Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1284 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C2×#16810795 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #20276732 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 3 | C2×#16810420 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×C53×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#13133265 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#13133243 | — | 2 | 3 | no | — |
| 7 | #16810795 | — | 2 | 6 | no | — |
| 8 | #16810795 | C22 | 1 | 12 | yes | complemented |
| 9 | C2×#13133145 | — | 2 | 3 | no | — |
| 10 | C2×#13133136 | C22 | 1 | 3 | yes | complemented |
| 11 | #16810420 | C22 | 1 | 12 | yes | complemented |
| 12 | C53×H(3,𝔽4) | — | 2 | 6 | no | — |
| 13 | C53×H(3,𝔽4) | C22 | 1 | 4 | yes | complemented |
| 14 | C2×#124×C53 | — | 2 | 3 | no | — |
| 15 | C2×#124×C53 | C22 | 1 | 3 | yes | — |
| 16 | C2×#120×C53 | — | 2 | 3 | no | — |
| 17 | #13133265 | — | 2 | 3 | no | — |
| 18 | #13133265 | — | 4 | 2 | no | — |
| 19 | #13133265 | C23 | 1 | 2 | yes | complemented |
| 20 | C2×#11719949 | — | 4 | 1 | no | — |
| 21 | C2×#11719945 | — | 2 | 3 | no | — |
| 22 | C2×Q8×D53 | — | 4 | 1 | no | — |
| 23 | #13133243 | — | 2 | 12 | no | — |
| 24 | #13133243 | — | 4 | 6 | no | — |
| 25 | C2×#11719923 | — | 4 | 3 | no | — |
| 26 | C2×#11719922 | — | 4 | 3 | no | — |
| 27 | C2×#11719917 | — | 2 | 3 | no | — |
| 28 | #13133145 | — | 2 | 12 | no | — |
| 29 | #13133136 | — | 2 | 3 | no | — |
| 30 | #13133136 | C23 | 1 | 6 | yes | complemented |
| 31 | C25×C53 | — | 2 | 1 | no | — |
| 32 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 33 | C22×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 34 | C22×D4×C53 | — | 2 | 3 | no | — |
| 35 | C22×D4×C53 | D4 | 1 | 3 | yes | — |
| 36 | #124×C53 | — | 2 | 12 | no | — |
| 37 | #124×C53 | — | 2 | 12 | no | — |
| 38 | #124×C53 | — | 2 | 6 | no | — |
| 39 | #124×C53 | D4 | 1 | 6 | yes | — |
| 40 | #124×C53 | C23 | 1 | 6 | yes | — |
| 41 | #120×C53 | — | 2 | 12 | no | — |
| 42 | #120×C53 | — | 2 | 12 | no | — |
| 43 | C2×C22⋊C4×C53 | — | 2 | 3 | no | — |
| 44 | C2×C22⋊C4×C53 | — | 2 | 6 | no | — |
| 45 | C2×C42×C53 | D4 | 1 | 3 | yes | — |
| 46 | C24×C53 | — | 2 | 4 | no | — |
| 47 | C24×C53 | — | 2 | 3 | no | — |
| 48 | C24×C53 | — | 8 | 4 | no | — |
| 49 | C24×C53 | — | 8 | 1 | no | — |
| 50 | C24×C53 | — | 8 | 1 | no | — |
| 51 | C2×Q8×C53 | — | 2 | 4 | no | — |
| 52 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 53 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 54 | C2×Q8×C53 | — | 8 | 2 | no | — |
