Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
146 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 17 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 | C7×C22⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #89999 | — | 3 | 1 | no | maximal, Hall |
| 3 | C2×C7⋊S4 | — | 4 | 1 | no | maximal |
| 4 | C7⋊GL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 5 | C22×C7×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#1143 | — | 3 | 2 | no | — |
| 7 | #6493 | — | 3 | 1 | no | — |
| 8 | #6409 | — | 3 | 2 | no | — |
| 9 | C22⋊(C7⋊C8) | — | 3 | 1 | no | — |
| 10 | #5931 | — | 7 | 1 | no | maximal, Hall |
| 11 | C2×C7×A4 | — | 4 | 1 | no | — |
| 12 | C7⋊S4 | — | 4 | 2 | no | — |
| 13 | C7×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 14 | C7×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 15 | C2×C7×Q8 | — | 3 | 2 | no | — |
| 16 | C2×C7×Q8 | — | 6 | 1 | no | — |
| 17 | C22×C4×C7 | — | 3 | 1 | no | — |
| 18 | C2×C22⋊D7 | — | 6 | 1 | no | — |
| 19 | C2×D28 | — | 3 | 1 | no | — |
| 20 | #1143 | — | 6 | 2 | no | — |
| 21 | #1143 | — | 6 | 2 | no | — |
| 22 | (C2×D7)⋊C4 | — | 6 | 1 | no | — |
| 23 | C4⋊Dic7 | — | 3 | 1 | no | — |
| 24 | C2×C7⋊C8 | — | 3 | 2 | no | — |
| 25 | C22⋊SL(2,𝔽3) | D7 | 1 | 1 | yes | characteristic, complemented |
| 26 | C2×D21 | — | 16 | 1 | no | — |
| 27 | C7×A4 | — | 4 | 1 | no | — |
| 28 | #448 | — | 21 | 1 | no | Sylow |
| 29 | C23×C7 | S4 | 1 | 1 | yes | characteristic |
| 30 | C22×D7 | — | 6 | 1 | no | — |
| 31 | C7×Q8 | — | 2 | 1 | no | — |
| 32 | C7×Q8 | — | 3 | 2 | no | — |
| 33 | C7×Q8 | — | 6 | 1 | no | — |
| 34 | C7×Q8 | S4 | 1 | 2 | yes | complemented |
| 35 | C2×C4×C7 | — | 3 | 1 | no | — |
| 36 | C2×C4×C7 | — | 3 | 1 | no | — |
| 37 | C2×C4×C7 | — | 6 | 2 | no | — |
| 38 | C22⋊D7 | — | 12 | 1 | no | — |
| 39 | C22⋊D7 | — | 12 | 1 | no | — |
| 40 | C2×Dic7 | — | 6 | 1 | no | — |
| 41 | D28 | — | 6 | 2 | no | — |
| 42 | C7⋊C8 | — | 6 | 2 | no | — |
| 43 | C2×S4 | — | 28 | 1 | no | — |
| 44 | GL(2,𝔽3) | — | 28 | 2 | no | — |
| 45 | C2×C3×C7 | — | 16 | 1 | no | — |
| 46 | D21 | — | 16 | 2 | no | — |
| 47 | C22×Q8 | D21 | 1 | 1 | yes | characteristic, complemented |
| 48 | C2×SD16 | — | 21 | 2 | no | — |
| 49 | #121 | — | 21 | 1 | no | — |
| 50 | #103 | — | 21 | 2 | no | — |
| 51 | C22⋊C8 | — | 21 | 1 | no | — |
| 52 | C22×C7 | — | 3 | 1 | no | — |
| 53 | C22×C7 | — | 3 | 1 | no | — |
| 54 | C22×C7 | GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 55 | C2×D7 | — | 12 | 1 | no | — |
| 56 | C2×D7 | — | 12 | 1 | no | — |
| 57 | C2×D7 | — | 12 | 1 | no | — |
| 58 | C4×C7 | — | 3 | 2 | no | — |
| 59 | C4×C7 | — | 6 | 1 | no | — |
| 60 | Dic7 | — | 12 | 1 | no | — |
