Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
644 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3567 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#1521 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C7×#1482 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C7×#564 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C7×#562 | C4 | 1 | 3 | yes | complemented |
| 6 | C7×#553 | — | 2 | 3 | no | — |
| 7 | C7×#552 | — | 2 | 3 | no | — |
| 8 | C7×#540 | — | 2 | 3 | no | — |
| 9 | C7×#539 | — | 2 | 3 | no | — |
| 10 | C7×#529 | C22 | 1 | 3 | yes | — |
| 11 | C7×#431 | C22 | 1 | 3 | yes | — |
| 12 | C7×#383 | C22 | 1 | 1 | yes | characteristic |
| 13 | C7×#328 | — | 2 | 3 | no | — |
| 14 | C7×#327 | — | 2 | 3 | no | — |
| 15 | C7×Q8⋊C8 | — | 2 | 3 | no | — |
| 16 | C7×D4⋊C8 | — | 2 | 3 | no | — |
| 17 | #8515 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 18 | C7×#127 | — | 2 | 1 | no | — |
| 19 | C7×#126 | — | 2 | 1 | no | — |
| 20 | C7×#126 | — | 2 | 3 | no | — |
| 21 | C7×#126 | — | 2 | 3 | no | — |
| 22 | C7×#125 | — | 2 | 3 | no | — |
| 23 | C7×#125 | — | 2 | 3 | no | — |
| 24 | C7×#125 | — | 2 | 3 | no | — |
| 25 | C7×#125 | C2×C4 | 1 | 3 | yes | — |
| 26 | C7×#124 | C2×C4 | 1 | 3 | yes | — |
| 27 | C7×#123 | — | 2 | 3 | no | — |
| 28 | C7×#123 | — | 2 | 3 | no | — |
| 29 | C7×#123 | — | 2 | 3 | no | — |
| 30 | C7×#123 | — | 2 | 3 | no | — |
| 31 | C7×#123 | — | 2 | 3 | no | — |
| 32 | C7×C22⋊Q8 | — | 2 | 3 | no | — |
| 33 | C7×C22⋊Q8 | — | 2 | 3 | no | — |
| 34 | C7×C22⋊Q8 | D4 | 1 | 3 | yes | — |
| 35 | C7×#121 | — | 2 | 3 | no | — |
| 36 | C7×#121 | — | 2 | 3 | no | — |
| 37 | C7×#121 | — | 2 | 3 | no | — |
| 38 | C7×#121 | — | 2 | 3 | no | — |
| 39 | C7×#121 | D4 | 1 | 3 | yes | — |
| 40 | C4×C7×Q8 | D4 | 1 | 3 | yes | — |
| 41 | C4×C7×D4 | — | 2 | 3 | no | — |
| 42 | C4×C7×D4 | — | 2 | 3 | no | — |
| 43 | C4×C7×D4 | — | 2 | 3 | no | — |
| 44 | C4×C7×D4 | D4 | 1 | 3 | yes | — |
| 45 | C7×#117 | — | 2 | 3 | no | — |
| 46 | C7×#117 | — | 2 | 3 | no | — |
| 47 | C2×C7×C4⋊C4 | — | 2 | 3 | no | — |
| 48 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 49 | C7×C4⋊C8 | — | 2 | 3 | no | — |
| 50 | C7×C22⋊C8 | — | 2 | 3 | no | — |
| 51 | C4×C7×C8 | — | 2 | 3 | no | — |
| 52 | C7×#95 | — | 2 | 3 | no | — |
| 53 | #3567 | C2×C7 | 1 | 1 | yes | characteristic |
| 54 | #1521 | C2×C7 | 1 | 3 | yes | complemented |
| 55 | #1482 | C2×C7 | 1 | 3 | yes | complemented |
| 56 | C2×C7×Q8 | C22⋊C4 | 1 | 3 | yes | — |
| 57 | C2×C7×D4 | — | 2 | 3 | no | — |
| 58 | C2×C7×D4 | — | 2 | 3 | no | — |
| 59 | C2×C7×D4 | — | 2 | 3 | no | — |
