Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
398 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3291 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C7×#2204 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C7×#1624 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C7×#1410 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C7×#1405 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C7×#577 | — | 2 | 2 | no | — |
| 7 | C7×#555 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C7×#492 | C22 | 1 | 1 | yes | characteristic |
| 9 | C7×#471 | — | 2 | 2 | no | — |
| 10 | C7×#459 | — | 2 | 1 | no | — |
| 11 | C7×#456 | C22 | 1 | 2 | yes | — |
| 12 | C7×#454 | — | 2 | 2 | no | — |
| 13 | C7×#453 | — | 2 | 1 | no | — |
| 14 | C7×#443 | — | 2 | 1 | no | — |
| 15 | C7×#356 | C22 | 1 | 2 | yes | complemented |
| 16 | C7×#355 | — | 2 | 2 | no | — |
| 17 | C7×#333 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | C7×D4⋊C8 | — | 2 | 1 | no | — |
| 19 | #13681 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 20 | C7×2−1+4 | D4 | 1 | 2 | yes | — |
| 21 | C7×H(5,𝔽2) | — | 2 | 1 | no | — |
| 22 | C7×#137 | — | 2 | 2 | no | — |
| 23 | C7×#137 | — | 4 | 2 | no | — |
| 24 | C7×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 25 | C7×AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 26 | C7×AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 27 | C7×#135 | — | 2 | 1 | no | — |
| 28 | C7×#135 | — | 4 | 2 | no | — |
| 29 | C2×C7×SD16 | — | 2 | 2 | no | — |
| 30 | C2×C7×SD16 | — | 4 | 2 | no | — |
| 31 | C2×C7×D8 | — | 4 | 1 | no | — |
| 32 | C7×#131 | — | 2 | 2 | no | — |
| 33 | C7×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 34 | C7×#127 | D4 | 1 | 1 | yes | characteristic |
| 35 | C7×#124 | — | 2 | 2 | no | — |
| 36 | C7×#121 | — | 4 | 1 | no | — |
| 37 | C4×C7×D4 | — | 2 | 1 | no | — |
| 38 | C7×#117 | — | 4 | 1 | no | — |
| 39 | C7×#108 | — | 2 | 1 | no | — |
| 40 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 41 | C7×#104 | — | 2 | 2 | no | — |
| 42 | C7×#104 | — | 4 | 2 | no | — |
| 43 | C7×#102 | — | 4 | 1 | no | — |
| 44 | C7×#101 | — | 2 | 2 | no | — |
| 45 | C7×#101 | D4 | 1 | 2 | yes | — |
| 46 | C7×#100 | — | 2 | 1 | no | — |
| 47 | C4×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 48 | #3291 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 49 | #2204 | C2×C7 | 1 | 2 | yes | complemented |
| 50 | #1624 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 51 | #1410 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 52 | #1405 | C2×C7 | 1 | 2 | yes | complemented |
| 53 | C7×#41 | — | 2 | 2 | no | — |
| 54 | C7×#41 | — | 4 | 2 | no | — |
| 55 | C7×#41 | — | 4 | 2 | no | — |
| 56 | C7×#41 | — | 4 | 2 | no | — |
| 57 | C2×C7×Q8 | — | 4 | 2 | no | — |
| 58 | C2×C7×Q8 | C2×D4 | 1 | 2 | yes | — |
| 59 | C2×C7×D4 | — | 2 | 1 | no | — |
| 60 | C2×C7×D4 | — | 2 | 2 | no | — |
| 61 | C2×C7×D4 | — | 2 | 1 | no | — |
| 62 | C2×C7×D4 | — | 4 | 1 | no | — |
