← C53×#7217 #27214729
Subgroups — C53×#7217 #27214729
Order 13568.
226 conjugacy classes of subgroups.
The subgroup lattice has 188 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C53×#2162 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×#1834 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C53×#1567 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C53×#423 | — | 2 | 1 | no | — |
| 5 | C2×C4×C8×C53 | C22 | 1 | 1 | yes | characteristic |
| 6 | C53×#384 | — | 2 | 1 | no | — |
| 7 | C53×C4⋊C16 | — | 2 | 1 | no | — |
| 8 | C53×C22⋊C16 | — | 2 | 1 | no | — |
| 9 | C4×C16×C53 | C4 | 1 | 2 | yes | complemented |
| 10 | C53×#340 | — | 2 | 1 | no | — |
| 11 | C53×#340 | C4 | 1 | 2 | yes | — |
| 12 | C53×#335 | — | 2 | 1 | no | — |
| 13 | C53×#335 | C4 | 1 | 2 | yes | complemented |
| 14 | C2×M16×C53 | — | 4 | 1 | no | — |
| 15 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C22×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 17 | C2×C4⋊C4×C53 | — | 2 | 2 | no | — |
| 18 | C2×C4⋊C4×C53 | — | 4 | 1 | no | — |
| 19 | C2×C4⋊C4×C53 | — | 4 | 1 | no | — |
| 20 | C2×C42×C53 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×C16×C53 | — | 2 | 2 | no | — |
| 22 | C4⋊C8×C53 | — | 2 | 2 | no | — |
| 23 | C4⋊C8×C53 | — | 4 | 1 | no | — |
| 24 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C4×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 27 | C4×C8×C53 | Q8 | 1 | 1 | yes | characteristic |
| 28 | C22×C4×C53 | — | 4 | 2 | no | — |
| 29 | C22×C4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 30 | C22×C4×C53 | D8 | 1 | 1 | yes | characteristic |
| 31 | C22×C4×C53 | SD16 | 1 | 1 | yes | characteristic |
| 32 | M16×C53 | — | 8 | 1 | no | — |
| 33 | M16×C53 | — | 8 | 1 | no | — |
| 34 | C2×C8×C53 | — | 4 | 1 | no | — |
| 35 | C2×C8×C53 | — | 4 | 1 | no | — |
| 36 | C2×C8×C53 | — | 4 | 2 | no | — |
| 37 | C2×C8×C53 | C42 | 1 | 1 | yes | characteristic |
| 38 | C2×C8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 39 | C2×C8×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 40 | C2×C8×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 41 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 42 | C4⋊C4×C53 | — | 4 | 2 | no | — |
| 43 | C4⋊C4×C53 | — | 8 | 2 | no | — |
| 44 | C4⋊C4×C53 | — | 8 | 1 | no | — |
| 45 | C4⋊C4×C53 | — | 8 | 1 | no | — |
| 46 | C42×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 47 | C42×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 48 | C42×C53 | SD16 | 1 | 1 | yes | characteristic |
| 49 | C42×C53 | Q16 | 1 | 1 | yes | characteristic |
| 50 | C16×C53 | — | 4 | 2 | no | — |
| 51 | C23×C53 | #102 | 1 | 1 | yes | characteristic |
| 52 | C2×C4×C53 | — | 2 | 1 | no | — |
| 53 | C2×C4×C53 | — | 2 | 1 | no | — |
| 54 | C2×C4×C53 | — | 2 | 2 | no | — |
| 55 | C2×C4×C53 | — | 8 | 2 | no | — |
| 56 | C2×C4×C53 | — | 8 | 2 | no | — |
| 57 | C2×C4×C53 | — | 8 | 2 | no | — |
| 58 | C2×C4×C53 | #95 | 1 | 1 | yes | characteristic |
| 59 | C2×C4×C53 | #102 | 1 | 1 | yes | characteristic |
