Finite Group Databasebeta

← C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7)) #17984391

Subgroups — C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7)) #17984391

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
2C3⋊(C5⋊(C7⋊D7))C311yesmaximal, characteristic, complemented
3C5⋊(ΓL(1,𝔽8)⋊D7)—31nomaximal
4C3⋊(ΓL(1,𝔽8)⋊D7)—51nomaximal, Hall
5C3×C5×C72C2×C311yescharacteristic, complemented
6C5×C7×ΓL(1,𝔽8)—21no—
7C5×C7×ΓL(1,𝔽8)D311yescharacteristic, complemented
8C3×D105—71nomaximal
9C3⋊(C5⋊AGL(1,𝔽7))—71nomaximal
10C5⋊(C7⋊D7)—31noHall
11C3×C7×ΓL(1,𝔽8)D511yescharacteristic, complemented, Hall
12C32×C5×C7—71no—
13C3×C5×ΓL(1,𝔽8)D711yescharacteristic, complemented
14C3⋊(C7⋊D7)—51no—
15ΓL(1,𝔽8)⋊D7—151no—
16C5×C72C3×D311yescharacteristic, complemented, Hall
17D105—71no—
18D105—71no—
19D105—212no—
20C3×D35—211no—
21C5⋊AGL(1,𝔽7)—211no—
22C3×C72C3×D511yescharacteristic, complemented
23C7×ΓL(1,𝔽8)—21no—
24C7×ΓL(1,𝔽8)D1511yescharacteristic, complemented
25C3×D21—351no—
26C3⋊AGL(1,𝔽7)—351no—
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—151noHall
35C3×D15—491noHall
36D35—211no—
37D35—211no—
38D35—632no—
39C32×C7—71no—
40C3×ΓL(1,𝔽8)D3511yescharacteristic, complemented
41C72C3×D1511yescharacteristic, complemented, Sylow
42C32×C5—71noHall
43D21—351no—
44D21—351no—
45D21—1052no—
46C3×D7—1051no—
47AGL(1,𝔽7)—1051no—
48C5×C7—32no—
49C5×C7C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
50C5×C7C3×D2111yescharacteristic, complemented
51D15—491no—
52C3×D5—1471no—
53C3×C7—32no—
54C3×C7—71no—
55C3×C7—141no—
56C3×C7C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
57C3×C7C3×D3511yescharacteristic, complemented
58ΓL(1,𝔽8)—21no—
59ΓL(1,𝔽8)D10511yescharacteristic, complemented
60C3×D3—2451noHall
61C3×C5—71no—
62C3×C5—141no—
63C3×C5ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
64D7—3152no—
65D7—1051no—
66D7—1051no—
67D5—1471noHall
68C32—71noSylow
69C7—32no—
70C7C3⋊(C5⋊AGL(1,𝔽7))11yescharacteristic, complemented
71C7C3×D10511yescharacteristic, complemented
72C2×C3—7351no—
73D3—2451no—
74C5C3⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic, complemented, Sylow
75C3—71no—
76C3—141no—
77C3C5⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic, complemented
78C2—7351noSylow
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C72C2×C3
LCSG ▷ C3×C5×C72
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3×C5×C7×ΓL(1,𝔽8)C2semidirect
2. C3⋊(C5⋊(C7⋊D7))C3semidirect
5. C3×C5×C72C2×C3semidirect
7. C5×C7×ΓL(1,𝔽8)D3semidirect
11. C3×C7×ΓL(1,𝔽8)D5semidirect
13. C3×C5×ΓL(1,𝔽8)D7semidirect
16. C5×C72C3×D3semidirect
22. C3×C72C3×D5semidirect
24. C7×ΓL(1,𝔽8)D15semidirect
30. C3×C5×C7AGL(1,𝔽7)semidirect
31. C3×C5×C7C3×D7semidirect
33. C5×ΓL(1,𝔽8)D21semidirect
40. C3×ΓL(1,𝔽8)D35semidirect
41. C72C3×D15semidirect
49. C5×C7C3⋊AGL(1,𝔽7)semidirect
50. C5×C7C3×D21semidirect
56. C3×C7C5⋊AGL(1,𝔽7)semidirect
57. C3×C7C3×D35semidirect
59. ΓL(1,𝔽8)D105semidirect
63. C3×C5ΓL(1,𝔽8)⋊D7semidirect
70. C7C3⋊(C5⋊AGL(1,𝔽7))semidirect
71. C7C3×D105semidirect
74. C5C3⋊(ΓL(1,𝔽8)⋊D7)semidirect
77. C3C5⋊(ΓL(1,𝔽8)⋊D7)semidirect

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
C3⋊(C5⋊(C7⋊D7))3
C5⋊(ΓL(1,𝔽8)⋊D7)3
C3⋊(ΓL(1,𝔽8)⋊D7)5
C3×D1057
C3⋊(C5⋊AGL(1,𝔽7))7