Finite Group Databasebeta

← C3×C7×ΓL(1,𝔽8)×C5⋊D5 #36347980

Subgroups — C3×C7×ΓL(1,𝔽8)×C5⋊D5 #36347980

Order 22050.

448 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 144 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×C52×C7×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented, Hall
2C3×C72×C5⋊D5C311yesmaximal, characteristic, complemented
3C7×ΓL(1,𝔽8)×C5⋊D5C313yesmaximal, direct product factor
4C3×C7×D5×ΓL(1,𝔽8)—56nomaximal
5C52×C7×ΓL(1,𝔽8)C2×C313yescomplemented
6C3×C52×C72C2×C311yescharacteristic, complemented
7C32×C7×C5⋊D5—71nomaximal
8C3×ΓL(1,𝔽8)×C5⋊D5C711yesmaximal, characteristic, direct product factor
9C72×C5⋊D5C3211yescharacteristic, complemented, Hall
10C3×C5×C7×ΓL(1,𝔽8)D516yescomplemented
11C32×C52×C7—71no—
12C3×C52×ΓL(1,𝔽8)C2×C711yescharacteristic, complemented
13C3×C72×D5—56no—
14C7×D5×ΓL(1,𝔽8)—518no—
15C52×C72C2×C3211yescharacteristic, complemented, Hall
16C3×C7×C5⋊D5—32no—
17C3×C7×C5⋊D5—73no—
18C3×C7×C5⋊D5ΓL(1,𝔽8)11yescharacteristic, direct product factor
19C3×C7×C5⋊D5C3×C711yescharacteristic, complemented
20ΓL(1,𝔽8)×C5⋊D5C3×C713yesdirect product factor
21C2×C3×C7×ΓL(1,𝔽8)—251noHall
22C3×C5×C72C3×D516yescomplemented
23C5×C7×ΓL(1,𝔽8)C3×D5118yescomplemented
24C32×C7×D5—356no—
25C3×D5×ΓL(1,𝔽8)—56no—
26C3×C52×C7—32no—
27C3×C52×C7—73no—
28C3×C52×C7C2×ΓL(1,𝔽8)11yescharacteristic, complemented
29C3×C52×C7C2×C3×C711yescharacteristic, complemented
30C52×ΓL(1,𝔽8)C2×C3×C713yescomplemented
31C72×D5—56no—
32C32×C5⋊D5—71noHall
33C3×C7×ΓL(1,𝔽8)C5⋊D511yescharacteristic, direct product factor, Hall
34C7×C5⋊D5—32no—
35C7×C5⋊D5C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
36C7×C5⋊D5C32×C711yescharacteristic, complemented
37C32×C5×C7—76no—
38C3×C5×ΓL(1,𝔽8)C7×D516yescomplemented
39C2×C3×C72—251no—
40C2×C7×ΓL(1,𝔽8)—253no—
41C5×C72C32×D516yescomplemented
42C32×C52—71noHall
43C3×C7×D5—56no—
44C3×C7×D5—56no—
45C3×C7×D5—1512no—
46C3×C7×D5—3518no—
47D5×ΓL(1,𝔽8)—518no—
48C52×C7—32no—
49C52×C7C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
50C52×C7C2×C32×C711yescharacteristic, complemented
51C3×C5⋊D5—73no—
52C3×C5⋊D5C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor
53C3×C72C3×C5⋊D511yescharacteristic, complemented
54C7×ΓL(1,𝔽8)C3×C5⋊D513yesdirect product factor
55C2×C32×C7—1751no—
56C2×C3×ΓL(1,𝔽8)—251no—
57C3×C5×C7—312no—
58C3×C5×C7—718no—
59C3×C5×C7D5×ΓL(1,𝔽8)16yescomplemented
60C3×C5×C7C3×C7×D516yescomplemented
61C5×ΓL(1,𝔽8)C3×C7×D5118yescomplemented
62C2×C72—251noHall
63C32×D5—356no—
64C3×C52—73no—
65C3×C52C2×C7×ΓL(1,𝔽8)11yescharacteristic, complemented
66C7×D5—56no—
67C7×D5—56no—
68C7×D5—1512no—
69C32×C7—71no—
70C3×ΓL(1,𝔽8)C7×C5⋊D511yescharacteristic, direct product factor
71C5⋊D5C3×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
72C72C32×C5⋊D511yescharacteristic, complemented, Sylow
73C32×C5—76no—
74C2×C3×C7—251no—
75C2×C3×C7—251no—
76C2×C3×C7—752no—
77C2×C3×C7—1753no—
78C2×ΓL(1,𝔽8)—253no—
79C5×C7—312no—
80C5×C7C3×D5×ΓL(1,𝔽8)16yescomplemented
81C5×C7C32×C7×D516yescomplemented
82C3×D5—56no—
83C3×D5—3518no—
84C52C2×C3×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
85C3×C7—32no—
86C3×C7—73no—
87C3×C7ΓL(1,𝔽8)×C5⋊D511yescharacteristic, direct product factor
88C3×C7C3×C7×C5⋊D511yescharacteristic, complemented
89ΓL(1,𝔽8)C3×C7×C5⋊D513yesdirect product factor
90C2×C32—1751noHall
91C3×C5—718no—
92C3×C5C7×D5×ΓL(1,𝔽8)16yescomplemented
93C2×C7—251no—
94C2×C7—251no—
95C2×C7—752no—
96D5—56no—
97C32—71noSylow
98C7—32no—
99C7C3×ΓL(1,𝔽8)×C5⋊D511yescharacteristic, direct product factor
100C7C32×C7×C5⋊D511yescharacteristic, complemented
101C2×C3—251no—
102C2×C3—1753no—
103C5C3×C7×D5×ΓL(1,𝔽8)16yescomplemented
104C3—73no—
105C3C7×ΓL(1,𝔽8)×C5⋊D511yescharacteristic, direct product factor
106C2—251noSylow
subgroupG/H
center, hypercenterC3×C7ΓL(1,𝔽8)×C5⋊D5
commutator, cosocle, hypocenterC52×C7C2×C32×C7
FrattiniC1G
Fitting, socleC3×C52×C72C2×C3
LCSG ▷ C52×C7
UCSC1 ◁ C3×C7

