Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
960 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 205 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 | #430×D53 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#16810604 | C2 | 1 | 3 | yes | maximal |
| 3 | #20276117 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20276115 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #20276113 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #338×D53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C53×#2069 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×#11719947 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×M16×D53 | C22 | 1 | 3 | yes | — |
| 10 | C2×M16×D53 | C22 | 1 | 3 | yes | — |
| 11 | #16810672 | C22 | 1 | 3 | yes | — |
| 12 | #16810671 | C22 | 1 | 3 | yes | — |
| 13 | #16810671 | C22 | 1 | 3 | yes | — |
| 14 | #16810670 | — | 2 | 6 | no | — |
| 15 | #108×D53 | — | 2 | 6 | no | — |
| 16 | #16810604 | — | 2 | 6 | no | — |
| 17 | #16810604 | — | 2 | 6 | no | — |
| 18 | C2×#13133135 | C22 | 1 | 3 | yes | — |
| 19 | C2×#13133135 | C22 | 1 | 3 | yes | — |
| 20 | C2×#13133134 | C22 | 1 | 3 | yes | — |
| 21 | #16810381 | C22 | 1 | 3 | yes | — |
| 22 | C53×#430 | C22 | 1 | 3 | yes | — |
| 23 | C2×#100×C53 | C22 | 1 | 3 | yes | — |
| 24 | C53×#338 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#11719947 | — | 2 | 3 | no | — |
| 26 | C2×#11719947 | — | 2 | 3 | no | — |
| 27 | C2×#11719947 | — | 2 | 3 | no | — |
| 28 | C2×#11719947 | — | 2 | 3 | no | — |
| 29 | C22×Dic106 | C23 | 1 | 1 | yes | characteristic |
| 30 | C22×C22⋊D53 | Q8 | 1 | 2 | yes | — |
| 31 | C23×Dic53 | Q8 | 1 | 2 | yes | — |
| 32 | C22×C4×D53 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×#11719917 | C23 | 1 | 3 | yes | — |
| 34 | C2×#11719917 | C23 | 1 | 3 | yes | — |
| 35 | C2×#11719912 | — | 2 | 3 | no | — |
| 36 | C2×#11719912 | — | 2 | 3 | no | — |
| 37 | M16×D53 | — | 2 | 3 | no | — |
| 38 | M16×D53 | — | 2 | 3 | no | — |
| 39 | M16×D53 | — | 2 | 3 | no | — |
| 40 | M16×D53 | — | 2 | 3 | no | — |
| 41 | M16×D53 | — | 2 | 3 | no | — |
| 42 | M16×D53 | — | 2 | 3 | no | — |
| 43 | M16×D53 | D4 | 1 | 6 | yes | — |
| 44 | M16×D53 | D4 | 1 | 6 | yes | — |
| 45 | C2×C8×D53 | — | 2 | 3 | no | — |
| 46 | C2×C8×D53 | — | 2 | 3 | no | — |
| 47 | #13133153 | — | 2 | 6 | no | — |
| 48 | #13133152 | — | 2 | 6 | no | — |
| 49 | #13133152 | — | 2 | 6 | no | — |
| 50 | #13133135 | — | 2 | 6 | no | — |
| 51 | #13133135 | — | 2 | 6 | no | — |
| 52 | #13133134 | — | 2 | 6 | no | — |
| 53 | C22×D4×C53 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×M16×C53 | C23 | 1 | 3 | yes | — |
