← C53×#9836 #27217348
Subgroups — C53×#9836 #27217348
Order 13568.
510 conjugacy classes of subgroups.
The subgroup lattice has 218 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C53×#500 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×#1990 | C2 | 1 | 2 | yes | maximal |
| 3 | C53×#1956 | C2 | 1 | 2 | yes | maximal |
| 4 | C53×#1844 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×C53×#322 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×C4⋊Q8×C53 | C22 | 1 | 2 | yes | — |
| 7 | C53×#500 | — | 2 | 2 | no | — |
| 8 | C53×#500 | C4 | 1 | 4 | yes | complemented |
| 9 | C2×#107×C53 | — | 2 | 2 | no | — |
| 10 | C2×#106×C53 | — | 2 | 2 | no | — |
| 11 | C2×#103×C53 | — | 2 | 4 | no | — |
| 12 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 15 | C53×#384 | C22 | 1 | 2 | yes | — |
| 16 | C53×#340 | — | 2 | 4 | no | — |
| 17 | C53×#322 | — | 2 | 2 | no | — |
| 18 | C22×Q8×C53 | — | 2 | 2 | no | — |
| 19 | C22×C8×C53 | D4 | 1 | 2 | yes | — |
| 20 | C22×C8×C53 | D4 | 1 | 2 | yes | — |
| 21 | C22×C8×C53 | D4 | 1 | 2 | yes | — |
| 22 | C4⋊Q8×C53 | — | 2 | 2 | no | — |
| 23 | C4⋊Q8×C53 | — | 2 | 2 | no | — |
| 24 | C4⋊Q8×C53 | — | 2 | 2 | no | — |
| 25 | C4⋊Q8×C53 | C2×C4 | 1 | 4 | yes | — |
| 26 | C2×C4⋊C4×C53 | — | 2 | 2 | no | — |
| 27 | C2×C4⋊C4×C53 | — | 2 | 2 | no | — |
| 28 | C2×C4⋊C4×C53 | — | 2 | 2 | no | — |
| 29 | C2×C4⋊C4×C53 | — | 2 | 2 | no | — |
| 30 | C2×C4⋊C4×C53 | — | 2 | 2 | no | — |
| 31 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 32 | #107×C53 | — | 2 | 8 | no | — |
| 33 | #106×C53 | — | 4 | 4 | no | — |
| 34 | #103×C53 | — | 4 | 8 | no | — |
| 35 | C4×C8×C53 | — | 2 | 1 | no | — |
| 36 | C4×C8×C53 | — | 2 | 1 | no | — |
| 37 | C4×C8×C53 | — | 2 | 1 | no | — |
| 38 | C4×C8×C53 | — | 2 | 1 | no | — |
| 39 | C4×C8×C53 | C2×C4 | 1 | 2 | yes | — |
| 40 | C4×C8×C53 | D4 | 1 | 2 | yes | — |
| 41 | C2×Q8×C53 | — | 2 | 4 | no | — |
| 42 | C2×Q8×C53 | — | 4 | 2 | no | — |
| 43 | C2×Q8×C53 | — | 4 | 4 | no | — |
| 44 | C2×Q8×C53 | — | 4 | 4 | no | — |
| 45 | C22×C4×C53 | — | 4 | 2 | no | — |
| 46 | C22×C4×C53 | — | 4 | 2 | no | — |
| 47 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 49 | C22×C4×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 50 | C2×C8×C53 | — | 2 | 2 | no | — |
| 51 | C2×C8×C53 | — | 2 | 2 | no | — |
| 52 | C2×C8×C53 | — | 2 | 4 | no | — |
| 53 | C2×C8×C53 | — | 2 | 4 | no | — |
| 54 | C2×C8×C53 | C22⋊C4 | 1 | 4 | yes | — |
| 55 | C2×C8×C53 | SD16 | 1 | 4 | yes | — |
| 56 | C2×C8×C53 | Q16 | 1 | 4 | yes | — |
| 57 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 58 | C4⋊C4×C53 | — | 2 | 4 | no | — |
| 59 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 60 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 61 | C4⋊C4×C53 | — | 4 | 4 | no | — |
| 62 | C4⋊C4×C53 | — | 4 | 4 | no | — |
| 63 | C4⋊C4×C53 | — | 4 | 4 | no | — |
| 64 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 65 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 67 | C42×C53 | #41 | 1 | 1 | yes | characteristic |
| 68 | C23×C53 | #127 | 1 | 1 | yes | characteristic |
| 69 | Q8×C53 | — | 4 | 8 | no | — |
| 70 | Q8×C53 | — | 8 | 4 | no | — |
| 71 | C2×C4×C53 | — | 4 | 4 | no | — |
| 72 | C2×C4×C53 | — | 8 | 2 | no | — |
| 73 | C2×C4×C53 | — | 8 | 2 | no | — |
