Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
364 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #11739236 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#1715 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11729104 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11725597 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11725249 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #90448 | — | 2 | 2 | no | — |
| 7 | C7×#498 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C7×#493 | C22 | 1 | 1 | yes | characteristic |
| 9 | C7×#492 | — | 2 | 1 | no | — |
| 10 | C7×#486 | — | 2 | 1 | no | — |
| 11 | #89932 | — | 2 | 2 | no | — |
| 12 | #89849 | — | 2 | 1 | no | — |
| 13 | #89849 | — | 2 | 1 | no | — |
| 14 | #89472 | C22 | 1 | 2 | yes | — |
| 15 | C7×#322 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | #89283 | — | 2 | 2 | no | — |
| 17 | #89282 | — | 2 | 2 | no | — |
| 18 | #89258 | C22 | 1 | 2 | yes | complemented |
| 19 | #12366 | — | 7 | 1 | no | maximal, Sylow |
| 20 | #6548 | — | 2 | 1 | no | — |
| 21 | #6548 | — | 2 | 1 | no | — |
| 22 | C2×C7×SD16 | — | 4 | 2 | no | — |
| 23 | C2×C7×D8 | — | 2 | 2 | no | — |
| 24 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 25 | C7×#127 | D4 | 1 | 1 | yes | characteristic |
| 26 | C7×#127 | D4 | 1 | 1 | yes | characteristic |
| 27 | #6495 | — | 2 | 2 | no | — |
| 28 | #6495 | — | 2 | 2 | no | — |
| 29 | #6475 | — | 4 | 2 | no | — |
| 30 | #6474 | — | 4 | 2 | no | — |
| 31 | #6474 | — | 4 | 2 | no | — |
| 32 | D7×D8 | — | 4 | 2 | no | — |
| 33 | #6467 | D4 | 1 | 2 | yes | — |
| 34 | #6464 | — | 2 | 2 | no | — |
| 35 | C7×#106 | — | 2 | 2 | no | — |
| 36 | C7×#102 | — | 4 | 2 | no | — |
| 37 | C4×C7×C8 | — | 2 | 1 | no | — |
| 38 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 39 | #6407 | — | 2 | 1 | no | — |
| 40 | #6407 | — | 2 | 1 | no | — |
| 41 | #6393 | D4 | 1 | 2 | yes | — |
| 42 | #6386 | — | 2 | 2 | no | — |
| 43 | #6379 | — | 2 | 2 | no | — |
| 44 | #3292 | — | 7 | 1 | no | — |
| 45 | #2230 | — | 7 | 2 | no | — |
| 46 | #1715 | D7 | 1 | 1 | yes | characteristic, complemented |
| 47 | #1357 | — | 7 | 1 | no | — |
| 48 | #1336 | — | 7 | 2 | no | — |
| 49 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 50 | C2×C7×D4 | — | 2 | 1 | no | — |
| 51 | C2×C7×D4 | — | 2 | 1 | no | — |
| 52 | C2×C7×D4 | — | 4 | 1 | no | — |
| 53 | C2×C7×D4 | — | 4 | 1 | no | — |
| 54 | C2×C22⋊D7 | — | 8 | 1 | no | — |
| 55 | C2×C22⋊D7 | — | 8 | 1 | no | — |
| 56 | #1159 | — | 4 | 2 | no | — |
| 57 | #1159 | — | 4 | 2 | no | — |
| 58 | D4×D7 | — | 4 | 2 | no | — |
| 59 | D4×D7 | — | 4 | 2 | no | — |
| 60 | #1157 | — | 2 | 2 | no | — |
| 61 | C7×SD16 | — | 8 | 4 | no | — |
| 62 | C7×D8 | — | 4 | 2 | no | — |
| 63 | C7×D8 | — | 4 | 2 | no | — |
| 64 | C2×C7×C8 | — | 2 | 2 | no | — |
| 65 | C2×C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 66 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 67 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 68 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | #1142 | — | 4 | 2 | no | — |
| 70 | #1142 | — | 4 | 2 | no | — |
| 71 | #1141 | — | 4 | 2 | no | — |
| 72 | #1141 | — | 4 | 2 | no | — |
| 73 | #1136 | — | 2 | 2 | no | — |
| 74 | Dic28 | SD16 | 1 | 2 | yes | complemented |
| 75 | D56 | SD16 | 1 | 2 | yes | complemented |
