Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
572 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 75 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 | #11734049 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C7×#1793 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11729897 | C2 | 1 | 2 | yes | maximal |
| 4 | #11725505 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11725450 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×C7×2−1+4 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C7×#104 | — | 2 | 1 | no | — |
| 8 | C7×#417 | — | 2 | 1 | no | — |
| 9 | C2×C7×#101 | C22 | 1 | 1 | yes | characteristic |
| 10 | C7×#391 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#6411 | — | 2 | 2 | no | — |
| 12 | #90004 | — | 2 | 2 | no | — |
| 13 | #90002 | C22 | 1 | 2 | yes | — |
| 14 | C2×#6410 | C22 | 1 | 2 | yes | — |
| 15 | #89751 | — | 2 | 1 | no | — |
| 16 | #89701 | — | 2 | 1 | no | — |
| 17 | #89428 | — | 2 | 2 | no | — |
| 18 | #89418 | — | 2 | 2 | no | — |
| 19 | #13378 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C7×2−1+4 | — | 2 | 1 | no | — |
| 21 | C7×2−1+4 | — | 2 | 2 | no | — |
| 22 | C7×2−1+4 | — | 4 | 2 | no | — |
| 23 | C7×2−1+4 | Q8 | 1 | 2 | yes | complemented |
| 24 | C2×C7×#41 | — | 2 | 2 | no | — |
| 25 | C2×C7×#41 | — | 2 | 1 | no | — |
| 26 | C2×C7×#41 | — | 4 | 1 | no | — |
| 27 | C22×C7×Q8 | — | 4 | 1 | no | — |
| 28 | C22×C7×Q8 | C23 | 1 | 1 | yes | characteristic |
| 29 | C22×Dic14 | — | 2 | 1 | no | — |
| 30 | C22×Dic14 | — | 2 | 1 | no | — |
| 31 | C2×C7×M16 | — | 2 | 1 | no | — |
| 32 | C2×C7×C4⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×C42×C7 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×C4⋊Dic7 | — | 2 | 1 | no | — |
| 35 | C2×C4⋊Dic7 | D4 | 1 | 2 | yes | — |
| 36 | C2×Dic7⋊C4 | — | 2 | 2 | no | — |
| 37 | C2×C4×Dic7 | D4 | 1 | 2 | yes | — |
| 38 | C2×#1136 | — | 2 | 2 | no | — |
| 39 | C7×#104 | — | 4 | 2 | no | — |
| 40 | C7×#103 | — | 4 | 1 | no | — |
| 41 | C7×#102 | — | 4 | 1 | no | — |
| 42 | C7×#101 | — | 2 | 2 | no | — |
| 43 | C7×#95 | — | 2 | 2 | no | — |
| 44 | #6411 | — | 4 | 4 | no | — |
| 45 | #6410 | — | 2 | 2 | no | — |
| 46 | #6410 | — | 2 | 2 | no | — |
| 47 | #6409 | — | 4 | 2 | no | — |
| 48 | #6406 | — | 4 | 2 | no | — |
| 49 | C7⋊#95 | — | 2 | 1 | no | — |
| 50 | C7⋊#95 | — | 2 | 2 | no | — |
| 51 | C7⋊#95 | — | 2 | 2 | no | — |
| 52 | C7⋊#95 | — | 2 | 2 | no | — |
| 53 | #2811 | — | 7 | 1 | no | — |
| 54 | #1793 | — | 7 | 2 | no | — |
| 55 | #1793 | D7 | 1 | 1 | yes | characteristic |
| 56 | #1394 | — | 7 | 2 | no | — |
| 57 | #1394 | — | 7 | 1 | no | — |
| 58 | C7×#41 | — | 2 | 4 | no | — |
| 59 | C7×#41 | — | 2 | 2 | no | — |
| 60 | C7×#41 | — | 2 | 4 | no | — |
| 61 | C7×#41 | — | 2 | 2 | no | — |
| 62 | C7×#41 | — | 4 | 2 | no | — |
