Finite Group Databasebeta

← C7×C3⋊((C5⋊D5)⋊ΓL(1,𝔽8)) #36348008

Subgroups — C7×C3⋊((C5⋊D5)⋊ΓL(1,𝔽8)) #36348008

Order 22050.

168 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×C7⋊(C52⋊C3)C211yesmaximal, characteristic, complemented, Hall
2C72×C3⋊(C5⋊D5)C311yesmaximal, characteristic, complemented
3C7×(C5⋊D5)⋊ΓL(1,𝔽8)—31nomaximal
4C7×C7⋊(C52⋊C3)—21no—
5C7×C7⋊(C52⋊C3)D311yescharacteristic, complemented
6C3×C52×C72C2×C311yescharacteristic, complemented
7C3⋊((C5⋊D5)⋊ΓL(1,𝔽8))C711yesmaximal, characteristic, direct product factor
8C7×C3⋊((C5⋊D5)⋊C3)—71nomaximal
9C72×C5⋊D5—31noHall
10C3×C7⋊(C52⋊C3)C2×C711yescharacteristic, complemented
11C3×C7×C52⋊C3—71no—
12C72×D15—152no—
13C52×C72C3×D311yescharacteristic, complemented, Hall
14C7×C3⋊(C5⋊D5)—32no—
15C7×C3⋊(C5⋊D5)ΓL(1,𝔽8)11yescharacteristic, complemented
16C7×C3⋊(C5⋊D5)C3×C711yescharacteristic, complemented
17(C5⋊D5)⋊ΓL(1,𝔽8)—31no—
18C7×(C5⋊D5)⋊C3—211no—
19D3×C7×ΓL(1,𝔽8)—251nomaximal, Hall
20C3×C5×C72—32no—
21C3×C52×C7—32no—
22C3×C52×C7C2×ΓL(1,𝔽8)11yescharacteristic, complemented
23C3×C52×C7C2×C3×C711yescharacteristic, complemented
24C7⋊(C52⋊C3)—21no—
25C7⋊(C52⋊C3)D3×C711yescharacteristic, complemented
26C7×C52⋊C3—71no—
27C7×C52⋊C3—141no—
28C72×D5—452no—
29C3⋊((C5⋊D5)⋊C3)—71noHall
30C3×C7×ΓL(1,𝔽8)—501noHall
31C7×C5⋊D5—31no—
32C7×C5⋊D5—31no—
33C7×C5⋊D5—92no—
34D3×C72—251no—
35C2×C7×ΓL(1,𝔽8)—751no—
36C5×C72—32no—
37C3×C52⋊C3—71noHall
38C7×D15—152no—
39C7×D15—152no—
40C7×D15—1512no—
41C52×C7—32no—
42C52×C7D3×ΓL(1,𝔽8)11yescharacteristic, complemented
43C52×C7C3×D3×C711yescharacteristic, complemented
44C3⋊(C5⋊D5)C7×ΓL(1,𝔽8)11yescharacteristic, complemented
45(C5⋊D5)⋊C3—211no—
46C3×C72(C5⋊D5)⋊C311yescharacteristic, complemented
47C7×ΓL(1,𝔽8)—501no—
48C7×ΓL(1,𝔽8)—501no—
49C3×D3×C7—1751no—
50D3×ΓL(1,𝔽8)—251no—
51C3×C5×C7—312no—
52C3×C5×C7—32no—
53C3×C5×C7—32no—
54C2×C72—751noHall
55C3×C52C2×C7×ΓL(1,𝔽8)11yescharacteristic, complemented
56C52⋊C3—71no—
57C52⋊C3—141no—
58C7×D5—452no—
59C7×D5—452no—
60C7×D5—4512no—
61C32×C7—3501no—
62C3×ΓL(1,𝔽8)—501no—
