← #13509340
Subgroups — #13509340
Order 1792.
514 conjugacy classes of subgroups.
The subgroup lattice has 228 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #11735920 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×C4×C7×C16 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11729170 | C2 | 1 | 4 | yes | maximal |
| 4 | #11726820 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11726819 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C4×Dic56 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #11726798 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #90161 | — | 2 | 2 | no | — |
| 9 | C22×C7×C16 | C22 | 1 | 2 | yes | — |
| 10 | C2×C4×C7×C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | #89888 | — | 2 | 2 | no | — |
| 12 | #89888 | C22 | 1 | 4 | yes | — |
| 13 | C2×Dic56 | C22 | 1 | 4 | yes | — |
| 14 | #89471 | C22 | 1 | 2 | yes | — |
| 15 | C4×Dic28 | — | 2 | 2 | no | — |
| 16 | C4×Dic28 | C22 | 1 | 4 | yes | — |
| 17 | #89454 | C22 | 1 | 2 | yes | — |
| 18 | #89453 | — | 2 | 2 | no | — |
| 19 | C4×C7×C16 | C22 | 1 | 1 | yes | characteristic |
| 20 | C4×C7×C16 | C22 | 1 | 2 | yes | — |
| 21 | C4×C7×C16 | C22 | 1 | 1 | yes | characteristic |
| 22 | #89299 | C22 | 1 | 4 | yes | — |
| 23 | #89298 | C22 | 1 | 8 | yes | — |
| 24 | #33734 | — | 7 | 1 | no | maximal, Sylow |
| 25 | C22×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×C42×C7 | D4 | 1 | 1 | yes | characteristic |
| 28 | #6489 | — | 4 | 2 | no | — |
| 29 | #6487 | — | 2 | 4 | no | — |
| 30 | #6487 | — | 4 | 2 | no | — |
| 31 | C2×Dic28 | — | 2 | 2 | no | — |
| 32 | C2×Dic28 | C23 | 1 | 4 | yes | — |
| 33 | #6433 | — | 4 | 2 | no | — |
| 34 | #6432 | — | 2 | 2 | no | — |
| 35 | #6432 | — | 2 | 2 | no | — |
| 36 | C4×Dic14 | — | 2 | 4 | no | — |
| 37 | C4×Dic14 | — | 4 | 2 | no | — |
| 38 | C2×C7×C16 | — | 2 | 4 | no | — |
| 39 | C2×C7×C16 | C23 | 1 | 2 | yes | — |
| 40 | C2×C7×C16 | C23 | 1 | 2 | yes | — |
| 41 | C4×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 42 | C4×C7×C8 | D4 | 1 | 1 | yes | characteristic |
| 43 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C4×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 45 | #6392 | — | 2 | 2 | no | — |
| 46 | #6392 | C23 | 1 | 4 | yes | — |
| 47 | #6390 | — | 2 | 4 | no | — |
| 48 | #6390 | — | 2 | 4 | no | — |
| 49 | Dic56 | — | 2 | 8 | no | — |
| 50 | #3170 | — | 7 | 2 | no | — |
| 51 | #2252 | — | 7 | 1 | no | — |
| 52 | #2247 | — | 7 | 1 | no | — |
| 53 | #2241 | — | 7 | 2 | no | — |
| 54 | #2216 | — | 7 | 4 | no | — |
| 55 | C4×Q32 | — | 7 | 4 | no | — |
| 56 | C2×C4×C16 | D7 | 1 | 1 | yes | characteristic |
| 57 | C22×C4×C7 | D8 | 1 | 1 | yes | characteristic |
| 58 | C22×C4×C7 | D8 | 1 | 1 | yes | characteristic |
| 59 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 60 | C2×Dic14 | — | 2 | 4 | no | — |
| 61 | C2×Dic14 | — | 4 | 2 | no | — |
| 62 | C2×C7×C8 | — | 2 | 2 | no | — |
| 63 | C2×C7×C8 | — | 2 | 2 | no | — |
| 64 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C2×C7×C8 | C24 | 1 | 1 | yes | characteristic |
| 68 | C42×C7 | D8 | 1 | 1 | yes | characteristic |
| 69 | C42×C7 | D8 | 1 | 1 | yes | characteristic |
| 70 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 72 | C22⋊Dic7 | — | 4 | 4 | no | — |
