Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
442 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 88 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 | D53⋊#364 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20289726 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20289725 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #492×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | D53⋊M32 | — | 2 | 2 | no | — |
| 6 | #16811546 | C22 | 1 | 4 | yes | complemented |
| 7 | #16811545 | — | 2 | 2 | no | — |
| 8 | C2×SD16×D53 | — | 2 | 2 | no | — |
| 9 | C4⋊Q8×D53 | C4 | 1 | 1 | yes | characteristic |
| 10 | #127×D53 | C4 | 1 | 1 | yes | characteristic |
| 11 | #16810997 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16810996 | C4 | 1 | 1 | yes | characteristic |
| 13 | #16810995 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16810994 | — | 2 | 2 | no | — |
| 15 | C4×C8×D53 | C22 | 1 | 1 | yes | characteristic |
| 16 | C53×#492 | C4 | 1 | 1 | yes | characteristic |
| 17 | #13133292 | — | 4 | 2 | no | — |
| 18 | #13133291 | — | 2 | 4 | no | — |
| 19 | D53⋊C16 | — | 4 | 2 | no | — |
| 20 | C2×Q8×D53 | — | 2 | 1 | no | — |
| 21 | C2×D4×D53 | — | 2 | 1 | no | — |
| 22 | C2×D4×D53 | — | 4 | 1 | no | — |
| 23 | C2×#11719913 | — | 2 | 2 | no | — |
| 24 | C2×#11719923 | — | 2 | 2 | no | — |
| 25 | C2×#11719924 | — | 2 | 2 | no | — |
| 26 | SD16×D53 | — | 4 | 2 | no | — |
| 27 | SD16×D53 | — | 4 | 2 | no | — |
| 28 | SD16×D53 | — | 4 | 2 | no | — |
| 29 | SD16×D53 | — | 4 | 2 | no | — |
| 30 | C2×C8×D53 | D4 | 1 | 2 | yes | — |
| 31 | #13133217 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | #13133216 | — | 4 | 1 | no | — |
| 33 | #13133215 | — | 2 | 1 | no | — |
| 34 | #13133214 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | #13133213 | — | 2 | 1 | no | — |
| 36 | #13133213 | — | 4 | 1 | no | — |
| 37 | C4⋊C4×D53 | — | 4 | 1 | no | — |
| 38 | C42×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C4×#88322 | C23 | 1 | 1 | yes | characteristic |
| 40 | C8×Dic53 | D4 | 1 | 2 | yes | — |
| 41 | C2×SD16×C53 | — | 2 | 2 | no | — |
| 42 | C4⋊Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | #127×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C23×D53 | — | 4 | 1 | no | — |
| 46 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 47 | C2×D4×C53 | — | 2 | 1 | no | — |
| 48 | C2×D4×C53 | — | 4 | 1 | no | — |
| 49 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 50 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 51 | Q8×D53 | — | 4 | 2 | no | — |
| 52 | Q8×D53 | — | 4 | 2 | no | — |
| 53 | D4×D53 | — | 4 | 2 | no | — |
| 54 | D4×D53 | — | 4 | 2 | no | — |
| 55 | D4×D53 | — | 8 | 4 | no | — |
| 56 | C2×D212 | — | 2 | 1 | no | — |
| 57 | C2×D212 | — | 4 | 1 | no | — |
| 58 | C2×C4×D53 | — | 2 | 1 | no | — |
| 59 | C2×C4×D53 | — | 4 | 1 | no | — |
| 60 | C2×C4×D53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C2×Dic106 | — | 2 | 1 | no | — |
