Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
548 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #16811610 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #16811596 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #16811458 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#11719940 | C22 | 1 | 3 | yes | complemented |
| 5 | C22×#88324 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | #13133278 | — | 2 | 3 | no | — |
| 7 | #13133277 | — | 2 | 3 | no | — |
| 8 | #13133274 | — | 2 | 3 | no | — |
| 9 | C22⋊#88324 | — | 2 | 3 | no | — |
| 10 | #13133264 | — | 2 | 3 | no | — |
| 11 | #13133262 | — | 2 | 3 | no | — |
| 12 | #13133262 | C4 | 1 | 3 | yes | — |
| 13 | C2×#11719947 | — | 2 | 3 | no | — |
| 14 | C2×#11719947 | C22 | 1 | 3 | yes | — |
| 15 | #41×D53 | — | 2 | 3 | no | — |
| 16 | H(5,𝔽2)×C53 | C4 | 1 | 1 | yes | characteristic |
| 17 | #41×C53 | — | 2 | 3 | no | — |
| 18 | C2×D4×C53 | — | 2 | 3 | no | — |
| 19 | C2×D4×C53 | C2×C4 | 1 | 3 | yes | — |
| 20 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 21 | C2×C22⋊D53 | — | 2 | 3 | no | — |
| 22 | C2×C22⋊D53 | C2×C4 | 1 | 3 | yes | — |
| 23 | C2×C22⋊D53 | D4 | 1 | 3 | yes | — |
| 24 | C22×Dic53 | — | 2 | 1 | no | — |
| 25 | C22×Dic53 | D4 | 1 | 3 | yes | — |
| 26 | C22×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 27 | #11719949 | — | 2 | 1 | no | — |
| 28 | Q8×D53 | — | 2 | 3 | no | — |
| 29 | #11719947 | — | 2 | 3 | no | — |
| 30 | #11719947 | — | 2 | 6 | no | — |
| 31 | #11719947 | — | 2 | 6 | no | — |
| 32 | #11719947 | — | 2 | 6 | no | — |
| 33 | #11719947 | — | 2 | 6 | no | — |
| 34 | D4×D53 | — | 2 | 3 | no | — |
| 35 | D4×D53 | — | 2 | 6 | no | — |
| 36 | #11719945 | — | 2 | 6 | no | — |
| 37 | #11719945 | — | 2 | 3 | no | — |
| 38 | C2×C4×D53 | — | 2 | 3 | no | — |
| 39 | C2×C4×D53 | D4 | 1 | 3 | yes | — |
| 40 | C2×Dic106 | — | 2 | 3 | no | — |
| 41 | C2×Dic106 | D4 | 1 | 3 | yes | — |
| 42 | #11719940 | — | 2 | 6 | no | — |
| 43 | C2×#88324 | — | 2 | 3 | no | — |
| 44 | C2×#88324 | — | 2 | 3 | no | — |
| 45 | C23×C53 | — | 2 | 1 | no | — |
| 46 | C23×C53 | C22⋊C4 | 1 | 3 | yes | — |
| 47 | C23×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 48 | C22×D53 | — | 2 | 3 | no | — |
| 49 | C22×D53 | C22⋊C4 | 1 | 3 | yes | — |
| 50 | Q8×C53 | — | 2 | 1 | no | — |
| 51 | D4×C53 | — | 2 | 3 | no | — |
| 52 | D4×C53 | — | 2 | 6 | no | — |
| 53 | C2×C4×C53 | — | 2 | 3 | no | — |
| 54 | C2×C4×C53 | C22⋊C4 | 1 | 3 | yes | — |