| 73 | C4⋊Dic7 | — | 2 | 4 | no | — |
| 74 | C4⋊Dic7 | — | 4 | 2 | no | — |
| 75 | C4⋊Dic7 | — | 4 | 2 | no | — |
| 76 | C4⋊Dic7 | — | 4 | 2 | no | — |
| 77 | Dic7⋊C4 | — | 4 | 4 | no | — |
| 78 | Dic7⋊C4 | — | 4 | 4 | no | — |
| 79 | C4×Dic7 | — | 4 | 4 | no | — |
| 80 | Dic28 | — | 2 | 8 | no | — |
| 81 | Dic28 | — | 4 | 4 | no | — |
| 82 | C7×C16 | — | 2 | 2 | no | — |
| 83 | C7×C16 | #41 | 1 | 4 | yes | — |
| 84 | #533 | — | 14 | 2 | no | — |
| 85 | C2×Q32 | — | 7 | 4 | no | — |
| 86 | C22×C16 | C2×D7 | 1 | 2 | yes | — |
| 87 | #500 | — | 7 | 2 | no | — |
| 88 | #494 | — | 7 | 2 | no | — |
| 89 | #487 | — | 14 | 2 | no | — |
| 90 | #467 | — | 7 | 4 | no | — |
| 91 | #467 | — | 14 | 2 | no | — |
| 92 | C4×Q16 | — | 7 | 4 | no | — |
| 93 | C4×Q16 | — | 14 | 2 | no | — |
| 94 | C2×C4×C8 | C2×D7 | 1 | 1 | yes | characteristic |
| 95 | #366 | — | 7 | 4 | no | — |
| 96 | #358 | — | 7 | 8 | no | — |
| 97 | C4×C16 | C2×D7 | 1 | 2 | yes | — |
| 98 | C4×C16 | C2×D7 | 1 | 1 | yes | characteristic |
| 99 | C4×C16 | C2×D7 | 1 | 1 | yes | characteristic |
| 100 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 101 | C2×C4×C7 | — | 2 | 1 | no | — |
| 102 | C2×C4×C7 | — | 2 | 1 | no | — |
| 103 | C2×C4×C7 | — | 2 | 2 | no | — |
| 104 | C2×C4×C7 | Q32 | 1 | 4 | yes | complemented |
| 105 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 108 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 109 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 111 | C2×Dic7 | — | 4 | 4 | no | — |
| 112 | C2×Dic7 | — | 4 | 4 | no | — |
| 113 | Dic14 | — | 4 | 4 | no | — |
| 114 | Dic14 | — | 4 | 4 | no | — |
| 115 | Dic14 | — | 8 | 4 | no | — |
| 116 | C7×C8 | — | 2 | 1 | no | — |
| 117 | C7×C8 | — | 2 | 1 | no | — |
| 118 | C7×C8 | #135 | 1 | 2 | yes | — |
| 119 | C7×C8 | C2×#41 | 1 | 2 | yes | — |
| 120 | C2×Q16 | — | 7 | 4 | no | — |
| 121 | C2×Q16 | — | 14 | 2 | no | — |
| 122 | C22×C8 | D28 | 1 | 1 | yes | characteristic |
| 123 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 124 | C4⋊Q8 | — | 14 | 2 | no | — |
| 125 | C4⋊Q8 | — | 14 | 2 | no | — |
| 126 | #125 | — | 28 | 2 | no | — |
| 127 | C22⋊Q8 | — | 14 | 4 | no | — |
| 128 | C22⋊Q8 | — | 28 | 2 | no | — |
| 129 | C4×Q8 | — | 14 | 4 | no | — |
| 130 | C4×Q8 | — | 28 | 2 | no | — |
| 131 | #117 | — | 28 | 2 | no | — |
| 132 | C2×C42 | D28 | 1 | 1 | yes | characteristic |
| 133 | Q32 | — | 14 | 8 | no | — |
| 134 | C2×C16 | — | 2 | 4 | no | — |
| 135 | C2×C16 | C22×D7 | 1 | 2 | yes | — |
| 136 | C2×C16 | C22×D7 | 1 | 2 | yes | — |
| 137 | #107 | — | 7 | 4 | no | — |
| 138 | #107 | — | 14 | 2 | no | — |
| 139 | #103 | — | 14 | 4 | no | — |
| 140 | #103 | — | 14 | 4 | no | — |
| 141 | C4×C8 | D28 | 1 | 1 | yes | characteristic |
| 142 | C4×C8 | D28 | 1 | 1 | yes | characteristic |
| 143 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 144 | C4×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 145 | C22×C7 | — | 2 | 1 | no | — |
| 146 | C22×C7 | — | 2 | 1 | no | — |
| 147 | C22×C7 | C2×Q32 | 1 | 2 | yes | — |
| 148 | C22×C7 | C22×D8 | 1 | 1 | yes | characteristic |
| 149 | C4×C7 | — | 2 | 1 | no | — |
| 150 | C4×C7 | — | 2 | 1 | no | — |
| 151 | C4×C7 | C2×Q32 | 1 | 2 | yes | — |