| 55 | C2×Q8×C53 | C2×D4 | 1 | 3 | yes | — |
| 56 | C2×Q8×C53 | C24 | 1 | 1 | yes | characteristic |
| 57 | C2×D4×C53 | — | 2 | 12 | no | — |
| 58 | C2×D4×C53 | — | 2 | 3 | no | — |
| 59 | C2×D4×C53 | — | 4 | 3 | no | — |
| 60 | C2×D4×C53 | — | 4 | 3 | no | — |
| 61 | C2×D4×C53 | — | 8 | 6 | no | — |
| 62 | C2×D4×C53 | C2×D4 | 1 | 6 | yes | — |
| 63 | C22×C4×C53 | — | 2 | 3 | no | — |
| 64 | C22×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 65 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 66 | #11719949 | — | 4 | 2 | no | — |
| 67 | #11719949 | — | 4 | 6 | no | — |
| 68 | Q8×D53 | — | 4 | 2 | no | — |
| 69 | Q8×D53 | — | 4 | 6 | no | — |
| 70 | #11719945 | — | 2 | 6 | no | — |
| 71 | #11719945 | — | 2 | 6 | no | — |
| 72 | #11719945 | — | 4 | 3 | no | — |
| 73 | #11719945 | — | 4 | 3 | no | — |
| 74 | C2×D212 | — | 2 | 3 | no | — |
| 75 | C2×C4×D53 | — | 4 | 3 | no | — |
| 76 | C2×Dic106 | — | 2 | 3 | no | — |
| 77 | C22⋊C4×C53 | — | 2 | 12 | no | — |
| 78 | C22⋊C4×C53 | — | 2 | 12 | no | — |
| 79 | C22⋊C4×C53 | — | 2 | 12 | no | — |
| 80 | C42×C53 | — | 2 | 3 | no | — |
| 81 | C42×C53 | C2×D4 | 1 | 6 | yes | — |
| 82 | #11719923 | — | 4 | 12 | no | — |
| 83 | #11719922 | — | 4 | 12 | no | — |
| 84 | #11719917 | — | 2 | 6 | no | — |
| 85 | #11719917 | — | 4 | 3 | no | — |
| 86 | C2×#88322 | — | 4 | 3 | no | — |
| 87 | C23×C53 | — | 2 | 6 | no | — |
| 88 | C23×C53 | — | 4 | 12 | no | — |
| 89 | C23×C53 | — | 4 | 3 | no | — |
| 90 | C23×C53 | — | 4 | 3 | no | — |
| 91 | C23×C53 | — | 8 | 8 | no | — |
| 92 | C23×C53 | — | 8 | 4 | no | — |
| 93 | C23×C53 | — | 8 | 8 | no | — |
| 94 | C23×C53 | — | 8 | 8 | no | — |
| 95 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 96 | C22×D53 | — | 4 | 1 | no | — |
| 97 | Q8×C53 | — | 4 | 3 | no | — |
| 98 | Q8×C53 | — | 4 | 1 | no | — |
| 99 | Q8×C53 | — | 8 | 4 | no | — |
| 100 | D4×C53 | — | 4 | 6 | no | — |
| 101 | D4×C53 | — | 8 | 3 | no | — |
| 102 | D4×C53 | — | 8 | 12 | no | — |
| 103 | C2×C4×C53 | — | 2 | 3 | no | — |
| 104 | C2×C4×C53 | — | 2 | 3 | no | — |
| 105 | C2×C4×C53 | — | 2 | 6 | no | — |
| 106 | C2×C4×C53 | — | 4 | 6 | no | — |
| 107 | C2×C4×C53 | #120 | 1 | 3 | yes | — |
| 108 | C2×C4×C53 | C22×D4 | 1 | 3 | yes | — |
| 109 | C22⋊D53 | — | 4 | 2 | no | — |
| 110 | C22⋊D53 | — | 4 | 2 | no | — |
| 111 | C2×Dic53 | — | 4 | 1 | no | — |
| 112 | D212 | — | 2 | 6 | no | — |
| 113 | D212 | — | 4 | 3 | no | — |
| 114 | C4×D53 | — | 4 | 6 | no | — |
