Finite Group Databasebeta

← C22×C5×C7×C11×ΓL(1,𝔽8) #44568929

Subgroups — C22×C5×C7×C11×ΓL(1,𝔽8) #44568929

Order 32340.

200 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×C5×C7×C11×ΓL(1,𝔽8)C213yesmaximal, direct product factor
2C22×C5×C72×C11C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C11×ΓL(1,𝔽8)C2211yescharacteristic, direct product factor, Hall
4C22×C7×C11×ΓL(1,𝔽8)C511yesmaximal, characteristic, direct product factor, Hall
5C2×C5×C72×C11C2×C313yescomplemented
6C22×C5×C11×ΓL(1,𝔽8)C711yesmaximal, characteristic, direct product factor
7C22×C3×C5×C7×C11—71nomaximal
8C2×C7×C11×ΓL(1,𝔽8)C2×C513yesdirect product factor
9C22×C5×C7×ΓL(1,𝔽8)C1111yesmaximal, characteristic, direct product factor, Hall
10C5×C72×C11C22×C311yescharacteristic, complemented, Hall
11C2×C5×C11×ΓL(1,𝔽8)C2×C713yesdirect product factor
12C2×C3×C5×C7×C11—73no—
13C22×C72×C11C3×C511yescharacteristic, complemented, Hall
14C7×C11×ΓL(1,𝔽8)C22×C511yescharacteristic, direct product factor, Hall
15C22×C5×C7×C11—32no—
16C22×C5×C7×C11ΓL(1,𝔽8)11yescharacteristic, direct product factor
17C22×C5×C7×C11C3×C711yescharacteristic, complemented
18C2×C5×C7×ΓL(1,𝔽8)C2×C1113yesdirect product factor
19C3×C5×C7×C11—71no—
20C5×C11×ΓL(1,𝔽8)C22×C711yescharacteristic, direct product factor
21C2×C72×C11C2×C3×C513yescomplemented
22C22×C5×C72C3×C1111yescharacteristic, complemented, Hall
23C22×C3×C7×C11—71no—
24C22×C11×ΓL(1,𝔽8)C5×C711yescharacteristic, direct product factor
25C2×C5×C7×C11—36no—
26C2×C5×C7×C11C2×ΓL(1,𝔽8)13yesdirect product factor
27C2×C5×C7×C11C2×C3×C713yescomplemented
28C5×C7×ΓL(1,𝔽8)C22×C1111yescharacteristic, direct product factor, Hall
29C22×C3×C5×C11—71noHall
30C22×C7×ΓL(1,𝔽8)C5×C1111yescharacteristic, direct product factor, Hall
31C72×C11C22×C3×C511yescharacteristic, complemented, Hall
32C2×C5×C72C2×C3×C1113yescomplemented
33C2×C3×C7×C11—73no—
34C2×C11×ΓL(1,𝔽8)C2×C5×C713yesdirect product factor
35C22×C3×C5×C7—71no—
36C22×C5×ΓL(1,𝔽8)C7×C1111yescharacteristic, direct product factor
37C5×C7×C11—32no—
38C5×C7×C11C22×ΓL(1,𝔽8)11yescharacteristic, direct product factor
39C5×C7×C11C22×C3×C711yescharacteristic, complemented
40C2×C3×C5×C11—73no—
41C22×C7×C11—32no—
42C22×C7×C11C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
43C22×C7×C11C3×C5×C711yescharacteristic, complemented
44C2×C7×ΓL(1,𝔽8)C2×C5×C1113yesdirect product factor
45C5×C72C22×C3×C1111yescharacteristic, complemented, Hall
46C3×C7×C11—71no—
47C11×ΓL(1,𝔽8)C22×C5×C711yescharacteristic, direct product factor
48C22×C5×C11C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
49C2×C3×C5×C7—73no—
50C2×C5×ΓL(1,𝔽8)C2×C7×C1113yesdirect product factor
51C22×C72C3×C5×C1111yescharacteristic, complemented, Hall
52C3×C5×C11—71noHall
53C2×C7×C11—36no—
54C2×C7×C11C2×C5×ΓL(1,𝔽8)13yesdirect product factor
55C2×C7×C11C2×C3×C5×C713yescomplemented
56C7×ΓL(1,𝔽8)C22×C5×C1111yescharacteristic, direct product factor, Hall
57C22×C5×C7—32no—
58C22×C5×C7C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
59C22×C5×C7C3×C7×C1111yescharacteristic, complemented
60C22×C3×C11—71noHall
61C2×C5×C11C2×C7×ΓL(1,𝔽8)13yesdirect product factor
62C3×C5×C7—71no—
63C5×ΓL(1,𝔽8)C22×C7×C1111yescharacteristic, direct product factor
64C2×C72C2×C3×C5×C1113yescomplemented
65C22×C3×C7—71no—
66C22×ΓL(1,𝔽8)C5×C7×C1111yescharacteristic, direct product factor
67C7×C11—32no—
68C7×C11C22×C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
69C7×C11C22×C3×C5×C711yescharacteristic, complemented
70C2×C5×C7—36no—
71C2×C5×C7C2×C11×ΓL(1,𝔽8)13yesdirect product factor
72C2×C5×C7C2×C3×C7×C1113yescomplemented
73C2×C3×C11—73no—
74C22×C3×C5—71noHall
75C5×C11C22×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
76C72C22×C3×C5×C1111yescharacteristic, complemented, Sylow
77C22×C11C5×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
78C2×C3×C7—73no—
79C2×ΓL(1,𝔽8)C2×C5×C7×C1113yesdirect product factor
80C5×C7—32no—
81C5×C7C22×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
82C5×C7C22×C3×C7×C1111yescharacteristic, complemented
83C3×C11—71noHall
84C2×C3×C5—73no—
85C22×C7—32no—
86C22×C7C5×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
87C22×C7C3×C5×C7×C1111yescharacteristic, complemented
88C2×C11C2×C5×C7×ΓL(1,𝔽8)13yesdirect product factor
89C3×C7—71no—
90ΓL(1,𝔽8)C22×C5×C7×C1111yescharacteristic, direct product factor
91C22×C5C7×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
92C3×C5—71noHall
93C2×C7—36no—
94C2×C7C2×C3×C5×C7×C1113yescomplemented
95C2×C7C2×C5×C11×ΓL(1,𝔽8)13yesdirect product factor
96C22×C3—71noHall
97C11C22×C5×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
98C2×C5C2×C7×C11×ΓL(1,𝔽8)13yesdirect product factor
99C7—32no—
100C7C22×C3×C5×C7×C1111yescharacteristic, complemented
101C7C22×C5×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
102C2×C3—73no—
103C5C22×C7×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
104C22C5×C7×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
105C3—71noSylow
106C2C2×C5×C7×C11×ΓL(1,𝔽8)13yesdirect product factor
subgroupG/H
center, hypercenterC22×C5×C7×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C5×C7×C11
FrattiniC1G
Fitting, socleC22×C5×C72×C11C3
LCSG ▷ C7
UCSC1 ◁ C22×C5×C7×C11

