Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
950 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 449 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 | #11763557 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #11763512 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #11763491 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #11763415 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #11763407 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #11763344 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C17×#564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C17×#125 | C22 | 1 | 6 | yes | — |
| 9 | C17×#125 | C22 | 1 | 6 | yes | — |
| 10 | C17×#125 | C22 | 1 | 3 | yes | — |
| 11 | #10587584 | C22 | 1 | 6 | yes | — |
| 12 | #10587584 | C22 | 1 | 6 | yes | — |
| 13 | #10587583 | C22 | 1 | 3 | yes | — |
| 14 | #10587582 | C22 | 1 | 6 | yes | — |
| 15 | #10587582 | C22 | 1 | 6 | yes | — |
| 16 | #10587581 | C22 | 1 | 6 | yes | — |
| 17 | #10587581 | C22 | 1 | 6 | yes | — |
| 18 | #10587581 | C22 | 1 | 6 | yes | — |
| 19 | #10587581 | C22 | 1 | 3 | yes | — |
| 20 | #10587581 | C22 | 1 | 6 | yes | — |
| 21 | #10587580 | C22 | 1 | 6 | yes | — |
| 22 | #10587580 | C22 | 1 | 6 | yes | — |
| 23 | #10587580 | C22 | 1 | 6 | yes | — |
| 24 | #10587580 | C22 | 1 | 6 | yes | — |
| 25 | #10587580 | C22 | 1 | 6 | yes | — |
| 26 | #10587579 | C22 | 1 | 6 | yes | — |
| 27 | #10587579 | C22 | 1 | 3 | yes | — |
| 28 | #10587579 | C22 | 1 | 6 | yes | — |
| 29 | #10587578 | C22 | 1 | 6 | yes | — |
| 30 | C4⋊C4×D17 | C22 | 1 | 6 | yes | — |
| 31 | #10587575 | C22 | 1 | 6 | yes | — |
| 32 | #10587573 | C22 | 1 | 3 | yes | — |
| 33 | #10587562 | C22 | 1 | 2 | yes | — |
| 34 | #10587561 | C22 | 1 | 3 | yes | — |
| 35 | #10587560 | C22 | 1 | 2 | yes | — |
| 36 | C4×D68 | C22 | 1 | 6 | yes | — |
| 37 | #10587558 | C22 | 1 | 6 | yes | — |
| 38 | C42×D17 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×D68 | C23 | 1 | 6 | yes | — |
| 40 | C2×D68 | C23 | 1 | 6 | yes | — |
| 41 | C2×D68 | C23 | 1 | 6 | yes | — |
| 42 | C2×D68 | C23 | 1 | 3 | yes | — |
| 43 | C2×C4×D17 | C23 | 1 | 3 | yes | — |
| 44 | C2×C4×D17 | C23 | 1 | 6 | yes | — |
| 45 | C2×C4×D17 | C23 | 1 | 6 | yes | — |
| 46 | C2×C4×D17 | C23 | 1 | 6 | yes | — |
| 47 | C2×Dic34 | C23 | 1 | 3 | yes | — |
| 48 | C4⋊C4×C17 | C23 | 1 | 3 | yes | — |
| 49 | C4⋊C4×C17 | C23 | 1 | 6 | yes | — |
| 50 | C4⋊C4×C17 | C23 | 1 | 6 | yes | — |
| 51 | C4⋊C4×C17 | C23 | 1 | 6 | yes | — |
| 52 | C4⋊C4×C17 | C23 | 1 | 3 | yes | — |
| 53 | C4⋊C4×C17 | C23 | 1 | 6 | yes | — |
| 54 | C42×C17 | C23 | 1 | 1 | yes | characteristic |
| 55 | C42×C17 | C23 | 1 | 2 | yes | — |
| 56 | C42×C17 | C23 | 1 | 2 | yes | — |
| 57 | #63227 | C23 | 1 | 6 | yes | — |
| 58 | #63227 | C23 | 1 | 6 | yes | — |
| 59 | #63227 | C23 | 1 | 6 | yes | — |
| 60 | #63227 | C23 | 1 | 6 | yes | — |
| 61 | #63227 | C23 | 1 | 6 | yes | — |
| 62 | #63227 | C23 | 1 | 6 | yes | — |
| 63 | #63227 | C23 | 1 | 6 | yes | — |