| 76 | #1132 | — | 2 | 2 | no | — |
| 77 | #1131 | — | 4 | 2 | no | — |
| 78 | C8×D7 | — | 4 | 2 | no | — |
| 79 | C7⋊C16 | — | 4 | 2 | no | — |
| 80 | #576 | — | 14 | 2 | no | — |
| 81 | #498 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 82 | #493 | C2×D7 | 1 | 1 | yes | characteristic |
| 83 | #492 | — | 2 | 1 | no | — |
| 84 | #486 | — | 2 | 1 | no | — |
| 85 | #472 | — | 14 | 2 | no | — |
| 86 | #453 | — | 14 | 1 | no | — |
| 87 | #453 | — | 14 | 1 | no | — |
| 88 | #443 | — | 7 | 2 | no | — |
| 89 | #365 | — | 14 | 2 | no | — |
| 90 | #364 | — | 7 | 2 | no | — |
| 91 | #360 | — | 14 | 2 | no | — |
| 92 | #322 | C2×D7 | 1 | 1 | yes | characteristic, complemented |
| 93 | C23×C7 | — | 4 | 1 | no | — |
| 94 | C23×C7 | — | 4 | 1 | no | — |
| 95 | C22×D7 | — | 8 | 1 | no | — |
| 96 | C22×D7 | — | 8 | 1 | no | — |
| 97 | C7×Q8 | — | 8 | 2 | no | — |
| 98 | C7×D4 | — | 4 | 2 | no | — |
| 99 | C7×D4 | — | 4 | 2 | no | — |
| 100 | C7×D4 | — | 8 | 2 | no | — |
| 101 | C7×D4 | — | 8 | 2 | no | — |
| 102 | C2×C4×C7 | — | 2 | 1 | no | — |
| 103 | C2×C4×C7 | — | 8 | 1 | no | — |
| 104 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 105 | C22⋊D7 | — | 4 | 2 | no | — |
| 106 | C22⋊D7 | — | 8 | 2 | no | — |
| 107 | C22⋊D7 | — | 8 | 2 | no | — |
| 108 | C2×Dic7 | — | 8 | 1 | no | — |
| 109 | C2×Dic7 | — | 8 | 1 | no | — |
| 110 | D28 | — | 2 | 2 | no | — |
| 111 | C4×D7 | — | 4 | 2 | no | — |
| 112 | Dic14 | — | 2 | 2 | no | — |
| 113 | C7×C8 | — | 4 | 2 | no | — |
| 114 | C7×C8 | C2×SD16 | 1 | 2 | yes | — |
| 115 | C7×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 116 | C7⋊C8 | — | 4 | 2 | no | — |
| 117 | H(5,𝔽2) | — | 14 | 1 | no | — |
| 118 | H(5,𝔽2) | — | 14 | 1 | no | — |
| 119 | AGL(1,ℤ/8) | — | 14 | 2 | no | — |
| 120 | AGL(1,ℤ/8) | — | 14 | 2 | no | — |
| 121 | AGL(1,ℤ/8) | — | 28 | 2 | no | — |
| 122 | AGL(1,ℤ/8) | — | 28 | 2 | no | — |
| 123 | #135 | — | 7 | 2 | no | — |
| 124 | #135 | — | 28 | 2 | no | — |
| 125 | C2×SD16 | — | 4 | 2 | no | — |
| 126 | C2×D8 | — | 2 | 2 | no | — |
| 127 | C2×D8 | — | 28 | 2 | no | — |
| 128 | #131 | — | 14 | 2 | no | — |
| 129 | C4⋊Q8 | — | 2 | 1 | no | — |
| 130 | #127 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 131 | #127 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 132 | M32 | — | 14 | 2 | no | — |
| 133 | #108 | — | 7 | 2 | no | — |
| 134 | #106 | — | 2 | 2 | no | — |
| 135 | #104 | — | 14 | 2 | no | — |
| 136 | #102 | — | 4 | 2 | no | — |
| 137 | #100 | — | 14 | 1 | no | — |
| 138 | #100 | — | 14 | 1 | no | — |
| 139 | C4×C8 | — | 2 | 1 | no | — |
| 140 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 141 | C22×C7 | — | 8 | 1 | no | — |
| 142 | C22×C7 | — | 8 | 1 | no | — |
| 143 | C22×C7 | — | 16 | 1 | no | — |
| 144 | C22×C7 | — | 16 | 1 | no | — |
| 145 | C22×C7 | #453 | 1 | 1 | yes | characteristic |
| 146 | C2×D7 | — | 4 | 2 | no | — |
| 147 | C2×D7 | — | 32 | 1 | no | — |
| 148 | C2×D7 | — | 32 | 1 | no | — |
| 149 | C4×C7 | — | 2 | 2 | no | — |
| 150 | C4×C7 | — | 16 | 1 | no | — |
| 151 | C4×C7 | #450 | 1 | 2 | yes | — |
| 152 | Dic7 | — | 4 | 2 | no | — |
| 153 | #41 | — | 14 | 2 | no | — |
| 154 | #41 | — | 28 | 2 | no | — |
| 155 | #41 | — | 28 | 2 | no | — |
| 156 | C2×Q8 | — | 4 | 1 | no | — |
| 157 | C2×D4 | — | 2 | 1 | no | — |
| 158 | C2×D4 | — | 2 | 1 | no | — |