| 63 | C7×#41 | — | 4 | 1 | no | — |
| 64 | C7×#41 | — | 4 | 2 | no | — |
| 65 | C7×#41 | — | 4 | 2 | no | — |
| 66 | C7×#41 | — | 4 | 4 | no | — |
| 67 | C7×#41 | — | 4 | 2 | no | — |
| 68 | C7×#41 | — | 4 | 1 | no | — |
| 69 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 70 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 71 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 72 | C2×C7×Q8 | — | 4 | 1 | no | — |
| 73 | C2×C7×Q8 | — | 4 | 2 | no | — |
| 74 | C2×C7×Q8 | — | 4 | 2 | no | — |
| 75 | C2×C7×Q8 | — | 4 | 4 | no | — |
| 76 | C2×C7×Q8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 77 | C2×C7×Q8 | #41 | 1 | 1 | yes | characteristic |
| 78 | C2×C7×Q8 | #41 | 1 | 2 | yes | — |
| 79 | C2×C7×D4 | — | 2 | 2 | no | — |
| 80 | C2×C7×D4 | — | 2 | 1 | no | — |
| 81 | C2×C7×D4 | — | 4 | 1 | no | — |
| 82 | C22×C4×C7 | — | 2 | 1 | no | — |
| 83 | C22×C4×C7 | — | 2 | 1 | no | — |
| 84 | C22×C4×C7 | — | 4 | 2 | no | — |
| 85 | C22×C4×C7 | — | 4 | 1 | no | — |
| 86 | C22×C4×C7 | C2×D4 | 1 | 2 | yes | — |
| 87 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C22×Dic7 | — | 2 | 2 | no | — |
| 89 | C22×Dic7 | — | 2 | 2 | no | — |
| 90 | C2×Dic14 | — | 2 | 4 | no | — |
| 91 | C2×Dic14 | — | 8 | 1 | no | — |
| 92 | C2×Dic14 | — | 8 | 1 | no | — |
| 93 | C2×Dic14 | — | 8 | 2 | no | — |
| 94 | C2×Dic14 | — | 8 | 1 | no | — |
| 95 | C7×M16 | — | 4 | 1 | no | — |
| 96 | C7×M16 | — | 4 | 1 | no | — |
| 97 | C2×C7×C8 | — | 4 | 1 | no | — |
| 98 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 99 | C42×C7 | — | 4 | 1 | no | — |
| 100 | C4⋊Dic7 | — | 4 | 2 | no | — |
| 101 | C4⋊Dic7 | — | 8 | 1 | no | — |
| 102 | Dic7⋊C4 | — | 8 | 2 | no | — |
| 103 | C4×Dic7 | — | 4 | 2 | no | — |
| 104 | #1136 | — | 4 | 2 | no | — |
| 105 | #1136 | — | 4 | 2 | no | — |
| 106 | C2×C7⋊C8 | — | 4 | 2 | no | — |
| 107 | C2×2−1+4 | C2×D7 | 1 | 1 | yes | characteristic |
| 108 | C2×#104 | — | 2 | 1 | no | — |
| 109 | C2×#104 | — | 14 | 2 | no | — |
| 110 | #417 | — | 2 | 1 | no | — |
| 111 | #417 | — | 14 | 2 | no | — |
| 112 | C2×#101 | — | 7 | 2 | no | — |
| 113 | C2×#101 | C2×D7 | 1 | 1 | yes | characteristic |
| 114 | #395 | — | 14 | 2 | no | — |
| 115 | #395 | — | 14 | 1 | no | — |
| 116 | #391 | — | 7 | 2 | no | — |
| 117 | #391 | — | 14 | 2 | no | — |
| 118 | #391 | — | 14 | 1 | no | — |
| 119 | #391 | C2×D7 | 1 | 1 | yes | characteristic |
| 120 | C23×C7 | — | 4 | 1 | no | — |
| 121 | C23×C7 | #120 | 1 | 1 | yes | characteristic |
| 122 | C7×Q8 | — | 2 | 4 | no | — |
| 123 | C7×Q8 | — | 2 | 2 | no | — |
| 124 | C7×Q8 | — | 4 | 1 | no | — |
| 125 | C7×Q8 | — | 4 | 1 | no | — |
| 126 | C7×Q8 | — | 4 | 2 | no | — |
| 127 | C7×Q8 | — | 4 | 2 | no | — |
| 128 | C7×Q8 | — | 4 | 1 | no | — |
| 129 | C7×D4 | — | 2 | 4 | no | — |
| 130 | C7×D4 | — | 2 | 2 | no | — |