| 63 | C2×C7×D4 | — | 4 | 2 | no | — |
| 64 | C2×C7×D4 | — | 8 | 1 | no | — |
| 65 | C22×C4×C7 | — | 4 | 1 | no | — |
| 66 | C7×Q16 | — | 2 | 1 | no | — |
| 67 | C7×Q16 | — | 4 | 2 | no | — |
| 68 | C7×SD16 | — | 2 | 2 | no | — |
| 69 | C7×SD16 | — | 4 | 2 | no | — |
| 70 | C7×SD16 | — | 4 | 2 | no | — |
| 71 | C7×SD16 | — | 4 | 2 | no | — |
| 72 | C7×SD16 | — | 4 | 2 | no | — |
| 73 | C7×SD16 | — | 8 | 2 | no | — |
| 74 | C7×D8 | — | 2 | 1 | no | — |
| 75 | C7×D8 | — | 4 | 2 | no | — |
| 76 | C7×D8 | — | 8 | 2 | no | — |
| 77 | C7×D8 | — | 8 | 2 | no | — |
| 78 | C7×M16 | — | 2 | 2 | no | — |
| 79 | C7×M16 | — | 4 | 2 | no | — |
| 80 | C7×M16 | — | 4 | 2 | no | — |
| 81 | C2×C7×C8 | — | 2 | 1 | no | — |
| 82 | C2×C7×C8 | — | 4 | 2 | no | — |
| 83 | C2×C7×C8 | — | 4 | 1 | no | — |
| 84 | C7×C4⋊C4 | — | 2 | 1 | no | — |
| 85 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 86 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 87 | C7×C22⋊C4 | — | 4 | 1 | no | — |
| 88 | C42×C7 | — | 2 | 2 | no | — |
| 89 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | #577 | — | 2 | 2 | no | — |
| 91 | #555 | C22×C7 | 1 | 1 | yes | characteristic, complemented |
| 92 | #492 | C22×C7 | 1 | 1 | yes | characteristic |
| 93 | #471 | — | 2 | 2 | no | — |
| 94 | #459 | — | 2 | 1 | no | — |
| 95 | #456 | C22×C7 | 1 | 2 | yes | — |
| 96 | #454 | — | 2 | 2 | no | — |
| 97 | #453 | — | 2 | 1 | no | — |
| 98 | #443 | — | 2 | 1 | no | — |
| 99 | #356 | C22×C7 | 1 | 2 | yes | complemented |
| 100 | #355 | — | 2 | 2 | no | — |
| 101 | #333 | C22×C7 | 1 | 1 | yes | characteristic, complemented |
| 102 | D4⋊C8 | — | 2 | 1 | no | — |
| 103 | C23×C7 | — | 4 | 1 | no | — |
| 104 | C23×C7 | — | 4 | 1 | no | — |
| 105 | C23×C7 | — | 8 | 1 | no | — |
| 106 | C7×Q8 | — | 2 | 2 | no | — |
| 107 | C7×Q8 | — | 4 | 2 | no | — |
| 108 | C7×Q8 | — | 4 | 2 | no | — |
| 109 | C7×D4 | — | 2 | 2 | no | — |
| 110 | C7×D4 | — | 4 | 2 | no | — |
| 111 | C7×D4 | — | 4 | 1 | no | — |
| 112 | C7×D4 | — | 4 | 1 | no | — |
| 113 | C7×D4 | — | 4 | 2 | no | — |
| 114 | C7×D4 | — | 4 | 2 | no | — |
| 115 | C7×D4 | — | 8 | 1 | no | — |
| 116 | C7×D4 | — | 8 | 1 | no | — |
| 117 | C7×D4 | — | 8 | 2 | no | — |
| 118 | C7×D4 | — | 8 | 2 | no | — |
| 119 | C2×C4×C7 | — | 2 | 1 | no | — |
| 120 | C2×C4×C7 | — | 2 | 2 | no | — |
| 121 | C2×C4×C7 | — | 4 | 2 | no | — |
| 122 | C2×C4×C7 | — | 4 | 1 | no | — |
| 123 | C2×C4×C7 | — | 8 | 1 | no | — |
| 124 | C2×C4×C7 | — | 8 | 1 | no | — |
| 125 | C2×C4×C7 | — | 8 | 2 | no | — |
| 126 | C2×C4×C7 | — | 8 | 1 | no | — |
| 127 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 128 | C7×C8 | — | 2 | 1 | no | — |
| 129 | C7×C8 | — | 2 | 1 | no | — |
| 130 | C7×C8 | — | 4 | 2 | no | — |
| 131 | C7×C8 | — | 8 | 1 | no | — |
| 132 | C7×C8 | — | 8 | 2 | no | — |
| 133 | 2−1+4 | C7×D4 | 1 | 2 | yes | — |
| 134 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 135 | #137 | — | 2 | 2 | no | — |