| 62 | C2×Dic106 | — | 4 | 1 | no | — |
| 63 | SD16×C53 | — | 4 | 2 | no | — |
| 64 | SD16×C53 | — | 4 | 2 | no | — |
| 65 | C2×C8×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 66 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 67 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | #11719924 | — | 4 | 2 | no | — |
| 69 | #11719924 | — | 4 | 2 | no | — |
| 70 | #11719923 | — | 4 | 2 | no | — |
| 71 | #11719923 | — | 4 | 2 | no | — |
| 72 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 73 | #11719919 | — | 4 | 2 | no | — |
| 74 | C4×Dic53 | — | 2 | 1 | no | — |
| 75 | C4×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C2×#88322 | C2×D4 | 1 | 2 | yes | — |
| 77 | #11719913 | — | 4 | 2 | no | — |
| 78 | #11719913 | — | 4 | 2 | no | — |
| 79 | C8×D53 | D8 | 1 | 2 | yes | — |
| 80 | C8×D53 | D8 | 1 | 2 | yes | — |
| 81 | C8×D53 | SD16 | 1 | 2 | yes | — |
| 82 | C8×D53 | SD16 | 1 | 2 | yes | — |
| 83 | #11719910 | — | 4 | 4 | no | — |
| 84 | C23×C53 | — | 4 | 1 | no | — |
| 85 | C22×D53 | — | 4 | 1 | no | — |
| 86 | C22×D53 | — | 8 | 4 | no | — |
| 87 | C22×D53 | — | 8 | 1 | no | — |
| 88 | C22×D53 | — | 8 | 1 | no | — |
| 89 | C22×D53 | #100 | 1 | 1 | yes | characteristic |
| 90 | Q8×C53 | — | 4 | 2 | no | — |
| 91 | D4×C53 | — | 4 | 2 | no | — |
| 92 | D4×C53 | — | 8 | 2 | no | — |
| 93 | C2×C4×C53 | — | 2 | 1 | no | — |
| 94 | C2×C4×C53 | — | 4 | 1 | no | — |
| 95 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C22⋊D53 | — | 8 | 4 | no | — |
| 97 | C22⋊D53 | — | 8 | 2 | no | — |
| 98 | C2×Dic53 | — | 2 | 1 | no | — |
| 99 | C2×Dic53 | — | 4 | 1 | no | — |
| 100 | C2×Dic53 | #100 | 1 | 1 | yes | characteristic |
| 101 | D212 | — | 4 | 2 | no | — |
| 102 | D212 | — | 8 | 2 | no | — |
| 103 | C4×D53 | — | 2 | 4 | no | — |
| 104 | C4×D53 | — | 8 | 1 | no | — |
| 105 | C4×D53 | — | 8 | 1 | no | — |
| 106 | C4×D53 | #102 | 1 | 2 | yes | — |
| 107 | C4×D53 | #102 | 1 | 2 | yes | — |
| 108 | Dic106 | — | 4 | 2 | no | — |
| 109 | Dic106 | — | 8 | 2 | no | — |
| 110 | C8×C53 | #102 | 1 | 2 | yes | — |
| 111 | C8×C53 | #102 | 1 | 2 | yes | — |
| 112 | #88322 | C2×D8 | 1 | 2 | yes | — |
| 113 | #88322 | C2×SD16 | 1 | 2 | yes | — |
| 114 | C2×#1359 | — | 53 | 1 | no | maximal, Sylow |
| 115 | C22×C53 | — | 8 | 2 | no | — |
| 116 | C22×C53 | — | 8 | 1 | no | — |
| 117 | C22×C53 | C2×#100 | 1 | 1 | yes | characteristic |
| 118 | C2×D53 | — | 2 | 2 | no | — |
| 119 | C2×D53 | — | 8 | 2 | no | — |
| 120 | C2×D53 | — | 8 | 1 | no | — |
| 121 | C2×D53 | — | 16 | 2 | no | — |
| 122 | C2×D53 | — | 16 | 2 | no | — |
| 123 | C2×D53 | #331 | 1 | 1 | yes | characteristic |
| 124 | C2×D53 | #331 | 1 | 1 | yes | characteristic |
| 125 | C4×C53 | — | 2 | 2 | no | — |
| 126 | C4×C53 | — | 8 | 1 | no | — |
| 127 | C4×C53 | C2×#102 | 1 | 2 | yes | — |
| 128 | Dic53 | — | 2 | 2 | no | — |
| 129 | Dic53 | — | 8 | 1 | no | — |
| 130 | Dic53 | #331 | 1 | 2 | yes | — |
| 131 | C2×#492 | — | 53 | 1 | no | — |
| 132 | C2×#364 | — | 53 | 2 | no | — |