| 60 | C2×C7×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 61 | C22×C4×C7 | — | 2 | 3 | no | — |
| 62 | C22×C4×C7 | — | 2 | 3 | no | — |
| 63 | C22×C4×C7 | — | 2 | 3 | no | — |
| 64 | C22×C4×C7 | — | 4 | 1 | no | — |
| 65 | C22×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 66 | C2×C7×C8 | — | 4 | 3 | no | — |
| 67 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 68 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 69 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 70 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 71 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 72 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 73 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 74 | C7×C4⋊C4 | — | 2 | 3 | no | — |
| 75 | C7×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 76 | C7×C4⋊C4 | C22⋊C4 | 1 | 3 | yes | — |
| 77 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 78 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 79 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 80 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 81 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 82 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 83 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 84 | C42×C7 | — | 2 | 1 | no | — |
| 85 | C42×C7 | — | 2 | 1 | no | — |
| 86 | C42×C7 | — | 2 | 3 | no | — |
| 87 | C42×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C42×C7 | C2×D4 | 1 | 3 | yes | — |
| 89 | #564 | C4×C7 | 1 | 1 | yes | characteristic, complemented |
| 90 | #562 | C4×C7 | 1 | 3 | yes | complemented |
| 91 | #553 | — | 2 | 3 | no | — |
| 92 | #552 | — | 2 | 3 | no | — |
| 93 | #540 | — | 2 | 3 | no | — |
| 94 | #539 | — | 2 | 3 | no | — |
| 95 | #529 | C22×C7 | 1 | 3 | yes | — |
| 96 | #431 | C22×C7 | 1 | 3 | yes | — |
| 97 | #383 | C22×C7 | 1 | 1 | yes | characteristic |
| 98 | #328 | — | 2 | 3 | no | — |
| 99 | #327 | — | 2 | 3 | no | — |
| 100 | Q8⋊C8 | — | 2 | 3 | no | — |
| 101 | D4⋊C8 | — | 2 | 3 | no | — |
| 102 | C23×C7 | — | 2 | 3 | no | — |
| 103 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 104 | C7×Q8 | — | 4 | 3 | no | — |
| 105 | C7×D4 | — | 4 | 3 | no | — |
| 106 | C7×D4 | — | 8 | 3 | no | — |
| 107 | C7×D4 | — | 8 | 3 | no | — |
| 108 | C7×D4 | — | 8 | 3 | no | — |
| 109 | C2×C4×C7 | — | 2 | 3 | no | — |
| 110 | C2×C4×C7 | — | 2 | 3 | no | — |
| 111 | C2×C4×C7 | — | 2 | 3 | no | — |
| 112 | C2×C4×C7 | — | 4 | 3 | no | — |
| 113 | C2×C4×C7 | — | 8 | 3 | no | — |
| 114 | C2×C4×C7 | — | 8 | 3 | no | — |
| 115 | C2×C4×C7 | — | 8 | 3 | no | — |