| 64 | C4⋊Dic17 | C23 | 1 | 6 | yes | — |
| 65 | #63225 | C23 | 1 | 6 | yes | — |
| 66 | #63225 | C23 | 1 | 6 | yes | — |
| 67 | #63225 | C23 | 1 | 6 | yes | — |
| 68 | C4×Dic17 | C23 | 1 | 3 | yes | — |
| 69 | C4×Dic17 | C23 | 1 | 6 | yes | — |
| 70 | C22×D17 | C24 | 1 | 6 | yes | — |
| 71 | C22×D17 | C24 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×C17 | C24 | 1 | 3 | yes | — |
| 73 | C2×C4×C17 | C24 | 1 | 6 | yes | — |
| 74 | C2×C4×C17 | C24 | 1 | 6 | yes | — |
| 75 | C2×Dic17 | C24 | 1 | 6 | yes | — |
| 76 | C2×Dic17 | C24 | 1 | 3 | yes | — |
| 77 | D68 | — | 4 | 3 | no | — |
| 78 | D68 | — | 4 | 6 | no | — |
| 79 | D68 | — | 4 | 6 | no | — |
| 80 | D68 | — | 4 | 6 | no | — |
| 81 | C4×D17 | — | 4 | 6 | no | — |
| 82 | C4×D17 | — | 4 | 6 | no | — |
| 83 | C4×D17 | — | 4 | 3 | no | — |
| 84 | C4×D17 | — | 4 | 6 | no | — |
| 85 | Dic34 | — | 4 | 3 | no | — |
| 86 | #3567 | — | 17 | 1 | no | maximal, Sylow |
| 87 | C22×C17 | C25 | 1 | 1 | yes | characteristic |
| 88 | C2×D17 | — | 2 | 6 | no | — |
| 89 | C2×D17 | — | 2 | 6 | no | — |
| 90 | C2×D17 | — | 2 | 3 | no | — |
| 91 | C2×D17 | — | 2 | 6 | no | — |
| 92 | C4×C17 | — | 2 | 3 | no | — |
| 93 | C4×C17 | — | 2 | 6 | no | — |
| 94 | C4×C17 | — | 2 | 6 | no | — |
| 95 | Dic17 | — | 2 | 6 | no | — |
| 96 | Dic17 | — | 2 | 3 | no | — |
| 97 | #564 | D17 | 1 | 1 | yes | characteristic, complemented |
| 98 | #562 | — | 17 | 3 | no | — |
| 99 | #553 | — | 17 | 6 | no | — |
| 100 | #552 | — | 17 | 6 | no | — |
| 101 | #540 | — | 17 | 6 | no | — |
| 102 | #539 | — | 17 | 6 | no | — |
| 103 | #529 | — | 17 | 3 | no | — |
| 104 | C2×C17 | #585 | 1 | 3 | yes | — |
| 105 | D17 | — | 4 | 6 | no | — |
| 106 | D17 | — | 4 | 1 | no | — |
| 107 | #127 | — | 17 | 2 | no | — |
| 108 | #126 | — | 17 | 6 | no | — |
| 109 | #126 | — | 17 | 6 | no | — |
| 110 | #126 | — | 17 | 2 | no | — |
| 111 | #125 | — | 17 | 6 | no | — |
| 112 | #125 | C2×D17 | 1 | 3 | yes | — |
| 113 | #125 | C2×D17 | 1 | 6 | yes | — |
| 114 | #125 | C2×D17 | 1 | 6 | yes | — |
| 115 | #124 | — | 17 | 3 | no | — |
| 116 | #123 | — | 17 | 6 | no | — |
| 117 | #123 | — | 17 | 6 | no | — |
| 118 | #123 | — | 17 | 6 | no | — |
| 119 | #123 | — | 17 | 6 | no | — |
| 120 | #123 | — | 17 | 6 | no | — |
| 121 | C22⋊Q8 | — | 17 | 3 | no | — |
| 122 | C22⋊Q8 | — | 17 | 6 | no | — |
| 123 | C22⋊Q8 | — | 17 | 6 | no | — |
| 124 | #121 | — | 17 | 3 | no | — |
| 125 | #121 | — | 17 | 6 | no | — |
| 126 | #121 | — | 17 | 6 | no | — |
| 127 | #121 | — | 17 | 6 | no | — |
| 128 | #121 | — | 17 | 6 | no | — |
| 129 | C4×Q8 | — | 17 | 3 | no | — |
| 130 | C4×D4 | — | 17 | 6 | no | — |
| 131 | C4×D4 | — | 17 | 3 | no | — |
| 132 | C4×D4 | — | 17 | 6 | no | — |
| 133 | C4×D4 | — | 17 | 6 | no | — |
| 134 | #117 | — | 17 | 6 | no | — |
| 135 | #117 | — | 17 | 6 | no | — |
| 136 | C2×C4⋊C4 | — | 17 | 6 | no | — |
| 137 | C2×C42 | — | 17 | 1 | no | — |
| 138 | C17 | #3567 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 139 | C2×Q8 | — | 17 | 3 | no | — |
| 140 | C2×D4 | — | 17 | 6 | no | — |
