156 | C2×D4 | — | 28 | 1 | no | — |
| 157 | C2×D4 | — | 28 | 1 | no | — |
| 158 | C2×D4 | — | 28 | 1 | no | — |
| 159 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 160 | Q16 | — | 4 | 2 | no | — |
| 161 | Q16 | — | 8 | 1 | no | — |
| 162 | D8 | — | 14 | 4 | no | — |
| 163 | D8 | — | 28 | 2 | no | — |
| 164 | M16 | — | 4 | 1 | no | — |
| 165 | M16 | — | 28 | 1 | no | — |
| 166 | M16 | — | 28 | 1 | no | — |
| 167 | C2×C8 | — | 2 | 1 | no | — |
| 168 | C2×C8 | — | 4 | 1 | no | — |
| 169 | C2×C8 | — | 28 | 1 | no | — |
| 170 | C2×C8 | C2×C22⋊D7 | 1 | 2 | yes | — |
| 171 | C4⋊C4 | — | 2 | 2 | no | — |
| 172 | C4⋊C4 | — | 4 | 2 | no | — |
| 173 | C4⋊C4 | — | 4 | 2 | no | — |
| 174 | C4⋊C4 | — | 4 | 2 | no | — |
| 175 | C4⋊C4 | — | 4 | 2 | no | — |
| 176 | C4⋊C4 | #1141 | 1 | 2 | yes | — |
| 177 | C22⋊C4 | — | 2 | 1 | no | — |
| 178 | C22⋊C4 | — | 4 | 2 | no | — |
| 179 | C42 | — | 2 | 1 | no | — |
| 180 | C42 | — | 4 | 2 | no | — |
| 181 | C16 | — | 28 | 2 | no | — |
| 182 | C2×C7 | — | 4 | 1 | no | — |
| 183 | C2×C7 | #1598 | 1 | 1 | yes | characteristic |
| 184 | C2×C7 | #2193 | 1 | 2 | yes | — |
| 185 | D7 | — | 16 | 2 | no | — |
| 186 | C23 | — | 28 | 2 | no | — |
| 187 | C23 | — | 56 | 1 | no | — |
| 188 | C23 | — | 56 | 1 | no | — |
| 189 | C23 | — | 112 | 2 | no | — |
| 190 | C23 | #6495 | 1 | 1 | yes | characteristic |
| 191 | Q8 | — | 4 | 2 | no | — |
| 192 | Q8 | — | 4 | 2 | no | — |
| 193 | D4 | — | 28 | 2 | no | — |
| 194 | D4 | — | 28 | 2 | no | — |
| 195 | D4 | — | 56 | 1 | no | — |
| 196 | D4 | — | 56 | 1 | no | — |
| 197 | C2×C4 | — | 2 | 1 | no | — |
| 198 | C2×C4 | — | 2 | 1 | no | — |
| 199 | C2×C4 | — | 2 | 2 | no | — |
| 200 | C2×C4 | — | 4 | 2 | no | — |
| 201 | C2×C4 | — | 4 | 2 | no | — |
| 202 | C2×C4 | C2×#1141 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | #6516 | 1 | 1 | yes | characteristic |
| 204 | C8 | — | 2 | 2 | no | — |
| 205 | C8 | — | 4 | 1 | no | — |
| 206 | C8 | — | 56 | 1 | no | — |
| 207 | C7 | #12165 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 208 | C22 | — | 2 | 2 | no | — |
| 209 | C22 | — | 2 | 1 | no | — |
| 210 | C22 | — | 56 | 2 | no | — |
| 211 | C22 | — | 112 | 2 | no | — |
| 212 | C22 | — | 112 | 2 | no | — |
| 213 | C22 | — | 112 | 2 | no | — |
| 214 | C22 | #89989 | 1 | 1 | yes | characteristic |
| 215 | C4 | — | 2 | 1 | no | — |
| 216 | C4 | — | 4 | 2 | no | — |
| 217 | C4 | — | 8 | 2 | no | — |
| 218 | C4 | — | 8 | 2 | no | — |
| 219 | C4 | #90016 | 1 | 1 | yes | characteristic |
| 220 | C4 | #90017 | 1 | 1 | yes | characteristic |
| 221 | C2 | — | 4 | 1 | no | — |
| 222 | C2 | — | 112 | 2 | no | — |
| 223 | C2 | #11729632 | 1 | 1 | yes | characteristic |
| 224 | C2 | #11729878 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3261 | C2 | semidirect |
| 4. #11725583 | C2 | semidirect |
| 5. #11725524 | C2 | semidirect |
| 44. #3261 | D7 | semidirect |
| 207. C7 | #12165 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3261 | 2 |
| #11728839 | 2 |
| #11728294 | 2 |
| #11725583 | 2 |
| #11725524 | 2 |
| #12165 | 7 |