| 55 | C2×M16×C53 | C23 | 1 | 3 | yes | — |
| 56 | #108×C53 | — | 2 | 6 | no | — |
| 57 | #100×C53 | — | 2 | 6 | no | — |
| 58 | C24×C53 | C2×Q8 | 1 | 2 | yes | — |
| 59 | C23×D53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 60 | C2×D4×C53 | — | 2 | 3 | no | — |
| 61 | C2×D4×C53 | — | 2 | 3 | no | — |
| 62 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 63 | C2×C22⋊D53 | — | 2 | 6 | no | — |
| 64 | C2×C22⋊D53 | — | 2 | 6 | no | — |
| 65 | C22×Dic53 | — | 2 | 6 | no | — |
| 66 | C22×Dic53 | — | 2 | 6 | no | — |
| 67 | C22×Dic53 | C2×Q8 | 1 | 2 | yes | — |
| 68 | C22×Dic53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 69 | #11719947 | — | 4 | 3 | no | — |
| 70 | #11719947 | — | 4 | 3 | no | — |
| 71 | #11719947 | — | 4 | 3 | no | — |
| 72 | #11719947 | — | 4 | 3 | no | — |
| 73 | #11719947 | — | 4 | 1 | no | — |
| 74 | #11719947 | — | 4 | 3 | no | — |
| 75 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 76 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 77 | C2×C4×D53 | #41 | 1 | 3 | yes | — |
| 78 | C2×C4×D53 | #41 | 1 | 3 | yes | — |
| 79 | C2×Dic106 | — | 2 | 3 | no | — |
| 80 | C2×Dic106 | — | 2 | 3 | no | — |
| 81 | M16×C53 | — | 2 | 3 | no | — |
| 82 | M16×C53 | — | 2 | 3 | no | — |
| 83 | M16×C53 | — | 2 | 3 | no | — |
| 84 | M16×C53 | C2×D4 | 1 | 6 | yes | — |
| 85 | C2×C8×C53 | — | 2 | 3 | no | — |
| 86 | C2×C8×C53 | — | 2 | 3 | no | — |
| 87 | #11719917 | — | 2 | 3 | no | — |
| 88 | #11719917 | — | 2 | 3 | no | — |
| 89 | #11719917 | — | 2 | 3 | no | — |
| 90 | #11719917 | C2×D4 | 1 | 6 | yes | — |
| 91 | C2×#88322 | — | 2 | 3 | no | — |
| 92 | C2×#88322 | — | 2 | 3 | no | — |
| 93 | #11719912 | — | 2 | 6 | no | — |
| 94 | #11719912 | — | 2 | 6 | no | — |
| 95 | #11719912 | — | 4 | 3 | no | — |
| 96 | #11719912 | — | 4 | 3 | no | — |
| 97 | C8×D53 | — | 2 | 6 | no | — |
| 98 | C8×D53 | — | 2 | 6 | no | — |
| 99 | C8×D53 | — | 4 | 3 | no | — |
| 100 | C8×D53 | — | 4 | 3 | no | — |
| 101 | C23×C53 | — | 2 | 6 | no | — |
| 102 | C23×C53 | — | 8 | 2 | no | — |
| 103 | C23×C53 | C22×Q8 | 1 | 1 | yes | characteristic |
| 104 | C22×D53 | — | 8 | 1 | no | — |
| 105 | C22×D53 | C22⋊Q8 | 1 | 3 | yes | — |
| 106 | C22×D53 | C22⋊Q8 | 1 | 3 | yes | — |
| 107 | D4×C53 | — | 4 | 3 | no | — |
| 108 | D4×C53 | — | 4 | 1 | no | — |
| 109 | C2×C4×C53 | C22×D4 | 1 | 3 | yes | — |
| 110 | C2×C4×C53 | C2×#41 | 1 | 3 | yes | — |
| 111 | C22⋊D53 | — | 4 | 6 | no | — |
| 112 | C22⋊D53 | — | 4 | 2 | no | — |
| 113 | C2×Dic53 | — | 2 | 6 | no | — |
| 114 | C2×Dic53 | — | 4 | 2 | no | — |