| 116 | C2×C4×C7 | — | 8 | 3 | no | — |
| 117 | C2×C4×C7 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 118 | C2×C4×C7 | #120 | 1 | 3 | yes | — |
| 119 | C7×C8 | — | 8 | 3 | no | — |
| 120 | #127 | — | 2 | 1 | no | — |
| 121 | #126 | — | 2 | 1 | no | — |
| 122 | #126 | — | 2 | 3 | no | — |
| 123 | #126 | — | 2 | 3 | no | — |
| 124 | #125 | — | 2 | 3 | no | — |
| 125 | #125 | — | 2 | 3 | no | — |
| 126 | #125 | — | 2 | 3 | no | — |
| 127 | #125 | C2×C4×C7 | 1 | 3 | yes | — |
| 128 | #124 | C2×C4×C7 | 1 | 3 | yes | — |
| 129 | #123 | — | 2 | 3 | no | — |
| 130 | #123 | — | 2 | 3 | no | — |
| 131 | #123 | — | 2 | 3 | no | — |
| 132 | #123 | — | 2 | 3 | no | — |
| 133 | #123 | — | 2 | 3 | no | — |
| 134 | C22⋊Q8 | — | 2 | 3 | no | — |
| 135 | C22⋊Q8 | — | 2 | 3 | no | — |
| 136 | C22⋊Q8 | C7×D4 | 1 | 3 | yes | — |
| 137 | #121 | — | 2 | 3 | no | — |
| 138 | #121 | — | 2 | 3 | no | — |
| 139 | #121 | — | 2 | 3 | no | — |
| 140 | #121 | — | 2 | 3 | no | — |
| 141 | #121 | C7×D4 | 1 | 3 | yes | — |
| 142 | C4×Q8 | C7×D4 | 1 | 3 | yes | — |
| 143 | C4×D4 | — | 2 | 3 | no | — |
| 144 | C4×D4 | — | 2 | 3 | no | — |
| 145 | C4×D4 | — | 2 | 3 | no | — |
| 146 | C4×D4 | C7×D4 | 1 | 3 | yes | — |
| 147 | #117 | — | 2 | 3 | no | — |
| 148 | #117 | — | 2 | 3 | no | — |
| 149 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 150 | C2×C42 | C23×C7 | 1 | 1 | yes | characteristic |
| 151 | C4⋊C8 | — | 2 | 3 | no | — |
| 152 | C22⋊C8 | — | 2 | 3 | no | — |
| 153 | C4×C8 | — | 2 | 3 | no | — |
| 154 | #95 | — | 2 | 3 | no | — |
| 155 | C22×C7 | — | 2 | 3 | no | — |
| 156 | C22×C7 | — | 4 | 3 | no | — |
| 157 | C22×C7 | — | 4 | 3 | no | — |
| 158 | C22×C7 | — | 4 | 3 | no | — |
| 159 | C22×C7 | #379 | 1 | 1 | yes | characteristic |
| 160 | C4×C7 | — | 2 | 3 | no | — |
| 161 | C4×C7 | — | 2 | 3 | no | — |
| 162 | C4×C7 | — | 4 | 3 | no | — |
| 163 | C4×C7 | — | 4 | 3 | no | — |
| 164 | C4×C7 | — | 4 | 3 | no | — |
| 165 | C4×C7 | — | 16 | 1 | no | — |
| 166 | C2×Q8 | C7×C22⋊C4 | 1 | 3 | yes | — |
| 167 | C2×D4 | — | 2 | 3 | no | — |
| 168 | C2×D4 | — | 2 | 3 | no | — |
| 169 | C2×D4 | — | 2 | 3 | no | — |
| 170 | C2×D4 | C7×C22⋊C4 | 1 | 3 | yes | — |
| 171 | C22×C4 | — | 2 | 3 | no | — |
| 172 | C22×C4 | — | 2 | 3 | no | — |
| 173 | C22×C4 | — | 2 | 3 | no | — |
| 174 | C22×C4 | — | 4 | 1 | no | — |
| 175 | C22×C4 | C2×C7×D4 | 1 | 3 | yes | — |
| 176 | C2×C8 | — | 4 | 3 | no | — |
| 177 | C4⋊C4 | — | 2 | 3 | no | — |
| 178 | C4⋊C4 | — | 2 | 3 | no | — |
| 179 | C4⋊C4 | — | 2 | 3 | no | — |
| 180 | C4⋊C4 | — | 2 | 3 | no | — |
| 181 | C4⋊C4 | — | 2 | 3 | no | — |