| 74 | C2×C4×C53 | — | 8 | 2 | no | — |
| 75 | C2×C4×C53 | — | 8 | 2 | no | — |
| 76 | C2×C4×C53 | — | 8 | 2 | no | — |
| 77 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 78 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 80 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 81 | C2×C4×C53 | #121 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×C53 | #124 | 1 | 1 | yes | characteristic |
| 83 | C2×C4×C53 | C2×SD16 | 1 | 2 | yes | — |
| 84 | C2×C4×C53 | C2×Q16 | 1 | 2 | yes | — |
| 85 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 86 | C2×C4×C53 | #137 | 1 | 4 | yes | — |
| 87 | C8×C53 | — | 4 | 2 | no | — |
| 88 | C8×C53 | — | 4 | 2 | no | — |
| 89 | C8×C53 | #103 | 1 | 8 | yes | complemented |
| 90 | #9836 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 91 | C22×C53 | #390 | 1 | 1 | yes | characteristic |
| 92 | C22×C53 | #492 | 1 | 1 | yes | characteristic |
| 93 | C22×C53 | #494 | 1 | 1 | yes | characteristic |
| 94 | C22×C53 | #496 | 1 | 2 | yes | — |
| 95 | C22×C53 | #497 | 1 | 2 | yes | — |
| 96 | C4×C53 | — | 8 | 4 | no | — |
| 97 | C4×C53 | — | 16 | 2 | no | — |
| 98 | C4×C53 | C2×#103 | 1 | 2 | yes | — |
| 99 | C4×C53 | #441 | 1 | 2 | yes | — |
| 100 | C4×C53 | #442 | 1 | 2 | yes | — |
| 101 | C4×C53 | #469 | 1 | 2 | yes | — |
| 102 | C4×C53 | #470 | 1 | 2 | yes | — |
| 103 | C4×C53 | #487 | 1 | 2 | yes | — |
| 104 | C2×#500 | C2×C53 | 1 | 1 | yes | characteristic |
| 105 | #1990 | C2×C53 | 1 | 2 | yes | — |
| 106 | #1956 | C2×C53 | 1 | 2 | yes | — |
| 107 | #1844 | C2×C53 | 1 | 1 | yes | characteristic |
| 108 | C2×#322 | C2×C53 | 1 | 1 | yes | characteristic |
| 109 | C2×C53 | #1714 | 1 | 2 | yes | — |
| 110 | C2×C53 | #1723 | 1 | 2 | yes | — |
| 111 | C2×C53 | #1956 | 1 | 1 | yes | characteristic |
| 112 | C2×C53 | #1971 | 1 | 1 | yes | characteristic |
| 113 | C2×C53 | #1972 | 1 | 1 | yes | characteristic |
| 114 | C2×C4⋊Q8 | C22×C53 | 1 | 2 | yes | — |
| 115 | #500 | — | 2 | 2 | no | — |
| 116 | #500 | C4×C53 | 1 | 4 | yes | complemented |
| 117 | C2×#107 | — | 2 | 2 | no | — |
| 118 | C2×#106 | — | 2 | 2 | no | — |
| 119 | C2×#103 | — | 2 | 4 | no | — |
| 120 | C2×C4×C8 | C22×C53 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C8 | C22×C53 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C8 | C22×C53 | 1 | 1 | yes | characteristic |
| 123 | #384 | C22×C53 | 1 | 2 | yes | — |
| 124 | #340 | — | 2 | 4 | no | — |
| 125 | #322 | — | 2 | 2 | no | — |
| 126 | C53 | #9836 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 127 | C22×Q8 | — | 2 | 2 | no | — |
| 128 | C22×C8 | D4×C53 | 1 | 2 | yes | — |
| 129 | C22×C8 | D4×C53 | 1 | 2 | yes | — |
| 130 | C22×C8 | D4×C53 | 1 | 2 | yes | — |
| 131 | C4⋊Q8 | — | 2 | 2 | no | — |
| 132 | C4⋊Q8 | — | 2 | 2 | no | — |
| 133 | C4⋊Q8 | — | 2 | 2 | no | — |
| 134 | C4⋊Q8 | C2×C4×C53 | 1 | 4 | yes | — |
| 135 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 136 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 137 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 138 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 139 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 140 | C2×C42 | C23×C53 | 1 | 1 | yes | characteristic |
| 141 | #107 | — | 2 | 8 | no | — |
| 142 | #106 | — | 4 | 4 | no | — |
| 143 | #103 | — | 4 | 8 | no | — |