| 115 | C2×Dic53 | — | 4 | 6 | no | — |
| 116 | C2×Dic53 | C22⋊Q8 | 1 | 3 | yes | — |
| 117 | C2×Dic53 | C22⋊Q8 | 1 | 3 | yes | — |
| 118 | C4×D53 | #121 | 1 | 3 | yes | — |
| 119 | C4×D53 | #121 | 1 | 3 | yes | — |
| 120 | C4×D53 | #121 | 1 | 3 | yes | — |
| 121 | C4×D53 | #121 | 1 | 3 | yes | — |
| 122 | C4×D53 | #126 | 1 | 1 | yes | characteristic |
| 123 | C4×D53 | #126 | 1 | 3 | yes | — |
| 124 | Dic106 | — | 4 | 3 | no | — |
| 125 | Dic106 | — | 4 | 1 | no | — |
| 126 | C8×C53 | — | 2 | 6 | no | — |
| 127 | C8×C53 | — | 4 | 3 | no | — |
| 128 | #88322 | — | 2 | 6 | no | — |
| 129 | #88322 | — | 4 | 3 | no | — |
| 130 | #23622 | — | 53 | 1 | no | maximal, Sylow |
| 131 | C22×C53 | — | 4 | 1 | no | — |
| 132 | C22×C53 | — | 4 | 2 | no | — |
| 133 | C22×C53 | — | 8 | 6 | no | — |
| 134 | C22×C53 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 135 | C2×D53 | — | 4 | 3 | no | — |
| 136 | C2×D53 | — | 4 | 3 | no | — |
| 137 | C2×D53 | #396 | 1 | 3 | yes | — |
| 138 | C2×D53 | #396 | 1 | 1 | yes | characteristic |
| 139 | C4×C53 | C2×#121 | 1 | 3 | yes | — |
| 140 | C4×C53 | C2×#126 | 1 | 1 | yes | characteristic |
| 141 | Dic53 | — | 4 | 2 | no | — |
| 142 | Dic53 | #396 | 1 | 1 | yes | characteristic |
| 143 | Dic53 | #396 | 1 | 3 | yes | — |
| 144 | C2×#430 | — | 53 | 3 | no | — |
| 145 | C2×#413 | — | 53 | 3 | no | — |
| 146 | #2070 | — | 53 | 3 | no | — |
| 147 | #2070 | — | 53 | 1 | no | — |
| 148 | #2069 | — | 53 | 3 | no | — |
| 149 | #2069 | D53 | 1 | 1 | yes | characteristic, complemented |
| 150 | C2×#338 | — | 53 | 1 | no | — |
| 151 | C2×C53 | — | 2 | 3 | no | — |
| 152 | C2×C53 | — | 8 | 2 | no | — |
| 153 | C2×C53 | C2×#396 | 1 | 1 | yes | characteristic |
| 154 | D53 | — | 2 | 3 | no | — |
| 155 | D53 | — | 2 | 1 | no | — |
| 156 | C22×#41 | — | 53 | 1 | no | — |
| 157 | C22×M16 | — | 53 | 3 | no | — |
| 158 | C22×M16 | — | 53 | 3 | no | — |
| 159 | #430 | — | 53 | 3 | no | — |
| 160 | #430 | — | 53 | 3 | no | — |
| 161 | #430 | — | 53 | 3 | no | — |
| 162 | #430 | — | 106 | 6 | no | — |
| 163 | #430 | C2×D53 | 1 | 3 | yes | — |
| 164 | C2×#108 | — | 106 | 6 | no | — |
| 165 | #413 | — | 106 | 6 | no | — |
| 166 | #413 | — | 106 | 6 | no | — |
| 167 | C2×#101 | — | 53 | 3 | no | — |
| 168 | C2×#101 | — | 53 | 3 | no | — |
| 169 | C2×#100 | — | 53 | 3 | no | — |
| 170 | C2×#100 | C2×D53 | 1 | 3 | yes | — |
| 171 | #338 | — | 53 | 3 | no | — |
| 172 | #338 | C2×D53 | 1 | 1 | yes | characteristic |
| 173 | C53 | #23622 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 174 | C2×#41 | — | 106 | 3 | no | — |