| 182 | C4⋊C4 | — | 2 | 3 | no | — |
| 183 | C4⋊C4 | — | 2 | 3 | no | — |
| 184 | C4⋊C4 | — | 2 | 3 | no | — |
| 185 | C4⋊C4 | C7×C22⋊C4 | 1 | 3 | yes | — |
| 186 | C4⋊C4 | C7×C22⋊C4 | 1 | 3 | yes | — |
| 187 | C22⋊C4 | — | 2 | 3 | no | — |
| 188 | C22⋊C4 | — | 2 | 3 | no | — |
| 189 | C22⋊C4 | — | 2 | 3 | no | — |
| 190 | C22⋊C4 | — | 2 | 3 | no | — |
| 191 | C22⋊C4 | — | 2 | 3 | no | — |
| 192 | C22⋊C4 | — | 2 | 3 | no | — |
| 193 | C22⋊C4 | — | 2 | 3 | no | — |
| 194 | C42 | — | 2 | 1 | no | — |
| 195 | C42 | — | 2 | 1 | no | — |
| 196 | C42 | — | 2 | 3 | no | — |
| 197 | C42 | C22×C4×C7 | 1 | 1 | yes | characteristic |
| 198 | C42 | C2×C7×D4 | 1 | 3 | yes | — |
| 199 | C2×C7 | — | 4 | 1 | no | — |
| 200 | C2×C7 | — | 8 | 3 | no | — |
| 201 | C2×C7 | #1794 | 1 | 3 | yes | — |
| 202 | C23 | — | 2 | 3 | no | — |
| 203 | C23 | C7×#120 | 1 | 1 | yes | characteristic |
| 204 | Q8 | — | 4 | 3 | no | — |
| 205 | D4 | — | 4 | 3 | no | — |
| 206 | D4 | — | 8 | 3 | no | — |
| 207 | D4 | — | 8 | 3 | no | — |
| 208 | D4 | — | 8 | 3 | no | — |
| 209 | C2×C4 | — | 2 | 3 | no | — |
| 210 | C2×C4 | — | 2 | 3 | no | — |
| 211 | C2×C4 | — | 2 | 3 | no | — |
| 212 | C2×C4 | — | 4 | 3 | no | — |
| 213 | C2×C4 | — | 8 | 3 | no | — |
| 214 | C2×C4 | — | 8 | 3 | no | — |
| 215 | C2×C4 | — | 8 | 3 | no | — |
| 216 | C2×C4 | — | 8 | 3 | no | — |
| 217 | C2×C4 | C2×C7×C22⋊C4 | 1 | 3 | yes | — |
| 218 | C2×C4 | C7×#120 | 1 | 3 | yes | — |
| 219 | C8 | — | 8 | 3 | no | — |
| 220 | C7 | #8515 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 221 | C22 | — | 2 | 3 | no | — |
| 222 | C22 | — | 4 | 3 | no | — |
| 223 | C22 | — | 4 | 3 | no | — |
| 224 | C22 | — | 4 | 3 | no | — |
| 225 | C22 | C7×#379 | 1 | 1 | yes | characteristic |
| 226 | C4 | — | 2 | 3 | no | — |
| 227 | C4 | — | 2 | 3 | no | — |
| 228 | C4 | — | 4 | 3 | no | — |
| 229 | C4 | — | 4 | 3 | no | — |
| 230 | C4 | — | 4 | 3 | no | — |
| 231 | C4 | — | 16 | 1 | no | — |
| 232 | C2 | — | 4 | 1 | no | — |
| 233 | C2 | — | 8 | 3 | no | — |
| 234 | C2 | C7×#1794 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#3567 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1521 | C2 | semidirect |
| 3. C7×#1482 | C2 | semidirect |
| 4. C7×#564 | C4 | semidirect |
| 5. C7×#562 | C4 | semidirect |
| 17. #8515 | C7 | direct (N × H) |
| 54. #1521 | C2×C7 | semidirect |
| 55. #1482 | C2×C7 | semidirect |
| 89. #564 | C4×C7 | semidirect |
| 90. #562 | C4×C7 | semidirect |
| 220. C7 | #8515 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3567 | 2 |
| C7×#1521 | 2 |
| C7×#1482 | 2 |
| #8515 | 7 |