| 131 | C7×D4 | — | 4 | 2 | no | — |
| 132 | C7×D4 | — | 4 | 1 | no | — |
| 133 | C7×D4 | — | 4 | 2 | no | — |
| 134 | C7×D4 | — | 4 | 2 | no | — |
| 135 | C2×C4×C7 | — | 2 | 2 | no | — |
| 136 | C2×C4×C7 | — | 2 | 2 | no | — |
| 137 | C2×C4×C7 | — | 2 | 1 | no | — |
| 138 | C2×C4×C7 | — | 2 | 1 | no | — |
| 139 | C2×C4×C7 | — | 2 | 2 | no | — |
| 140 | C2×C4×C7 | — | 4 | 4 | no | — |
| 141 | C2×C4×C7 | — | 4 | 2 | no | — |
| 142 | C2×C4×C7 | — | 4 | 4 | no | — |
| 143 | C2×C4×C7 | — | 4 | 2 | no | — |
| 144 | C2×C4×C7 | — | 4 | 1 | no | — |
| 145 | C2×C4×C7 | — | 4 | 1 | no | — |
| 146 | C2×C4×C7 | — | 4 | 1 | no | — |
| 147 | C2×C4×C7 | — | 4 | 2 | no | — |
| 148 | C2×C4×C7 | — | 4 | 1 | no | — |
| 149 | C2×C4×C7 | C22⋊Q8 | 1 | 2 | yes | — |
| 150 | C2×C4×C7 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 151 | C2×C4×C7 | #124 | 1 | 2 | yes | — |
| 152 | C2×C4×C7 | #124 | 1 | 1 | yes | characteristic |
| 153 | C2×Dic7 | — | 2 | 4 | no | — |
| 154 | C2×Dic7 | — | 4 | 2 | no | — |
| 155 | C2×Dic7 | — | 4 | 2 | no | — |
| 156 | C2×Dic7 | — | 4 | 2 | no | — |
| 157 | C2×Dic7 | — | 4 | 2 | no | — |
| 158 | C2×Dic7 | — | 4 | 2 | no | — |
| 159 | Dic14 | — | 8 | 4 | no | — |
| 160 | Dic14 | — | 8 | 4 | no | — |
| 161 | C7×C8 | — | 8 | 1 | no | — |
| 162 | C7⋊C8 | — | 8 | 2 | no | — |
| 163 | 2−1+4 | — | 2 | 1 | no | — |
| 164 | 2−1+4 | — | 2 | 2 | no | — |
| 165 | 2−1+4 | — | 4 | 2 | no | — |
| 166 | 2−1+4 | Dic14 | 1 | 2 | yes | complemented |
| 167 | C2×#41 | — | 2 | 2 | no | — |
| 168 | C2×#41 | — | 2 | 1 | no | — |
| 169 | C2×#41 | — | 4 | 1 | no | — |
| 170 | C22×Q8 | — | 4 | 1 | no | — |
| 171 | C22×Q8 | — | 14 | 1 | no | — |
| 172 | C22×Q8 | — | 14 | 1 | no | — |
| 173 | C22×Q8 | C22×D7 | 1 | 1 | yes | characteristic |
| 174 | C2×M16 | — | 2 | 1 | no | — |
| 175 | C2×M16 | — | 14 | 2 | no | — |
| 176 | C2×C4⋊C4 | — | 7 | 2 | no | — |
| 177 | C2×C4⋊C4 | — | 14 | 2 | no | — |
| 178 | C2×C4⋊C4 | — | 14 | 1 | no | — |
| 179 | C2×C4⋊C4 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 180 | C2×C42 | — | 7 | 2 | no | — |
| 181 | C2×C42 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 182 | #104 | — | 4 | 2 | no | — |
| 183 | #104 | — | 28 | 4 | no | — |
| 184 | #103 | — | 4 | 1 | no | — |
| 185 | #103 | — | 28 | 2 | no | — |
| 186 | #102 | — | 4 | 1 | no | — |
| 187 | #102 | — | 28 | 2 | no | — |
| 188 | #101 | — | 2 | 2 | no | — |
| 189 | #101 | — | 14 | 2 | no | — |
| 190 | #101 | — | 14 | 2 | no | — |
| 191 | #95 | — | 2 | 2 | no | — |
| 192 | #95 | — | 14 | 2 | no | — |
| 193 | #95 | — | 14 | 1 | no | — |
| 194 | #95 | — | 14 | 2 | no | — |
| 195 | #95 | — | 14 | 2 | no | — |
| 196 | C22×C7 | — | 2 | 1 | no | — |
| 197 | C22×C7 | — | 2 | 1 | no | — |
| 198 | C22×C7 | — | 4 | 2 | no | — |
| 199 | C22×C7 | — | 8 | 1 | no | — |