| 60 | C2×C4×C53 | #103 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C53 | #103 | 1 | 1 | yes | characteristic |
| 62 | C2×C4×C53 | #106 | 1 | 1 | yes | characteristic |
| 63 | C2×C4×C53 | #107 | 1 | 1 | yes | characteristic |
| 64 | C2×C4×C53 | D16 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×C53 | SD32 | 1 | 1 | yes | characteristic |
| 66 | C2×C4×C53 | SD32 | 1 | 1 | yes | characteristic |
| 67 | C2×C4×C53 | Q32 | 1 | 1 | yes | characteristic |
| 68 | C8×C53 | — | 2 | 1 | no | — |
| 69 | C8×C53 | — | 2 | 1 | no | — |
| 70 | C8×C53 | — | 2 | 1 | no | — |
| 71 | C8×C53 | — | 2 | 1 | no | — |
| 72 | C8×C53 | — | 8 | 2 | no | — |
| 73 | #7217 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 74 | C22×C53 | — | 2 | 1 | no | — |
| 75 | C22×C53 | — | 2 | 1 | no | — |
| 76 | C22×C53 | #340 | 1 | 1 | yes | characteristic |
| 77 | C22×C53 | #357 | 1 | 1 | yes | characteristic |
| 78 | C22×C53 | #358 | 1 | 1 | yes | characteristic |
| 79 | C4×C53 | — | 2 | 1 | no | — |
| 80 | C4×C53 | — | 2 | 1 | no | — |
| 81 | C4×C53 | — | 16 | 2 | no | — |
| 82 | C4×C53 | H(3,ℤ/4) | 1 | 1 | yes | characteristic |
| 83 | C4×C53 | #344 | 1 | 1 | yes | characteristic |
| 84 | C4×C53 | #357 | 1 | 1 | yes | characteristic |
| 85 | C4×C53 | #358 | 1 | 1 | yes | characteristic |
| 86 | C4×C53 | #365 | 1 | 2 | yes | — |
| 87 | C4×C53 | #366 | 1 | 1 | yes | characteristic |
| 88 | C4×C53 | #367 | 1 | 1 | yes | characteristic |
| 89 | #2162 | C2×C53 | 1 | 1 | yes | characteristic |
| 90 | #1834 | C2×C53 | 1 | 1 | yes | characteristic |
| 91 | #1567 | C2×C53 | 1 | 1 | yes | characteristic |
| 92 | C2×C53 | — | 2 | 1 | no | — |
| 93 | C2×C53 | — | 2 | 1 | no | — |
| 94 | C2×C53 | #1286 | 1 | 1 | yes | characteristic |
| 95 | C2×C53 | #1380 | 1 | 1 | yes | characteristic |
| 96 | C2×C53 | #1383 | 1 | 1 | yes | characteristic |
| 97 | #423 | — | 2 | 1 | no | — |
| 98 | C2×C4×C8 | C22×C53 | 1 | 1 | yes | characteristic |
| 99 | #384 | — | 2 | 1 | no | — |
| 100 | C4⋊C16 | — | 2 | 1 | no | — |
| 101 | C22⋊C16 | — | 2 | 1 | no | — |
| 102 | C4×C16 | C4×C53 | 1 | 2 | yes | complemented |
| 103 | #340 | — | 2 | 1 | no | — |
| 104 | #340 | C4×C53 | 1 | 2 | yes | — |
| 105 | #335 | — | 2 | 1 | no | — |
| 106 | #335 | C4×C53 | 1 | 2 | yes | complemented |
| 107 | C53 | #7217 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 108 | C2×M16 | — | 4 | 1 | no | — |
| 109 | C22×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 110 | C22×C8 | D4×C53 | 1 | 1 | yes | characteristic |
| 111 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 112 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 113 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 114 | C2×C42 | D4×C53 | 1 | 1 | yes | characteristic |
| 115 | C2×C16 | — | 2 | 2 | no | — |
| 116 | C4⋊C8 | — | 2 | 2 | no | — |
| 117 | C4⋊C8 | — | 4 | 1 | no | — |
| 118 | C4×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 119 | C4×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 120 | C4×C8 | D4×C53 | 1 | 1 | yes | characteristic |
| 121 | C4×C8 | Q8×C53 | 1 | 1 | yes | characteristic |
| 122 | C22×C4 | — | 4 | 2 | no | — |
