Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
458 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 458 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 | #22620382 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22611593 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22608094 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22608093 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C73×#1408 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#14556615 | — | 2 | 2 | no | — |
| 7 | #17989135 | C22 | 1 | 1 | yes | characteristic |
| 8 | #127×D73 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | #17988975 | — | 2 | 2 | no | — |
| 10 | #17988963 | C22 | 1 | 1 | yes | characteristic |
| 11 | #17988960 | C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | #17988959 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | #17988559 | — | 2 | 2 | no | — |
| 14 | C4×#11924011 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | #17988381 | — | 2 | 2 | no | — |
| 16 | #17988380 | C22 | 1 | 2 | yes | complemented |
| 17 | #493×C73 | C22 | 1 | 1 | yes | characteristic |
| 18 | #353×C73 | C22 | 1 | 2 | yes | complemented |
| 19 | C2×#11924027 | D4 | 1 | 2 | yes | — |
| 20 | C2×D4×D73 | — | 2 | 1 | no | — |
| 21 | C2×D4×D73 | — | 4 | 1 | no | — |
| 22 | #14556615 | — | 4 | 4 | no | — |
| 23 | #14556615 | — | 4 | 4 | no | — |
| 24 | C2×#11924017 | — | 2 | 2 | no | — |
| 25 | C2×#11924016 | — | 2 | 2 | no | — |
| 26 | C2×#11924013 | — | 2 | 2 | no | — |
| 27 | C2×#11924011 | — | 2 | 1 | no | — |
| 28 | #14556591 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #14556588 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | #14556587 | — | 2 | 1 | no | — |
| 31 | #14556587 | — | 4 | 1 | no | — |
| 32 | #14556578 | D4 | 1 | 2 | yes | — |
| 33 | #14556562 | — | 4 | 1 | no | — |
| 34 | D4×Dic73 | — | 4 | 1 | no | — |
| 35 | #14556547 | — | 4 | 2 | no | — |
| 36 | C42×D73 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | #14556506 | — | 2 | 2 | no | — |
| 38 | #14556505 | — | 4 | 2 | no | — |
| 39 | #14556501 | — | 2 | 1 | no | — |
| 40 | C4×#10596831 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×D8×C73 | — | 2 | 2 | no | — |
| 42 | #127×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | #127×C73 | C23 | 1 | 1 | yes | characteristic |
| 44 | #99×C73 | — | 2 | 2 | no | — |
| 45 | C4×C8×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 46 | C23×D73 | — | 4 | 1 | no | — |
| 47 | #11924027 | — | 4 | 2 | no | — |
| 48 | #11924027 | — | 4 | 2 | no | — |
| 49 | C2×Dic146 | C22⋊C4 | 1 | 2 | yes | — |
| 50 | C2×D292 | — | 2 | 1 | no | — |
| 51 | C2×D292 | — | 4 | 1 | no | — |
| 52 | C2×C22⋊D73 | — | 2 | 2 | no | — |
| 53 | C2×C22⋊D73 | — | 4 | 2 | no | — |
| 54 | C2×C22⋊D73 | — | 4 | 1 | no | — |
| 55 | D4×D73 | — | 4 | 2 | no | — |
| 56 | D4×D73 | — | 4 | 2 | no | — |
| 57 | D4×D73 | — | 8 | 4 | no | — |
| 58 | C22×Dic73 | — | 2 | 2 | no | — |
| 59 | C2×C4×D73 | — | 2 | 1 | no | — |