Complements

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

NH ≅ G/Ntype
1. C3×C52×C7×ΓL(1,𝔽8)C2semidirect
2. C3×C72×C5⋊D5C3semidirect
3. C7×ΓL(1,𝔽8)×C5⋊D5C3direct (N × H)
5. C52×C7×ΓL(1,𝔽8)C2×C3semidirect
6. C3×C52×C72C2×C3semidirect
8. C3×ΓL(1,𝔽8)×C5⋊D5C7direct (N × H)
9. C72×C5⋊D5C32semidirect
10. C3×C5×C7×ΓL(1,𝔽8)D5semidirect
12. C3×C52×ΓL(1,𝔽8)C2×C7semidirect
15. C52×C72C2×C32semidirect
18. C3×C7×C5⋊D5ΓL(1,𝔽8)direct (N × H)
19. C3×C7×C5⋊D5C3×C7semidirect
20. ΓL(1,𝔽8)×C5⋊D5C3×C7direct (N × H)
22. C3×C5×C72C3×D5semidirect
23. C5×C7×ΓL(1,𝔽8)C3×D5semidirect
28. C3×C52×C7C2×ΓL(1,𝔽8)semidirect
29. C3×C52×C7C2×C3×C7semidirect
30. C52×ΓL(1,𝔽8)C2×C3×C7semidirect
33. C3×C7×ΓL(1,𝔽8)C5⋊D5direct (N × H)
35. C7×C5⋊D5C3×ΓL(1,𝔽8)direct (N × H)
36. C7×C5⋊D5C32×C7semidirect
38. C3×C5×ΓL(1,𝔽8)C7×D5semidirect
41. C5×C72C32×D5semidirect
49. C52×C7C2×C3×ΓL(1,𝔽8)semidirect
50. C52×C7C2×C32×C7semidirect
52. C3×C5⋊D5C7×ΓL(1,𝔽8)direct (N × H)
53. C3×C72C3×C5⋊D5semidirect
54. C7×ΓL(1,𝔽8)C3×C5⋊D5direct (N × H)
59. C3×C5×C7D5×ΓL(1,𝔽8)semidirect
60. C3×C5×C7C3×C7×D5semidirect
61. C5×ΓL(1,𝔽8)C3×C7×D5semidirect
65. C3×C52C2×C7×ΓL(1,𝔽8)semidirect
70. C3×ΓL(1,𝔽8)C7×C5⋊D5direct (N × H)
71. C5⋊D5C3×C7×ΓL(1,𝔽8)direct (N × H)
72. C72C32×C5⋊D5semidirect
80. C5×C7C3×D5×ΓL(1,𝔽8)semidirect
81. C5×C7C32×C7×D5semidirect
84. C52C2×C3×C7×ΓL(1,𝔽8)semidirect
87. C3×C7ΓL(1,𝔽8)×C5⋊D5direct (N × H)
88. C3×C7C3×C7×C5⋊D5semidirect
89. ΓL(1,𝔽8)C3×C7×C5⋊D5direct (N × H)
92. C3×C5C7×D5×ΓL(1,𝔽8)semidirect
99. C7C3×ΓL(1,𝔽8)×C5⋊D5direct (N × H)
100. C7C32×C7×C5⋊D5semidirect
103. C5C3×C7×D5×ΓL(1,𝔽8)semidirect
105. C3C7×ΓL(1,𝔽8)×C5⋊D5direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C52×C7×ΓL(1,𝔽8)2
C3×C72×C5⋊D53
C7×ΓL(1,𝔽8)×C5⋊D53
C3×C7×D5×ΓL(1,𝔽8)5
C32×C7×C5⋊D57
C3×ΓL(1,𝔽8)×C5⋊D57