| 133 | #1359 | — | 53 | 2 | no | — |
| 134 | #1359 | — | 53 | 2 | no | — |
| 135 | C2×C53 | — | 2 | 1 | no | — |
| 136 | C2×C53 | — | 16 | 1 | no | — |
| 137 | C2×C53 | C2×#331 | 1 | 1 | yes | characteristic |
| 138 | D53 | — | 2 | 1 | no | — |
| 139 | D53 | — | 16 | 1 | no | — |
| 140 | D53 | #1359 | 1 | 2 | yes | complemented |
| 141 | C22×SD16 | — | 106 | 2 | no | — |
| 142 | C2×C4⋊Q8 | — | 53 | 1 | no | — |
| 143 | C2×#127 | — | 53 | 1 | no | — |
| 144 | C2×M32 | — | 106 | 2 | no | — |
| 145 | #492 | — | 53 | 1 | no | — |
| 146 | #492 | — | 53 | 1 | no | — |
| 147 | #492 | — | 53 | 1 | no | — |
| 148 | #492 | — | 106 | 2 | no | — |
| 149 | #492 | #6157 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×C8 | — | 53 | 1 | no | — |
| 151 | #364 | — | 53 | 4 | no | — |
| 152 | #364 | — | 106 | 2 | no | — |
| 153 | C53 | C2×#1359 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 154 | C22×Q8 | — | 106 | 1 | no | — |
| 155 | C22×D4 | — | 106 | 1 | no | — |
| 156 | C22×D4 | — | 212 | 1 | no | — |
| 157 | C2×SD16 | — | 2 | 2 | no | — |
| 158 | C2×SD16 | — | 106 | 2 | no | — |
| 159 | C2×SD16 | — | 106 | 2 | no | — |
| 160 | C2×SD16 | — | 106 | 2 | no | — |
| 161 | C2×SD16 | — | 212 | 2 | no | — |
| 162 | C2×SD16 | — | 212 | 2 | no | — |
| 163 | C2×SD16 | — | 212 | 2 | no | — |
| 164 | C2×SD16 | — | 212 | 2 | no | — |
| 165 | C22×C8 | — | 53 | 2 | no | — |
| 166 | C4⋊Q8 | — | 53 | 1 | no | — |
| 167 | C4⋊Q8 | — | 106 | 1 | no | — |
| 168 | C4⋊Q8 | — | 212 | 1 | no | — |
| 169 | C4⋊Q8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 170 | #127 | — | 53 | 1 | no | — |
| 171 | #127 | — | 106 | 1 | no | — |
| 172 | #127 | — | 212 | 1 | no | — |
| 173 | #127 | C2×#6157 | 1 | 1 | yes | characteristic |
| 174 | C2×C4⋊C4 | — | 212 | 1 | no | — |
| 175 | C2×C42 | — | 53 | 1 | no | — |
| 176 | M32 | — | 106 | 4 | no | — |
| 177 | M32 | — | 212 | 2 | no | — |
| 178 | C2×C16 | — | 212 | 2 | no | — |
| 179 | C4×C8 | — | 53 | 2 | no | — |
| 180 | C4×C8 | — | 53 | 1 | no | — |
| 181 | C4×C8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 182 | C24 | — | 212 | 1 | no | — |
| 183 | C2×Q8 | — | 2 | 1 | no | — |
| 184 | C2×Q8 | — | 106 | 1 | no | — |
| 185 | C2×Q8 | — | 212 | 1 | no | — |
| 186 | C2×Q8 | — | 212 | 2 | no | — |
| 187 | C2×Q8 | — | 212 | 2 | no | — |
| 188 | C2×D4 | — | 2 | 1 | no | — |
| 189 | C2×D4 | — | 4 | 1 | no | — |
| 190 | C2×D4 | — | 106 | 1 | no | — |
| 191 | C2×D4 | — | 424 | 4 | no | — |
| 192 | C2×D4 | — | 212 | 1 | no | — |
| 193 | C2×D4 | — | 212 | 2 | no | — |
| 194 | C2×D4 | — | 212 | 2 | no | — |
| 195 | C2×D4 | — | 212 | 1 | no | — |
| 196 | C2×D4 | — | 212 | 2 | no | — |
| 197 | C22×C4 | — | 53 | 1 | no | — |
| 198 | C22×C4 | — | 106 | 1 | no | — |
| 199 | C22×C4 | — | 212 | 1 | no | — |
| 200 | SD16 | — | 4 | 2 | no | — |
| 201 | SD16 | — | 4 | 2 | no | — |
| 202 | SD16 | — | 212 | 2 | no | — |
| 203 | SD16 | — | 212 | 2 | no | — |
| 204 | SD16 | — | 212 | 2 | no | — |