| 60 | C2×C4×D73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 61 | #11924017 | — | 4 | 4 | no | — |
| 62 | #11924016 | — | 4 | 4 | no | — |
| 63 | #11924013 | — | 4 | 4 | no | — |
| 64 | #11924011 | — | 2 | 4 | no | — |
| 65 | C2×#10596831 | — | 2 | 1 | no | — |
| 66 | C4⋊Dic73 | — | 4 | 1 | no | — |
| 67 | #11924007 | — | 8 | 2 | no | — |
| 68 | C22⋊Dic73 | — | 4 | 2 | no | — |
| 69 | C22⋊Dic73 | — | 4 | 2 | no | — |
| 70 | C4×Dic73 | — | 2 | 1 | no | — |
| 71 | C4×Dic73 | — | 8 | 2 | no | — |
| 72 | C4×Dic73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C2×D4×C73 | — | 2 | 1 | no | — |
| 74 | C2×D4×C73 | — | 4 | 1 | no | — |
| 75 | C2×D4×C73 | — | 4 | 1 | no | — |
| 76 | C2×D4×C73 | C2×D4 | 1 | 2 | yes | — |
| 77 | D8×C73 | — | 4 | 4 | no | — |
| 78 | C2×C8×C73 | — | 2 | 1 | no | — |
| 79 | C22⋊C4×C73 | — | 4 | 2 | no | — |
| 80 | C42×C73 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C23×C73 | — | 2 | 2 | no | — |
| 82 | C23×C73 | — | 4 | 1 | no | — |
| 83 | C22×D73 | — | 4 | 1 | no | — |
| 84 | C22×D73 | — | 8 | 1 | no | — |
| 85 | C22×D73 | — | 8 | 1 | no | — |
| 86 | C22×D73 | — | 8 | 4 | no | — |
| 87 | C22×D73 | #99 | 1 | 1 | yes | characteristic |
| 88 | D4×C73 | — | 4 | 2 | no | — |
| 89 | D4×C73 | — | 4 | 2 | no | — |
| 90 | D4×C73 | — | 8 | 2 | no | — |
| 91 | D4×C73 | — | 8 | 2 | no | — |
| 92 | C2×C4×C73 | — | 2 | 1 | no | — |
| 93 | C2×C4×C73 | — | 8 | 2 | no | — |
| 94 | C2×C4×C73 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 95 | C22⋊D73 | — | 8 | 2 | no | — |
| 96 | C22⋊D73 | — | 8 | 4 | no | — |
| 97 | C22⋊D73 | — | 8 | 2 | no | — |
| 98 | C2×Dic73 | — | 2 | 1 | no | — |
| 99 | C2×Dic73 | — | 2 | 2 | no | — |
| 100 | C2×Dic73 | — | 4 | 2 | no | — |
| 101 | C2×Dic73 | — | 4 | 2 | no | — |
| 102 | C2×Dic73 | — | 8 | 2 | no | — |
| 103 | C2×Dic73 | #99 | 1 | 1 | yes | characteristic |
| 104 | D292 | — | 4 | 2 | no | — |
| 105 | D292 | — | 8 | 2 | no | — |
| 106 | C4×D73 | — | 2 | 2 | no | — |
| 107 | C4×D73 | — | 4 | 2 | no | — |
| 108 | Dic146 | — | 4 | 2 | no | — |
| 109 | C8×C73 | — | 2 | 2 | no | — |
| 110 | #10596831 | — | 2 | 2 | no | — |
| 111 | C22×C73 | — | 4 | 2 | no | — |
| 112 | C22×C73 | — | 8 | 2 | no | — |
| 113 | C22×C73 | — | 8 | 2 | no | — |
| 114 | C22×C73 | — | 8 | 1 | no | — |
| 115 | C22×C73 | C2×#99 | 1 | 1 | yes | characteristic |
| 116 | C2×D73 | — | 2 | 1 | no | — |
| 117 | C2×D73 | — | 4 | 1 | no | — |
| 118 | C2×D73 | — | 8 | 2 | no | — |
| 119 | C2×D73 | — | 8 | 1 | no | — |
| 120 | C2×D73 | — | 16 | 2 | no | — |
| 121 | C2×D73 | — | 16 | 2 | no | — |
| 122 | C4×C73 | — | 2 | 2 | no | — |
| 123 | C4×C73 | — | 2 | 1 | no | — |
| 124 | C4×C73 | — | 16 | 2 | no | — |
| 125 | Dic73 | — | 4 | 1 | no | — |
| 126 | Dic73 | — | 4 | 2 | no | — |
| 127 | Dic73 | — | 16 | 2 | no | — |
| 128 | Dic73 | #353 | 1 | 2 | yes | — |
| 129 | #13585 | — | 73 | 1 | no | maximal, Sylow |
| 130 | C2×C73 | — | 2 | 1 | no | — |
| 131 | C2×C73 | — | 8 | 2 | no | — |
| 132 | C2×C73 | — | 16 | 1 | no | — |
| 133 | C2×C73 | C2×#353 | 1 | 1 | yes | characteristic |
