← #13509286
Subgroups — #13509286
Order 1792.
490 conjugacy classes of subgroups.
The subgroup lattice has 224 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #11736615 | C2 | 1 | 2 | yes | maximal |
| 2 | C7×#2158 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11729158 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×C16×Dic7 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11728402 | C2 | 1 | 2 | yes | maximal |
| 6 | #11727097 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #11727094 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11727075 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #11727074 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #90208 | — | 2 | 2 | no | — |
| 11 | C22×C7×C16 | C22 | 1 | 1 | yes | characteristic |
| 12 | C7×#427 | C22 | 1 | 2 | yes | — |
| 13 | #89878 | C22 | 1 | 2 | yes | — |
| 14 | C2×C8×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×C7⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 16 | #89770 | — | 2 | 2 | no | — |
| 17 | #89650 | C22 | 1 | 4 | yes | — |
| 18 | #89646 | C22 | 1 | 4 | yes | — |
| 19 | #89630 | — | 2 | 2 | no | — |
| 20 | #89627 | — | 2 | 2 | no | — |
| 21 | C7×#367 | C22 | 1 | 2 | yes | — |
| 22 | C7×#366 | C22 | 1 | 2 | yes | — |
| 23 | #89300 | C22 | 1 | 2 | yes | — |
| 24 | #89299 | C22 | 1 | 2 | yes | — |
| 25 | C16×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 26 | C16×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 27 | C16×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 28 | C16×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 29 | #89285 | C22 | 1 | 4 | yes | — |
| 30 | #89284 | C22 | 1 | 4 | yes | — |
| 31 | #33726 | — | 7 | 1 | no | maximal, Sylow |
| 32 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C7×#117 | — | 2 | 2 | no | — |
| 34 | #6489 | — | 2 | 2 | no | — |
| 35 | C2×C4×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×C7⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 37 | #6453 | — | 2 | 4 | no | — |
| 38 | #6451 | — | 2 | 4 | no | — |
| 39 | #6441 | — | 4 | 4 | no | — |
| 40 | C4×Dic14 | — | 4 | 4 | no | — |
| 41 | C2×C7×C16 | Q8 | 1 | 4 | yes | complemented |
| 42 | C2×C7×C16 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×C7×C16 | C23 | 1 | 1 | yes | characteristic |
| 44 | C7×#107 | C23 | 1 | 4 | yes | — |
| 45 | C7×#106 | — | 2 | 2 | no | — |
| 46 | #6392 | C23 | 1 | 4 | yes | — |
| 47 | #6391 | — | 2 | 2 | no | — |
| 48 | C8×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 49 | C8×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 50 | C8×Dic7 | C23 | 1 | 1 | yes | characteristic |
| 51 | C8×Dic7 | C23 | 1 | 1 | yes | characteristic |
| 52 | C2×C7⋊C16 | Q8 | 1 | 4 | yes | complemented |
| 53 | C2×C7⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×C7⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 55 | #6382 | — | 2 | 4 | no | — |
| 56 | #6381 | — | 2 | 4 | no | — |
| 57 | #3160 | — | 7 | 2 | no | — |
| 58 | #2254 | — | 7 | 2 | no | — |
| 59 | #2253 | — | 7 | 2 | no | — |
| 60 | #2253 | — | 7 | 2 | no | — |
| 61 | #2252 | — | 7 | 2 | no | — |
| 62 | #2158 | — | 7 | 1 | no | — |
| 63 | #2158 | — | 7 | 2 | no | — |
| 64 | #2158 | D7 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×C16 | — | 7 | 1 | no | — |
| 66 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 67 | C22×Dic7 | D8 | 1 | 2 | yes | — |