| 55 | C22⋊D53 | — | 2 | 6 | no | — |
| 56 | C22⋊D53 | — | 2 | 6 | no | — |
| 57 | C22⋊D53 | — | 2 | 6 | no | — |
| 58 | C2×Dic53 | — | 2 | 3 | no | — |
| 59 | C2×Dic53 | — | 2 | 3 | no | — |
| 60 | C2×Dic53 | — | 2 | 3 | no | — |
| 61 | C2×Dic53 | C22⋊C4 | 1 | 3 | yes | — |
| 62 | C2×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 63 | C2×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 64 | D212 | — | 2 | 3 | no | — |
| 65 | C4×D53 | — | 2 | 6 | no | — |
| 66 | C4×D53 | — | 2 | 3 | no | — |
| 67 | C4×D53 | — | 2 | 3 | no | — |
| 68 | Dic106 | — | 2 | 3 | no | — |
| 69 | Dic106 | — | 2 | 6 | no | — |
| 70 | #88324 | — | 4 | 1 | no | — |
| 71 | #88324 | — | 4 | 3 | no | — |
| 72 | C22×C53 | — | 2 | 3 | no | — |
| 73 | C22×C53 | — | 4 | 1 | no | — |
| 74 | C22×C53 | — | 4 | 3 | no | — |
| 75 | C22×C53 | — | 8 | 1 | no | — |
| 76 | C22×C53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 77 | C2×D53 | — | 2 | 3 | no | — |
| 78 | C2×D53 | — | 4 | 3 | no | — |
| 79 | C2×D53 | — | 8 | 3 | no | — |
| 80 | C4×C53 | — | 2 | 3 | no | — |
| 81 | Dic53 | — | 2 | 3 | no | — |
| 82 | Dic53 | #120 | 1 | 3 | yes | — |
| 83 | Dic53 | #120 | 1 | 1 | yes | characteristic |
| 84 | #1791 | — | 53 | 1 | no | maximal, Sylow |
| 85 | C2×C53 | — | 2 | 3 | no | — |
| 86 | C2×C53 | — | 4 | 3 | no | — |
| 87 | C2×C53 | #379 | 1 | 1 | yes | characteristic |
| 88 | D53 | — | 4 | 3 | no | — |
| 89 | #585 | — | 53 | 1 | no | — |
| 90 | #413 | — | 53 | 3 | no | — |
| 91 | #408 | — | 53 | 3 | no | — |
| 92 | C53 | #1791 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 93 | 2−1+4 | — | 106 | 3 | no | — |
| 94 | H(5,𝔽2) | — | 53 | 3 | no | — |
| 95 | H(5,𝔽2) | — | 106 | 3 | no | — |
| 96 | H(5,𝔽2) | #6157 | 1 | 1 | yes | characteristic |
| 97 | C2×#41 | — | 53 | 3 | no | — |
| 98 | C2×#41 | — | 106 | 3 | no | — |
| 99 | C2×#41 | — | 106 | 3 | no | — |
| 100 | C2×M16 | — | 53 | 3 | no | — |
| 101 | C22×C8 | — | 53 | 1 | no | — |
| 102 | #101 | — | 106 | 3 | no | — |
| 103 | #100 | — | 106 | 3 | no | — |
| 104 | C22⋊C8 | — | 106 | 3 | no | — |
| 105 | C22⋊C8 | — | 106 | 3 | no | — |
| 106 | #41 | — | 2 | 3 | no | — |
| 107 | #41 | — | 106 | 6 | no | — |
| 108 | #41 | — | 106 | 6 | no | — |
| 109 | #41 | — | 106 | 6 | no | — |
| 110 | #41 | — | 106 | 6 | no | — |
| 111 | #41 | — | 106 | 3 | no | — |
| 112 | #41 | — | 106 | 6 | no | — |
| 113 | #41 | — | 106 | 1 | no | — |
| 114 | #41 | — | 106 | 3 | no | — |
| 115 | C2×Q8 | — | 53 | 3 | no | — |
| 116 | C2×Q8 | — | 106 | 3 | no | — |