63C5⋊D5—31noHall
64C72C3⋊((C5⋊D5)⋊C3)11yescharacteristic, complemented, Sylow
65C2×C3×C7—5251no—
66D3×C7—251no—
67D3×C7—251no—
68D3×C7—752no—
69C2×ΓL(1,𝔽8)—751no—
70C5×C7—312no—
71C5×C7—32no—
72C5×C7—32no—
73D15—152no—
74C52D3×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
75C3×C7—32no—
76C3×C7—3501no—
77C3×C7—3501no—
78C3×C7C7×(C5⋊D5)⋊C311yescharacteristic, complemented
79C3×C7(C5⋊D5)⋊ΓL(1,𝔽8)11yescharacteristic, complemented
80ΓL(1,𝔽8)—501no—
81ΓL(1,𝔽8)—501no—
82C3×D3—1751noHall
83C3×C5—32no—
84C2×C7—751no—
85C2×C7—751no—
86C2×C7—2252no—
87D5—452no—
88C32—3501noSylow
89C7—32no—
90C7C7×C3⋊((C5⋊D5)⋊C3)11yescharacteristic, complemented
91C7C3⋊((C5⋊D5)⋊ΓL(1,𝔽8))11yescharacteristic, direct product factor
92C2×C3—5251no—
93D3—251no—
94C5—32no—
95C3—3501no—
96C3—3501no—
97C3C7×(C5⋊D5)⋊ΓL(1,𝔽8)11yescharacteristic, complemented
98C2—751noSylow
subgroupG/H
center, hypercenterC7C3⋊((C5⋊D5)⋊ΓL(1,𝔽8))
commutator, cosocle, hypocenterC3×C52×C7C2×C3×C7
FrattiniC1G
Fitting, socleC3×C52×C72C2×C3
LCSG ▷ C3×C52×C7
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
1. C3×C7×C7⋊(C52⋊C3)C2semidirect
2. C72×C3⋊(C5⋊D5)C3semidirect
5. C7×C7⋊(C52⋊C3)D3semidirect
6. C3×C52×C72C2×C3semidirect
7. C3⋊((C5⋊D5)⋊ΓL(1,𝔽8))C7direct (N × H)
10. C3×C7⋊(C52⋊C3)C2×C7semidirect
13. C52×C72C3×D3semidirect
15. C7×C3⋊(C5⋊D5)ΓL(1,𝔽8)semidirect
16. C7×C3⋊(C5⋊D5)C3×C7semidirect
22. C3×C52×C7C2×ΓL(1,𝔽8)semidirect
23. C3×C52×C7C2×C3×C7semidirect
25. C7⋊(C52⋊C3)D3×C7semidirect
42. C52×C7D3×ΓL(1,𝔽8)semidirect
43. C52×C7C3×D3×C7semidirect
44. C3⋊(C5⋊D5)C7×ΓL(1,𝔽8)semidirect
46. C3×C72(C5⋊D5)⋊C3semidirect
55. C3×C52C2×C7×ΓL(1,𝔽8)semidirect
64. C72C3⋊((C5⋊D5)⋊C3)semidirect
74. C52D3×C7×ΓL(1,𝔽8)semidirect
78. C3×C7C7×(C5⋊D5)⋊C3semidirect
79. C3×C7(C5⋊D5)⋊ΓL(1,𝔽8)semidirect
90. C7C7×C3⋊((C5⋊D5)⋊C3)semidirect
91. C7C3⋊((C5⋊D5)⋊ΓL(1,𝔽8))direct (N × H)
97. C3C7×(C5⋊D5)⋊ΓL(1,𝔽8)semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C7⋊(C52⋊C3)2
C72×C3⋊(C5⋊D5)3
C7×(C5⋊D5)⋊ΓL(1,𝔽8)3
C3⋊((C5⋊D5)⋊ΓL(1,𝔽8))7
C7×C3⋊((C5⋊D5)⋊C3)7
D3×C7×ΓL(1,𝔽8)25