| 144 | C4×C8 | — | 2 | 1 | no | — |
| 145 | C4×C8 | — | 2 | 1 | no | — |
| 146 | C4×C8 | — | 2 | 1 | no | — |
| 147 | C4×C8 | — | 2 | 1 | no | — |
| 148 | C4×C8 | C2×C4×C53 | 1 | 2 | yes | — |
| 149 | C4×C8 | D4×C53 | 1 | 2 | yes | — |
| 150 | C2×Q8 | — | 2 | 4 | no | — |
| 151 | C2×Q8 | — | 4 | 2 | no | — |
| 152 | C2×Q8 | — | 4 | 4 | no | — |
| 153 | C2×Q8 | — | 4 | 4 | no | — |
| 154 | C22×C4 | — | 4 | 2 | no | — |
| 155 | C22×C4 | — | 4 | 2 | no | — |
| 156 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 157 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 158 | C22×C4 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 159 | C2×C8 | — | 2 | 2 | no | — |
| 160 | C2×C8 | — | 2 | 2 | no | — |
| 161 | C2×C8 | — | 2 | 4 | no | — |
| 162 | C2×C8 | — | 2 | 4 | no | — |
| 163 | C2×C8 | C22⋊C4×C53 | 1 | 4 | yes | — |
| 164 | C2×C8 | SD16×C53 | 1 | 4 | yes | — |
| 165 | C2×C8 | Q16×C53 | 1 | 4 | yes | — |
| 166 | C4⋊C4 | — | 2 | 4 | no | — |
| 167 | C4⋊C4 | — | 2 | 4 | no | — |
| 168 | C4⋊C4 | — | 4 | 2 | no | — |
| 169 | C4⋊C4 | — | 4 | 2 | no | — |
| 170 | C4⋊C4 | — | 4 | 4 | no | — |
| 171 | C4⋊C4 | — | 4 | 4 | no | — |
| 172 | C4⋊C4 | — | 4 | 4 | no | — |
| 173 | C42 | C22×C4×C53 | 1 | 1 | yes | characteristic |
| 174 | C42 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 175 | C42 | #41×C53 | 1 | 1 | yes | characteristic |
| 176 | C42 | #41×C53 | 1 | 1 | yes | characteristic |
| 177 | C23 | #127×C53 | 1 | 1 | yes | characteristic |
| 178 | Q8 | — | 4 | 8 | no | — |
| 179 | Q8 | — | 8 | 4 | no | — |
| 180 | C2×C4 | — | 4 | 4 | no | — |
| 181 | C2×C4 | — | 8 | 2 | no | — |
| 182 | C2×C4 | — | 8 | 2 | no | — |
| 183 | C2×C4 | — | 8 | 2 | no | — |
| 184 | C2×C4 | — | 8 | 2 | no | — |
| 185 | C2×C4 | — | 8 | 2 | no | — |
| 186 | C2×C4 | C2×C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 187 | C2×C4 | C4×D4×C53 | 1 | 1 | yes | characteristic |
| 188 | C2×C4 | C4×D4×C53 | 1 | 1 | yes | characteristic |
| 189 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 190 | C2×C4 | #121×C53 | 1 | 1 | yes | characteristic |
| 191 | C2×C4 | #124×C53 | 1 | 1 | yes | characteristic |
| 192 | C2×C4 | C2×SD16×C53 | 1 | 2 | yes | — |
| 193 | C2×C4 | C2×Q16×C53 | 1 | 2 | yes | — |
| 194 | C2×C4 | AGL(1,ℤ/8)×C53 | 1 | 4 | yes | — |
| 195 | C2×C4 | #137×C53 | 1 | 4 | yes | — |
| 196 | C8 | — | 4 | 2 | no | — |
| 197 | C8 | — | 4 | 2 | no | — |
| 198 | C8 | #103×C53 | 1 | 8 | yes | complemented |
| 199 | C22 | C53×#390 | 1 | 1 | yes | characteristic |
| 200 | C22 | C53×#492 | 1 | 1 | yes | characteristic |
| 201 | C22 | C53×#494 | 1 | 1 | yes | characteristic |
| 202 | C22 | C53×#496 | 1 | 2 | yes | — |
| 203 | C22 | C53×#497 | 1 | 2 | yes | — |
| 204 | C4 | — | 8 | 4 | no | — |
| 205 | C4 | — | 16 | 2 | no | — |
| 206 | C4 | C2×#103×C53 | 1 | 2 | yes | — |
| 207 | C4 | C53×#441 | 1 | 2 | yes | — |
| 208 | C4 | C53×#442 | 1 | 2 | yes | — |
| 209 | C4 | C53×#469 | 1 | 2 | yes | — |
| 210 | C4 | C53×#470 | 1 | 2 | yes | — |
| 211 | C4 | C53×#487 | 1 | 2 | yes | — |
| 212 | C2 | C53×#1714 | 1 | 2 | yes | — |
| 213 | C2 | C53×#1723 | 1 | 2 | yes | — |
| 214 | C2 | C53×#1956 | 1 | 1 | yes | characteristic |
| 215 | C2 | C53×#1971 | 1 | 1 | yes | characteristic |
| 216 | C2 | C53×#1972 | 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