Finite Group DatabasebetaFinite Group DatabasebetaOrder 2400.
112 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5⋊#6732 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×SL(2,𝔽5) | C4 | 1 | 1 | yes | characteristic, complemented |
| 3 | #91885 | — | 5 | 1 | no | maximal |
| 4 | GL(2,𝔽5) | — | 5 | 1 | no | maximal |
| 5 | #87830 | — | 6 | 1 | no | maximal |
| 6 | C5⋊(#41⋊C3) | — | 5 | 1 | no | — |
| 7 | #6740 | — | 10 | 1 | no | maximal |
| 8 | #6732 | — | 5 | 1 | no | — |
| 9 | C2×#1044 | — | 6 | 1 | no | — |
| 10 | C2×#1043 | — | 6 | 1 | no | — |
| 11 | #6036 | — | 6 | 1 | no | — |
| 12 | #4010 | — | 15 | 1 | no | — |
| 13 | C5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 14 | #1200 | — | 10 | 1 | no | — |
| 15 | C3⋊#170 | — | 10 | 1 | no | — |
| 16 | C3×#170 | — | 10 | 1 | no | — |
| 17 | SL(2,𝔽5) | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×C5⋊D5 | — | 6 | 1 | no | — |
| 19 | #1044 | — | 6 | 2 | no | — |
| 20 | #1043 | — | 6 | 2 | no | — |
| 21 | C5×Dic5 | — | 6 | 1 | no | — |
| 22 | C5×Dic5 | — | 6 | 1 | no | — |
| 23 | GU(2,𝔽3) | — | 25 | 1 | no | Hall |
| 24 | #713 | — | 5 | 1 | no | — |
| 25 | C4×AGL(1,𝔽5) | — | 15 | 1 | no | — |
| 26 | C4×AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 27 | #700 | — | 15 | 1 | no | — |
| 28 | C2×D15 | — | 10 | 1 | no | — |
| 29 | C3×Dic5 | — | 10 | 1 | no | — |
| 30 | C5×Dic3 | — | 10 | 1 | no | — |
| 31 | C2×C52 | — | 6 | 1 | no | — |
| 32 | C5⋊D5 | — | 6 | 1 | no | — |
| 33 | C5⋊D5 | — | 6 | 1 | no | — |
| 34 | #41⋊C3 | — | 25 | 1 | no | — |
| 35 | ΓL(1,𝔽25) | — | 50 | 1 | no | — |
| 36 | C2×AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 37 | C2×AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 38 | C2×AGL(1,𝔽5) | — | 30 | 1 | no | — |
| 39 | C2×AGL(1,𝔽5) | — | 60 | 2 | no | — |
| 40 | C5×Q8 | — | 5 | 1 | no | — |
| 41 | D20 | — | 15 | 1 | no | — |
| 42 | C4×D5 | — | 15 | 1 | no | — |
| 43 | C4×D5 | — | 30 | 1 | no | — |
| 44 | #170 | — | 10 | 1 | no | — |
| 45 | #104 | — | 75 | 1 | no | Sylow |
| 46 | C2×C3×C5 | — | 10 | 1 | no | — |
| 47 | D15 | — | 20 | 1 | no | — |
| 48 | C52 | — | 6 | 1 | no | Sylow |
| 49 | C4×D3 | — | 50 | 1 | no | — |
| 50 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 51 | C3×C8 | — | 50 | 1 | no | — |
| 52 | C3⋊C8 | — | 50 | 1 | no | — |
| 53 | C2×D5 | — | 15 | 1 | no | — |
| 54 | C2×D5 | — | 30 | 1 | no | — |
| 55 | C2×D5 | — | 60 | 2 | no | — |
| 56 | AGL(1,𝔽5) | — | 30 | 2 | no | — |
| 57 | AGL(1,𝔽5) | — | 30 | 2 | no | — |
| 58 | AGL(1,𝔽5) | — | 30 | 2 | no | — |
| 59 | AGL(1,𝔽5) | — | 60 | 4 | no | — |
| 60 | C4×C5 | — | 15 | 1 | no | — |
| 61 | C4×C5 | — | 30 | 1 | no | — |
| 62 | Dic5 | — | 6 | 1 | no | — |
| 63 | Dic5 | S5 | 1 | 1 | yes | characteristic |
| 64 | #41 | — | 25 | 1 | no | — |
| 65 | M16 | — | 75 | 1 | no | — |
| 66 | C42 | — | 75 | 1 | no | — |
| 67 | C3×C5 | — | 10 | 1 | no | — |
| 68 | C2×D3 | — | 50 | 1 | no | — |
| 69 | C3×C4 | — | 50 | 1 | no | — |
| 70 | Dic3 | — | 10 | 1 | no | — |
| 71 | C2×C5 | — | 6 | 1 | no | — |
| 72 | C2×C5 | — | 12 | 2 | no | — |
| 73 | C2×C5 | A5⋊C4 | 1 | 1 | yes | characteristic |
| 74 | D5 | — | 30 | 1 | no | — |
| 75 | D5 | — | 30 | 1 | no | — |
| 76 | D5 | — | 30 | 1 | no | — |
| 77 | D5 | — | 60 | 2 | no | — |
| 78 | D5 | — | 60 | 2 | no | — |
| 79 | Q8 | — | 5 | 1 | no | — |
| 80 | D4 | — | 75 | 1 | no | — |
| 81 | C2×C4 | — | 75 | 1 | no | — |
| 82 | C2×C4 | — | 150 | 1 | no | — |
| 83 | C8 | — | 50 | 1 | no | — |
| 84 | C2×C3 | — | 10 | 1 | no | — |
| 85 | D3 | — | 100 | 1 | no | — |
| 86 | C5 | — | 6 | 1 | no | — |
| 87 | C5 | — | 12 | 2 | no | — |
| 88 | C5 | GL(2,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 89 | C22 | — | 75 | 1 | no | — |
| 90 | C4 | — | 5 | 1 | no | — |
| 91 | C4 | — | 15 | 1 | no | — |
| 92 | C4 | — | 150 | 2 | no | — |
| 93 | C3 | — | 10 | 1 | no | Sylow |
| 94 | C2 | — | 150 | 1 | no | — |
| 95 | C2 | A5⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A5⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×SL(2,𝔽5) | C4 |
| Fitting, socle | C2×C5 | A5⋊C4 |
| prime radical, cosocle | C5⋊#6732 | C2 |
| LCS | G ▷ C5×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C5×SL(2,𝔽5) | C4 | semidirect |
| 17. SL(2,𝔽5) | AGL(1,𝔽5) | semidirect |
| 88. C5 | GL(2,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5⋊#6732 | 2 |
| #91885 | 5 |
| GL(2,𝔽5) | 5 |
| #87830 | 6 |
| #6740 | 10 |