| 115 | C4×D53 | — | 4 | 6 | no | — |
| 116 | Dic106 | — | 2 | 6 | no | — |
| 117 | Dic106 | — | 4 | 3 | no | — |
| 118 | #88322 | — | 4 | 6 | no | — |
| 119 | C2×#2202 | — | 53 | 1 | no | maximal, Sylow |
| 120 | C22×C53 | — | 2 | 2 | no | — |
| 121 | C22×C53 | — | 4 | 12 | no | — |
| 122 | C22×C53 | — | 4 | 6 | no | — |
| 123 | C22×C53 | — | 8 | 6 | no | — |
| 124 | C22×C53 | — | 8 | 16 | no | — |
| 125 | C22×C53 | — | 8 | 6 | no | — |
| 126 | C22×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 127 | C22×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 128 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 129 | C2×D53 | — | 4 | 2 | no | — |
| 130 | C2×D53 | — | 8 | 2 | no | — |
| 131 | C4×C53 | — | 2 | 3 | no | — |
| 132 | C4×C53 | — | 2 | 3 | no | — |
| 133 | C4×C53 | — | 4 | 6 | no | — |
| 134 | Dic53 | — | 4 | 2 | no | — |
| 135 | C2×H(3,𝔽4) | D53 | 1 | 1 | yes | characteristic, complemented |
| 136 | C2×#454 | — | 53 | 3 | no | — |
| 137 | #2202 | — | 53 | 8 | no | — |
| 138 | C2×#355 | — | 53 | 3 | no | — |
| 139 | C2×C53 | — | 2 | 1 | no | — |
| 140 | C2×C53 | — | 2 | 1 | no | — |
| 141 | C2×C53 | — | 8 | 6 | no | — |
| 142 | C2×C53 | #2202 | 1 | 2 | yes | complemented |
| 143 | C2×C53 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 144 | D53 | — | 8 | 2 | no | — |
| 145 | C2×2−1+4 | — | 53 | 1 | no | — |
| 146 | C2×AGL(1,ℤ/8) | — | 106 | 3 | no | — |
| 147 | H(3,𝔽4) | — | 2 | 6 | no | — |
| 148 | H(3,𝔽4) | C2×D53 | 1 | 4 | yes | complemented |
| 149 | C2×#124 | — | 2 | 3 | no | — |
| 150 | C2×#124 | C2×D53 | 1 | 3 | yes | — |
| 151 | C2×#120 | — | 2 | 3 | no | — |
| 152 | #454 | — | 53 | 12 | no | — |
| 153 | #454 | — | 106 | 6 | no | — |
| 154 | C2×#104 | — | 106 | 3 | no | — |
| 155 | C2×#101 | — | 53 | 3 | no | — |
| 156 | #355 | — | 53 | 12 | no | — |
| 157 | C53 | C2×#2202 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 158 | C25 | — | 2 | 1 | no | — |
| 159 | 2−1+4 | — | 53 | 2 | no | — |
| 160 | 2−1+4 | — | 106 | 3 | no | — |
| 161 | 2−1+4 | — | 212 | 2 | no | — |
| 162 | C2×#41 | — | 106 | 3 | no | — |
| 163 | C2×#41 | — | 212 | 1 | no | — |
| 164 | C22×Q8 | — | 2 | 1 | no | — |
| 165 | C22×Q8 | — | 212 | 1 | no | — |
| 166 | C22×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 167 | C22×D4 | — | 2 | 3 | no | — |
| 168 | C22×D4 | C22⋊D53 | 1 | 3 | yes | — |
| 169 | AGL(1,ℤ/8) | — | 106 | 12 | no | — |
| 170 | AGL(1,ℤ/8) | — | 212 | 6 | no | — |
| 171 | C2×SD16 | — | 212 | 3 | no | — |
| 172 | C2×D8 | — | 212 | 3 | no | — |
