Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 270 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 | #11763551 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11762997 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11762986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11762595 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C17×#354 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10587691 | C22 | 1 | 1 | yes | characteristic |
| 7 | C17×#124 | C22 | 1 | 1 | yes | characteristic |
| 8 | #10587633 | — | 2 | 1 | no | — |
| 9 | #10587628 | — | 2 | 1 | no | — |
| 10 | #10587623 | C22 | 1 | 2 | yes | — |
| 11 | #10587571 | — | 2 | 2 | no | — |
| 12 | #10587568 | — | 2 | 2 | no | — |
| 13 | #10587568 | — | 2 | 2 | no | — |
| 14 | #10587561 | — | 2 | 1 | no | — |
| 15 | C17×#99 | C22 | 1 | 2 | yes | — |
| 16 | #10587504 | C22 | 1 | 1 | yes | characteristic |
| 17 | #10587477 | — | 2 | 2 | no | — |
| 18 | #10587476 | — | 2 | 1 | no | — |
| 19 | C23×D17 | D4 | 1 | 2 | yes | complemented |
| 20 | C2×Q8×C17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C2×D4×C17 | C23 | 1 | 1 | yes | characteristic |
| 22 | C2×C22⋊D17 | — | 2 | 2 | no | — |
| 23 | C2×C22⋊D17 | — | 2 | 2 | no | — |
| 24 | #63254 | — | 4 | 1 | no | — |
| 25 | D4×D17 | — | 4 | 1 | no | — |
| 26 | #63252 | — | 2 | 1 | no | — |
| 27 | C2×D68 | — | 2 | 1 | no | — |
| 28 | C2×D68 | — | 2 | 1 | no | — |
| 29 | C2×D68 | — | 4 | 2 | no | — |
| 30 | C2×Dic34 | — | 2 | 1 | no | — |
| 31 | C22⋊C4×C17 | — | 2 | 2 | no | — |
| 32 | C22⋊C4×C17 | — | 2 | 2 | no | — |
| 33 | C42×C17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 34 | C22⋊Dic17 | D4 | 1 | 2 | yes | complemented |
| 35 | #63231 | — | 4 | 1 | no | — |
| 36 | #63230 | — | 4 | 1 | no | — |
| 37 | #63227 | — | 2 | 2 | no | — |
| 38 | #63227 | — | 2 | 2 | no | — |
| 39 | #63227 | — | 2 | 2 | no | — |
| 40 | #63227 | — | 4 | 2 | no | — |
| 41 | C4×Dic17 | — | 2 | 1 | no | — |
| 42 | #63223 | — | 2 | 1 | no | — |
| 43 | C23×C17 | C2×D4 | 1 | 2 | yes | — |
| 44 | C22×D17 | — | 2 | 2 | no | — |
| 45 | C22×D17 | — | 2 | 2 | no | — |
| 46 | C22×D17 | — | 4 | 2 | no | — |
| 47 | C22×D17 | — | 8 | 2 | no | — |
| 48 | Q8×C17 | — | 4 | 1 | no | — |
| 49 | D4×C17 | — | 4 | 1 | no | — |
| 50 | C2×C4×C17 | — | 2 | 1 | no | — |
| 51 | C2×C4×C17 | — | 2 | 1 | no | — |
| 52 | C2×C4×C17 | — | 4 | 2 | no | — |
| 53 | C2×C4×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C22⋊D17 | — | 4 | 2 | no | — |
| 55 | C22⋊D17 | — | 4 | 1 | no | — |
| 56 | C22⋊D17 | — | 8 | 2 | no | — |
| 57 | C2×Dic17 | — | 2 | 2 | no | — |