| 141 | C2×D4 | — | 17 | 3 | no | — |
| 142 | C2×D4 | — | 17 | 6 | no | — |
| 143 | C2×D4 | — | 17 | 6 | no | — |
| 144 | C22×C4 | — | 17 | 3 | no | — |
| 145 | C22×C4 | — | 17 | 6 | no | — |
| 146 | C22×C4 | — | 17 | 6 | no | — |
| 147 | C22×C4 | — | 17 | 6 | no | — |
| 148 | C4⋊C4 | — | 17 | 6 | no | — |
| 149 | C4⋊C4 | — | 17 | 6 | no | — |
| 150 | C4⋊C4 | — | 17 | 6 | no | — |
| 151 | C4⋊C4 | — | 17 | 6 | no | — |
| 152 | C4⋊C4 | C22×D17 | 1 | 6 | yes | — |
| 153 | C4⋊C4 | C22×D17 | 1 | 6 | yes | — |
| 154 | C4⋊C4 | C22×D17 | 1 | 6 | yes | — |
| 155 | C4⋊C4 | C22×D17 | 1 | 6 | yes | — |
| 156 | C4⋊C4 | C22×D17 | 1 | 3 | yes | — |
| 157 | C4⋊C4 | C22×D17 | 1 | 3 | yes | — |
| 158 | C22⋊C4 | — | 17 | 6 | no | — |
| 159 | C22⋊C4 | — | 17 | 6 | no | — |
| 160 | C22⋊C4 | — | 17 | 6 | no | — |
| 161 | C22⋊C4 | — | 17 | 6 | no | — |
| 162 | C22⋊C4 | — | 17 | 6 | no | — |
| 163 | C22⋊C4 | — | 17 | 6 | no | — |
| 164 | C22⋊C4 | — | 17 | 6 | no | — |
| 165 | C42 | — | 17 | 3 | no | — |
| 166 | C42 | — | 17 | 6 | no | — |
| 167 | C42 | C22×D17 | 1 | 1 | yes | characteristic |
| 168 | C42 | C22×D17 | 1 | 2 | yes | — |
| 169 | C42 | C22×D17 | 1 | 2 | yes | — |
| 170 | C23 | — | 17 | 6 | no | — |
| 171 | C23 | — | 17 | 1 | no | — |
| 172 | Q8 | — | 68 | 3 | no | — |
| 173 | D4 | — | 68 | 6 | no | — |
| 174 | D4 | — | 68 | 6 | no | — |
| 175 | D4 | — | 68 | 6 | no | — |
| 176 | D4 | — | 68 | 3 | no | — |
| 177 | C2×C4 | — | 17 | 3 | no | — |
| 178 | C2×C4 | — | 17 | 6 | no | — |
| 179 | C2×C4 | — | 68 | 6 | no | — |
| 180 | C2×C4 | — | 68 | 6 | no | — |
| 181 | C2×C4 | — | 68 | 6 | no | — |
| 182 | C2×C4 | — | 68 | 3 | no | — |
| 183 | C2×C4 | C23×D17 | 1 | 3 | yes | — |
| 184 | C2×C4 | C23×D17 | 1 | 6 | yes | — |
| 185 | C2×C4 | C23×D17 | 1 | 6 | yes | — |
| 186 | C22 | — | 34 | 6 | no | — |
| 187 | C22 | — | 34 | 6 | no | — |
| 188 | C22 | — | 34 | 3 | no | — |
| 189 | C22 | — | 34 | 6 | no | — |
| 190 | C22 | C24×D17 | 1 | 1 | yes | characteristic |
| 191 | C4 | — | 2 | 6 | no | — |
| 192 | C4 | — | 2 | 6 | no | — |
| 193 | C4 | — | 2 | 3 | no | — |
| 194 | C4 | — | 34 | 3 | no | — |
| 195 | C4 | — | 34 | 6 | no | — |
| 196 | C2 | — | 68 | 6 | no | — |
| 197 | C2 | — | 68 | 1 | no | — |
| 198 | C2 | #11763646 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D17 |
| commutator, socle, cosocle | C22×C17 | C25 |
| Fitting | C17×#564 | C2 |
| hypercenter | #564 | D17 |
| hypocenter | C17 | #3567 |
| LCS | G ▷ C22×C17 ▷ C17 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11763557 | C2 | semidirect |
| 2. #11763512 | C2 | semidirect |
| 3. #11763491 | C2 | semidirect |
| 4. #11763415 | C2 | semidirect |
| 5. #11763407 | C2 | semidirect |
| 6. #11763344 | C2 | semidirect |
| 7. C17×#564 | C2 | semidirect |
| 97. #564 | D17 | semidirect |
| 138. C17 | #3567 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11763557 | 2 |
| #11763512 | 2 |
| #11763491 | 2 |
| #11763415 | 2 |
| #11763407 | 2 |
| #11763344 | 2 |
| C17×#564 | 2 |
| #3567 | 17 |