Finite Group Databasebeta

← C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7)) #30633558

Subgroups — C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7)) #30633558

Order 16170.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×C7×C11×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented, Hall
2C5⋊(C7⋊D77)C311yesmaximal, characteristic, complemented, Hall
3C11⋊(ΓL(1,𝔽8)⋊D7)—51nomaximal, Hall
4C5×C72×C11C2×C311yescharacteristic, complemented, Hall
5C5⋊(C11⋊AGL(1,𝔽7))—71nomaximal
6C3×D385—71nomaximal
7C7×C11×ΓL(1,𝔽8)D511yescharacteristic, complemented, Hall
8C5⋊(ΓL(1,𝔽8)⋊D7)—111nomaximal, Hall
9C3×C5×C7×C11—71no—
10C5×C11×ΓL(1,𝔽8)D711yescharacteristic, complemented
11C7⋊D77—51noHall
12D385—71no—
13D385—71no—
14D385—212no—
15C5×C7×ΓL(1,𝔽8)D1111yescharacteristic, complemented, Hall
16C72×C11C3×D511yescharacteristic, complemented, Hall
17C5⋊(C7⋊D7)—111noHall
18C3×D77—351no—
19C11⋊AGL(1,𝔽7)—351no—
20C5×C7×C11—32no—
21C5×C7×C11AGL(1,𝔽7)11yescharacteristic, complemented
22C5×C7×C11C3×D711yescharacteristic, complemented
23C3×D55—491noHall
24ΓL(1,𝔽8)⋊D7—551noHall
25C5×C72C3×D1111yescharacteristic, complemented, Hall
26C3×C7×C11—71no—
27C11×ΓL(1,𝔽8)D3511yescharacteristic, complemented
28C3×D35—771no—
29C5⋊AGL(1,𝔽7)—771no—
30C3×C5×C11—71noHall
31D77—351no—
32D77—351no—
33D77—1052no—
34C7×ΓL(1,𝔽8)D5511yescharacteristic, complemented, Hall
35D55—491noHall
36C3×C5×C7—71no—
37C5×ΓL(1,𝔽8)D7711yescharacteristic, complemented
38C7⋊D7—551noHall
39C7×C11—32no—
40C7×C11C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
41C7×C11C3×D3511yescharacteristic, complemented
42D35—771no—
43D35—771no—
44D35—2312no—
45C3×D11—2451noHall
46C5×C11ΓL(1,𝔽8)⋊D711yescharacteristic, complemented, Hall
47C72C3×D5511yescharacteristic, complemented, Sylow
48C3×D7—3851no—
49AGL(1,𝔽7)—3851no—
50C5×C7—32no—
51C5×C7C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
52C5×C7C3×D7711yescharacteristic, complemented
53C3×C11—71noHall
54C3×D5—5391noHall
55D11—2451noHall
56C3×C7—71no—
57ΓL(1,𝔽8)D38511yescharacteristic, complemented
58C3×C5—71noHall
59D7—3851no—
60D7—3851no—
61D7—11552no—
62C11C5⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic, complemented, Sylow
63D5—5391noHall
64C7—32no—
65C7C3×D38511yescharacteristic, complemented
66C7C5⋊(C11⋊AGL(1,𝔽7))11yescharacteristic, complemented
67C2×C3—26951noHall
68C5C11⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic, complemented, Sylow
69C3—71noSylow
70C2—26951noSylow
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C11C2×C3
LCSG ▷ C5×C72×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5×C7×C11×ΓL(1,𝔽8)C2semidirect
2. C5⋊(C7⋊D77)C3semidirect
4. C5×C72×C11C2×C3semidirect
7. C7×C11×ΓL(1,𝔽8)D5semidirect
10. C5×C11×ΓL(1,𝔽8)D7semidirect
15. C5×C7×ΓL(1,𝔽8)D11semidirect
16. C72×C11C3×D5semidirect
21. C5×C7×C11AGL(1,𝔽7)semidirect
22. C5×C7×C11C3×D7semidirect
25. C5×C72C3×D11semidirect
27. C11×ΓL(1,𝔽8)D35semidirect
34. C7×ΓL(1,𝔽8)D55semidirect
37. C5×ΓL(1,𝔽8)D77semidirect
40. C7×C11C5⋊AGL(1,𝔽7)semidirect
41. C7×C11C3×D35semidirect
46. C5×C11ΓL(1,𝔽8)⋊D7semidirect
47. C72C3×D55semidirect
51. C5×C7C11⋊AGL(1,𝔽7)semidirect
52. C5×C7C3×D77semidirect
57. ΓL(1,𝔽8)D385semidirect
62. C11C5⋊(ΓL(1,𝔽8)⋊D7)semidirect
65. C7C3×D385semidirect
66. C7C5⋊(C11⋊AGL(1,𝔽7))semidirect
68. C5C11⋊(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
C5×C7×C11×ΓL(1,𝔽8)2
C5⋊(C7⋊D77)3
C11⋊(ΓL(1,𝔽8)⋊D7)5
C5⋊(C11⋊AGL(1,𝔽7))7
C3×D3857
C5⋊(ΓL(1,𝔽8)⋊D7)11