Finite Group Databasebeta

← AGL(1,𝔽7)×D55 #17987662

Subgroups — AGL(1,𝔽7)×D55 #17987662

Order 4620.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5⋊(C11⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)×D55C211yesmaximal, characteristic, complemented
3C5×C11×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4D7×D55C311yesmaximal, characteristic, complemented, Hall
5C5×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6D11×AGL(1,𝔽7)—51nomaximal, Hall
7D385C2×C311yescharacteristic, complemented
8C7×D55C2×C311yescharacteristic, complemented
9C5×C11×D7C2×C311yescharacteristic, complemented
10C2×C3×D55—71nomaximal, Hall
11C11⋊AGL(1,𝔽7)—51no—
12C11×AGL(1,𝔽7)D511yescharacteristic, complemented
13ΓL(1,𝔽8)×D11—51no—
14D5×AGL(1,𝔽7)—111nomaximal, Hall
15C5×C7×C11C22×C311yescharacteristic, complemented, Hall
16C2×C3×C5×C11—71no—
17C3×D55—71no—
18C3×D55—71no—
19D7×D11—51noHall
20C11×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented, Hall
21C2×D55—71noHall
22C5⋊AGL(1,𝔽7)—111no—
23D5×ΓL(1,𝔽8)—111no—
24C5×AGL(1,𝔽7)D1111yescharacteristic, complemented
25C3×C5×C11—71noHall
26D77—51no—
27C7×D11—51no—
28C11×D7C3×D511yescharacteristic, complemented
29D5×D7—111noHall
30C2×C3×D11—351noHall
31C2×C5×C11—71no—
32D55—71no—
33D55AGL(1,𝔽7)11yescharacteristic, direct product factor
34C5×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
35C2×AGL(1,𝔽7)—551noHall
36C7×C11C2×C3×D511yescharacteristic, complemented, Hall
37D35—111no—
38C5×D7C3×D1111yescharacteristic, complemented
39C7×D5—111no—
40C2×C3×C11—71no—
41C3×D11—351no—
42C3×D11—351no—
43C2×C3×D5—771noHall
44C5×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
45C2×D11—351noHall
46C2×ΓL(1,𝔽8)—551no—
47AGL(1,𝔽7)—551no—
48AGL(1,𝔽7)D5511yescharacteristic, direct product factor
49C5×C7C2×C3×D1111yescharacteristic, complemented, Hall
50C3×C11—71noHall
51C2×C3×C5—71no—
52C3×D5—771no—
53C3×D5—771no—
54C2×D7—551noHall
55C2×C11—71no—
56D11—51no—
57D11—351no—
58ΓL(1,𝔽8)C2×D5511yescharacteristic, complemented, Hall
59C2×D5—771noHall
60C3×C5—71noHall
61C2×C7—551no—
62D7—551no—
63D7C3×D5511yescharacteristic, complemented
64C22×C3—3851noHall
65C11D5×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
66C2×C5—71no—
67D5—111no—
68D5—771no—
69C7C2×C3×D5511yescharacteristic, complemented, Sylow
70C2×C3—71no—
71C2×C3—3851no—
72C2×C3—3851no—
73C5D11×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
74C22—3851noSylow
75C3—71noSylow
76C2—71no—
77C2—551no—
78C2—3851no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C11C22×C3
LCSG ▷ C5×C7×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5⋊(C11⋊AGL(1,𝔽7))C2semidirect
2. ΓL(1,𝔽8)×D55C2semidirect
3. C5×C11×AGL(1,𝔽7)C2semidirect
4. D7×D55C3semidirect
5. C5×C11×ΓL(1,𝔽8)C22semidirect
7. D385C2×C3semidirect
8. C7×D55C2×C3semidirect
9. C5×C11×D7C2×C3semidirect
12. C11×AGL(1,𝔽7)D5semidirect
15. C5×C7×C11C22×C3semidirect
20. C11×ΓL(1,𝔽8)C2×D5semidirect
24. C5×AGL(1,𝔽7)D11semidirect
28. C11×D7C3×D5semidirect
33. D55AGL(1,𝔽7)direct (N × H)
34. C5×ΓL(1,𝔽8)C2×D11semidirect
36. C7×C11C2×C3×D5semidirect
38. C5×D7C3×D11semidirect
44. C5×C11C2×AGL(1,𝔽7)semidirect
48. AGL(1,𝔽7)D55direct (N × H)
49. C5×C7C2×C3×D11semidirect
58. ΓL(1,𝔽8)C2×D55semidirect
63. D7C3×D55semidirect
65. C11D5×AGL(1,𝔽7)semidirect
69. C7C2×C3×D55semidirect
73. C5D11×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C5⋊(C11⋊AGL(1,𝔽7))2
ΓL(1,𝔽8)×D552
C5×C11×AGL(1,𝔽7)2
D7×D553
D11×AGL(1,𝔽7)5
C2×C3×D557
D5×AGL(1,𝔽7)11