← C53×#13641 #27221153
Subgroups — C53×#13641 #27221153
Order 13568.
180 conjugacy classes of subgroups.
The subgroup lattice has 140 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C53×#2108 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C53×C4⋊C32 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C53×C22⋊C32 | C2 | 1 | 2 | yes | maximal |
| 4 | C53×#1397 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C4×C32×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×M32×C53 | C4 | 1 | 1 | yes | characteristic |
| 7 | C22×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 8 | C53×#405 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×C32×C53 | C22 | 1 | 2 | yes | — |
| 10 | C2×C32×C53 | C22 | 1 | 2 | yes | — |
| 11 | C53×#346 | C4 | 1 | 1 | yes | characteristic |
| 12 | C53×#346 | C4 | 1 | 1 | yes | characteristic |
| 13 | C4×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×C16×C53 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×M16×C53 | C8 | 1 | 1 | yes | characteristic |
| 16 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | #117×C53 | C8 | 1 | 1 | yes | characteristic |
| 18 | M32×C53 | — | 2 | 2 | no | — |
| 19 | C2×C16×C53 | — | 2 | 2 | no | — |
| 20 | C2×C16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C2×C16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C2×C16×C53 | C23 | 1 | 1 | yes | characteristic |
| 24 | #97×C53 | C8 | 1 | 1 | yes | characteristic |
| 25 | #97×C53 | C8 | 1 | 1 | yes | characteristic |
| 26 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C32×C53 | — | 2 | 2 | no | — |
| 29 | C32×C53 | — | 2 | 2 | no | — |
| 30 | C22×C4×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 31 | M16×C53 | — | 2 | 2 | no | — |
| 32 | C2×C8×C53 | — | 2 | 1 | no | — |
| 33 | C2×C8×C53 | — | 2 | 1 | no | — |
| 34 | C2×C8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 35 | C2×C8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 36 | C2×C8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 37 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 38 | C4⋊C4×C53 | C16 | 1 | 2 | yes | — |
| 39 | C22⋊C4×C53 | C16 | 1 | 2 | yes | — |
| 40 | C42×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 41 | C42×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 42 | C16×C53 | — | 2 | 1 | no | — |
| 43 | C16×C53 | — | 2 | 1 | no | — |
| 44 | C16×C53 | M16 | 1 | 2 | yes | — |
| 45 | C16×C53 | #41 | 1 | 2 | yes | — |
| 46 | C23×C53 | C2×C16 | 1 | 1 | yes | characteristic |
| 47 | C2×C4×C53 | — | 2 | 1 | no | — |
| 48 | C2×C4×C53 | — | 2 | 1 | no | — |
| 49 | C2×C4×C53 | C2×C16 | 1 | 1 | yes | characteristic |
| 50 | C2×C4×C53 | C2×C16 | 1 | 2 | yes | — |
| 51 | C2×C4×C53 | C2×C16 | 1 | 2 | yes | — |
| 52 | C2×C4×C53 | C22×C8 | 1 | 1 | yes | characteristic |
| 53 | C8×C53 | — | 2 | 1 | no | — |
| 54 | C8×C53 | — | 2 | 1 | no | — |
| 55 | C8×C53 | M32 | 1 | 2 | yes | — |
| 56 | C8×C53 | #117 | 1 | 1 | yes | characteristic |
| 57 | C8×C53 | C2×M16 | 1 | 1 | yes | characteristic |
| 58 | #13641 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 59 | C22×C53 | — | 2 | 1 | no | — |
| 60 | C22×C53 | — | 2 | 2 | no | — |
| 61 | C22×C53 | C22×C16 | 1 | 1 | yes | characteristic |
| 62 | C4×C53 | — | 2 | 1 | no | — |
| 63 | C4×C53 | — | 2 | 2 | no | — |
| 64 | C4×C53 | — | 2 | 2 | no | — |
| 65 | C4×C53 | #431 | 1 | 1 | yes | characteristic |
| 66 | C4×C53 | C2×M32 | 1 | 1 | yes | characteristic |
| 67 | #2108 | C2×C53 | 1 | 1 | yes | characteristic |
| 68 | C4⋊C32 | C2×C53 | 1 | 2 | yes | complemented |