| 200 | C22×C7 | — | 8 | 1 | no | — |
| 201 | C22×C7 | #393 | 1 | 1 | yes | characteristic |
| 202 | C22×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 203 | C22×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 204 | C4×C7 | — | 2 | 2 | no | — |
| 205 | C4×C7 | — | 2 | 1 | no | — |
| 206 | C4×C7 | — | 2 | 2 | no | — |
| 207 | C4×C7 | — | 2 | 1 | no | — |
| 208 | C4×C7 | — | 4 | 2 | no | — |
| 209 | C4×C7 | — | 4 | 2 | no | — |
| 210 | C4×C7 | — | 8 | 1 | no | — |
| 211 | Dic7 | — | 4 | 4 | no | — |
| 212 | Dic7 | — | 8 | 2 | no | — |
| 213 | #41 | — | 2 | 2 | no | — |
| 214 | #41 | — | 2 | 4 | no | — |
| 215 | #41 | — | 2 | 4 | no | — |
| 216 | #41 | — | 2 | 2 | no | — |
| 217 | #41 | — | 4 | 1 | no | — |
| 218 | #41 | — | 4 | 2 | no | — |
| 219 | #41 | — | 4 | 2 | no | — |
| 220 | #41 | — | 4 | 2 | no | — |
| 221 | #41 | — | 4 | 4 | no | — |
| 222 | #41 | — | 4 | 2 | no | — |
| 223 | #41 | — | 4 | 1 | no | — |
| 224 | C2×Q8 | — | 2 | 1 | no | — |
| 225 | C2×Q8 | — | 2 | 2 | no | — |
| 226 | C2×Q8 | — | 4 | 1 | no | — |
| 227 | C2×Q8 | — | 4 | 1 | no | — |
| 228 | C2×Q8 | — | 4 | 2 | no | — |
| 229 | C2×Q8 | — | 4 | 2 | no | — |
| 230 | C2×Q8 | — | 4 | 4 | no | — |
| 231 | C2×Q8 | — | 14 | 4 | no | — |
| 232 | C2×Q8 | — | 56 | 1 | no | — |
| 233 | C2×Q8 | — | 56 | 1 | no | — |
| 234 | C2×Q8 | — | 56 | 2 | no | — |
| 235 | C2×Q8 | — | 56 | 1 | no | — |
| 236 | C2×Q8 | C2×Dic14 | 1 | 1 | yes | characteristic |
| 237 | C2×Q8 | #1157 | 1 | 1 | yes | characteristic |
| 238 | C2×Q8 | #1159 | 1 | 2 | yes | — |
| 239 | C2×D4 | — | 2 | 1 | no | — |
| 240 | C2×D4 | — | 2 | 2 | no | — |
| 241 | C2×D4 | — | 4 | 1 | no | — |
| 242 | C22×C4 | — | 2 | 1 | no | — |
| 243 | C22×C4 | — | 2 | 1 | no | — |
| 244 | C22×C4 | — | 4 | 1 | no | — |
| 245 | C22×C4 | — | 4 | 2 | no | — |
| 246 | C22×C4 | — | 14 | 2 | no | — |
| 247 | C22×C4 | — | 14 | 2 | no | — |
| 248 | C22×C4 | D4×D7 | 1 | 2 | yes | — |
| 249 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 250 | M16 | — | 4 | 1 | no | — |
| 251 | M16 | — | 4 | 1 | no | — |
| 252 | M16 | — | 28 | 2 | no | — |
| 253 | M16 | — | 28 | 2 | no | — |
| 254 | C2×C8 | — | 4 | 1 | no | — |
| 255 | C2×C8 | — | 28 | 2 | no | — |
| 256 | C4⋊C4 | — | 4 | 1 | no | — |
| 257 | C4⋊C4 | — | 28 | 2 | no | — |
| 258 | C4⋊C4 | — | 56 | 2 | no | — |
| 259 | C4⋊C4 | — | 56 | 1 | no | — |
| 260 | C42 | — | 4 | 1 | no | — |
| 261 | C42 | — | 28 | 2 | no | — |
| 262 | C2×C7 | — | 2 | 1 | no | — |
| 263 | C2×C7 | — | 2 | 1 | no | — |
| 264 | C2×C7 | — | 8 | 2 | no | — |
| 265 | C2×C7 | #2029 | 1 | 1 | yes | characteristic |
| 266 | C2×C7 | #2202 | 1 | 1 | yes | characteristic |
| 267 | C2×C7 | #2205 | 1 | 1 | yes | characteristic |
| 268 | C23 | — | 4 | 1 | no | — |
| 269 | C23 | #6500 | 1 | 1 | yes | characteristic |