| 175 | C2×#41 | — | 106 | 3 | no | — |
| 176 | C2×#41 | — | 106 | 3 | no | — |
| 177 | C2×#41 | — | 106 | 3 | no | — |
| 178 | C22×Q8 | — | 53 | 1 | no | — |
| 179 | C22×D4 | — | 53 | 2 | no | — |
| 180 | C22×D4 | C22×D53 | 1 | 1 | yes | characteristic |
| 181 | C23×C4 | — | 53 | 1 | no | — |
| 182 | C23×C4 | — | 53 | 2 | no | — |
| 183 | C2×M16 | — | 53 | 3 | no | — |
| 184 | C2×M16 | — | 53 | 6 | no | — |
| 185 | C2×M16 | — | 53 | 6 | no | — |
| 186 | C2×M16 | — | 53 | 3 | no | — |
| 187 | C2×M16 | — | 106 | 3 | no | — |
| 188 | C2×M16 | — | 106 | 3 | no | — |
| 189 | C2×M16 | — | 106 | 3 | no | — |
| 190 | C2×M16 | — | 106 | 3 | no | — |
| 191 | C2×M16 | — | 106 | 3 | no | — |
| 192 | C2×M16 | — | 106 | 3 | no | — |
| 193 | C2×M16 | — | 106 | 3 | no | — |
| 194 | C2×M16 | — | 106 | 3 | no | — |
| 195 | C2×M16 | C22×D53 | 1 | 3 | yes | — |
| 196 | C2×M16 | C22×D53 | 1 | 3 | yes | — |
| 197 | C22×C8 | — | 106 | 3 | no | — |
| 198 | C22×C8 | — | 106 | 3 | no | — |
| 199 | #108 | — | 2 | 6 | no | — |
| 200 | #108 | — | 106 | 6 | no | — |
| 201 | #108 | — | 106 | 6 | no | — |
| 202 | #108 | — | 106 | 6 | no | — |
| 203 | #101 | — | 106 | 6 | no | — |
| 204 | #101 | — | 106 | 6 | no | — |
| 205 | #100 | — | 2 | 6 | no | — |
| 206 | #100 | — | 106 | 6 | no | — |
| 207 | C24 | — | 53 | 1 | no | — |
| 208 | C24 | Q8×D53 | 1 | 2 | yes | — |
| 209 | #41 | — | 212 | 3 | no | — |
| 210 | #41 | — | 212 | 3 | no | — |
| 211 | #41 | — | 212 | 3 | no | — |
| 212 | #41 | — | 212 | 3 | no | — |
| 213 | #41 | — | 212 | 1 | no | — |
| 214 | #41 | — | 212 | 3 | no | — |
| 215 | C2×Q8 | — | 106 | 3 | no | — |
| 216 | C2×Q8 | — | 106 | 3 | no | — |
| 217 | C2×D4 | — | 2 | 3 | no | — |
| 218 | C2×D4 | — | 2 | 3 | no | — |
| 219 | C2×D4 | — | 106 | 6 | no | — |
| 220 | C2×D4 | — | 106 | 6 | no | — |
| 221 | C22×C4 | — | 53 | 1 | no | — |
| 222 | C22×C4 | — | 53 | 3 | no | — |
| 223 | C22×C4 | — | 53 | 3 | no | — |
| 224 | C22×C4 | — | 53 | 3 | no | — |
| 225 | C22×C4 | — | 53 | 3 | no | — |
| 226 | C22×C4 | — | 53 | 2 | no | — |
| 227 | C22×C4 | — | 106 | 6 | no | — |
| 228 | C22×C4 | — | 106 | 6 | no | — |
| 229 | C22×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 230 | M16 | — | 2 | 3 | no | — |
| 231 | M16 | — | 2 | 3 | no | — |
| 232 | M16 | — | 2 | 3 | no | — |
| 233 | M16 | — | 53 | 6 | no | — |
| 234 | M16 | — | 106 | 6 | no | — |
| 235 | M16 | — | 106 | 6 | no | — |
| 236 | M16 | — | 106 | 3 | no | — |
| 237 | M16 | — | 106 | 3 | no | — |
| 238 | M16 | — | 106 | 3 | no | — |
| 239 | M16 | — | 212 | 3 | no | — |
| 240 | M16 | — | 212 | 3 | no | — |
| 241 | M16 | D4×D53 | 1 | 6 | yes | — |
