Finite Group DatabasebetaFinite Group DatabasebetaOrder 5504.
246 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #14583635 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14583090 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #14583066 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #14582688 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C43×#353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13063370 | C22 | 1 | 1 | yes | characteristic |
| 7 | #13063351 | — | 2 | 1 | no | — |
| 8 | #13063312 | — | 2 | 2 | no | — |
| 9 | #13063309 | — | 2 | 2 | no | — |
| 10 | #13063306 | — | 2 | 1 | no | — |
| 11 | #13063306 | C22 | 1 | 2 | yes | — |
| 12 | C4×Dic86 | — | 2 | 1 | no | — |
| 13 | D4×Dic43 | — | 2 | 2 | no | — |
| 14 | #13063253 | — | 2 | 1 | no | — |
| 15 | #13063242 | C22 | 1 | 1 | yes | characteristic |
| 16 | #13063239 | — | 2 | 2 | no | — |
| 17 | #127×C43 | C22 | 1 | 1 | yes | characteristic |
| 18 | #99×C43 | C22 | 1 | 2 | yes | — |
| 19 | C2×D4×C43 | — | 2 | 2 | no | — |
| 20 | C2×D4×C43 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×D4×C43 | C23 | 1 | 1 | yes | characteristic |
| 22 | C2×C22⋊D43 | — | 2 | 2 | no | — |
| 23 | C22×Dic43 | — | 2 | 1 | no | — |
| 24 | C22×Dic43 | D4 | 1 | 2 | yes | — |
| 25 | #10621900 | — | 4 | 1 | no | — |
| 26 | D4×D43 | — | 4 | 1 | no | — |
| 27 | #10621898 | — | 2 | 1 | no | — |
| 28 | C2×Dic86 | — | 2 | 1 | no | — |
| 29 | C22⋊C4×C43 | — | 2 | 2 | no | — |
| 30 | C42×C43 | D4 | 1 | 1 | yes | characteristic |
| 31 | C22⋊Dic43 | — | 2 | 2 | no | — |
| 32 | C22⋊Dic43 | — | 2 | 1 | no | — |
| 33 | C22⋊Dic43 | — | 2 | 2 | no | — |
| 34 | C22⋊Dic43 | D4 | 1 | 2 | yes | — |
| 35 | #10621883 | — | 4 | 1 | no | — |
| 36 | #10621882 | — | 4 | 1 | no | — |
| 37 | (C2×D43)⋊C4 | — | 4 | 2 | no | — |
| 38 | C4⋊Dic43 | — | 2 | 1 | no | — |
| 39 | C4⋊Dic43 | — | 4 | 2 | no | — |
| 40 | Dic43⋊C4 | — | 2 | 2 | no | — |
| 41 | C4×Dic43 | — | 2 | 2 | no | — |
| 42 | #10621877 | — | 2 | 1 | no | — |
| 43 | C23×C43 | — | 2 | 1 | no | — |
| 44 | C23×C43 | C2×D4 | 1 | 2 | yes | — |
| 45 | C22×D43 | — | 2 | 2 | no | — |
| 46 | D4×C43 | — | 4 | 1 | no | — |
| 47 | D4×C43 | — | 4 | 1 | no | — |
| 48 | D4×C43 | — | 4 | 4 | no | — |
| 49 | C2×C4×C43 | — | 2 | 1 | no | — |
| 50 | C2×C4×C43 | — | 4 | 2 | no | — |
| 51 | C2×C4×C43 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C22⋊D43 | — | 4 | 2 | no | — |
| 53 | C22⋊D43 | — | 4 | 1 | no | — |
| 54 | C2×Dic43 | — | 2 | 2 | no | — |
| 55 | C2×Dic43 | — | 2 | 2 | no | — |
| 56 | C2×Dic43 | — | 2 | 2 | no | — |
| 57 | C2×Dic43 | — | 4 | 2 | no | — |
| 58 | C2×Dic43 | — | 8 | 1 | no | — |
| 59 | D172 | — | 2 | 1 | no | — |
| 60 | C4×D43 | — | 4 | 1 | no | — |
| 61 | Dic86 | — | 2 | 1 | no | — |
| 62 | Dic86 | — | 4 | 2 | no | — |
| 63 | C43⋊C8 | — | 4 | 1 | no | — |
| 64 | C22×C43 | — | 2 | 2 | no | — |
| 65 | C22×C43 | — | 4 | 2 | no | — |
| 66 | C22×C43 | — | 4 | 2 | no | — |
| 67 | C22×C43 | — | 4 | 1 | no | — |
| 68 | C22×C43 | #120 | 1 | 1 | yes | characteristic |
| 69 | C2×D43 | — | 4 | 1 | no | — |
| 70 | C2×D43 | — | 8 | 2 | no | — |
| 71 | C4×C43 | — | 2 | 2 | no | — |
| 72 | C4×C43 | — | 2 | 1 | no | — |
| 73 | C4×C43 | — | 8 | 2 | no | — |
| 74 | Dic43 | — | 4 | 2 | no | — |
| 75 | Dic43 | — | 4 | 1 | no | — |
| 76 | Dic43 | — | 4 | 2 | no | — |
| 77 | #2198 | — | 43 | 1 | no | maximal, Sylow |
| 78 | C2×C43 | — | 2 | 1 | no | — |
| 79 | C2×C43 | — | 4 | 2 | no | — |
| 80 | C2×C43 | — | 8 | 1 | no | — |