| 58 | C2×Dic17 | — | 2 | 2 | no | — |
| 59 | D68 | — | 2 | 1 | no | — |
| 60 | D68 | — | 8 | 1 | no | — |
| 61 | D68 | — | 8 | 1 | no | — |
| 62 | D68 | — | 8 | 4 | no | — |
| 63 | C4×D17 | — | 4 | 1 | no | — |
| 64 | Dic34 | — | 2 | 1 | no | — |
| 65 | Dic34 | — | 8 | 1 | no | — |
| 66 | #3622 | — | 4 | 1 | no | — |
| 67 | #2199 | — | 17 | 1 | no | maximal, Sylow |
| 68 | C22×C17 | — | 2 | 2 | no | — |
| 69 | C22×C17 | — | 4 | 2 | no | — |
| 70 | C22×C17 | #120 | 1 | 1 | yes | characteristic |
| 71 | C2×D17 | — | 4 | 2 | no | — |
| 72 | C2×D17 | — | 4 | 2 | no | — |
| 73 | C2×D17 | — | 4 | 4 | no | — |
| 74 | C2×D17 | — | 4 | 1 | no | — |
| 75 | C2×D17 | — | 4 | 2 | no | — |
| 76 | C2×D17 | — | 8 | 2 | no | — |
| 77 | C2×D17 | — | 8 | 2 | no | — |
| 78 | C4×C17 | — | 2 | 1 | no | — |
| 79 | C4×C17 | — | 4 | 1 | no | — |
| 80 | C4×C17 | — | 4 | 1 | no | — |
| 81 | C4×C17 | — | 8 | 2 | no | — |
| 82 | Dic17 | — | 4 | 2 | no | — |
| 83 | Dic17 | — | 4 | 1 | no | — |
| 84 | H(3,𝔽4) | — | 17 | 1 | no | — |
| 85 | B(4,𝔽2) | — | 17 | 2 | no | — |
| 86 | #455 | — | 17 | 1 | no | — |
| 87 | #354 | D17 | 1 | 1 | yes | characteristic, complemented |
| 88 | AΣL(1,𝔽16) | — | 17 | 2 | no | — |
| 89 | C2×C17 | — | 2 | 1 | no | — |
| 90 | C2×C17 | — | 4 | 2 | no | — |
| 91 | C2×C17 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 92 | D17 | — | 8 | 2 | no | — |
| 93 | D17 | — | 8 | 1 | no | — |
| 94 | H(5,𝔽2) | — | 17 | 1 | no | — |
| 95 | #137 | — | 34 | 1 | no | — |
| 96 | #124 | — | 34 | 2 | no | — |
| 97 | #124 | — | 34 | 1 | no | — |
| 98 | #124 | — | 34 | 1 | no | — |
| 99 | #124 | C2×D17 | 1 | 1 | yes | characteristic |
| 100 | #120 | — | 17 | 2 | no | — |
| 101 | #120 | — | 34 | 2 | no | — |
| 102 | #120 | — | 34 | 2 | no | — |
| 103 | #104 | — | 34 | 1 | no | — |
| 104 | #100 | — | 17 | 1 | no | — |
| 105 | #99 | — | 34 | 2 | no | — |
| 106 | #99 | C2×D17 | 1 | 2 | yes | — |
| 107 | C17 | #2199 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 108 | C24 | — | 17 | 2 | no | — |
| 109 | #41 | — | 34 | 1 | no | — |
| 110 | #41 | — | 68 | 1 | no | — |
| 111 | C2×Q8 | — | 34 | 1 | no | — |
| 112 | C2×Q8 | D68 | 1 | 1 | yes | characteristic, complemented |
| 113 | C2×D4 | — | 34 | 1 | no | — |
| 114 | C2×D4 | — | 34 | 1 | no | — |
| 115 | C2×D4 | — | 34 | 2 | no | — |
| 116 | C2×D4 | — | 34 | 2 | no | — |
| 117 | C2×D4 | — | 68 | 2 | no | — |
| 118 | C2×D4 | — | 68 | 1 | no | — |
| 119 | C2×D4 | C22×D17 | 1 | 1 | yes | characteristic |
| 120 | Q16 | — | 68 | 1 | no | — |
| 121 | SD16 | — | 68 | 1 | no | — |
| 122 | M16 | — | 34 | 1 | no | — |
| 123 | C22⋊C4 | — | 2 | 2 | no | — |
