Finite Group Databasebeta

← C5×C3⋊(ΓL(1,𝔽8)⋊D7) #17984380

Subgroups — C5×C3⋊(ΓL(1,𝔽8)⋊D7) #17984380

Order 4410.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×C7×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented, Hall
2C5×C3⋊(C7⋊D7)C311yesmaximal, characteristic, complemented
3C5×ΓL(1,𝔽8)⋊D7—31nomaximal
4C3⋊(ΓL(1,𝔽8)⋊D7)C511yesmaximal, characteristic, direct product factor, Hall
5C3×C5×C72C2×C311yescharacteristic, complemented
6C5×C7×ΓL(1,𝔽8)—21no—
7C5×C7×ΓL(1,𝔽8)D311yescharacteristic, complemented
8C3×C5×D21—71nomaximal
9C5×C3⋊AGL(1,𝔽7)—71nomaximal
10C5×C7⋊D7—31noHall
11C3×C7×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented, Hall
12C32×C5×C7—71no—
13C3×C5×ΓL(1,𝔽8)D711yescharacteristic, complemented
14C3⋊(C7⋊D7)C3×C511yescharacteristic, complemented
15ΓL(1,𝔽8)⋊D7—31no—
16C5×C72C3×D311yescharacteristic, complemented, Hall
17C5×D21—71no—
18C5×D21—71no—
19C5×D21—212no—
20C3×C5×D7—211no—
21C5×AGL(1,𝔽7)—211no—
22C3×C72C2×C3×C511yescharacteristic, complemented
23C7×ΓL(1,𝔽8)—21no—
24C7×ΓL(1,𝔽8)C5×D311yescharacteristic, complemented
25C3×D21—71no—
26C3⋊AGL(1,𝔽7)—71no—
27C3×C5×C7—32no—
28C3×C5×C7—71no—
29C3×C5×C7—141no—
30C3×C5×C7AGL(1,𝔽7)11yescharacteristic, complemented
31C3×C5×C7C3×D711yescharacteristic, complemented
32C5×ΓL(1,𝔽8)—21no—
33C5×ΓL(1,𝔽8)D2111yescharacteristic, complemented
34C7⋊D7—31noHall
35C3×C5×D3—491noHall
36C5×D7—211no—
37C5×D7—211no—
38C5×D7—632no—
39C32×C7—71no—
40C3×ΓL(1,𝔽8)C5×D711yescharacteristic, complemented
41C72C3×C5×D311yescharacteristic, complemented, Sylow
42C32×C5—71noHall
43D21—71no—
44D21—71no—
45D21—212no—
46C3×D7—211no—
47AGL(1,𝔽7)—211no—
48C5×C7—32no—
49C5×C7C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
50C5×C7C3×D2111yescharacteristic, complemented
51C2×C3×C5—1471no—
52C5×D3—491no—
53C3×C7—32no—
54C3×C7—71no—
55C3×C7—141no—
56C3×C7C5×AGL(1,𝔽7)11yescharacteristic, complemented
57C3×C7C3×C5×D711yescharacteristic, complemented
58ΓL(1,𝔽8)—21no—
59ΓL(1,𝔽8)C5×D2111yescharacteristic, complemented
60C3×D3—491noHall
61C3×C5—71no—
62C3×C5—141no—
63C3×C5ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
64D7—211no—
65D7—211no—
66D7—632no—
67C2×C5—1471noHall
68C32—71noSylow
69C7—32no—
70C7C5×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
71C7C3×C5×D2111yescharacteristic, complemented
72C2×C3—1471no—
73D3—491no—
74C5C3⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic, direct product factor, Sylow
75C3—71no—
76C3—141no—
77C3C5×ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
78C2—1471noSylow
subgroupG/H
center, hypercenterC5C3⋊(ΓL(1,𝔽8)⋊D7)
commutator, cosocle, hypocenterC3×C72C2×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C72C2×C3
LCSG ▷ C3×C72
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×C7×ΓL(1,𝔽8)C2semidirect
2. C5×C3⋊(C7⋊D7)C3semidirect
4. C3⋊(ΓL(1,𝔽8)⋊D7)C5direct (N × H)
5. C3×C5×C72C2×C3semidirect
7. C5×C7×ΓL(1,𝔽8)D3semidirect
11. C3×C7×ΓL(1,𝔽8)C2×C5semidirect
13. C3×C5×ΓL(1,𝔽8)D7semidirect
14. C3⋊(C7⋊D7)C3×C5semidirect
16. C5×C72C3×D3semidirect
22. C3×C72C2×C3×C5semidirect
24. C7×ΓL(1,𝔽8)C5×D3semidirect
30. C3×C5×C7AGL(1,𝔽7)semidirect
31. C3×C5×C7C3×D7semidirect
33. C5×ΓL(1,𝔽8)D21semidirect
40. C3×ΓL(1,𝔽8)C5×D7semidirect
41. C72C3×C5×D3semidirect
49. C5×C7C3⋊AGL(1,𝔽7)semidirect
50. C5×C7C3×D21semidirect
56. C3×C7C5×AGL(1,𝔽7)semidirect
57. C3×C7C3×C5×D7semidirect
59. ΓL(1,𝔽8)C5×D21semidirect
63. C3×C5ΓL(1,𝔽8)⋊D7semidirect
70. C7C5×C3⋊AGL(1,𝔽7)semidirect
71. C7C3×C5×D21semidirect
74. C5C3⋊(ΓL(1,𝔽8)⋊D7)direct (N × H)
77. C3C5×ΓL(1,𝔽8)⋊D7semidirect

Maximal subgroups

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

subgroupindex
C3×C5×C7×ΓL(1,𝔽8)2
C5×C3⋊(C7⋊D7)3
C5×ΓL(1,𝔽8)⋊D73
C3⋊(ΓL(1,𝔽8)⋊D7)5
C3×C5×D217
C5×C3⋊AGL(1,𝔽7)7