| 159 | C2×D4 | — | 4 | 1 | no | — |
| 160 | C2×D4 | — | 4 | 1 | no | — |
| 161 | C2×D4 | — | 28 | 2 | no | — |
| 162 | C2×D4 | — | 28 | 2 | no | — |
| 163 | C2×D4 | — | 56 | 1 | no | — |
| 164 | C2×D4 | — | 56 | 1 | no | — |
| 165 | Q16 | — | 7 | 2 | no | — |
| 166 | SD16 | — | 8 | 4 | no | — |
| 167 | SD16 | — | 14 | 2 | no | — |
| 168 | SD16 | — | 28 | 2 | no | — |
| 169 | SD16 | — | 28 | 2 | no | — |
| 170 | D8 | — | 4 | 2 | no | — |
| 171 | D8 | — | 4 | 2 | no | — |
| 172 | D8 | — | 7 | 2 | no | — |
| 173 | D8 | — | 28 | 2 | no | — |
| 174 | D8 | — | 28 | 2 | no | — |
| 175 | M16 | — | 14 | 2 | no | — |
| 176 | M16 | — | 28 | 2 | no | — |
| 177 | C2×C8 | — | 2 | 2 | no | — |
| 178 | C2×C8 | — | 28 | 2 | no | — |
| 179 | C2×C8 | D4×D7 | 1 | 2 | yes | — |
| 180 | C4⋊C4 | — | 4 | 1 | no | — |
| 181 | C4⋊C4 | — | 4 | 1 | no | — |
| 182 | C42 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 183 | C16 | — | 28 | 2 | no | — |
| 184 | C2×C7 | — | 2 | 1 | no | — |
| 185 | C2×C7 | — | 16 | 1 | no | — |
| 186 | C2×C7 | — | 16 | 1 | no | — |
| 187 | C2×C7 | #1629 | 1 | 1 | yes | characteristic |
| 188 | D7 | — | 8 | 2 | no | — |
| 189 | C23 | — | 4 | 1 | no | — |
| 190 | C23 | — | 4 | 1 | no | — |
| 191 | C23 | — | 56 | 1 | no | — |
| 192 | C23 | — | 56 | 1 | no | — |
| 193 | Q8 | — | 8 | 2 | no | — |
| 194 | Q8 | — | 14 | 2 | no | — |
| 195 | D4 | — | 4 | 2 | no | — |
| 196 | D4 | — | 4 | 2 | no | — |
| 197 | D4 | — | 8 | 2 | no | — |
| 198 | D4 | — | 8 | 2 | no | — |
| 199 | D4 | — | 14 | 2 | no | — |
| 200 | D4 | — | 28 | 2 | no | — |
| 201 | D4 | — | 56 | 2 | no | — |
| 202 | D4 | — | 56 | 2 | no | — |
| 203 | C2×C4 | — | 2 | 1 | no | — |
| 204 | C2×C4 | — | 8 | 1 | no | — |
| 205 | C2×C4 | — | 28 | 2 | no | — |
| 206 | C2×C4 | — | 56 | 1 | no | — |
| 207 | C2×C4 | — | 56 | 1 | no | — |
| 208 | C2×C4 | #6500 | 1 | 1 | yes | characteristic |
| 209 | C8 | — | 4 | 2 | no | — |
| 210 | C8 | — | 28 | 2 | no | — |
| 211 | C8 | D7×SD16 | 1 | 2 | yes | — |
| 212 | C8 | #6477 | 1 | 2 | yes | — |
| 213 | C7 | #12366 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 214 | C22 | — | 8 | 1 | no | — |
| 215 | C22 | — | 8 | 1 | no | — |
| 216 | C22 | — | 16 | 1 | no | — |
| 217 | C22 | — | 16 | 1 | no | — |
| 218 | C22 | — | 28 | 2 | no | — |
| 219 | C22 | — | 224 | 1 | no | — |
| 220 | C22 | — | 224 | 1 | no | — |
| 221 | C22 | #89849 | 1 | 1 | yes | characteristic |
| 222 | C4 | — | 2 | 2 | no | — |
| 223 | C4 | — | 16 | 1 | no | — |
| 224 | C4 | — | 28 | 2 | no | — |
| 225 | C4 | #89941 | 1 | 2 | yes | — |
| 226 | C2 | — | 2 | 1 | no | — |
| 227 | C2 | — | 16 | 1 | no | — |
| 228 | C2 | — | 16 | 1 | no | — |
| 229 | C2 | — | 56 | 2 | no | — |
| 230 | C2 | #11729020 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1715 | C2 | semidirect |
| 3. #11729104 | C2 | semidirect |
| 4. #11725597 | C2 | semidirect |
| 5. #11725249 | C2 | semidirect |
| 7. C7×#498 | C22 | semidirect |
| 15. C7×#322 | C22 | semidirect |
| 18. #89258 | C22 | semidirect |
| 46. #1715 | D7 | semidirect |
| 74. Dic28 | SD16 | semidirect |
| 75. D56 | SD16 | semidirect |
| 81. #498 | C2×D7 | semidirect |
| 92. #322 | C2×D7 | semidirect |
| 213. C7 | #12366 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11739236 | 2 |
| C7×#1715 | 2 |
| #11729104 | 2 |
| #11725597 | 2 |
| #11725249 | 2 |
| #12366 | 7 |