| 68 | C2×Dic14 | — | 4 | 4 | no | — |
| 69 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C2×C7×C8 | C2×Q8 | 1 | 2 | yes | — |
| 71 | C2×C7×C8 | C2×Q8 | 1 | 2 | yes | — |
| 72 | C2×C7×C8 | C24 | 1 | 1 | yes | characteristic |
| 73 | C7×C4⋊C4 | — | 2 | 4 | no | — |
| 74 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 75 | C42×C7 | — | 4 | 2 | no | — |
| 76 | C22⋊Dic7 | — | 4 | 2 | no | — |
| 77 | C4⋊Dic7 | — | 2 | 4 | no | — |
| 78 | C4⋊Dic7 | — | 4 | 4 | no | — |
| 79 | Dic7⋊C4 | — | 4 | 4 | no | — |
| 80 | Dic7⋊C4 | — | 4 | 4 | no | — |
| 81 | C4×Dic7 | — | 4 | 2 | no | — |
| 82 | C4×Dic7 | D8 | 1 | 2 | yes | — |
| 83 | C4×Dic7 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C4×Dic7 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C7⋊C8 | Q16 | 1 | 4 | yes | — |
| 86 | C2×C7⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C7⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | C7×C16 | C2×Q8 | 1 | 2 | yes | — |
| 89 | C7×C16 | C2×Q8 | 1 | 2 | yes | — |
| 90 | C7⋊C16 | C2×Q8 | 1 | 2 | yes | — |
| 91 | C7⋊C16 | C2×Q8 | 1 | 2 | yes | — |
| 92 | #533 | — | 14 | 2 | no | — |
| 93 | C22×C16 | — | 7 | 1 | no | — |
| 94 | C22×C16 | C2×D7 | 1 | 1 | yes | characteristic |
| 95 | #500 | — | 7 | 4 | no | — |
| 96 | #499 | — | 7 | 4 | no | — |
| 97 | #499 | — | 14 | 2 | no | — |
| 98 | #498 | — | 14 | 2 | no | — |
| 99 | #427 | — | 7 | 2 | no | — |
| 100 | #427 | — | 14 | 2 | no | — |
| 101 | #427 | C2×D7 | 1 | 2 | yes | — |
| 102 | C2×C4×C8 | — | 7 | 1 | no | — |
| 103 | #367 | — | 7 | 2 | no | — |
| 104 | #367 | — | 7 | 4 | no | — |
| 105 | #367 | C2×D7 | 1 | 2 | yes | — |
| 106 | #366 | — | 7 | 2 | no | — |
| 107 | #366 | — | 7 | 4 | no | — |
| 108 | #366 | C2×D7 | 1 | 2 | yes | — |
| 109 | C4×C16 | — | 7 | 1 | no | — |
| 110 | C4×C16 | — | 7 | 1 | no | — |
| 111 | C4×C16 | — | 7 | 1 | no | — |
| 112 | C4×C16 | — | 7 | 1 | no | — |
| 113 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C7 | — | 4 | 4 | no | — |
| 115 | C2×C4×C7 | C4⋊Q8 | 1 | 2 | yes | — |
| 116 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C7 | C2×Q16 | 1 | 2 | yes | — |
| 118 | C2×C4×C7 | C22×D4 | 1 | 1 | yes | characteristic |
| 119 | C2×Dic7 | — | 2 | 4 | no | — |
| 120 | C2×Dic7 | — | 4 | 4 | no | — |
| 121 | C2×Dic7 | C2×D8 | 1 | 2 | yes | — |
| 122 | C2×Dic7 | C2×D8 | 1 | 2 | yes | — |
| 123 | Dic14 | — | 8 | 8 | no | — |
| 124 | C7×C8 | C4⋊Q8 | 1 | 2 | yes | — |
| 125 | C7×C8 | C22×Q8 | 1 | 1 | yes | characteristic |
| 126 | C7×C8 | C22×Q8 | 1 | 1 | yes | characteristic |
| 127 | C7⋊C8 | C2×Q16 | 1 | 2 | yes | — |
| 128 | C7⋊C8 | C2×Q16 | 1 | 2 | yes | — |
| 129 | C22×C8 | — | 7 | 1 | no | — |
| 130 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 131 | C4⋊Q8 | — | 14 | 4 | no | — |
| 132 | #125 | — | 14 | 4 | no | — |
| 133 | C22⋊Q8 | — | 28 | 4 | no | — |
| 134 | C4×Q8 | — | 28 | 4 | no | — |
| 135 | #117 | — | 2 | 2 | no | — |
| 136 | #117 | — | 14 | 2 | no | — |
| 137 | C2×C42 | — | 7 | 1 | no | — |
| 138 | C2×C16 | — | 7 | 4 | no | — |
| 139 | C2×C16 | — | 7 | 1 | no | — |
| 140 | C2×C16 | — | 7 | 1 | no | — |
| 141 | C2×C16 | Dic14 | 1 | 4 | yes | complemented |
| 142 | C2×C16 | C22×D7 | 1 | 1 | yes | characteristic |
| 143 | C2×C16 | C22×D7 | 1 | 1 | yes | characteristic |
| 144 | #107 | — | 7 | 4 | no | — |
