Finite Group Databasebeta

← C2×C3⋊(ΓL(1,𝔽8)⋊D97) #37567941

Subgroups — C2×C3⋊(ΓL(1,𝔽8)⋊D97) #37567941

Order 24444.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×ΓL(1,𝔽8)×C97C211yesmaximal, characteristic, complemented
2C3⋊(ΓL(1,𝔽8)⋊D97)C212yesmaximal, direct product factor
3C2×D2037C311yesmaximal, characteristic, complemented
4C2×ΓL(1,𝔽8)⋊D97—31nomaximal
5C3×ΓL(1,𝔽8)×C97C2211yescharacteristic, complemented, Hall
6ΓL(1,𝔽8)⋊D97—32no—
7D2037C2×C312yescomplemented
8C2×ΓL(1,𝔽8)×C97—21no—
9C2×ΓL(1,𝔽8)×C97D311yescharacteristic, complemented
10C2×C3×C7×C97C2×C311yescharacteristic, complemented
11C2×C3×D291—71nomaximal, Hall
12C2×D679—31noHall
13C3×C7×C97C22×C311yescharacteristic, complemented
14ΓL(1,𝔽8)×C97—21no—
15ΓL(1,𝔽8)×C97C2×D311yescharacteristic, complemented
16C2×C32×C97—141no—
17C3×D291—72no—
18C2×C7×C97C3×D311yescharacteristic, complemented
19D679—32no—
20C2×D291—71no—
21C2×C3×D97—211no—
22C32×C97—141noHall
23C7×C97C2×C3×D311yescharacteristic, complemented, Hall
24C2×C3×C97—71no—
25C2×C3×C97—141no—
26C2×C3×C97AGL(1,𝔽7)11yescharacteristic, complemented
27D291—72no—
28C3×D97—212no—
29C2×D97—211noHall
30C3×C97—71no—
31C3×C97—141no—
32C3×C97C2×AGL(1,𝔽7)11yescharacteristic, complemented
33C2×C3⋊AGL(1,𝔽7)—971nomaximal, Hall
34C2×C97C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
35D97—212no—
36C2×C3×ΓL(1,𝔽8)D9711yescharacteristic, complemented
37C3⋊AGL(1,𝔽7)—972no—
38C97C2×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
39C2×D21—971no—
40C2×AGL(1,𝔽7)—2911no—
41C3×ΓL(1,𝔽8)C2×D9711yescharacteristic, complemented, Hall
42C2×C3×C7C3×D9711yescharacteristic, complemented
43D21—972no—
44C2×ΓL(1,𝔽8)—21no—
45C2×ΓL(1,𝔽8)D29111yescharacteristic, complemented
46AGL(1,𝔽7)—2912no—
47C2×C3×D3—6791noHall
48C2×D7—2911noHall
49C3×C7C2×C3×D9711yescharacteristic, complemented
50ΓL(1,𝔽8)—21no—
51ΓL(1,𝔽8)C2×D29111yescharacteristic, complemented
52C2×C32—141no—
53C3×D3—6792no—
54C2×C7C3×D29111yescharacteristic, complemented
55D7—2912no—
56C22×C3—20371no—
57C2×D3—6791no—
58C32—141noSylow
59C7C2×C3×D29111yescharacteristic, complemented, Sylow
60C2×C3—141no—
61C2×C3—141no—
62C2×C3—20372no—
63C2×C3ΓL(1,𝔽8)⋊D9711yescharacteristic, complemented
64D3—6792no—
65C22—20371noSylow
66C3—141no—
67C3—141no—
68C3C2×ΓL(1,𝔽8)⋊D9711yescharacteristic, complemented
69C2—20372no—
70C2C3⋊(ΓL(1,𝔽8)⋊D97)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊(ΓL(1,𝔽8)⋊D97)
commutator, cosocle, hypocenterC3×C7×C97C22×C3
FrattiniC1G
Fitting, socleC2×C3×C7×C97C2×C3
LCSG ▷ C3×C7×C97
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×ΓL(1,𝔽8)×C97C2semidirect
2. C3⋊(ΓL(1,𝔽8)⋊D97)C2direct (N × H)
3. C2×D2037C3semidirect
5. C3×ΓL(1,𝔽8)×C97C22semidirect
7. D2037C2×C3semidirect
9. C2×ΓL(1,𝔽8)×C97D3semidirect
10. C2×C3×C7×C97C2×C3semidirect
13. C3×C7×C97C22×C3semidirect
15. ΓL(1,𝔽8)×C97C2×D3semidirect
18. C2×C7×C97C3×D3semidirect
23. C7×C97C2×C3×D3semidirect
26. C2×C3×C97AGL(1,𝔽7)semidirect
32. C3×C97C2×AGL(1,𝔽7)semidirect
34. C2×C97C3⋊AGL(1,𝔽7)semidirect
36. C2×C3×ΓL(1,𝔽8)D97semidirect
38. C97C2×C3⋊AGL(1,𝔽7)semidirect
41. C3×ΓL(1,𝔽8)C2×D97semidirect
42. C2×C3×C7C3×D97semidirect
45. C2×ΓL(1,𝔽8)D291semidirect
49. C3×C7C2×C3×D97semidirect
51. ΓL(1,𝔽8)C2×D291semidirect
54. C2×C7C3×D291semidirect
59. C7C2×C3×D291semidirect
63. C2×C3ΓL(1,𝔽8)⋊D97semidirect
68. C3C2×ΓL(1,𝔽8)⋊D97semidirect
70. C2C3⋊(ΓL(1,𝔽8)⋊D97)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)×C972
C3⋊(ΓL(1,𝔽8)⋊D97)2
C2×D20373
C2×ΓL(1,𝔽8)⋊D973
C2×C3×D2917
C2×C3⋊AGL(1,𝔽7)97