| 69 | C22⋊C32 | C2×C53 | 1 | 2 | yes | — |
| 70 | #1397 | C2×C53 | 1 | 1 | yes | characteristic, complemented |
| 71 | C4×C32 | C2×C53 | 1 | 1 | yes | characteristic, complemented |
| 72 | C2×C53 | — | 4 | 1 | no | — |
| 73 | C2×C53 | #2162 | 1 | 1 | yes | characteristic |
| 74 | C2×C53 | #2258 | 1 | 2 | yes | — |
| 75 | C2×M32 | C4×C53 | 1 | 1 | yes | characteristic |
| 76 | C22×C16 | C22×C53 | 1 | 1 | yes | characteristic |
| 77 | #405 | C4×C53 | 1 | 1 | yes | characteristic |
| 78 | C2×C32 | C22×C53 | 1 | 2 | yes | — |
| 79 | C2×C32 | C22×C53 | 1 | 2 | yes | — |
| 80 | #346 | C4×C53 | 1 | 1 | yes | characteristic |
| 81 | #346 | C4×C53 | 1 | 1 | yes | characteristic |
| 82 | C4×C16 | C22×C53 | 1 | 1 | yes | characteristic |
| 83 | C4×C16 | C22×C53 | 1 | 1 | yes | characteristic |
| 84 | C53 | #13641 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 85 | C2×M16 | C8×C53 | 1 | 1 | yes | characteristic |
| 86 | C22×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 87 | #117 | C8×C53 | 1 | 1 | yes | characteristic |
| 88 | M32 | — | 2 | 2 | no | — |
| 89 | C2×C16 | — | 2 | 2 | no | — |
| 90 | C2×C16 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 91 | C2×C16 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 92 | C2×C16 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 93 | C2×C16 | C23×C53 | 1 | 1 | yes | characteristic |
| 94 | #97 | C8×C53 | 1 | 1 | yes | characteristic |
| 95 | #97 | C8×C53 | 1 | 1 | yes | characteristic |
| 96 | C4×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 97 | C4×C8 | C2×C4×C53 | 1 | 1 | yes | characteristic |
| 98 | C32 | — | 2 | 2 | no | — |
| 99 | C32 | — | 2 | 2 | no | — |
| 100 | C22×C4 | C2×C8×C53 | 1 | 1 | yes | characteristic |
| 101 | M16 | — | 2 | 2 | no | — |
| 102 | C2×C8 | — | 2 | 1 | no | — |
| 103 | C2×C8 | — | 2 | 1 | no | — |
| 104 | C2×C8 | C2×C8×C53 | 1 | 1 | yes | characteristic |
| 105 | C2×C8 | C2×C8×C53 | 1 | 1 | yes | characteristic |
| 106 | C2×C8 | C2×C8×C53 | 1 | 1 | yes | characteristic |
| 107 | C2×C8 | C22×C4×C53 | 1 | 1 | yes | characteristic |
| 108 | C4⋊C4 | C16×C53 | 1 | 2 | yes | — |
| 109 | C22⋊C4 | C16×C53 | 1 | 2 | yes | — |
| 110 | C42 | C2×C8×C53 | 1 | 1 | yes | characteristic |
| 111 | C42 | C2×C8×C53 | 1 | 1 | yes | characteristic |
| 112 | C16 | — | 2 | 1 | no | — |
| 113 | C16 | — | 2 | 1 | no | — |
| 114 | C16 | M16×C53 | 1 | 2 | yes | — |
| 115 | C16 | #41×C53 | 1 | 2 | yes | — |
| 116 | C23 | C2×C16×C53 | 1 | 1 | yes | characteristic |
| 117 | C2×C4 | — | 2 | 1 | no | — |
| 118 | C2×C4 | — | 2 | 1 | no | — |
| 119 | C2×C4 | C2×C16×C53 | 1 | 2 | yes | — |
| 120 | C2×C4 | C2×C16×C53 | 1 | 2 | yes | — |
| 121 | C2×C4 | C2×C16×C53 | 1 | 1 | yes | characteristic |
| 122 | C2×C4 | C22×C8×C53 | 1 | 1 | yes | characteristic |
| 123 | C8 | — | 2 | 1 | no | — |
| 124 | C8 | — | 2 | 1 | no | — |
| 125 | C8 | M32×C53 | 1 | 2 | yes | — |
| 126 | C8 | #117×C53 | 1 | 1 | yes | characteristic |
| 127 | C8 | C2×M16×C53 | 1 | 1 | yes | characteristic |
| 128 | C22 | — | 2 | 2 | no | — |
| 129 | C22 | — | 2 | 1 | no | — |
| 130 | C22 | C22×C16×C53 | 1 | 1 | yes | characteristic |
| 131 | C4 | — | 2 | 2 | no | — |
| 132 | C4 | — | 2 | 2 | no | — |
| 133 | C4 | — | 2 | 1 | no | — |
| 134 | C4 | C53×#431 | 1 | 1 | yes | characteristic |
| 135 | C4 | C2×M32×C53 | 1 | 1 | yes | characteristic |
| 136 | C2 | — | 4 | 1 | no | — |
| 137 | C2 | C53×#2162 | 1 | 1 | yes | characteristic |
| 138 | C2 | C53×#2258 | 1 | 2 | yes | — |
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