← C53×#12827 #27220339
Subgroups — C53×#12827 #27220339
Order 13568.
376 conjugacy classes of subgroups.
The subgroup lattice has 264 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C53×#2950 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×#2153 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C53×#362 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C53×#2141 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C53×#1833 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C53×#1389 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C53×#1384 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C22×Q16×C53 | C22 | 1 | 1 | yes | characteristic |
| 9 | C53×#519 | — | 2 | 1 | no | — |
| 10 | C2×M32×C53 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×M32×C53 | C22 | 1 | 1 | yes | characteristic |
| 12 | C53×#441 | C4 | 1 | 4 | yes | complemented |
| 13 | C4×Q16×C53 | — | 2 | 2 | no | — |
| 14 | C2×#108×C53 | C22 | 1 | 1 | yes | characteristic |
| 15 | C53×#427 | C22 | 1 | 1 | yes | characteristic |
| 16 | C53×#427 | C22 | 1 | 1 | yes | characteristic |
| 17 | C53×#419 | — | 2 | 1 | no | — |
| 18 | C53×#405 | C22 | 1 | 1 | yes | characteristic |
| 19 | C53×#364 | — | 2 | 2 | no | — |
| 20 | C53×#362 | — | 2 | 2 | no | — |
| 21 | C53×#358 | — | 2 | 2 | no | — |
| 22 | C53×#344 | — | 2 | 1 | no | — |
| 23 | C53×H(3,ℤ/4) | — | 2 | 1 | no | — |
| 24 | C22×Q8×C53 | — | 2 | 1 | no | — |
| 25 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 26 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 27 | C2×Q16×C53 | — | 2 | 1 | no | — |
| 28 | C2×Q16×C53 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×Q16×C53 | D4 | 1 | 2 | yes | — |
| 30 | C2×Q16×C53 | Q8 | 1 | 2 | yes | — |
| 31 | C2×M16×C53 | — | 2 | 1 | no | — |
| 32 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 33 | C22×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 34 | C4×Q8×C53 | — | 2 | 4 | no | — |
| 35 | C4×Q8×C53 | — | 4 | 2 | no | — |
| 36 | #117×C53 | — | 2 | 1 | no | — |
| 37 | #117×C53 | — | 2 | 1 | no | — |
| 38 | #117×C53 | — | 4 | 2 | no | — |
| 39 | #117×C53 | D4 | 1 | 1 | yes | characteristic |
| 40 | M32×C53 | — | 2 | 1 | no | — |
| 41 | M32×C53 | — | 2 | 1 | no | — |
| 42 | M32×C53 | — | 2 | 1 | no | — |
| 43 | M32×C53 | — | 2 | 1 | no | — |
| 44 | C2×C16×C53 | — | 2 | 1 | no | — |
| 45 | C2×C16×C53 | — | 2 | 1 | no | — |
| 46 | #108×C53 | — | 2 | 1 | no | — |
| 47 | #108×C53 | — | 2 | 1 | no | — |
| 48 | #107×C53 | — | 2 | 1 | no | — |
| 49 | #107×C53 | — | 2 | 1 | no | — |
| 50 | #106×C53 | — | 2 | 1 | no | — |
| 51 | #106×C53 | C2×C4 | 1 | 2 | yes | — |
| 52 | #103×C53 | — | 2 | 4 | no | — |
| 53 | #97×C53 | C2×C4 | 1 | 2 | yes | — |
| 54 | C4×C8×C53 | D4 | 1 | 2 | yes | — |
| 55 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 56 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 57 | C2×Q8×C53 | — | 2 | 2 | no | — |
| 58 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 59 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 60 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 61 | C22×C4×C53 | — | 4 | 1 | no | — |
| 62 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | Q16×C53 | — | 2 | 2 | no | — |
| 64 | Q16×C53 | — | 2 | 2 | no | — |
| 65 | Q16×C53 | — | 2 | 2 | no | — |
| 66 | Q16×C53 | C4⋊C4 | 1 | 4 | yes | — |
| 67 | M16×C53 | — | 2 | 1 | no | — |
| 68 | M16×C53 | — | 2 | 1 | no | — |
| 69 | M16×C53 | — | 4 | 1 | no | — |
| 70 | M16×C53 | — | 4 | 1 | no | — |
| 71 | C2×C8×C53 | — | 4 | 1 | no | — |
| 72 | C2×C8×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 73 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×C8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C2×C8×C53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 78 | C2×C8×C53 | #41 | 1 | 1 | yes | characteristic |
| 79 | C4⋊C4×C53 | — | 2 | 1 | no | — |
| 80 | C4⋊C4×C53 | — | 2 | 2 | no | — |
| 81 | C4⋊C4×C53 | — | 4 | 4 | no | — |
| 82 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 83 | C22⋊C4×C53 | — | 2 | 1 | no | — |
| 84 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 85 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 86 | C42×C53 | — | 4 | 4 | no | — |
| 87 | C42×C53 | — | 4 | 1 | no | — |
| 88 | C42×C53 | — | 4 | 1 | no | — |
| 89 | C42×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 90 | C16×C53 | — | 4 | 1 | no | — |
| 91 | C16×C53 | — | 4 | 1 | no | — |
| 92 | C23×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 93 | Q8×C53 | — | 2 | 4 | no | — |
| 94 | Q8×C53 | — | 4 | 2 | no | — |
| 95 | Q8×C53 | — | 4 | 2 | no | — |
| 96 | Q8×C53 | — | 4 | 2 | no | — |
| 97 | C2×C4×C53 | — | 2 | 2 | no | — |
| 98 | C2×C4×C53 | — | 4 | 2 | no | — |
| 99 | C2×C4×C53 | — | 4 | 2 | no | — |
| 100 | C2×C4×C53 | — | 4 | 2 | no | — |
| 101 | C2×C4×C53 | — | 4 | 2 | no | — |
| 102 | C2×C4×C53 | — | 8 | 1 | no | — |
| 103 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C53 | #135 | 1 | 1 | yes | characteristic |
| 108 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 109 | C8×C53 | — | 2 | 2 | no | — |
| 110 | C8×C53 | — | 8 | 1 | no | — |
| 111 | C8×C53 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 112 | C8×C53 | #117 | 1 | 1 | yes | characteristic |
| 113 | C8×C53 | #121 | 1 | 1 | yes | characteristic |
| 114 | C8×C53 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 115 | #12827 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 116 | C22×C53 | — | 4 | 1 | no | — |
| 117 | C22×C53 | #418 | 1 | 1 | yes | characteristic |
| 118 | C22×C53 | #463 | 1 | 1 | yes | characteristic |
| 119 | C22×C53 | #478 | 1 | 1 | yes | characteristic |
| 120 | C4×C53 | — | 4 | 2 | no | — |
| 121 | C4×C53 | — | 4 | 4 | no | — |
| 122 | C4×C53 | — | 8 | 2 | no | — |
| 123 | C4×C53 | — | 16 | 1 | no | — |
| 124 | C4×C53 | #380 | 1 | 1 | yes | characteristic |
| 125 | C4×C53 | #416 | 1 | 1 | yes | characteristic |
| 126 | C4×C53 | #463 | 1 | 1 | yes | characteristic |
| 127 | C4×C53 | #478 | 1 | 1 | yes | characteristic |
| 128 | #2950 | C2×C53 | 1 | 1 | yes | characteristic |
| 129 | #2153 | C2×C53 | 1 | 1 | yes | characteristic |
| 130 | C2×#362 | C2×C53 | 1 | 1 | yes | characteristic |
| 131 | #2141 | C2×C53 | 1 | 1 | yes | characteristic |
| 132 | #1833 | C2×C53 | 1 | 1 | yes | characteristic |
| 133 | #1389 | C2×C53 | 1 | 1 | yes | characteristic |
| 134 | #1384 | C2×C53 | 1 | 1 | yes | characteristic |
| 135 | C2×C53 | — | 2 | 1 | no | — |
| 136 | C2×C53 | — | 2 | 1 | no | — |
| 137 | C2×C53 | — | 2 | 1 | no | — |
| 138 | C2×C53 | #1804 | 1 | 1 | yes | characteristic |
| 139 | C22×Q16 | C22×C53 | 1 | 1 | yes | characteristic |
| 140 | #519 | — | 2 | 1 | no | — |
| 141 | C2×M32 | C22×C53 | 1 | 1 | yes | characteristic |
| 142 | C2×M32 | C22×C53 | 1 | 1 | yes | characteristic |
| 143 | #441 | C4×C53 | 1 | 4 | yes | complemented |
| 144 | C4×Q16 | — | 2 | 2 | no | — |
| 145 | C2×#108 | C22×C53 | 1 | 1 | yes | characteristic |
| 146 | #427 | C22×C53 | 1 | 1 | yes | characteristic |
| 147 | #427 | C22×C53 | 1 | 1 | yes | characteristic |
| 148 | #419 | — | 2 | 1 | no | — |
| 149 | #405 | C22×C53 | 1 | 1 | yes | characteristic |
| 150 | #364 | — | 2 | 2 | no | — |
| 151 | #362 | — | 2 | 2 | no | — |
| 152 | #358 | — | 2 | 2 | no | — |
| 153 | #344 | — | 2 | 1 | no | — |
| 154 | H(3,ℤ/4) | — | 2 | 1 | no | — |
| 155 | C53 | #12827 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 156 | C22×Q8 | — | 2 | 1 | no | — |
| 157 | C2×Q16 | — | 2 | 1 | no | — |
| 158 | C2×Q16 | — | 2 | 1 | no | — |
| 159 | C2×Q16 | — | 2 | 1 | no | — |
| 160 | C2×Q16 | C2×C4×C53 | 1 | 2 | yes | — |
| 161 | C2×Q16 | D4×C53 | 1 | 2 | yes | — |
| 162 | C2×Q16 | Q8×C53 | 1 | 2 | yes | — |
| 163 | C2×M16 | — | 2 | 1 | no | — |
| 164 | C2×M16 | D4×C53 | 1 | 1 | yes | characteristic |
| 165 | C22×C8 | C23×C53 | 1 | 1 | yes | characteristic |
| 166 | C4×Q8 | — | 2 | 4 | no | — |
| 167 | C4×Q8 | — | 4 | 2 | no | — |
| 168 | #117 | — | 2 | 1 | no | — |
| 169 | #117 | — | 2 | 1 | no | — |
| 170 | #117 | — | 4 | 2 | no | — |
| 171 | #117 | D4×C53 | 1 | 1 | yes | characteristic |
| 172 | M32 | — | 2 | 1 | no | — |
| 173 | M32 | — | 2 | 1 | no | — |
| 174 | M32 | — | 2 | 1 | no | — |
| 175 | M32 | — | 2 | 1 | no | — |
| 176 | C2×C16 | — | 2 | 1 | no | — |
| 177 | C2×C16 | — | 2 | 1 | no | — |
| 178 | #108 | — | 2 | 1 | no | — |
| 179 | #108 | — | 2 | 1 | no | — |
| 180 | #107 | — | 2 | 1 | no | — |
| 181 | #107 | — | 2 | 1 | no | — |
| 182 | #106 | — | 2 | 1 | no | — |
| 183 | #106 | C2×C4×C53 | 1 | 2 | yes | — |
| 184 | #103 | — | 2 | 4 | no | — |
| 185 | #97 | C2×C4×C53 | 1 | 2 | yes | — |
| 186 | C4×C8 | D4×C53 | 1 | 2 | yes | — |
| 187 | C2×Q8 | — | 2 | 2 | no | — |
| 188 | C2×Q8 | — | 2 | 2 | no | — |
| 189 | C2×Q8 | — | 2 | 2 | no | — |
| 190 | C2×Q8 | — | 4 | 1 | no | — |
| 191 | C2×Q8 | — | 4 | 1 | no | — |
| 192 | C2×Q8 | — | 4 | 1 | no | — |
| 193 | C22×C4 | — | 4 | 1 | no | — |
| 194 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 195 | Q16 | — | 2 | 2 | no | — |
| 196 | Q16 | — | 2 | 2 | no | — |
| 197 | Q16 | — | 2 | 2 | no | — |
| 198 | Q16 | C4⋊C4×C53 | 1 | 4 | yes | — |
| 199 | M16 | — | 2 | 1 | no | — |
| 200 | M16 | — | 2 | 1 | no | — |
| 201 | M16 | — | 4 | 1 | no | — |
| 202 | M16 | — | 4 | 1 | no | — |
| 203 | C2×C8 | — | 4 | 1 | no | — |
| 204 | C2×C8 | C22⋊C4×C53 | 1 | 2 | yes | — |
| 205 | C2×C8 | C22×C4×C53 | 1 | 1 | yes | characteristic |
| 206 | C2×C8 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 207 | C2×C8 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 208 | C2×C8 | C2×Q8×C53 | 1 | 1 | yes | characteristic |
| 209 | C2×C8 | #41×C53 | 1 | 1 | yes | characteristic |
| 210 | C2×C8 | #41×C53 | 1 | 1 | yes | characteristic |
| 211 | C4⋊C4 | — | 2 | 1 | no | — |
| 212 | C4⋊C4 | — | 2 | 2 | no | — |
| 213 | C4⋊C4 | — | 4 | 4 | no | — |
| 214 | C4⋊C4 | — | 4 | 1 | no | — |
| 215 | C22⋊C4 | — | 2 | 1 | no | — |
| 216 | C22⋊C4 | — | 4 | 1 | no | — |
| 217 | C22⋊C4 | — | 4 | 1 | no | — |
| 218 | C42 | — | 4 | 4 | no | — |
| 219 | C42 | — | 4 | 1 | no | — |
| 220 | C42 | — | 4 | 1 | no | — |
| 221 | C42 | C22⋊C4×C53 | 1 | 2 | yes | — |
| 222 | C16 | — | 4 | 1 | no | — |
| 223 | C16 | — | 4 | 1 | no | — |
| 224 | C23 | AGL(1,ℤ/8)×C53 | 1 | 1 | yes | characteristic |
| 225 | Q8 | — | 2 | 4 | no | — |
| 226 | Q8 | — | 4 | 2 | no | — |
| 227 | Q8 | — | 4 | 2 | no | — |
| 228 | Q8 | — | 4 | 2 | no | — |
| 229 | C2×C4 | — | 2 | 2 | no | — |
| 230 | C2×C4 | — | 4 | 2 | no | — |
| 231 | C2×C4 | — | 4 | 2 | no | — |
| 232 | C2×C4 | — | 4 | 2 | no | — |
| 233 | C2×C4 | — | 4 | 2 | no | — |
| 234 | C2×C4 | — | 8 | 1 | no | — |
| 235 | C2×C4 | C2×C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 236 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | C22⋊Q8×C53 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | #135×C53 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | #135×C53 | 1 | 1 | yes | characteristic |
| 240 | C2×C4 | AGL(1,ℤ/8)×C53 | 1 | 1 | yes | characteristic |
| 241 | C8 | — | 2 | 2 | no | — |
| 242 | C8 | — | 8 | 1 | no | — |
| 243 | C8 | C2×C4⋊C4×C53 | 1 | 1 | yes | characteristic |
| 244 | C8 | #117×C53 | 1 | 1 | yes | characteristic |
| 245 | C8 | #121×C53 | 1 | 1 | yes | characteristic |
| 246 | C8 | C22⋊Q8×C53 | 1 | 1 | yes | characteristic |
| 247 | C22 | — | 4 | 1 | no | — |
| 248 | C22 | C53×#418 | 1 | 1 | yes | characteristic |
| 249 | C22 | C53×#463 | 1 | 1 | yes | characteristic |
| 250 | C22 | C53×#478 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 4 | 2 | no | — |
| 252 | C4 | — | 4 | 4 | no | — |
| 253 | C4 | — | 8 | 2 | no | — |
| 254 | C4 | — | 16 | 1 | no | — |
| 255 | C4 | C53×#380 | 1 | 1 | yes | characteristic |
| 256 | C4 | C53×#416 | 1 | 1 | yes | characteristic |
| 257 | C4 | C53×#463 | 1 | 1 | yes | characteristic |
| 258 | C4 | C53×#478 | 1 | 1 | yes | characteristic |
| 259 | C2 | — | 2 | 1 | no | — |
| 260 | C2 | — | 2 | 1 | no | — |
| 261 | C2 | — | 2 | 1 | no | — |
| 262 | C2 | C53×#1804 | 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