← C53×#3504 #20272441
Subgroups — C53×#3504 #20272441
Order 6784.
988 conjugacy classes of subgroups.
The subgroup lattice has 150 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C53×#557 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | C53×#554 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | C53×#544 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C53×#533 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#125×C53 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C53×#520 | C2 | 1 | 2 | yes | maximal |
| 7 | C4×#41×C53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#41×C53 | C22 | 1 | 1 | yes | characteristic |
| 9 | C4⋊Q8×C53 | C22 | 1 | 6 | yes | — |
| 10 | C4⋊Q8×C53 | C22 | 1 | 12 | yes | — |
| 11 | #125×C53 | C22 | 1 | 3 | yes | — |
| 12 | #125×C53 | C22 | 1 | 6 | yes | — |
| 13 | #125×C53 | C22 | 1 | 12 | yes | — |
| 14 | #125×C53 | C22 | 1 | 1 | yes | characteristic |
| 15 | #125×C53 | C22 | 1 | 12 | yes | — |
| 16 | C22⋊Q8×C53 | C22 | 1 | 12 | yes | — |
| 17 | C22⋊Q8×C53 | C22 | 1 | 24 | yes | — |
| 18 | C4×Q8×C53 | C22 | 1 | 2 | yes | — |
| 19 | C4×Q8×C53 | C22 | 1 | 4 | yes | — |
| 20 | C4×Q8×C53 | C22 | 1 | 12 | yes | — |
| 21 | C4×D4×C53 | C22 | 1 | 6 | yes | — |
| 22 | C4×D4×C53 | C22 | 1 | 12 | yes | — |
| 23 | #117×C53 | C22 | 1 | 3 | yes | — |
| 24 | #117×C53 | C22 | 1 | 6 | yes | — |
| 25 | C2×C4⋊C4×C53 | C22 | 1 | 6 | yes | — |
| 26 | C2×C4⋊C4×C53 | C22 | 1 | 12 | yes | — |
| 27 | C2×C42×C53 | C22 | 1 | 3 | yes | — |
| 28 | #41×C53 | Q8 | 1 | 8 | yes | complemented |
| 29 | C2×Q8×C53 | C23 | 1 | 12 | yes | — |
| 30 | C2×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×D4×C53 | C23 | 1 | 3 | yes | — |
| 32 | C22×C4×C53 | C23 | 1 | 6 | yes | — |
| 33 | C22×C4×C53 | C23 | 1 | 12 | yes | — |
| 34 | C22×C4×C53 | C23 | 1 | 3 | yes | — |
| 35 | C4⋊C4×C53 | C23 | 1 | 6 | yes | — |
| 36 | C4⋊C4×C53 | C23 | 1 | 12 | yes | — |
| 37 | C4⋊C4×C53 | C23 | 1 | 12 | yes | — |
| 38 | C4⋊C4×C53 | C23 | 1 | 24 | yes | — |
| 39 | C4⋊C4×C53 | C23 | 1 | 2 | yes | — |
| 40 | C4⋊C4×C53 | C23 | 1 | 6 | yes | — |
| 41 | C4⋊C4×C53 | C23 | 1 | 4 | yes | — |
| 42 | C4⋊C4×C53 | C23 | 1 | 12 | yes | — |
| 43 | C22⋊C4×C53 | C23 | 1 | 12 | yes | — |
| 44 | C22⋊C4×C53 | C23 | 1 | 6 | yes | — |
| 45 | C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 46 | C42×C53 | C23 | 1 | 3 | yes | — |
| 47 | C42×C53 | C23 | 1 | 6 | yes | — |
| 48 | C42×C53 | C23 | 1 | 12 | yes | — |
| 49 | C23×C53 | C24 | 1 | 3 | yes | — |
| 50 | Q8×C53 | — | 4 | 12 | no | — |
| 51 | Q8×C53 | C2×Q8 | 1 | 4 | yes | — |
| 52 | D4×C53 | C2×Q8 | 1 | 12 | yes | — |
| 53 | C2×C4×C53 | — | 4 | 6 | no | — |
| 54 | C2×C4×C53 | — | 4 | 12 | no | — |
| 55 | C2×C4×C53 | C2×Q8 | 1 | 12 | yes | — |
| 56 | C2×C4×C53 | C24 | 1 | 2 | yes | — |
| 57 | C2×C4×C53 | C24 | 1 | 12 | yes | — |
| 58 | C2×C4×C53 | C24 | 1 | 6 | yes | — |
| 59 | C2×C4×C53 | C24 | 1 | 4 | yes | — |
| 60 | C2×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C53 | C24 | 1 | 3 | yes | — |
| 62 | C22×C53 | — | 2 | 6 | no | — |
| 63 | C22×C53 | C22×Q8 | 1 | 6 | yes | — |
| 64 | C22×C53 | C25 | 1 | 1 | yes | characteristic |
| 65 | C4×C53 | — | 2 | 2 | no | — |
| 66 | C4×C53 | — | 2 | 12 | no | — |
| 67 | C4×C53 | — | 2 | 6 | no | — |
| 68 | C4×C53 | — | 2 | 4 | no | — |
| 69 | C4×C53 | C22×Q8 | 1 | 2 | yes | — |
| 70 | C4×C53 | C22×Q8 | 1 | 6 | yes | — |
| 71 | #3504 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 72 | C2×C53 | — | 2 | 6 | no | — |