| 124 | C22⋊C4 | — | 2 | 2 | no | — |
| 125 | C22⋊C4 | — | 17 | 2 | no | — |
| 126 | C22⋊C4 | — | 34 | 2 | no | — |
| 127 | C22⋊C4 | — | 34 | 2 | no | — |
| 128 | C22⋊C4 | — | 34 | 2 | no | — |
| 129 | C22⋊C4 | — | 68 | 2 | no | — |
| 130 | C42 | — | 34 | 1 | no | — |
| 131 | C42 | D68 | 1 | 1 | yes | characteristic, complemented |
| 132 | C23 | — | 34 | 2 | no | — |
| 133 | C23 | — | 34 | 2 | no | — |
| 134 | C23 | — | 68 | 2 | no | — |
| 135 | C23 | — | 136 | 2 | no | — |
| 136 | C23 | D4×D17 | 1 | 2 | yes | — |
| 137 | Q8 | — | 4 | 1 | no | — |
| 138 | Q8 | — | 34 | 1 | no | — |
| 139 | Q8 | — | 136 | 1 | no | — |
| 140 | D4 | — | 4 | 1 | no | — |
| 141 | D4 | — | 34 | 1 | no | — |
| 142 | D4 | — | 68 | 2 | no | — |
| 143 | D4 | — | 68 | 1 | no | — |
| 144 | D4 | — | 136 | 1 | no | — |
| 145 | D4 | — | 136 | 1 | no | — |
| 146 | D4 | — | 136 | 4 | no | — |
| 147 | D4 | — | 136 | 2 | no | — |
| 148 | C2×C4 | — | 2 | 1 | no | — |
| 149 | C2×C4 | — | 2 | 1 | no | — |
| 150 | C2×C4 | — | 4 | 2 | no | — |
| 151 | C2×C4 | — | 34 | 2 | no | — |
| 152 | C2×C4 | — | 34 | 2 | no | — |
| 153 | C2×C4 | — | 68 | 1 | no | — |
| 154 | C2×C4 | C2×D68 | 1 | 1 | yes | characteristic |
| 155 | C8 | — | 68 | 1 | no | — |
| 156 | C22 | — | 2 | 2 | no | — |
| 157 | C22 | — | 4 | 2 | no | — |
| 158 | C22 | — | 68 | 2 | no | — |
| 159 | C22 | — | 68 | 2 | no | — |
| 160 | C22 | — | 68 | 4 | no | — |
| 161 | C22 | — | 68 | 1 | no | — |
| 162 | C22 | — | 68 | 2 | no | — |
| 163 | C22 | — | 136 | 2 | no | — |
| 164 | C22 | — | 136 | 2 | no | — |
| 165 | C22 | #10587568 | 1 | 1 | yes | characteristic |
| 166 | C4 | — | 2 | 1 | no | — |
| 167 | C4 | — | 4 | 1 | no | — |
| 168 | C4 | — | 4 | 1 | no | — |
| 169 | C4 | — | 8 | 2 | no | — |
| 170 | C4 | — | 68 | 2 | no | — |
| 171 | C4 | — | 68 | 1 | no | — |
| 172 | C2 | — | 2 | 1 | no | — |
| 173 | C2 | — | 4 | 2 | no | — |
| 174 | C2 | — | 136 | 2 | no | — |
| 175 | C2 | — | 136 | 1 | no | — |
| 176 | C2 | #11762997 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11763551 | C2 | semidirect |
| 2. #11762997 | C2 | semidirect |
| 3. #11762986 | C2 | semidirect |
| 4. #11762595 | C2 | semidirect |
| 5. C17×#354 | C2 | semidirect |
| 19. C23×D17 | D4 | semidirect |
| 20. C2×Q8×C17 | D4 | semidirect |
| 33. C42×C17 | D4 | semidirect |
| 34. C22⋊Dic17 | D4 | semidirect |
| 87. #354 | D17 | semidirect |
| 107. C17 | #2199 | semidirect |
| 112. C2×Q8 | D68 | semidirect |
| 131. C42 | D68 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11763551 | 2 |
| #11762997 | 2 |
| #11762986 | 2 |
| #11762595 | 2 |
| C17×#354 | 2 |
| #2199 | 17 |