Finite Group Databasebeta

← C2×C3⋊(ΓL(1,𝔽8)⋊D71) #31822081

Subgroups — C2×C3⋊(ΓL(1,𝔽8)⋊D71) #31822081

Order 17892.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×ΓL(1,𝔽8)×C71C211yesmaximal, characteristic, complemented
2C3⋊(ΓL(1,𝔽8)⋊D71)C212yesmaximal, direct product factor
3C2×D1491C311yesmaximal, characteristic, complemented
4C2×ΓL(1,𝔽8)⋊D71—31nomaximal
5C3×ΓL(1,𝔽8)×C71C2211yescharacteristic, complemented, Hall
6ΓL(1,𝔽8)⋊D71—32no—
7D1491C2×C312yescomplemented
8C2×ΓL(1,𝔽8)×C71—21no—
9C2×ΓL(1,𝔽8)×C71D311yescharacteristic, complemented
10C2×C3×C7×C71C2×C311yescharacteristic, complemented
11C2×C3×D213—71nomaximal, Hall
12C2×D497—31noHall
13C3×C7×C71C22×C311yescharacteristic, complemented
14ΓL(1,𝔽8)×C71—21no—
15ΓL(1,𝔽8)×C71C2×D311yescharacteristic, complemented
16C2×C32×C71—141no—
17C3×D213—72no—
18C2×C7×C71C3×D311yescharacteristic, complemented
19D497—32no—
20C2×D213—71no—
21C2×C3×D71—211no—
22C32×C71—141noHall
23C7×C71C2×C3×D311yescharacteristic, complemented, Hall
24C2×C3×C71—141no—
25C2×C3×C71—141no—
26C2×C3×C71AGL(1,𝔽7)11yescharacteristic, complemented
27D213—72no—
28C3×D71—212no—
29C2×D71—211noHall
30C2×C3⋊AGL(1,𝔽7)—711nomaximal, Hall
31C3×C71—141no—
32C3×C71—141no—
33C3×C71C2×AGL(1,𝔽7)11yescharacteristic, complemented
34C2×C71C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
35D71—212no—
36C2×C3×ΓL(1,𝔽8)D7111yescharacteristic, complemented
37C3⋊AGL(1,𝔽7)—712no—
38C2×D21—711no—
39C2×AGL(1,𝔽7)—2131no—
40C71C2×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
41C3×ΓL(1,𝔽8)C2×D7111yescharacteristic, complemented, Hall
42C2×C3×C7C3×D7111yescharacteristic, complemented
43D21—712no—
44C2×ΓL(1,𝔽8)—21no—
45C2×ΓL(1,𝔽8)D21311yescharacteristic, complemented
46AGL(1,𝔽7)—2132no—
47C2×C3×D3—4971noHall
48C2×D7—2131noHall
49C3×C7C2×C3×D7111yescharacteristic, complemented
50ΓL(1,𝔽8)—21no—
51ΓL(1,𝔽8)C2×D21311yescharacteristic, complemented
52C2×C32—141no—
53C3×D3—4972no—
54C2×C7C3×D21311yescharacteristic, complemented
55D7—2132no—
56C22×C3—14911no—
57C2×D3—4971no—
58C32—141noSylow
59C7C2×C3×D21311yescharacteristic, complemented, Sylow
60C2×C3—141no—
61C2×C3—141no—
62C2×C3—14912no—
63C2×C3ΓL(1,𝔽8)⋊D7111yescharacteristic, complemented
64D3—4972no—
65C22—14911noSylow
66C3—141no—
67C3—141no—
68C3C2×ΓL(1,𝔽8)⋊D7111yescharacteristic, complemented
69C2—14912no—
70C2C3⋊(ΓL(1,𝔽8)⋊D71)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊(ΓL(1,𝔽8)⋊D71)
commutator, cosocle, hypocenterC3×C7×C71C22×C3
FrattiniC1G
Fitting, socleC2×C3×C7×C71C2×C3
LCSG ▷ C3×C7×C71
UCSC1 ◁ C2

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×ΓL(1,𝔽8)×C71C2semidirect
2. C3⋊(ΓL(1,𝔽8)⋊D71)C2direct (N × H)
3. C2×D1491C3semidirect
5. C3×ΓL(1,𝔽8)×C71C22semidirect
7. D1491C2×C3semidirect
9. C2×ΓL(1,𝔽8)×C71D3semidirect
10. C2×C3×C7×C71C2×C3semidirect
13. C3×C7×C71C22×C3semidirect
15. ΓL(1,𝔽8)×C71C2×D3semidirect
18. C2×C7×C71C3×D3semidirect
23. C7×C71C2×C3×D3semidirect
26. C2×C3×C71AGL(1,𝔽7)semidirect
33. C3×C71C2×AGL(1,𝔽7)semidirect
34. C2×C71C3⋊AGL(1,𝔽7)semidirect
36. C2×C3×ΓL(1,𝔽8)D71semidirect
40. C71C2×C3⋊AGL(1,𝔽7)semidirect
41. C3×ΓL(1,𝔽8)C2×D71semidirect
42. C2×C3×C7C3×D71semidirect
45. C2×ΓL(1,𝔽8)D213semidirect
49. C3×C7C2×C3×D71semidirect
51. ΓL(1,𝔽8)C2×D213semidirect
54. C2×C7C3×D213semidirect
59. C7C2×C3×D213semidirect
63. C2×C3ΓL(1,𝔽8)⋊D71semidirect
68. C3C2×ΓL(1,𝔽8)⋊D71semidirect
70. C2C3⋊(ΓL(1,𝔽8)⋊D71)direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×ΓL(1,𝔽8)×C712
C3⋊(ΓL(1,𝔽8)⋊D71)2
C2×D14913
C2×ΓL(1,𝔽8)⋊D713
C2×C3×D2137
C2×C3⋊AGL(1,𝔽7)71