| 136 | #137 | — | 4 | 2 | no | — |
| 137 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 138 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 139 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 140 | #135 | — | 2 | 1 | no | — |
| 141 | #135 | — | 4 | 2 | no | — |
| 142 | C2×SD16 | — | 2 | 2 | no | — |
| 143 | C2×SD16 | — | 4 | 2 | no | — |
| 144 | C2×D8 | — | 4 | 1 | no | — |
| 145 | #131 | — | 2 | 2 | no | — |
| 146 | C4⋊Q8 | C23×C7 | 1 | 1 | yes | characteristic |
| 147 | #127 | C7×D4 | 1 | 1 | yes | characteristic |
| 148 | #124 | — | 2 | 2 | no | — |
| 149 | #121 | — | 4 | 1 | no | — |
| 150 | C4×D4 | — | 2 | 1 | no | — |
| 151 | #117 | — | 4 | 1 | no | — |
| 152 | #108 | — | 2 | 1 | no | — |
| 153 | C4⋊C8 | — | 2 | 1 | no | — |
| 154 | #104 | — | 2 | 2 | no | — |
| 155 | #104 | — | 4 | 2 | no | — |
| 156 | #102 | — | 4 | 1 | no | — |
| 157 | #101 | — | 2 | 2 | no | — |
| 158 | #101 | C7×D4 | 1 | 2 | yes | — |
| 159 | #100 | — | 2 | 1 | no | — |
| 160 | C4×C8 | C7×D4 | 1 | 1 | yes | characteristic |
| 161 | C22×C7 | — | 4 | 2 | no | — |
| 162 | C22×C7 | — | 8 | 1 | no | — |
| 163 | C22×C7 | — | 8 | 1 | no | — |
| 164 | C22×C7 | — | 8 | 1 | no | — |
| 165 | C22×C7 | — | 8 | 1 | no | — |
| 166 | C22×C7 | — | 16 | 1 | no | — |
| 167 | C22×C7 | — | 32 | 1 | no | — |
| 168 | C22×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 169 | C4×C7 | — | 2 | 1 | no | — |
| 170 | C4×C7 | — | 2 | 1 | no | — |
| 171 | C4×C7 | — | 2 | 1 | no | — |
| 172 | C4×C7 | — | 4 | 2 | no | — |
| 173 | C4×C7 | — | 4 | 2 | no | — |
| 174 | C4×C7 | — | 8 | 1 | no | — |
| 175 | #41 | — | 2 | 2 | no | — |
| 176 | #41 | — | 4 | 2 | no | — |
| 177 | #41 | — | 4 | 2 | no | — |
| 178 | #41 | — | 4 | 2 | no | — |
| 179 | C2×Q8 | — | 4 | 2 | no | — |
| 180 | C2×Q8 | C2×C7×D4 | 1 | 2 | yes | — |
| 181 | C2×D4 | — | 2 | 1 | no | — |
| 182 | C2×D4 | — | 2 | 2 | no | — |
| 183 | C2×D4 | — | 2 | 1 | no | — |
| 184 | C2×D4 | — | 4 | 1 | no | — |
| 185 | C2×D4 | — | 4 | 2 | no | — |
| 186 | C2×D4 | — | 8 | 1 | no | — |
| 187 | C22×C4 | — | 4 | 1 | no | — |
| 188 | Q16 | — | 2 | 1 | no | — |
| 189 | Q16 | — | 4 | 2 | no | — |
| 190 | SD16 | — | 2 | 2 | no | — |
| 191 | SD16 | — | 4 | 2 | no | — |
| 192 | SD16 | — | 4 | 2 | no | — |
| 193 | SD16 | — | 4 | 2 | no | — |
| 194 | SD16 | — | 4 | 2 | no | — |
| 195 | SD16 | — | 8 | 2 | no | — |
| 196 | D8 | — | 2 | 1 | no | — |
| 197 | D8 | — | 4 | 2 | no | — |
| 198 | D8 | — | 8 | 2 | no | — |
| 199 | D8 | — | 8 | 2 | no | — |
| 200 | M16 | — | 2 | 2 | no | — |
| 201 | M16 | — | 4 | 2 | no | — |
| 202 | M16 | — | 4 | 2 | no | — |
| 203 | C2×C8 | — | 2 | 1 | no | — |
| 204 | C2×C8 | — | 4 | 2 | no | — |
| 205 | C2×C8 | — | 4 | 1 | no | — |
| 206 | C4⋊C4 | — | 2 | 1 | no | — |
| 207 | C4⋊C4 | — | 4 | 1 | no | — |
| 208 | C22⋊C4 | — | 4 | 2 | no | — |
| 209 | C22⋊C4 | — | 4 | 1 | no | — |
| 210 | C42 | — | 2 | 2 | no | — |
| 211 | C42 | C2×C7×D4 | 1 | 1 | yes | characteristic |
| 212 | C2×C7 | — | 2 | 1 | no | — |