| 73 | C2×C53 | C23×Q8 | 1 | 1 | yes | characteristic |
| 74 | C2×C53 | #585 | 1 | 2 | yes | — |
| 75 | #557 | C2×C53 | 1 | 4 | yes | complemented |
| 76 | #554 | C2×C53 | 1 | 12 | yes | complemented |
| 77 | #544 | C2×C53 | 1 | 6 | yes | complemented |
| 78 | #533 | C2×C53 | 1 | 3 | yes | complemented |
| 79 | C2×#125 | C2×C53 | 1 | 3 | yes | complemented |
| 80 | #520 | C2×C53 | 1 | 2 | yes | — |
| 81 | C4×#41 | C2×C53 | 1 | 1 | yes | characteristic |
| 82 | C53 | #3504 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 83 | C2×#41 | C22×C53 | 1 | 1 | yes | characteristic |
| 84 | C4⋊Q8 | C22×C53 | 1 | 6 | yes | — |
| 85 | C4⋊Q8 | C22×C53 | 1 | 12 | yes | — |
| 86 | #125 | C22×C53 | 1 | 3 | yes | — |
| 87 | #125 | C22×C53 | 1 | 6 | yes | — |
| 88 | #125 | C22×C53 | 1 | 12 | yes | — |
| 89 | #125 | C22×C53 | 1 | 1 | yes | characteristic |
| 90 | #125 | C22×C53 | 1 | 12 | yes | — |
| 91 | C22⋊Q8 | C22×C53 | 1 | 12 | yes | — |
| 92 | C22⋊Q8 | C22×C53 | 1 | 24 | yes | — |
| 93 | C4×Q8 | C22×C53 | 1 | 2 | yes | — |
| 94 | C4×Q8 | C22×C53 | 1 | 4 | yes | — |
| 95 | C4×Q8 | C22×C53 | 1 | 12 | yes | — |
| 96 | C4×D4 | C22×C53 | 1 | 6 | yes | — |
| 97 | C4×D4 | C22×C53 | 1 | 12 | yes | — |
| 98 | #117 | C22×C53 | 1 | 3 | yes | — |
| 99 | #117 | C22×C53 | 1 | 6 | yes | — |
| 100 | C2×C4⋊C4 | C22×C53 | 1 | 6 | yes | — |
| 101 | C2×C4⋊C4 | C22×C53 | 1 | 12 | yes | — |
| 102 | C2×C42 | C22×C53 | 1 | 3 | yes | — |
| 103 | #41 | Q8×C53 | 1 | 8 | yes | complemented |
| 104 | C2×Q8 | C23×C53 | 1 | 12 | yes | — |
| 105 | C2×Q8 | C23×C53 | 1 | 1 | yes | characteristic |
| 106 | C2×D4 | C23×C53 | 1 | 3 | yes | — |
| 107 | C22×C4 | C23×C53 | 1 | 6 | yes | — |
| 108 | C22×C4 | C23×C53 | 1 | 12 | yes | — |
| 109 | C22×C4 | C23×C53 | 1 | 3 | yes | — |
| 110 | C4⋊C4 | C23×C53 | 1 | 6 | yes | — |
| 111 | C4⋊C4 | C23×C53 | 1 | 12 | yes | — |
| 112 | C4⋊C4 | C23×C53 | 1 | 12 | yes | — |
| 113 | C4⋊C4 | C23×C53 | 1 | 24 | yes | — |
| 114 | C4⋊C4 | C23×C53 | 1 | 2 | yes | — |
| 115 | C4⋊C4 | C23×C53 | 1 | 6 | yes | — |
| 116 | C4⋊C4 | C23×C53 | 1 | 4 | yes | — |
| 117 | C4⋊C4 | C23×C53 | 1 | 12 | yes | — |
| 118 | C22⋊C4 | C23×C53 | 1 | 12 | yes | — |
| 119 | C22⋊C4 | C23×C53 | 1 | 6 | yes | — |
| 120 | C42 | C23×C53 | 1 | 1 | yes | characteristic |
| 121 | C42 | C23×C53 | 1 | 3 | yes | — |
| 122 | C42 | C23×C53 | 1 | 6 | yes | — |
| 123 | C42 | C23×C53 | 1 | 12 | yes | — |
| 124 | C23 | C24×C53 | 1 | 3 | yes | — |
| 125 | Q8 | — | 4 | 12 | no | — |
| 126 | Q8 | C2×Q8×C53 | 1 | 4 | yes | — |
| 127 | D4 | C2×Q8×C53 | 1 | 12 | yes | — |
| 128 | C2×C4 | — | 4 | 6 | no | — |
| 129 | C2×C4 | — | 4 | 12 | no | — |
| 130 | C2×C4 | C2×Q8×C53 | 1 | 12 | yes | — |
| 131 | C2×C4 | C24×C53 | 1 | 2 | yes | — |
| 132 | C2×C4 | C24×C53 | 1 | 4 | yes | — |
| 133 | C2×C4 | C24×C53 | 1 | 6 | yes | — |
| 134 | C2×C4 | C24×C53 | 1 | 12 | yes | — |
| 135 | C2×C4 | C24×C53 | 1 | 3 | yes | — |
| 136 | C2×C4 | C24×C53 | 1 | 1 | yes | characteristic |
| 137 | C22 | — | 2 | 6 | no | — |
| 138 | C22 | C22×Q8×C53 | 1 | 6 | yes | — |
| 139 | C22 | C25×C53 | 1 | 1 | yes | characteristic |
| 140 | C4 | — | 2 | 2 | no | — |
| 141 | C4 | — | 2 | 12 | no | — |
| 142 | C4 | — | 2 | 6 | no | — |
| 143 | C4 | — | 2 | 4 | no | — |
| 144 | C4 | C22×Q8×C53 | 1 | 2 | yes | — |
| 145 | C4 | C22×Q8×C53 | 1 | 6 | yes | — |
| 146 | C2 | — | 2 | 6 | no | — |
| 147 | C2 | C23×Q8×C53 | 1 | 1 | yes | characteristic |
| 148 | C2 | C53×#585 | 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