| 242 | C2×C8 | — | 2 | 3 | no | — |
| 243 | C2×C8 | — | 2 | 3 | no | — |
| 244 | C2×C8 | — | 106 | 6 | no | — |
| 245 | C2×C8 | — | 106 | 6 | no | — |
| 246 | C2×C8 | — | 106 | 3 | no | — |
| 247 | C2×C8 | — | 106 | 3 | no | — |
| 248 | C2×C8 | — | 212 | 3 | no | — |
| 249 | C2×C8 | — | 212 | 3 | no | — |
| 250 | C23 | — | 2 | 6 | no | — |
| 251 | C23 | — | 8 | 2 | no | — |
| 252 | C23 | — | 53 | 3 | no | — |
| 253 | C23 | — | 53 | 3 | no | — |
| 254 | C23 | — | 424 | 1 | no | — |
| 255 | C23 | C2×Q8×D53 | 1 | 1 | yes | characteristic |
| 256 | Q8 | — | 212 | 3 | no | — |
| 257 | Q8 | — | 212 | 1 | no | — |
| 258 | D4 | — | 4 | 3 | no | — |
| 259 | D4 | — | 4 | 1 | no | — |
| 260 | D4 | — | 212 | 6 | no | — |
| 261 | D4 | — | 212 | 2 | no | — |
| 262 | C2×C4 | — | 53 | 3 | no | — |
| 263 | C2×C4 | — | 53 | 3 | no | — |
| 264 | C2×C4 | — | 53 | 3 | no | — |
| 265 | C2×C4 | — | 53 | 3 | no | — |
| 266 | C2×C4 | — | 53 | 3 | no | — |
| 267 | C2×C4 | — | 53 | 3 | no | — |
| 268 | C2×C4 | — | 53 | 1 | no | — |
| 269 | C2×C4 | — | 53 | 3 | no | — |
| 270 | C2×C4 | — | 106 | 6 | no | — |
| 271 | C2×C4 | — | 212 | 2 | no | — |
| 272 | C2×C4 | — | 212 | 6 | no | — |
| 273 | C2×C4 | C2×D4×D53 | 1 | 3 | yes | — |
| 274 | C2×C4 | #41×D53 | 1 | 3 | yes | — |
| 275 | C8 | — | 2 | 6 | no | — |
| 276 | C8 | — | 4 | 3 | no | — |
| 277 | C8 | — | 106 | 6 | no | — |
| 278 | C8 | — | 212 | 3 | no | — |
| 279 | C22 | — | 4 | 1 | no | — |
| 280 | C22 | — | 4 | 2 | no | — |
| 281 | C22 | — | 8 | 6 | no | — |
| 282 | C22 | — | 53 | 3 | no | — |
| 283 | C22 | — | 53 | 1 | no | — |
| 284 | C22 | — | 212 | 3 | no | — |
| 285 | C22 | — | 212 | 3 | no | — |
| 286 | C22 | C22⋊Q8×D53 | 1 | 3 | yes | — |
| 287 | C4 | — | 53 | 1 | no | — |
| 288 | C4 | — | 53 | 3 | no | — |
| 289 | C4 | — | 212 | 2 | no | — |
| 290 | C4 | #121×D53 | 1 | 3 | yes | — |
| 291 | C4 | #126×D53 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 2 | 3 | no | — |
| 293 | C2 | — | 8 | 2 | no | — |
| 294 | C2 | — | 106 | 3 | no | — |
| 295 | C2 | — | 106 | 1 | no | — |
| 296 | C2 | #396×D53 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #430×D53 | C2 | semidirect |
| 3. #20276117 | C2 | semidirect |
| 4. #20276115 | C2 | semidirect |
| 5. #20276113 | C2 | semidirect |
| 6. #338×D53 | C2 | semidirect |
| 7. C53×#2069 | C2 | semidirect |
| 149. #2069 | D53 | semidirect |
| 173. C53 | #23622 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #430×D53 | 2 |
| C2×#16810604 | 2 |
| #20276117 | 2 |
| #20276115 | 2 |
| #20276113 | 2 |
| #338×D53 | 2 |
| C53×#2069 | 2 |
| #23622 | 53 |