| 61 | C2×A4 | — | 4 | 1 | no | — |
| 62 | S4 | — | 28 | 2 | no | — |
| 63 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 64 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 65 | C3×C7 | — | 16 | 1 | no | Hall |
| 66 | C2×Q8 | — | 3 | 2 | no | — |
| 67 | C2×Q8 | — | 6 | 1 | no | — |
| 68 | C2×D4 | — | 21 | 1 | no | — |
| 69 | C2×D4 | — | 42 | 1 | no | — |
| 70 | C22×C4 | — | 3 | 1 | no | — |
| 71 | SD16 | — | 42 | 2 | no | — |
| 72 | SD16 | — | 42 | 2 | no | — |
| 73 | C2×C8 | — | 21 | 2 | no | — |
| 74 | C4⋊C4 | — | 21 | 1 | no | — |
| 75 | C22⋊C4 | — | 42 | 1 | no | — |
| 76 | C2×C7 | — | 3 | 1 | no | — |
| 77 | C2×C7 | — | 3 | 1 | no | — |
| 78 | C2×C7 | #1020 | 1 | 1 | yes | characteristic |
| 79 | D7 | — | 24 | 1 | no | — |
| 80 | C2×D3 | — | 112 | 1 | no | — |
| 81 | A4 | — | 4 | 1 | no | — |
| 82 | C23 | — | 42 | 1 | no | — |
| 83 | C23 | C7⋊S4 | 1 | 1 | yes | characteristic |
| 84 | Q8 | — | 2 | 1 | no | — |
| 85 | Q8 | — | 3 | 2 | no | — |
| 86 | Q8 | — | 6 | 1 | no | — |
| 87 | Q8 | C7⋊S4 | 1 | 2 | yes | complemented |
| 88 | D4 | — | 42 | 2 | no | — |
| 89 | D4 | — | 84 | 1 | no | — |
| 90 | D4 | — | 84 | 1 | no | — |
| 91 | C2×C4 | — | 3 | 1 | no | — |
| 92 | C2×C4 | — | 3 | 1 | no | — |
| 93 | C2×C4 | — | 6 | 2 | no | — |
| 94 | C2×C4 | — | 42 | 1 | no | — |
| 95 | C8 | — | 42 | 2 | no | — |
| 96 | C7 | #5931 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 97 | C2×C3 | — | 16 | 1 | no | — |
| 98 | D3 | — | 112 | 2 | no | — |
| 99 | C22 | — | 3 | 1 | no | — |
| 100 | C22 | — | 3 | 1 | no | — |
| 101 | C22 | — | 84 | 1 | no | — |
| 102 | C22 | — | 84 | 1 | no | — |
| 103 | C22 | — | 84 | 1 | no | — |
| 104 | C22 | C7⋊GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 105 | C4 | — | 3 | 2 | no | — |
| 106 | C4 | — | 6 | 1 | no | — |
| 107 | C4 | — | 84 | 1 | no | — |
| 108 | C3 | — | 16 | 1 | no | Sylow |
| 109 | C2 | — | 3 | 1 | no | — |
| 110 | C2 | — | 3 | 1 | no | — |
| 111 | C2 | — | 168 | 1 | no | — |
| 112 | C2 | C7⋊#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊#1020 |
| commutator, cosocle, hypocenter | C7×C22⋊SL(2,𝔽3) | C2 |
| Fitting | C22×C7×Q8 | D3 |
| socle | C23×C7 | S4 |
| LCS | G ▷ C7×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C22⋊SL(2,𝔽3) | C2 | semidirect |
| 5. C22×C7×Q8 | D3 | semidirect |
| 25. C22⋊SL(2,𝔽3) | D7 | semidirect |
| 34. C7×Q8 | S4 | semidirect |
| 47. C22×Q8 | D21 | semidirect |
| 54. C22×C7 | GL(2,𝔽3) | semidirect |
| 87. Q8 | C7⋊S4 | semidirect |
| 96. C7 | #5931 | semidirect |
| 104. C22 | C7⋊GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C22⋊SL(2,𝔽3) | 2 |
| #89999 | 3 |
| C2×C7⋊S4 | 4 |
| C7⋊GL(2,𝔽3) | 4 |
| #5931 | 7 |