| 81 | C2×C43 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 82 | D43 | — | 8 | 1 | no | — |
| 83 | #559 | — | 43 | 1 | no | — |
| 84 | #458 | — | 43 | 2 | no | — |
| 85 | #453 | — | 43 | 1 | no | — |
| 86 | #353 | D43 | 1 | 1 | yes | characteristic, complemented |
| 87 | #352 | — | 43 | 2 | no | — |
| 88 | C43 | #2198 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 89 | H(5,𝔽2) | — | 43 | 1 | no | — |
| 90 | AGL(1,ℤ/8) | — | 86 | 1 | no | — |
| 91 | #127 | C2×D43 | 1 | 1 | yes | characteristic |
| 92 | #124 | — | 86 | 2 | no | — |
| 93 | #123 | — | 43 | 2 | no | — |
| 94 | #123 | — | 86 | 2 | no | — |
| 95 | #123 | — | 86 | 1 | no | — |
| 96 | C4×Q8 | — | 86 | 1 | no | — |
| 97 | C4×D4 | — | 86 | 2 | no | — |
| 98 | #104 | — | 86 | 1 | no | — |
| 99 | #100 | — | 43 | 1 | no | — |
| 100 | #99 | — | 86 | 2 | no | — |
| 101 | #99 | C2×D43 | 1 | 2 | yes | — |
| 102 | #41 | — | 86 | 1 | no | — |
| 103 | #41 | — | 172 | 1 | no | — |
| 104 | C2×Q8 | — | 86 | 1 | no | — |
| 105 | C2×D4 | — | 2 | 2 | no | — |
| 106 | C2×D4 | — | 86 | 2 | no | — |
| 107 | C2×D4 | — | 172 | 1 | no | — |
| 108 | C2×D4 | D172 | 1 | 1 | yes | characteristic |
| 109 | C2×D4 | C22×D43 | 1 | 1 | yes | characteristic |
| 110 | C22×C4 | — | 43 | 2 | no | — |
| 111 | C22×C4 | — | 86 | 1 | no | — |
| 112 | SD16 | — | 172 | 1 | no | — |
| 113 | D8 | — | 172 | 1 | no | — |
| 114 | M16 | — | 86 | 1 | no | — |
| 115 | C4⋊C4 | — | 86 | 1 | no | — |
| 116 | C4⋊C4 | — | 86 | 2 | no | — |
| 117 | C4⋊C4 | — | 172 | 2 | no | — |
| 118 | C22⋊C4 | — | 2 | 2 | no | — |
| 119 | C22⋊C4 | — | 43 | 2 | no | — |
| 120 | C22⋊C4 | — | 86 | 2 | no | — |
| 121 | C22⋊C4 | — | 86 | 1 | no | — |
| 122 | C22⋊C4 | — | 86 | 2 | no | — |
| 123 | C22⋊C4 | — | 172 | 2 | no | — |
| 124 | C42 | — | 86 | 2 | no | — |
| 125 | C42 | D172 | 1 | 1 | yes | characteristic |
| 126 | C23 | — | 2 | 1 | no | — |
| 127 | C23 | — | 86 | 2 | no | — |
| 128 | C23 | D4×D43 | 1 | 2 | yes | — |
| 129 | Q8 | — | 86 | 1 | no | — |
| 130 | Q8 | — | 172 | 2 | no | — |
| 131 | D4 | — | 4 | 1 | no | — |
| 132 | D4 | — | 4 | 1 | no | — |
| 133 | D4 | — | 4 | 4 | no | — |
| 134 | D4 | — | 86 | 1 | no | — |
| 135 | D4 | — | 172 | 2 | no | — |
| 136 | D4 | — | 172 | 1 | no | — |
| 137 | C2×C4 | — | 2 | 1 | no | — |
| 138 | C2×C4 | — | 4 | 2 | no | — |
| 139 | C2×C4 | — | 86 | 2 | no | — |
| 140 | C2×C4 | — | 86 | 2 | no | — |
| 141 | C2×C4 | — | 86 | 2 | no | — |
| 142 | C2×C4 | — | 344 | 1 | no | — |
| 143 | C2×C4 | — | 172 | 1 | no | — |
| 144 | C2×C4 | — | 172 | 2 | no | — |
| 145 | C2×C4 | C2×D172 | 1 | 1 | yes | characteristic |
| 146 | C8 | — | 172 | 1 | no | — |
| 147 | C22 | — | 2 | 2 | no | — |
| 148 | C22 | — | 4 | 2 | no | — |
| 149 | C22 | — | 4 | 2 | no | — |
| 150 | C22 | — | 4 | 1 | no | — |
| 151 | C22 | — | 344 | 2 | no | — |
| 152 | C22 | — | 172 | 1 | no | — |
| 153 | C22 | #13063295 | 1 | 1 | yes | characteristic |
| 154 | C4 | — | 2 | 2 | no | — |
| 155 | C4 | — | 2 | 1 | no | — |
| 156 | C4 | — | 8 | 2 | no | — |
| 157 | C4 | — | 172 | 2 | no | — |
| 158 | C4 | — | 172 | 1 | no | — |
| 159 | C4 | — | 172 | 2 | no | — |
| 160 | C2 | — | 2 | 1 | no | — |
| 161 | C2 | — | 4 | 2 | no | — |
| 162 | C2 | — | 8 | 1 | no | — |
| 163 | C2 | — | 344 | 1 | no | — |
| 164 | C2 | #14583087 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14583635 | C2 | semidirect |
| 2. #14583090 | C2 | semidirect |
| 4. #14582688 | C2 | semidirect |
| 5. C43×#353 | C2 | semidirect |
| 86. #353 | D43 | semidirect |
| 88. C43 | #2198 | semidirect |