| 134 | D73 | — | 4 | 1 | no | — |
| 135 | D73 | — | 16 | 1 | no | — |
| 136 | #3151 | — | 73 | 1 | no | — |
| 137 | #2126 | — | 73 | 2 | no | — |
| 138 | #1408 | — | 73 | 1 | no | — |
| 139 | #1408 | — | 73 | 2 | no | — |
| 140 | #1408 | D73 | 1 | 1 | yes | characteristic, complemented |
| 141 | C73 | #13585 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 142 | C2×AGL(1,ℤ/8) | — | 146 | 2 | no | — |
| 143 | #532 | — | 73 | 1 | no | — |
| 144 | C2×#127 | — | 73 | 1 | no | — |
| 145 | #496 | — | 146 | 2 | no | — |
| 146 | #493 | — | 73 | 1 | no | — |
| 147 | #493 | C2×D73 | 1 | 1 | yes | characteristic |
| 148 | #492 | — | 73 | 1 | no | — |
| 149 | #492 | — | 73 | 1 | no | — |
| 150 | #410 | — | 146 | 2 | no | — |
| 151 | C4×M16 | — | 73 | 1 | no | — |
| 152 | #354 | — | 146 | 2 | no | — |
| 153 | #353 | — | 73 | 2 | no | — |
| 154 | #353 | C2×D73 | 1 | 2 | yes | complemented |
| 155 | C2×#41 | — | 73 | 2 | no | — |
| 156 | C22×D4 | — | 292 | 1 | no | — |
| 157 | C22×D4 | — | 146 | 1 | no | — |
| 158 | AGL(1,ℤ/8) | — | 292 | 4 | no | — |
| 159 | AGL(1,ℤ/8) | — | 292 | 4 | no | — |
| 160 | C2×SD16 | — | 146 | 2 | no | — |
| 161 | C2×SD16 | — | 146 | 2 | no | — |
| 162 | C2×D8 | — | 2 | 2 | no | — |
| 163 | C2×D8 | — | 146 | 2 | no | — |
| 164 | C2×M16 | — | 146 | 1 | no | — |
| 165 | C4⋊Q8 | — | 73 | 1 | no | — |
| 166 | #127 | — | 292 | 1 | no | — |
| 167 | #127 | — | 73 | 1 | no | — |
| 168 | #127 | — | 146 | 1 | no | — |
| 169 | #127 | C4×D73 | 1 | 1 | yes | characteristic |
| 170 | #127 | C22×D73 | 1 | 1 | yes | characteristic |
| 171 | #124 | — | 73 | 2 | no | — |
| 172 | #121 | — | 292 | 1 | no | — |
| 173 | C4×D4 | — | 292 | 1 | no | — |
| 174 | #117 | — | 292 | 2 | no | — |
| 175 | C2×C42 | — | 73 | 1 | no | — |
| 176 | #99 | — | 2 | 2 | no | — |
| 177 | #99 | — | 292 | 2 | no | — |
| 178 | #99 | — | 146 | 2 | no | — |
| 179 | #97 | — | 146 | 1 | no | — |
| 180 | C4×C8 | — | 73 | 1 | no | — |
| 181 | C4×C8 | C4×D73 | 1 | 1 | yes | characteristic |
| 182 | C24 | — | 292 | 1 | no | — |
| 183 | #41 | — | 292 | 2 | no | — |
| 184 | #41 | — | 292 | 2 | no | — |
| 185 | C2×Q8 | — | 73 | 2 | no | — |
| 186 | C2×D4 | — | 2 | 1 | no | — |
| 187 | C2×D4 | — | 4 | 1 | no | — |
| 188 | C2×D4 | — | 4 | 1 | no | — |
| 189 | C2×D4 | — | 292 | 1 | no | — |
| 190 | C2×D4 | — | 292 | 1 | no | — |
| 191 | C2×D4 | — | 292 | 2 | no | — |
| 192 | C2×D4 | — | 292 | 2 | no | — |
| 193 | C2×D4 | — | 292 | 2 | no | — |
| 194 | C2×D4 | — | 584 | 4 | no | — |
| 195 | C2×D4 | — | 146 | 2 | no | — |
| 196 | C2×D4 | — | 146 | 1 | no | — |
| 197 | C2×D4 | D4×D73 | 1 | 2 | yes | — |
| 198 | C22×C4 | — | 73 | 1 | no | — |
| 199 | C22×C4 | — | 146 | 1 | no | — |
| 200 | C22×C4 | — | 146 | 2 | no | — |
| 201 | SD16 | — | 292 | 4 | no | — |
| 202 | SD16 | — | 292 | 4 | no | — |
| 203 | D8 | — | 4 | 4 | no | — |
| 204 | D8 | — | 292 | 4 | no | — |
| 205 | M16 | — | 146 | 4 | no | — |
| 206 | C2×C8 | — | 2 | 1 | no | — |
| 207 | C2×C8 | — | 146 | 1 | no | — |