| 213 | C2×C7 | — | 8 | 2 | no | — |
| 214 | C2×C7 | — | 16 | 1 | no | — |
| 215 | C2×C7 | — | 16 | 1 | no | — |
| 216 | C2×C7 | #2196 | 1 | 1 | yes | characteristic |
| 217 | C23 | — | 4 | 1 | no | — |
| 218 | C23 | — | 4 | 1 | no | — |
| 219 | C23 | — | 8 | 1 | no | — |
| 220 | Q8 | — | 2 | 2 | no | — |
| 221 | Q8 | — | 4 | 2 | no | — |
| 222 | Q8 | — | 4 | 2 | no | — |
| 223 | D4 | — | 2 | 2 | no | — |
| 224 | D4 | — | 4 | 2 | no | — |
| 225 | D4 | — | 4 | 1 | no | — |
| 226 | D4 | — | 4 | 1 | no | — |
| 227 | D4 | — | 4 | 2 | no | — |
| 228 | D4 | — | 4 | 2 | no | — |
| 229 | D4 | — | 8 | 1 | no | — |
| 230 | D4 | — | 8 | 1 | no | — |
| 231 | D4 | — | 8 | 2 | no | — |
| 232 | D4 | — | 8 | 2 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 2 | 2 | no | — |
| 235 | C2×C4 | — | 4 | 2 | no | — |
| 236 | C2×C4 | — | 4 | 1 | no | — |
| 237 | C2×C4 | — | 8 | 1 | no | — |
| 238 | C2×C4 | — | 8 | 1 | no | — |
| 239 | C2×C4 | — | 8 | 2 | no | — |
| 240 | C2×C4 | — | 8 | 1 | no | — |
| 241 | C2×C4 | C7×#120 | 1 | 1 | yes | characteristic |
| 242 | C8 | — | 2 | 1 | no | — |
| 243 | C8 | — | 2 | 1 | no | — |
| 244 | C8 | — | 4 | 2 | no | — |
| 245 | C8 | — | 8 | 1 | no | — |
| 246 | C8 | — | 8 | 2 | no | — |
| 247 | C7 | #13681 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 248 | C22 | — | 4 | 2 | no | — |
| 249 | C22 | — | 8 | 1 | no | — |
| 250 | C22 | — | 8 | 1 | no | — |
| 251 | C22 | — | 8 | 1 | no | — |
| 252 | C22 | — | 8 | 1 | no | — |
| 253 | C22 | — | 16 | 1 | no | — |
| 254 | C22 | — | 32 | 1 | no | — |
| 255 | C22 | C7×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 256 | C4 | — | 2 | 1 | no | — |
| 257 | C4 | — | 2 | 1 | no | — |
| 258 | C4 | — | 2 | 1 | no | — |
| 259 | C4 | — | 4 | 2 | no | — |
| 260 | C4 | — | 4 | 2 | no | — |
| 261 | C4 | — | 8 | 1 | no | — |
| 262 | C2 | — | 2 | 1 | no | — |
| 263 | C2 | — | 8 | 2 | no | — |
| 264 | C2 | — | 16 | 1 | no | — |
| 265 | C2 | — | 16 | 1 | no | — |
| 266 | C2 | C7×#2196 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2196 |
| commutator, Frattini, cosocle | C4⋊Q8 | C23×C7 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C7×C4⋊Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3291 | C2 | semidirect |
| 2. C7×#2204 | C2 | semidirect |
| 3. C7×#1624 | C2 | semidirect |
| 4. C7×#1410 | C2 | semidirect |
| 5. C7×#1405 | C2 | semidirect |
| 7. C7×#555 | C22 | semidirect |
| 15. C7×#356 | C22 | semidirect |
| 17. C7×#333 | C22 | semidirect |
| 19. #13681 | C7 | direct (N × H) |
| 48. #3291 | C2×C7 | semidirect |
| 49. #2204 | C2×C7 | semidirect |
| 50. #1624 | C2×C7 | semidirect |
| 51. #1410 | C2×C7 | semidirect |
| 52. #1405 | C2×C7 | semidirect |
| 91. #555 | C22×C7 | semidirect |
| 99. #356 | C22×C7 | semidirect |
| 101. #333 | C22×C7 | semidirect |
| 247. C7 | #13681 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3291 | 2 |
| C7×#2204 | 2 |
| C7×#1624 | 2 |
| C7×#1410 | 2 |
| C7×#1405 | 2 |
| #13681 | 7 |