| 173 | C2×M16 | — | 106 | 3 | no | — |
| 174 | #124 | — | 2 | 12 | no | — |
| 175 | #124 | — | 2 | 6 | no | — |
| 176 | #124 | — | 2 | 12 | no | — |
| 177 | #124 | C22⋊D53 | 1 | 6 | yes | — |
| 178 | #124 | C22×D53 | 1 | 6 | yes | — |
| 179 | #120 | — | 2 | 12 | no | — |
| 180 | #120 | — | 2 | 12 | no | — |
| 181 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 182 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 183 | C2×C42 | C22⋊D53 | 1 | 3 | yes | — |
| 184 | #104 | — | 106 | 12 | no | — |
| 185 | #101 | — | 53 | 6 | no | — |
| 186 | #101 | — | 106 | 3 | no | — |
| 187 | C24 | — | 2 | 4 | no | — |
| 188 | C24 | — | 2 | 3 | no | — |
| 189 | C24 | — | 8 | 4 | no | — |
| 190 | C24 | — | 8 | 1 | no | — |
| 191 | C24 | — | 8 | 1 | no | — |
| 192 | #41 | — | 106 | 6 | no | — |
| 193 | #41 | — | 106 | 6 | no | — |
| 194 | #41 | — | 212 | 3 | no | — |
| 195 | #41 | — | 212 | 2 | no | — |
| 196 | #41 | — | 212 | 6 | no | — |
| 197 | #41 | — | 212 | 3 | no | — |
| 198 | C2×Q8 | — | 2 | 4 | no | — |
| 199 | C2×Q8 | — | 4 | 1 | no | — |
| 200 | C2×Q8 | — | 4 | 1 | no | — |
| 201 | C2×Q8 | — | 8 | 2 | no | — |
| 202 | C2×Q8 | — | 106 | 3 | no | — |
| 203 | C2×Q8 | — | 212 | 2 | no | — |
| 204 | C2×Q8 | — | 212 | 6 | no | — |
| 205 | C2×Q8 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 206 | C2×Q8 | C23×D53 | 1 | 1 | yes | characteristic |
| 207 | C2×D4 | — | 2 | 12 | no | — |
| 208 | C2×D4 | — | 2 | 3 | no | — |
| 209 | C2×D4 | — | 4 | 3 | no | — |
| 210 | C2×D4 | — | 4 | 3 | no | — |
| 211 | C2×D4 | — | 8 | 6 | no | — |
| 212 | C2×D4 | — | 106 | 3 | no | — |
| 213 | C2×D4 | — | 212 | 1 | no | — |
| 214 | C2×D4 | C2×C22⋊D53 | 1 | 6 | yes | — |
| 215 | C22×C4 | — | 2 | 3 | no | — |
| 216 | C22×C4 | — | 212 | 3 | no | — |
| 217 | C22×C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 218 | SD16 | — | 212 | 12 | no | — |
| 219 | D8 | — | 212 | 12 | no | — |
| 220 | M16 | — | 106 | 6 | no | — |
| 221 | M16 | — | 212 | 3 | no | — |
| 222 | C2×C8 | — | 212 | 3 | no | — |
| 223 | C22⋊C4 | — | 2 | 12 | no | — |
| 224 | C22⋊C4 | — | 2 | 12 | no | — |
| 225 | C22⋊C4 | — | 2 | 12 | no | — |
| 226 | C42 | — | 2 | 3 | no | — |
| 227 | C42 | C2×C22⋊D53 | 1 | 6 | yes | — |
| 228 | C23 | — | 2 | 6 | no | — |
| 229 | C23 | — | 4 | 12 | no | — |
| 230 | C23 | — | 4 | 3 | no | — |
| 231 | C23 | — | 4 | 3 | no | — |
| 232 | C23 | — | 8 | 8 | no | — |
| 233 | C23 | — | 8 | 4 | no | — |
| 234 | C23 | — | 8 | 8 | no | — |
| 235 | C23 | — | 8 | 8 | no | — |