| 123 | C22×C4 | C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 124 | C22×C4 | D8×C53 | 1 | 1 | yes | characteristic |
| 125 | C22×C4 | SD16×C53 | 1 | 1 | yes | characteristic |
| 126 | M16 | — | 8 | 1 | no | — |
| 127 | M16 | — | 8 | 1 | no | — |
| 128 | C2×C8 | — | 4 | 1 | no | — |
| 129 | C2×C8 | — | 4 | 1 | no | — |
| 130 | C2×C8 | — | 4 | 2 | no | — |
| 131 | C2×C8 | C42×C53 | 1 | 1 | yes | characteristic |
| 132 | C2×C8 | C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 133 | C2×C8 | C4⋊C4×C53 | 1 | 1 | yes | characteristic |
| 134 | C2×C8 | C4⋊C4×C53 | 1 | 1 | yes | characteristic |
| 135 | C4⋊C4 | — | 4 | 2 | no | — |
| 136 | C4⋊C4 | — | 4 | 2 | no | — |
| 137 | C4⋊C4 | — | 8 | 2 | no | — |
| 138 | C4⋊C4 | — | 8 | 1 | no | — |
| 139 | C4⋊C4 | — | 8 | 1 | no | — |
| 140 | C42 | C22⋊C4×C53 | 1 | 1 | yes | characteristic |
| 141 | C42 | C4⋊C4×C53 | 1 | 1 | yes | characteristic |
| 142 | C42 | SD16×C53 | 1 | 1 | yes | characteristic |
| 143 | C42 | Q16×C53 | 1 | 1 | yes | characteristic |
| 144 | C16 | — | 4 | 2 | no | — |
| 145 | C23 | #102×C53 | 1 | 1 | yes | characteristic |
| 146 | C2×C4 | — | 2 | 1 | no | — |
| 147 | C2×C4 | — | 2 | 1 | no | — |
| 148 | C2×C4 | — | 2 | 2 | no | — |
| 149 | C2×C4 | — | 8 | 2 | no | — |
| 150 | C2×C4 | — | 8 | 2 | no | — |
| 151 | C2×C4 | — | 8 | 2 | no | — |
| 152 | C2×C4 | #95×C53 | 1 | 1 | yes | characteristic |
| 153 | C2×C4 | #102×C53 | 1 | 1 | yes | characteristic |
| 154 | C2×C4 | #103×C53 | 1 | 1 | yes | characteristic |
| 155 | C2×C4 | #103×C53 | 1 | 1 | yes | characteristic |
| 156 | C2×C4 | #106×C53 | 1 | 1 | yes | characteristic |
| 157 | C2×C4 | #107×C53 | 1 | 1 | yes | characteristic |
| 158 | C2×C4 | D16×C53 | 1 | 1 | yes | characteristic |
| 159 | C2×C4 | SD32×C53 | 1 | 1 | yes | characteristic |
| 160 | C2×C4 | SD32×C53 | 1 | 1 | yes | characteristic |
| 161 | C2×C4 | Q32×C53 | 1 | 1 | yes | characteristic |
| 162 | C8 | — | 2 | 1 | no | — |
| 163 | C8 | — | 2 | 1 | no | — |
| 164 | C8 | — | 2 | 1 | no | — |
| 165 | C8 | — | 2 | 1 | no | — |
| 166 | C8 | — | 8 | 2 | no | — |
| 167 | C22 | — | 2 | 1 | no | — |
| 168 | C22 | — | 2 | 1 | no | — |
| 169 | C22 | C53×#340 | 1 | 1 | yes | characteristic |
| 170 | C22 | C53×#357 | 1 | 1 | yes | characteristic |
| 171 | C22 | C53×#358 | 1 | 1 | yes | characteristic |
| 172 | C4 | — | 2 | 1 | no | — |
| 173 | C4 | — | 2 | 1 | no | — |
| 174 | C4 | — | 16 | 2 | no | — |
| 175 | C4 | C53×H(3,ℤ/4) | 1 | 1 | yes | characteristic |
| 176 | C4 | C53×#344 | 1 | 1 | yes | characteristic |
| 177 | C4 | C53×#357 | 1 | 1 | yes | characteristic |
| 178 | C4 | C53×#358 | 1 | 1 | yes | characteristic |
| 179 | C4 | C53×#365 | 1 | 2 | yes | — |
| 180 | C4 | C53×#366 | 1 | 1 | yes | characteristic |
| 181 | C4 | C53×#367 | 1 | 1 | yes | characteristic |
| 182 | C2 | — | 2 | 1 | no | — |
| 183 | C2 | — | 2 | 1 | no | — |
| 184 | C2 | C53×#1286 | 1 | 1 | yes | characteristic |
| 185 | C2 | C53×#1380 | 1 | 1 | yes | characteristic |
| 186 | C2 | C53×#1383 | 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