Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
132 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 16 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×#997 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#455 | — | 3 | 1 | no | maximal, Hall |
| 3 | C7×SL(2,ℤ/4) | — | 4 | 1 | no | maximal |
| 4 | C7×#226 | — | 4 | 2 | no | maximal |
| 5 | C7×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 6 | C7×#137 | — | 3 | 2 | no | — |
| 7 | C7×#124 | — | 3 | 1 | no | — |
| 8 | C7×#104 | — | 3 | 2 | no | — |
| 9 | C7×#100 | — | 3 | 1 | no | — |
| 10 | #5933 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 11 | C2×C7×A4 | — | 4 | 1 | no | — |
| 12 | C2×C7×A4 | — | 8 | 1 | no | — |
| 13 | C7×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 14 | C7×#41 | — | 3 | 2 | no | — |
| 15 | C2×C7×Q8 | — | 3 | 1 | no | — |
| 16 | C2×C7×D4 | — | 3 | 1 | no | — |
| 17 | C2×C7×D4 | — | 6 | 1 | no | — |
| 18 | C7×Q16 | — | 6 | 2 | no | — |
| 19 | C7×SD16 | — | 6 | 2 | no | — |
| 20 | C7×M16 | — | 3 | 2 | no | — |
| 21 | C7×C22⋊C4 | — | 6 | 1 | no | — |
| 22 | C7×C22⋊C4 | — | 6 | 1 | no | — |
| 23 | C42×C7 | — | 3 | 1 | no | — |
| 24 | #997 | C2×C7 | 1 | 1 | yes | characteristic |
| 25 | C7×A4 | — | 4 | 1 | no | — |
| 26 | C7×A4 | — | 8 | 1 | no | — |
| 27 | C7×Dic3 | — | 16 | 1 | no | — |
| 28 | #455 | — | 3 | 1 | no | Sylow |
| 29 | C23×C7 | — | 2 | 1 | no | — |
| 30 | C23×C7 | — | 3 | 1 | no | — |
| 31 | C23×C7 | S4 | 1 | 1 | yes | characteristic |
| 32 | C7×Q8 | — | 6 | 2 | no | — |
| 33 | C7×Q8 | S4 | 1 | 2 | yes | — |
| 34 | C7×D4 | — | 3 | 2 | no | — |
| 35 | C7×D4 | — | 6 | 2 | no | — |
| 36 | C2×C4×C7 | — | 3 | 1 | no | — |
| 37 | C2×C4×C7 | — | 6 | 1 | no | — |
| 38 | C2×C4×C7 | — | 6 | 1 | no | — |
| 39 | C2×C4×C7 | — | 6 | 1 | no | — |
| 40 | C7×C8 | — | 6 | 2 | no | — |
| 41 | SL(2,ℤ/4) | — | 4 | 1 | no | — |
| 42 | #226 | — | 4 | 2 | no | — |
| 43 | C2×C3×C7 | — | 16 | 1 | no | — |
| 44 | H(5,𝔽2) | D3×C7 | 1 | 1 | yes | characteristic |
| 45 | #137 | — | 3 | 2 | no | — |
| 46 | #124 | — | 3 | 1 | no | — |
| 47 | #104 | — | 3 | 2 | no | — |
| 48 | #100 | — | 3 | 1 | no | — |
| 49 | C22×C7 | — | 3 | 1 | no | — |
| 50 | C22×C7 | — | 4 | 1 | no | — |
| 51 | C22×C7 | — | 6 | 1 | no | — |
| 52 | C22×C7 | — | 8 | 1 | no | — |
| 53 | C22×C7 | — | 12 | 1 | no | — |
| 54 | C4×C7 | — | 3 | 2 | no | — |
| 55 | C4×C7 | — | 12 | 1 | no | — |
| 56 | C4×C7 | — | 12 | 1 | no | — |
| 57 | C2×A4 | — | 4 | 1 | no | — |
| 58 | C2×A4 | — | 8 | 1 | no | — |
| 59 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 60 | C3×C7 | — | 16 | 1 | no | Hall |
| 61 | #41 | — | 3 | 2 | no | — |
| 62 | C2×Q8 | — | 3 | 1 | no | — |
| 63 | C2×D4 | — | 3 | 1 | no | — |
| 64 | C2×D4 | — | 6 | 1 | no | — |
| 65 | Q16 | — | 6 | 2 | no | — |
| 66 | SD16 | — | 6 | 2 | no | — |
| 67 | M16 | — | 3 | 2 | no | — |
| 68 | C22⋊C4 | — | 6 | 1 | no | — |
| 69 | C22⋊C4 | — | 6 | 1 | no | — |
| 70 | C42 | — | 3 | 1 | no | — |
| 71 | C2×C7 | — | 6 | 1 | no | — |
| 72 | C2×C7 | — | 12 | 1 | no | — |
| 73 | C2×C7 | #1020 | 1 | 1 | yes | characteristic |
| 74 | A4 | — | 4 | 1 | no | — |
| 75 | A4 | — | 8 | 1 | no | — |
| 76 | Dic3 | — | 16 | 1 | no | — |
| 77 | C23 | — | 2 | 1 | no | — |
| 78 | C23 | — | 3 | 1 | no | — |
| 79 | C23 | C7×S4 | 1 | 1 | yes | characteristic |
| 80 | Q8 | — | 6 | 2 | no | — |
| 81 | Q8 | C7×S4 | 1 | 2 | yes | — |
| 82 | D4 | — | 3 | 2 | no | — |
| 83 | D4 | — | 6 | 2 | no | — |
| 84 | C2×C4 | — | 3 | 1 | no | — |
| 85 | C2×C4 | — | 6 | 1 | no | — |
| 86 | C2×C4 | — | 6 | 1 | no | — |
| 87 | C2×C4 | — | 6 | 1 | no | — |
| 88 | C8 | — | 6 | 2 | no | — |
| 89 | C7 | #5933 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 90 | C2×C3 | — | 16 | 1 | no | — |
| 91 | C22 | — | 3 | 1 | no | — |
| 92 | C22 | — | 4 | 1 | no | — |
| 93 | C22 | — | 6 | 1 | no | — |
| 94 | C22 | — | 8 | 1 | no | — |
| 95 | C22 | — | 12 | 1 | no | — |
| 96 | C4 | — | 3 | 2 | no | — |
| 97 | C4 | — | 12 | 1 | no | — |
| 98 | C4 | — | 12 | 1 | no | — |
| 99 | C3 | — | 16 | 1 | no | Sylow |
| 100 | C2 | — | 6 | 1 | no | — |
| 101 | C2 | — | 12 | 1 | no | — |
| 102 | C2 | C7×#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C7 | #1020 |
| commutator, cosocle, hypocenter | #997 | C2×C7 |
| Frattini | C2 | C7×#1020 |
| Fitting | C7×H(5,𝔽2) | D3 |
| prime radical | C7×#997 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. #5933 | C7 | direct (N × H) |
| 89. C7 | #5933 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#997 | 2 |
| C7×#455 | 3 |
| C7×SL(2,ℤ/4) | 4 |
| C7×#226 | 4 |
| #5933 | 7 |