| 270 | Q8 | — | 2 | 4 | no | — |
| 271 | Q8 | — | 2 | 2 | no | — |
| 272 | Q8 | — | 4 | 1 | no | — |
| 273 | Q8 | — | 4 | 1 | no | — |
| 274 | Q8 | — | 4 | 2 | no | — |
| 275 | Q8 | — | 4 | 1 | no | — |
| 276 | Q8 | — | 4 | 2 | no | — |
| 277 | Q8 | — | 56 | 4 | no | — |
| 278 | Q8 | — | 56 | 4 | no | — |
| 279 | D4 | — | 2 | 4 | no | — |
| 280 | D4 | — | 2 | 2 | no | — |
| 281 | D4 | — | 4 | 1 | no | — |
| 282 | D4 | — | 4 | 2 | no | — |
| 283 | D4 | — | 4 | 2 | no | — |
| 284 | D4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 2 | 1 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 2 | 1 | no | — |
| 288 | C2×C4 | — | 2 | 2 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 4 | 2 | no | — |
| 291 | C2×C4 | — | 4 | 4 | no | — |
| 292 | C2×C4 | — | 4 | 4 | no | — |
| 293 | C2×C4 | — | 4 | 2 | no | — |
| 294 | C2×C4 | — | 4 | 1 | no | — |
| 295 | C2×C4 | — | 4 | 1 | no | — |
| 296 | C2×C4 | — | 4 | 1 | no | — |
| 297 | C2×C4 | — | 4 | 2 | no | — |
| 298 | C2×C4 | — | 4 | 1 | no | — |
| 299 | C2×C4 | — | 14 | 4 | no | — |
| 300 | C2×C4 | — | 28 | 2 | no | — |
| 301 | C2×C4 | — | 28 | 2 | no | — |
| 302 | C2×C4 | — | 28 | 2 | no | — |
| 303 | C2×C4 | — | 28 | 2 | no | — |
| 304 | C2×C4 | — | 28 | 2 | no | — |
| 305 | C2×C4 | #6441 | 1 | 2 | yes | — |
| 306 | C2×C4 | #6448 | 1 | 2 | yes | — |
| 307 | C2×C4 | #6487 | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | #6499 | 1 | 1 | yes | characteristic |
| 309 | C8 | — | 8 | 1 | no | — |
| 310 | C8 | — | 56 | 2 | no | — |
| 311 | C7 | #13378 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 312 | C22 | — | 2 | 1 | no | — |
| 313 | C22 | — | 2 | 1 | no | — |
| 314 | C22 | — | 4 | 2 | no | — |
| 315 | C22 | — | 8 | 1 | no | — |
| 316 | C22 | — | 8 | 1 | no | — |
| 317 | C22 | #89719 | 1 | 1 | yes | characteristic |
| 318 | C22 | #89804 | 1 | 1 | yes | characteristic |
| 319 | C22 | #89804 | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 2 | 1 | no | — |
| 321 | C4 | — | 2 | 2 | no | — |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 4 | 2 | no | — |
| 325 | C4 | — | 4 | 2 | no | — |
| 326 | C4 | — | 8 | 1 | no | — |
| 327 | C4 | — | 28 | 4 | no | — |
| 328 | C4 | — | 56 | 2 | no | — |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | — | 8 | 2 | no | — |
| 332 | C2 | #11727251 | 1 | 1 | yes | characteristic |
| 333 | C2 | #11728741 | 1 | 1 | yes | characteristic |
| 334 | C2 | #11728749 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11734049 | C2 | semidirect |
| 4. #11725505 | C2 | semidirect |
| 5. #11725450 | C2 | semidirect |
| 23. C7×2−1+4 | Q8 | semidirect |
| 166. 2−1+4 | Dic14 | semidirect |
| 311. C7 | #13378 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11734049 | 2 |
| C7×#1793 | 2 |
| #11729897 | 2 |
| #11725505 | 2 |
| #11725450 | 2 |
| #13378 | 7 |