| 117 | C2×Q8 | — | 106 | 3 | no | — |
| 118 | C2×D4 | — | 2 | 3 | no | — |
| 119 | C2×D4 | — | 53 | 3 | no | — |
| 120 | C2×D4 | — | 53 | 3 | no | — |
| 121 | C2×D4 | — | 106 | 6 | no | — |
| 122 | C2×D4 | — | 106 | 3 | no | — |
| 123 | C2×D4 | — | 106 | 3 | no | — |
| 124 | C2×D4 | — | 106 | 3 | no | — |
| 125 | C2×D4 | C2×#6157 | 1 | 3 | yes | — |
| 126 | C22×C4 | — | 53 | 1 | no | — |
| 127 | C22×C4 | — | 53 | 3 | no | — |
| 128 | C22×C4 | — | 53 | 3 | no | — |
| 129 | C22×C4 | — | 106 | 3 | no | — |
| 130 | C22×C4 | — | 106 | 1 | no | — |
| 131 | M16 | — | 106 | 6 | no | — |
| 132 | C2×C8 | — | 106 | 3 | no | — |
| 133 | C2×C8 | — | 106 | 3 | no | — |
| 134 | C23 | — | 2 | 1 | no | — |
| 135 | C23 | — | 53 | 3 | no | — |
| 136 | C23 | — | 106 | 3 | no | — |
| 137 | C23 | C22⋊#6157 | 1 | 3 | yes | — |
| 138 | C23 | C22×#6157 | 1 | 1 | yes | characteristic |
| 139 | Q8 | — | 2 | 1 | no | — |
| 140 | Q8 | — | 106 | 6 | no | — |
| 141 | Q8 | — | 106 | 3 | no | — |
| 142 | D4 | — | 2 | 3 | no | — |
| 143 | D4 | — | 2 | 6 | no | — |
| 144 | D4 | — | 106 | 3 | no | — |
| 145 | D4 | — | 106 | 6 | no | — |
| 146 | D4 | — | 106 | 6 | no | — |
| 147 | D4 | — | 106 | 6 | no | — |
| 148 | C2×C4 | — | 2 | 3 | no | — |
| 149 | C2×C4 | — | 53 | 3 | no | — |
| 150 | C2×C4 | — | 53 | 3 | no | — |
| 151 | C2×C4 | — | 53 | 3 | no | — |
| 152 | C2×C4 | — | 106 | 3 | no | — |
| 153 | C2×C4 | — | 106 | 3 | no | — |
| 154 | C2×C4 | — | 106 | 6 | no | — |
| 155 | C2×C4 | — | 106 | 3 | no | — |
| 156 | C2×C4 | — | 106 | 3 | no | — |
| 157 | C2×C4 | — | 106 | 3 | no | — |
| 158 | C2×C4 | C22⋊#6157 | 1 | 3 | yes | — |
| 159 | C8 | — | 212 | 1 | no | — |
| 160 | C8 | — | 212 | 3 | no | — |
| 161 | C22 | — | 2 | 3 | no | — |
| 162 | C22 | — | 4 | 1 | no | — |
| 163 | C22 | — | 4 | 3 | no | — |
| 164 | C22 | — | 8 | 1 | no | — |
| 165 | C22 | — | 106 | 3 | no | — |
| 166 | C22 | — | 424 | 3 | no | — |
| 167 | C22 | — | 212 | 3 | no | — |
| 168 | C22 | C2×C22⋊#6157 | 1 | 3 | yes | — |
| 169 | C4 | — | 2 | 3 | no | — |
| 170 | C4 | — | 53 | 1 | no | — |
| 171 | C4 | — | 53 | 3 | no | — |
| 172 | C4 | — | 106 | 3 | no | — |
| 173 | C2 | — | 2 | 3 | no | — |
| 174 | C2 | — | 4 | 3 | no | — |
| 175 | C2 | — | 212 | 3 | no | — |
| 176 | C2 | #16811564 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16811610 | C2 | semidirect |
| 2. #16811596 | C2 | semidirect |
| 4. C2×#11719940 | C22 | semidirect |
| 5. C22×#88324 | C22 | semidirect |
| 92. C53 | #1791 | semidirect |