| 152 | C4×C7 | C2×Q32 | 1 | 2 | yes | — |
| 153 | C4×C7 | #508 | 1 | 2 | yes | — |
| 154 | C4×C7 | #532 | 1 | 1 | yes | characteristic |
| 155 | C4×C7 | C2×#135 | 1 | 1 | yes | characteristic |
| 156 | Dic7 | — | 8 | 4 | no | — |
| 157 | Dic7 | — | 8 | 4 | no | — |
| 158 | C2×Q8 | — | 14 | 4 | no | — |
| 159 | C2×Q8 | — | 28 | 2 | no | — |
| 160 | C22×C4 | D56 | 1 | 1 | yes | characteristic |
| 161 | C22×C4 | D56 | 1 | 1 | yes | characteristic |
| 162 | C22×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 163 | Q16 | — | 14 | 8 | no | — |
| 164 | Q16 | — | 28 | 4 | no | — |
| 165 | C2×C8 | — | 2 | 2 | no | — |
| 166 | C2×C8 | — | 2 | 2 | no | — |
| 167 | C2×C8 | C2×D28 | 1 | 1 | yes | characteristic |
| 168 | C2×C8 | C2×D28 | 1 | 1 | yes | characteristic |
| 169 | C2×C8 | C2×D28 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | C23×D7 | 1 | 1 | yes | characteristic |
| 171 | C4⋊C4 | — | 14 | 4 | no | — |
| 172 | C4⋊C4 | — | 28 | 4 | no | — |
| 173 | C4⋊C4 | — | 28 | 4 | no | — |
| 174 | C4⋊C4 | — | 28 | 2 | no | — |
| 175 | C4⋊C4 | — | 28 | 2 | no | — |
| 176 | C4⋊C4 | — | 28 | 2 | no | — |
| 177 | C22⋊C4 | — | 28 | 4 | no | — |
| 178 | C42 | — | 28 | 4 | no | — |
| 179 | C42 | D56 | 1 | 1 | yes | characteristic |
| 180 | C42 | D56 | 1 | 1 | yes | characteristic |
| 181 | C42 | C2×D28 | 1 | 1 | yes | characteristic |
| 182 | C42 | C2×D28 | 1 | 1 | yes | characteristic |
| 183 | C16 | — | 2 | 2 | no | — |
| 184 | C16 | #1157 | 1 | 4 | yes | — |
| 185 | C2×C7 | — | 2 | 2 | no | — |
| 186 | C2×C7 | #3169 | 1 | 1 | yes | characteristic |
| 187 | C2×C7 | C22×Q32 | 1 | 1 | yes | characteristic |
| 188 | C2×C7 | C2×#508 | 1 | 1 | yes | characteristic |
| 189 | C23 | C2×D56 | 1 | 1 | yes | characteristic |
| 190 | Q8 | — | 28 | 4 | no | — |
| 191 | Q8 | — | 28 | 4 | no | — |
| 192 | Q8 | — | 56 | 4 | no | — |
| 193 | C2×C4 | — | 2 | 1 | no | — |
| 194 | C2×C4 | — | 2 | 1 | no | — |
| 195 | C2×C4 | — | 2 | 2 | no | — |
| 196 | C2×C4 | — | 28 | 4 | no | — |
| 197 | C2×C4 | — | 28 | 4 | no | — |
| 198 | C2×C4 | Dic56 | 1 | 4 | yes | complemented |
| 199 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | C2×D56 | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | C22×D28 | 1 | 1 | yes | characteristic |
| 205 | C8 | — | 2 | 1 | no | — |
| 206 | C8 | — | 2 | 1 | no | — |
| 207 | C8 | #6467 | 1 | 2 | yes | — |
| 208 | C8 | C2×#1157 | 1 | 2 | yes | — |
| 209 | C7 | #33734 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 210 | C22 | — | 2 | 1 | no | — |
| 211 | C22 | — | 2 | 1 | no | — |
| 212 | C22 | C2×Dic56 | 1 | 2 | yes | — |
| 213 | C22 | C22×D56 | 1 | 1 | yes | characteristic |
| 214 | C4 | — | 2 | 1 | no | — |
| 215 | C4 | — | 2 | 1 | no | — |
| 216 | C4 | — | 56 | 4 | no | — |
| 217 | C4 | — | 56 | 4 | no | — |
| 218 | C4 | #89676 | 1 | 2 | yes | — |
| 219 | C4 | C2×Dic56 | 1 | 2 | yes | — |
| 220 | C4 | C2×Dic56 | 1 | 2 | yes | — |
| 221 | C4 | #90168 | 1 | 1 | yes | characteristic |
| 222 | C4 | C2×#6467 | 1 | 1 | yes | characteristic |
| 223 | C2 | — | 2 | 2 | no | — |
| 224 | C2 | #11735915 | 1 | 1 | yes | characteristic |
| 225 | C2 | C2×#89676 | 1 | 1 | yes | characteristic |
| 226 | C2 | C22×Dic56 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta