← #504×#6157 #28327906
Subgroups — #504×#6157 #28327906
Order 13568.
834 conjugacy classes of subgroups.
The subgroup lattice has 270 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | D53⋊#504 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #131×#6157 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20290927 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20290923 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | M32×#6157 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×C16×#6157 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #504×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | D53⋊#131 | C4 | 1 | 1 | yes | characteristic |
| 9 | #41×#6157 | C4 | 1 | 1 | yes | characteristic |
| 10 | #16811682 | C4 | 1 | 6 | yes | complemented |
| 11 | #16811681 | C4 | 1 | 2 | yes | complemented |
| 12 | D53⋊M32 | C22 | 1 | 3 | yes | — |
| 13 | C2×D53⋊C16 | C22 | 1 | 3 | yes | — |
| 14 | #16811589 | C4 | 1 | 3 | yes | — |
| 15 | #16811586 | C22 | 1 | 3 | yes | — |
| 16 | M16×#6157 | C4 | 1 | 3 | yes | — |
| 17 | C2×C8×#6157 | C22 | 1 | 3 | yes | — |
| 18 | #16811510 | C22 | 1 | 6 | yes | — |
| 19 | C16×#6157 | C22 | 1 | 1 | yes | characteristic |
| 20 | C16×#6157 | C22 | 1 | 3 | yes | — |
| 21 | C16×#6157 | C22 | 1 | 3 | yes | — |
| 22 | C16×#6157 | C22 | 1 | 3 | yes | — |
| 23 | #131×D53 | C22 | 1 | 1 | yes | characteristic |
| 24 | #16811041 | C4 | 1 | 3 | yes | complemented |
| 25 | #16811040 | C4 | 1 | 1 | yes | characteristic, complemented |
| 26 | #16811039 | C4 | 1 | 3 | yes | complemented |
| 27 | M32×D53 | C22 | 1 | 3 | yes | — |
| 28 | C2×C16×D53 | C22 | 1 | 3 | yes | — |
| 29 | C53×#504 | C4 | 1 | 1 | yes | characteristic, complemented |
| 30 | #13133305 | C8 | 1 | 2 | yes | — |
| 31 | #13133304 | C8 | 1 | 6 | yes | — |
| 32 | D53⋊M16 | C2×C4 | 1 | 3 | yes | — |
| 33 | C2×D53⋊C8 | C2×C4 | 1 | 3 | yes | — |
| 34 | Q8×#6157 | C8 | 1 | 2 | yes | — |
| 35 | D4×#6157 | C8 | 1 | 6 | yes | — |
| 36 | #13133296 | C2×C4 | 1 | 3 | yes | — |
| 37 | C2×C4×#6157 | C2×C4 | 1 | 3 | yes | — |
| 38 | #13133292 | C2×C4 | 1 | 6 | yes | complemented |
| 39 | #13133291 | C2×C4 | 1 | 6 | yes | complemented |
| 40 | D53⋊C16 | C2×C4 | 1 | 6 | yes | complemented |
| 41 | D53⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 42 | D53⋊C16 | C23 | 1 | 3 | yes | — |
| 43 | C2×#11719910 | C2×C4 | 1 | 6 | yes | complemented |
| 44 | #13133271 | C2×C4 | 1 | 6 | yes | — |
| 45 | C8×#6157 | C2×C4 | 1 | 3 | yes | — |
| 46 | C8×#6157 | C2×C4 | 1 | 3 | yes | — |
| 47 | C8×#6157 | C23 | 1 | 1 | yes | characteristic |
| 48 | C8×#6157 | C23 | 1 | 3 | yes | — |
| 49 | #41×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #13133225 | C2×C4 | 1 | 3 | yes | — |
| 51 | #13133224 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #13133223 | C2×C4 | 1 | 3 | yes | — |
| 53 | M16×D53 | C2×C4 | 1 | 3 | yes | — |
| 54 | C2×C8×D53 | C23 | 1 | 3 | yes | — |
| 55 | #13133157 | C2×C4 | 1 | 3 | yes | complemented |
| 56 | #13133156 | C2×C4 | 1 | 6 | yes | complemented |
| 57 | C2×#11719908 | C2×C4 | 1 | 3 | yes | complemented |
| 58 | C16×D53 | C2×C4 | 1 | 3 | yes | complemented |
| 59 | C16×D53 | C2×C4 | 1 | 3 | yes | complemented |
| 60 | C16×D53 | C23 | 1 | 1 | yes | characteristic |
| 61 | C16×D53 | C23 | 1 | 3 | yes | — |
| 62 | #131×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | M32×C53 | C2×C4 | 1 | 3 | yes | complemented |
| 64 | C2×C16×C53 | C2×C4 | 1 | 3 | yes | complemented |
| 65 | C22×#6157 | C2×C8 | 1 | 6 | yes | — |
| 66 | #41×C53 | C42 | 1 | 1 | yes | characteristic |
| 67 | #11719949 | C2×C8 | 1 | 1 | yes | characteristic |
| 68 | Q8×D53 | C2×C8 | 1 | 1 | yes | characteristic |
| 69 | #11719947 | C2×C8 | 1 | 3 | yes | — |
| 70 | D4×D53 | C2×C8 | 1 | 3 | yes | — |
| 71 | #11719945 | C42 | 1 | 3 | yes | — |
| 72 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 73 | C22⋊#6157 | C2×C8 | 1 | 6 | yes | — |
| 74 | #11719940 | C2×C8 | 1 | 6 | yes | — |
| 75 | C2×#88324 | C2×C8 | 1 | 6 | yes | — |
| 76 | C4⋊#6157 | C2×C8 | 1 | 6 | yes | — |
| 77 | C4×#6157 | C2×C8 | 1 | 6 | yes | — |
| 78 | C4×#6157 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C4×#6157 | C22×C4 | 1 | 3 | yes | — |
| 80 | #11719936 | C2×C8 | 1 | 6 | yes | — |
| 81 | D53⋊C8 | C2×C8 | 1 | 6 | yes | — |
| 82 | D53⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | D53⋊C8 | C22×C4 | 1 | 3 | yes | — |
| 84 | M16×C53 | C42 | 1 | 3 | yes | — |
| 85 | C2×C8×C53 | C22×C4 | 1 | 3 | yes | — |
| 86 | #11719917 | C42 | 1 | 3 | yes | — |
| 87 | C2×#88322 | C22×C4 | 1 | 3 | yes | — |
| 88 | #11719912 | C42 | 1 | 6 | yes | — |
| 89 | C8×D53 | C22×C4 | 1 | 3 | yes | — |
| 90 | C8×D53 | C22×C4 | 1 | 3 | yes | — |
| 91 | C8×D53 | C22×C4 | 1 | 3 | yes | — |
| 92 | C8×D53 | C24 | 1 | 1 | yes | characteristic |
| 93 | #11719910 | C22×C4 | 1 | 2 | yes | — |
| 94 | #11719910 | C22×C4 | 1 | 6 | yes | — |
| 95 | C16×C53 | C22×C4 | 1 | 3 | yes | — |
| 96 | C16×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 97 | #11719908 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | #11719908 | C22×C4 | 1 | 3 | yes | — |
| 99 | C22×D53 | C22×C8 | 1 | 3 | yes | — |
| 100 | C2×#6157 | — | 2 | 12 | no | — |
| 101 | C2×#6157 | C22×C8 | 1 | 6 | yes | — |
| 102 | C2×#6157 | C22×C8 | 1 | 2 | yes | — |
| 103 | Q8×C53 | C4×C8 | 1 | 1 | yes | characteristic |
| 104 | D4×C53 | C4×C8 | 1 | 3 | yes | — |
| 105 | C2×C4×C53 | C2×C42 | 1 | 3 | yes | — |
| 106 | C22⋊D53 | C4×C8 | 1 | 6 | yes | — |
| 107 | C2×Dic53 | C22×C8 | 1 | 3 | yes | — |
| 108 | D212 | C4×C8 | 1 | 3 | yes | — |
| 109 | C4×D53 | C2×C42 | 1 | 3 | yes | — |
| 110 | C4×D53 | C22×C8 | 1 | 3 | yes | — |
| 111 | C4×D53 | C22×C8 | 1 | 3 | yes | — |
| 112 | C4×D53 | C23×C4 | 1 | 1 | yes | characteristic |
| 113 | Dic106 | C4×C8 | 1 | 3 | yes | — |
| 114 | #88324 | C22×C8 | 1 | 6 | yes | — |
| 115 | #88324 | C22×C8 | 1 | 2 | yes | — |
| 116 | C8×C53 | C2×C42 | 1 | 3 | yes | — |
| 117 | C8×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 118 | #88322 | C2×C42 | 1 | 3 | yes | — |
| 119 | #88322 | C23×C4 | 1 | 1 | yes | characteristic |
| 120 | C4×#504 | — | 53 | 1 | no | maximal, Sylow |
| 121 | C22×C53 | C2×C4×C8 | 1 | 3 | yes | — |
| 122 | C2×D53 | — | 2 | 6 | no | — |
| 123 | C2×D53 | C2×C4×C8 | 1 | 3 | yes | — |
| 124 | C2×D53 | C23×C8 | 1 | 1 | yes | characteristic |
| 125 | #6157 | — | 2 | 6 | no | — |
| 126 | #6157 | #504 | 1 | 4 | yes | direct product factor |
| 127 | C4×C53 | C2×C4×C8 | 1 | 3 | yes | — |
| 128 | C4×C53 | C22×C42 | 1 | 1 | yes | characteristic |
| 129 | Dic53 | C2×C4×C8 | 1 | 3 | yes | — |
| 130 | Dic53 | C23×C8 | 1 | 1 | yes | characteristic |
| 131 | C2×#504 | — | 53 | 1 | no | — |
| 132 | C2×#504 | — | 53 | 1 | no | — |
| 133 | C4×#131 | — | 53 | 1 | no | — |
| 134 | #2108 | — | 53 | 3 | no | — |
| 135 | #2108 | — | 53 | 3 | no | — |
| 136 | C4×M32 | — | 53 | 3 | no | — |
| 137 | C2×C4×C16 | — | 53 | 3 | no | — |
| 138 | C2×C53 | — | 2 | 3 | no | — |
| 139 | C2×C53 | C22×C4×C8 | 1 | 1 | yes | characteristic |
| 140 | D53 | — | 2 | 3 | no | — |
| 141 | D53 | C2×#504 | 1 | 2 | yes | — |
| 142 | C2×#131 | — | 53 | 1 | no | — |
| 143 | C2×#131 | — | 53 | 1 | no | — |
| 144 | C4×#41 | — | 53 | 1 | no | — |
| 145 | #504 | — | 53 | 2 | no | — |
| 146 | #504 | — | 53 | 6 | no | — |
| 147 | #504 | — | 53 | 1 | no | — |
| 148 | #504 | — | 53 | 3 | no | — |
| 149 | #504 | — | 53 | 3 | no | — |
| 150 | #504 | #6157 | 1 | 1 | yes | characteristic, direct product factor |
| 151 | C2×M32 | — | 53 | 3 | no | — |
| 152 | C2×M32 | — | 53 | 3 | no | — |
| 153 | C22×C16 | — | 53 | 3 | no | — |
| 154 | C22×C16 | — | 53 | 3 | no | — |
| 155 | #405 | — | 53 | 3 | no | — |
| 156 | #405 | — | 53 | 3 | no | — |
| 157 | C4×M16 | — | 53 | 3 | no | — |
| 158 | C2×C4×C8 | — | 53 | 3 | no | — |
| 159 | #346 | — | 53 | 6 | no | — |
| 160 | C4×C16 | — | 53 | 1 | no | — |
| 161 | C4×C16 | — | 53 | 3 | no | — |
| 162 | C4×C16 | — | 53 | 3 | no | — |
| 163 | C4×C16 | — | 53 | 3 | no | — |
| 164 | C53 | C4×#504 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 165 | C2×#41 | — | 53 | 1 | no | — |
| 166 | #131 | — | 53 | 6 | no | — |
| 167 | #131 | — | 53 | 2 | no | — |
| 168 | #131 | — | 53 | 1 | no | — |
| 169 | #131 | — | 53 | 3 | no | — |
| 170 | #131 | — | 53 | 3 | no | — |
| 171 | #131 | C2×#6157 | 1 | 1 | yes | characteristic |
| 172 | C2×M16 | — | 53 | 3 | no | — |
| 173 | C2×M16 | — | 53 | 3 | no | — |
| 174 | C22×C8 | — | 53 | 3 | no | — |
| 175 | C22×C8 | — | 53 | 3 | no | — |
| 176 | C4×Q8 | — | 53 | 2 | no | — |
| 177 | C4×D4 | — | 53 | 6 | no | — |
| 178 | #117 | — | 53 | 3 | no | — |
| 179 | C2×C42 | — | 53 | 3 | no | — |
| 180 | M32 | — | 53 | 6 | no | — |
| 181 | M32 | — | 53 | 6 | no | — |
| 182 | M32 | — | 53 | 3 | no | — |
| 183 | M32 | — | 53 | 6 | no | — |
| 184 | M32 | C2×#6157 | 1 | 3 | yes | complemented |
| 185 | C2×C16 | — | 53 | 1 | no | — |
| 186 | C2×C16 | — | 53 | 3 | no | — |
| 187 | C2×C16 | — | 53 | 6 | no | — |
| 188 | C2×C16 | — | 53 | 6 | no | — |
| 189 | C2×C16 | — | 53 | 3 | no | — |
| 190 | C2×C16 | — | 53 | 3 | no | — |
| 191 | C2×C16 | — | 53 | 3 | no | — |
| 192 | C2×C16 | — | 53 | 1 | no | — |
| 193 | C2×C16 | — | 53 | 3 | no | — |
| 194 | C2×C16 | C2×#6157 | 1 | 3 | yes | complemented |
| 195 | #97 | — | 53 | 6 | no | — |
| 196 | C4×C8 | — | 53 | 1 | no | — |
| 197 | C4×C8 | — | 53 | 3 | no | — |
| 198 | C4×C8 | — | 53 | 3 | no | — |
| 199 | C4×C8 | — | 53 | 3 | no | — |
| 200 | #41 | — | 53 | 1 | no | — |
| 201 | #41 | — | 53 | 3 | no | — |
| 202 | #41 | — | 53 | 3 | no | — |
| 203 | #41 | C4×#6157 | 1 | 1 | yes | characteristic |
| 204 | C2×Q8 | — | 53 | 1 | no | — |
| 205 | C2×D4 | — | 53 | 3 | no | — |
| 206 | C22×C4 | — | 53 | 3 | no | — |
| 207 | C22×C4 | — | 53 | 6 | no | — |
| 208 | M16 | — | 53 | 6 | no | — |
| 209 | M16 | — | 53 | 6 | no | — |
| 210 | M16 | — | 53 | 3 | no | — |
| 211 | M16 | — | 53 | 6 | no | — |
| 212 | M16 | C4×#6157 | 1 | 3 | yes | — |
| 213 | C2×C8 | — | 53 | 6 | no | — |
| 214 | C2×C8 | — | 53 | 1 | no | — |
| 215 | C2×C8 | — | 53 | 3 | no | — |
| 216 | C2×C8 | — | 53 | 6 | no | — |
| 217 | C2×C8 | — | 53 | 3 | no | — |
| 218 | C2×C8 | — | 53 | 3 | no | — |
| 219 | C2×C8 | — | 53 | 3 | no | — |
| 220 | C2×C8 | — | 53 | 1 | no | — |
| 221 | C2×C8 | — | 53 | 3 | no | — |
| 222 | C2×C8 | C22×#6157 | 1 | 3 | yes | — |
| 223 | C4⋊C4 | — | 53 | 6 | no | — |
| 224 | C22⋊C4 | — | 53 | 6 | no | — |
| 225 | C42 | — | 53 | 6 | no | — |
| 226 | C42 | — | 53 | 1 | no | — |
| 227 | C42 | — | 53 | 3 | no | — |
| 228 | C16 | — | 53 | 2 | no | — |
| 229 | C16 | — | 53 | 6 | no | — |
| 230 | C16 | — | 53 | 1 | no | — |
| 231 | C16 | — | 53 | 3 | no | — |
| 232 | C16 | C22×#6157 | 1 | 3 | yes | — |
| 233 | C16 | C22×#6157 | 1 | 1 | yes | characteristic |
| 234 | C23 | — | 53 | 3 | no | — |
| 235 | Q8 | — | 53 | 3 | no | — |
| 236 | Q8 | C8×#6157 | 1 | 1 | yes | characteristic |
| 237 | D4 | — | 53 | 6 | no | — |
| 238 | D4 | — | 53 | 3 | no | — |
| 239 | D4 | C8×#6157 | 1 | 3 | yes | — |
| 240 | C2×C4 | — | 53 | 2 | no | — |
| 241 | C2×C4 | — | 53 | 3 | no | — |
| 242 | C2×C4 | — | 53 | 3 | no | — |
| 243 | C2×C4 | — | 53 | 3 | no | — |
| 244 | C2×C4 | — | 53 | 3 | no | — |
| 245 | C2×C4 | — | 53 | 1 | no | — |
| 246 | C2×C4 | — | 53 | 6 | no | — |
| 247 | C2×C4 | — | 106 | 12 | no | — |
| 248 | C2×C4 | C2×C4×#6157 | 1 | 3 | yes | — |
| 249 | C8 | — | 53 | 2 | no | — |
| 250 | C8 | — | 53 | 6 | no | — |
| 251 | C8 | — | 53 | 3 | no | — |
| 252 | C8 | — | 53 | 1 | no | — |
| 253 | C8 | C2×C4×#6157 | 1 | 3 | yes | — |
| 254 | C8 | C23×#6157 | 1 | 1 | yes | characteristic |
| 255 | C22 | — | 53 | 1 | no | — |
| 256 | C22 | — | 53 | 3 | no | — |
| 257 | C22 | — | 106 | 6 | no | — |
| 258 | C22 | C2×C8×#6157 | 1 | 3 | yes | — |
| 259 | C4 | — | 53 | 4 | no | — |
| 260 | C4 | — | 53 | 1 | no | — |
| 261 | C4 | — | 53 | 3 | no | — |
| 262 | C4 | — | 106 | 6 | no | — |
| 263 | C4 | C2×C8×#6157 | 1 | 3 | yes | — |
| 264 | C4 | C22×C4×#6157 | 1 | 1 | yes | characteristic |
| 265 | C2 | — | 2 | 3 | no | — |
| 266 | C2 | — | 53 | 2 | no | — |
| 267 | C2 | — | 106 | 3 | no | — |
| 268 | C2 | C22×C8×#6157 | 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