| 145 | #107 | — | 14 | 4 | no | — |
| 146 | #107 | C22×D7 | 1 | 4 | yes | — |
| 147 | #106 | — | 2 | 2 | no | — |
| 148 | #106 | — | 14 | 4 | no | — |
| 149 | #106 | — | 14 | 2 | no | — |
| 150 | C4×C8 | — | 7 | 1 | no | — |
| 151 | C4×C8 | — | 7 | 1 | no | — |
| 152 | C4×C8 | — | 7 | 1 | no | — |
| 153 | C4×C8 | — | 7 | 1 | no | — |
| 154 | C22×C7 | — | 2 | 2 | no | — |
| 155 | C22×C7 | #500 | 1 | 2 | yes | — |
| 156 | C22×C7 | C22×D8 | 1 | 1 | yes | characteristic |
| 157 | C4×C7 | — | 8 | 4 | no | — |
| 158 | C4×C7 | #500 | 1 | 2 | yes | — |
| 159 | C4×C7 | C2×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 160 | C4×C7 | C22×Q16 | 1 | 1 | yes | characteristic |
| 161 | Dic7 | — | 2 | 2 | no | — |
| 162 | Dic7 | — | 8 | 4 | no | — |
| 163 | Dic7 | #508 | 1 | 4 | yes | — |
| 164 | C2×Q8 | — | 28 | 4 | no | — |
| 165 | C22×C4 | — | 7 | 2 | no | — |
| 166 | C22×C4 | D4×D7 | 1 | 1 | yes | characteristic |
| 167 | C2×C8 | — | 7 | 4 | no | — |
| 168 | C2×C8 | — | 7 | 1 | no | — |
| 169 | C2×C8 | — | 7 | 1 | no | — |
| 170 | C2×C8 | C2×Dic14 | 1 | 2 | yes | — |
| 171 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | Q8×D7 | 1 | 2 | yes | — |
| 173 | C2×C8 | C23×D7 | 1 | 1 | yes | characteristic |
| 174 | C4⋊C4 | — | 2 | 4 | no | — |
| 175 | C4⋊C4 | — | 14 | 4 | no | — |
| 176 | C4⋊C4 | — | 28 | 4 | no | — |
| 177 | C4⋊C4 | — | 28 | 4 | no | — |
| 178 | C4⋊C4 | — | 28 | 4 | no | — |
| 179 | C22⋊C4 | — | 4 | 2 | no | — |
| 180 | C22⋊C4 | — | 28 | 2 | no | — |
| 181 | C42 | — | 4 | 2 | no | — |
| 182 | C42 | — | 7 | 2 | no | — |
| 183 | C42 | — | 7 | 1 | no | — |
| 184 | C42 | — | 7 | 1 | no | — |
| 185 | C42 | — | 28 | 2 | no | — |
| 186 | C16 | — | 7 | 2 | no | — |
| 187 | C16 | — | 7 | 2 | no | — |
| 188 | C16 | C2×Dic14 | 1 | 2 | yes | — |
| 189 | C16 | C2×Dic14 | 1 | 2 | yes | — |
| 190 | C2×C7 | — | 2 | 2 | no | — |
| 191 | C2×C7 | C2×#500 | 1 | 1 | yes | characteristic |
| 192 | C2×C7 | C2×#508 | 1 | 2 | yes | — |
| 193 | C23 | D7×D8 | 1 | 1 | yes | characteristic |
| 194 | Q8 | — | 56 | 8 | no | — |
| 195 | C2×C4 | — | 4 | 4 | no | — |
| 196 | C2×C4 | — | 7 | 2 | no | — |
| 197 | C2×C4 | — | 7 | 2 | no | — |
| 198 | C2×C4 | — | 14 | 4 | no | — |
| 199 | C2×C4 | — | 28 | 4 | no | — |
| 200 | C2×C4 | #6451 | 1 | 2 | yes | — |
| 201 | C2×C4 | D7×D8 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | D7×Q16 | 1 | 2 | yes | — |
| 203 | C2×C4 | C2×D4×D7 | 1 | 1 | yes | characteristic |
| 204 | C8 | — | 7 | 2 | no | — |
| 205 | C8 | — | 7 | 2 | no | — |
| 206 | C8 | #6451 | 1 | 2 | yes | — |
| 207 | C8 | C22×Dic14 | 1 | 1 | yes | characteristic |
| 208 | C8 | C2×Q8×D7 | 1 | 1 | yes | characteristic |
| 209 | C7 | #33726 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 210 | C22 | — | 2 | 2 | no | — |
| 211 | C22 | #89646 | 1 | 2 | yes | — |
| 212 | C22 | C2×D7×D8 | 1 | 1 | yes | characteristic |
| 213 | C4 | — | 7 | 4 | no | — |
| 214 | C4 | — | 8 | 4 | no | — |
| 215 | C4 | — | 14 | 2 | no | — |
| 216 | C4 | — | 56 | 4 | no | — |
| 217 | C4 | #89646 | 1 | 2 | yes | — |
| 218 | C4 | C2×#6451 | 1 | 1 | yes | characteristic |
| 219 | C4 | C2×D7×Q16 | 1 | 1 | yes | characteristic |
| 220 | C2 | — | 2 | 2 | no | — |
| 221 | C2 | C2×#89646 | 1 | 1 | yes | characteristic |
| 222 | C2 | D7×#508 | 1 | 2 | yes | — |
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