| 236 | C23 | — | 212 | 1 | no | — |
| 237 | C23 | #13133186 | 1 | 1 | yes | characteristic |
| 238 | Q8 | — | 4 | 3 | no | — |
| 239 | Q8 | — | 4 | 1 | no | — |
| 240 | Q8 | — | 8 | 4 | no | — |
| 241 | Q8 | — | 106 | 6 | no | — |
| 242 | Q8 | — | 212 | 3 | no | — |
| 243 | D4 | — | 4 | 6 | no | — |
| 244 | D4 | — | 8 | 3 | no | — |
| 245 | D4 | — | 8 | 12 | no | — |
| 246 | D4 | — | 106 | 6 | no | — |
| 247 | D4 | — | 212 | 3 | no | — |
| 248 | D4 | — | 212 | 2 | no | — |
| 249 | D4 | — | 212 | 2 | no | — |
| 250 | C2×C4 | — | 2 | 3 | no | — |
| 251 | C2×C4 | — | 2 | 3 | no | — |
| 252 | C2×C4 | — | 2 | 6 | no | — |
| 253 | C2×C4 | — | 4 | 6 | no | — |
| 254 | C2×C4 | — | 212 | 6 | no | — |
| 255 | C2×C4 | — | 212 | 6 | no | — |
| 256 | C2×C4 | — | 212 | 1 | no | — |
| 257 | C2×C4 | #13133186 | 1 | 3 | yes | — |
| 258 | C2×C4 | C22×C22⋊D53 | 1 | 3 | yes | — |
| 259 | C8 | — | 212 | 6 | no | — |
| 260 | C22 | — | 2 | 2 | no | — |
| 261 | C22 | — | 4 | 12 | no | — |
| 262 | C22 | — | 4 | 6 | no | — |
| 263 | C22 | — | 8 | 6 | no | — |
| 264 | C22 | — | 8 | 16 | no | — |
| 265 | C22 | — | 8 | 6 | no | — |
| 266 | C22 | — | 424 | 2 | no | — |
| 267 | C22 | — | 212 | 2 | no | — |
| 268 | C22 | #16810810 | 1 | 1 | yes | characteristic |
| 269 | C22 | #16810810 | 1 | 1 | yes | characteristic |
| 270 | C22 | C2×#13133186 | 1 | 1 | yes | characteristic |
| 271 | C4 | — | 2 | 3 | no | — |
| 272 | C4 | — | 2 | 3 | no | — |
| 273 | C4 | — | 4 | 6 | no | — |
| 274 | C4 | — | 212 | 2 | no | — |
| 275 | C2 | — | 2 | 1 | no | — |
| 276 | C2 | — | 2 | 1 | no | — |
| 277 | C2 | — | 8 | 6 | no | — |
| 278 | C2 | — | 424 | 2 | no | — |
| 279 | C2 | #20276732 | 1 | 2 | yes | direct product factor |
| 280 | C2 | C2×#16810810 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16810795 | C2 | semidirect |
| 2. #20276732 | C2 | direct (N × H) |
| 3. C2×#16810420 | C2 | semidirect |
| 4. C2×C53×H(3,𝔽4) | C2 | semidirect |
| 5. C2×#13133265 | C22 | semidirect |
| 8. #16810795 | C22 | semidirect |
| 10. C2×#13133136 | C22 | semidirect |
| 11. #16810420 | C22 | semidirect |
| 13. C53×H(3,𝔽4) | C22 | semidirect |
| 19. #13133265 | C23 | semidirect |
| 30. #13133136 | C23 | semidirect |
| 135. C2×H(3,𝔽4) | D53 | semidirect |
| 142. C2×C53 | #2202 | semidirect |
| 148. H(3,𝔽4) | C2×D53 | semidirect |
| 157. C53 | C2×#2202 | semidirect |
| 279. C2 | #20276732 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16810795 | 2 |
| #20276732 | 2 |
| C2×#16810420 | 2 |
| C2×C53×H(3,𝔽4) | 2 |
| C2×#2202 | 53 |