| 205 | SD16 | — | 212 | 2 | no | — |
| 206 | SD16 | — | 212 | 2 | no | — |
| 207 | SD16 | — | 212 | 2 | no | — |
| 208 | C2×C8 | — | 53 | 2 | no | — |
| 209 | C2×C8 | — | 53 | 2 | no | — |
| 210 | C2×C8 | — | 53 | 2 | no | — |
| 211 | C2×C8 | — | 53 | 2 | no | — |
| 212 | C2×C8 | — | 53 | 2 | no | — |
| 213 | C2×C8 | C22⋊#6157 | 1 | 2 | yes | — |
| 214 | C4⋊C4 | — | 4 | 1 | no | — |
| 215 | C4⋊C4 | — | 212 | 1 | no | — |
| 216 | C4⋊C4 | — | 212 | 2 | no | — |
| 217 | C42 | — | 53 | 1 | no | — |
| 218 | C42 | — | 106 | 1 | no | — |
| 219 | C42 | C22×#6157 | 1 | 1 | yes | characteristic |
| 220 | C16 | — | 212 | 4 | no | — |
| 221 | C23 | — | 4 | 1 | no | — |
| 222 | C23 | — | 53 | 1 | no | — |
| 223 | C23 | — | 424 | 1 | no | — |
| 224 | C23 | — | 424 | 1 | no | — |
| 225 | C23 | — | 424 | 4 | no | — |
| 226 | C23 | — | 212 | 1 | no | — |
| 227 | Q8 | — | 4 | 2 | no | — |
| 228 | Q8 | — | 424 | 2 | no | — |
| 229 | Q8 | — | 212 | 2 | no | — |
| 230 | D4 | — | 4 | 2 | no | — |
| 231 | D4 | — | 8 | 2 | no | — |
| 232 | D4 | — | 424 | 2 | no | — |
| 233 | D4 | — | 424 | 2 | no | — |
| 234 | D4 | — | 424 | 4 | no | — |
| 235 | D4 | — | 212 | 2 | no | — |
| 236 | C2×C4 | — | 2 | 1 | no | — |
| 237 | C2×C4 | — | 4 | 1 | no | — |
| 238 | C2×C4 | — | 53 | 1 | no | — |
| 239 | C2×C4 | — | 53 | 2 | no | — |
| 240 | C2×C4 | — | 53 | 2 | no | — |
| 241 | C2×C4 | — | 106 | 4 | no | — |
| 242 | C2×C4 | — | 106 | 1 | no | — |
| 243 | C2×C4 | — | 424 | 1 | no | — |
| 244 | C2×C4 | — | 424 | 1 | no | — |
| 245 | C2×C4 | — | 212 | 1 | no | — |
| 246 | C2×C4 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 247 | C8 | — | 53 | 2 | no | — |
| 248 | C8 | — | 53 | 2 | no | — |
| 249 | C8 | #13133279 | 1 | 2 | yes | — |
| 250 | C8 | #13133279 | 1 | 2 | yes | — |
| 251 | C22 | — | 8 | 2 | no | — |
| 252 | C22 | — | 8 | 1 | no | — |
| 253 | C22 | — | 53 | 1 | no | — |
| 254 | C22 | — | 53 | 1 | no | — |
| 255 | C22 | — | 848 | 2 | no | — |
| 256 | C22 | — | 848 | 2 | no | — |
| 257 | C22 | — | 106 | 2 | no | — |
| 258 | C22 | — | 424 | 2 | no | — |
| 259 | C22 | — | 424 | 1 | no | — |
| 260 | C22 | D53⋊#100 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 2 | 2 | no | — |
| 262 | C4 | — | 8 | 1 | no | — |
| 263 | C4 | — | 53 | 2 | no | — |
| 264 | C4 | — | 106 | 2 | no | — |
| 265 | C4 | — | 424 | 1 | no | — |
| 266 | C4 | C2×#13133279 | 1 | 2 | yes | — |
| 267 | C2 | — | 2 | 1 | no | — |
| 268 | C2 | — | 16 | 1 | no | — |
| 269 | C2 | — | 53 | 2 | no | — |
| 270 | C2 | — | 848 | 1 | no | — |
| 271 | C2 | — | 106 | 1 | no | — |
| 272 | C2 | D53⋊#331 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D53⋊#364 | C2 | semidirect |
| 2. #20289726 | C2 | semidirect |
| 3. #20289725 | C2 | semidirect |
| 6. #16811546 | C22 | semidirect |
| 140. D53 | #1359 | semidirect |
| 153. C53 | C2×#1359 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D53⋊#364 | 2 |
| #20289726 | 2 |
| #20289725 | 2 |
| #492×D53 | 2 |
| C2×#1359 | 53 |