| 208 | C4⋊C4 | — | 292 | 1 | no | — |
| 209 | C4⋊C4 | — | 584 | 2 | no | — |
| 210 | C22⋊C4 | — | 4 | 2 | no | — |
| 211 | C22⋊C4 | — | 292 | 2 | no | — |
| 212 | C22⋊C4 | — | 292 | 2 | no | — |
| 213 | C42 | — | 584 | 2 | no | — |
| 214 | C42 | — | 73 | 1 | no | — |
| 215 | C42 | — | 146 | 1 | no | — |
| 216 | C42 | C2×C4×D73 | 1 | 1 | yes | characteristic |
| 217 | C23 | — | 2 | 2 | no | — |
| 218 | C23 | — | 4 | 1 | no | — |
| 219 | C23 | — | 292 | 1 | no | — |
| 220 | C23 | — | 584 | 4 | no | — |
| 221 | C23 | — | 584 | 1 | no | — |
| 222 | C23 | — | 584 | 1 | no | — |
| 223 | C23 | — | 73 | 1 | no | — |
| 224 | Q8 | — | 292 | 2 | no | — |
| 225 | D4 | — | 4 | 2 | no | — |
| 226 | D4 | — | 4 | 2 | no | — |
| 227 | D4 | — | 8 | 2 | no | — |
| 228 | D4 | — | 8 | 2 | no | — |
| 229 | D4 | — | 292 | 2 | no | — |
| 230 | D4 | — | 584 | 2 | no | — |
| 231 | D4 | — | 584 | 2 | no | — |
| 232 | D4 | — | 584 | 4 | no | — |
| 233 | D4 | — | 584 | 2 | no | — |
| 234 | C2×C4 | — | 2 | 1 | no | — |
| 235 | C2×C4 | — | 8 | 2 | no | — |
| 236 | C2×C4 | — | 292 | 2 | no | — |
| 237 | C2×C4 | — | 292 | 2 | no | — |
| 238 | C2×C4 | — | 292 | 2 | no | — |
| 239 | C2×C4 | — | 584 | 2 | no | — |
| 240 | C2×C4 | — | 73 | 1 | no | — |
| 241 | C2×C4 | — | 146 | 2 | no | — |
| 242 | C2×C4 | — | 146 | 1 | no | — |
| 243 | C2×C4 | — | 146 | 2 | no | — |
| 244 | C2×C4 | C22⋊C4×D73 | 1 | 1 | yes | characteristic |
| 245 | C8 | — | 2 | 2 | no | — |
| 246 | C8 | — | 146 | 2 | no | — |
| 247 | C22 | — | 4 | 2 | no | — |
| 248 | C22 | — | 8 | 2 | no | — |
| 249 | C22 | — | 8 | 2 | no | — |
| 250 | C22 | — | 8 | 1 | no | — |
| 251 | C22 | — | 292 | 1 | no | — |
| 252 | C22 | — | 584 | 2 | no | — |
| 253 | C22 | — | 584 | 1 | no | — |
| 254 | C22 | — | 1168 | 2 | no | — |
| 255 | C22 | — | 1168 | 2 | no | — |
| 256 | C22 | — | 146 | 1 | no | — |
| 257 | C22 | #99×D73 | 1 | 1 | yes | characteristic |
| 258 | C4 | — | 2 | 2 | no | — |
| 259 | C4 | — | 2 | 1 | no | — |
| 260 | C4 | — | 16 | 2 | no | — |
| 261 | C4 | — | 292 | 1 | no | — |
| 262 | C4 | — | 292 | 2 | no | — |
| 263 | C4 | — | 73 | 2 | no | — |
| 264 | C4 | — | 1168 | 2 | no | — |
| 265 | C2 | — | 2 | 1 | no | — |
| 266 | C2 | — | 8 | 2 | no | — |
| 267 | C2 | — | 16 | 1 | no | — |
| 268 | C2 | — | 292 | 1 | no | — |
| 269 | C2 | — | 1168 | 1 | no | — |
| 270 | C2 | #353×D73 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22611593 | C2 | semidirect |
| 3. #22608094 | C2 | semidirect |
| 4. #22608093 | C2 | semidirect |
| 5. C73×#1408 | C2 | semidirect |
| 8. #127×D73 | C4 | semidirect |
| 11. #17988960 | C4 | semidirect |
| 12. #17988959 | C4 | semidirect |
| 14. C4×#11924011 | C4 | semidirect |
| 16. #17988380 | C22 | semidirect |
| 18. #353×C73 | C22 | semidirect |
| 140. #1408 | D73 | semidirect |
| 141. C73 | #13585 | semidirect |
| 154. #353 | C2×D73 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22620382 | 2 |
| #22611593 | 2 |
| #22608094 | 2 |
| #22608093 | 2 |
| C73×#1408 | 2 |
| #13585 | 73 |