Complements

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

NH ≅ G/Ntype
1. C2×C5×C7×C11×ΓL(1,𝔽8)C2direct (N × H)
2. C22×C5×C72×C11C3semidirect
3. C5×C7×C11×ΓL(1,𝔽8)C22direct (N × H)
4. C22×C7×C11×ΓL(1,𝔽8)C5direct (N × H)
5. C2×C5×C72×C11C2×C3semidirect
6. C22×C5×C11×ΓL(1,𝔽8)C7direct (N × H)
8. C2×C7×C11×ΓL(1,𝔽8)C2×C5direct (N × H)
9. C22×C5×C7×ΓL(1,𝔽8)C11direct (N × H)
10. C5×C72×C11C22×C3semidirect
11. C2×C5×C11×ΓL(1,𝔽8)C2×C7direct (N × H)
13. C22×C72×C11C3×C5semidirect
14. C7×C11×ΓL(1,𝔽8)C22×C5direct (N × H)
16. C22×C5×C7×C11ΓL(1,𝔽8)direct (N × H)
17. C22×C5×C7×C11C3×C7semidirect
18. C2×C5×C7×ΓL(1,𝔽8)C2×C11direct (N × H)
20. C5×C11×ΓL(1,𝔽8)C22×C7direct (N × H)
21. C2×C72×C11C2×C3×C5semidirect
22. C22×C5×C72C3×C11semidirect
24. C22×C11×ΓL(1,𝔽8)C5×C7direct (N × H)
26. C2×C5×C7×C11C2×ΓL(1,𝔽8)direct (N × H)
27. C2×C5×C7×C11C2×C3×C7semidirect
28. C5×C7×ΓL(1,𝔽8)C22×C11direct (N × H)
30. C22×C7×ΓL(1,𝔽8)C5×C11direct (N × H)
31. C72×C11C22×C3×C5semidirect
32. C2×C5×C72C2×C3×C11semidirect
34. C2×C11×ΓL(1,𝔽8)C2×C5×C7direct (N × H)
36. C22×C5×ΓL(1,𝔽8)C7×C11direct (N × H)
38. C5×C7×C11C22×ΓL(1,𝔽8)direct (N × H)
39. C5×C7×C11C22×C3×C7semidirect
42. C22×C7×C11C5×ΓL(1,𝔽8)direct (N × H)
43. C22×C7×C11C3×C5×C7semidirect
44. C2×C7×ΓL(1,𝔽8)C2×C5×C11direct (N × H)
45. C5×C72C22×C3×C11semidirect
47. C11×ΓL(1,𝔽8)C22×C5×C7direct (N × H)
48. C22×C5×C11C7×ΓL(1,𝔽8)direct (N × H)
50. C2×C5×ΓL(1,𝔽8)C2×C7×C11direct (N × H)
51. C22×C72C3×C5×C11semidirect
54. C2×C7×C11C2×C5×ΓL(1,𝔽8)direct (N × H)
55. C2×C7×C11C2×C3×C5×C7semidirect
56. C7×ΓL(1,𝔽8)C22×C5×C11direct (N × H)
58. C22×C5×C7C11×ΓL(1,𝔽8)direct (N × H)
59. C22×C5×C7C3×C7×C11semidirect
61. C2×C5×C11C2×C7×ΓL(1,𝔽8)direct (N × H)
63. C5×ΓL(1,𝔽8)C22×C7×C11direct (N × H)
64. C2×C72C2×C3×C5×C11semidirect
66. C22×ΓL(1,𝔽8)C5×C7×C11direct (N × H)
68. C7×C11C22×C5×ΓL(1,𝔽8)direct (N × H)
69. C7×C11C22×C3×C5×C7semidirect
71. C2×C5×C7C2×C11×ΓL(1,𝔽8)direct (N × H)
72. C2×C5×C7C2×C3×C7×C11semidirect
75. C5×C11C22×C7×ΓL(1,𝔽8)direct (N × H)
76. C72C22×C3×C5×C11semidirect
77. C22×C11C5×C7×ΓL(1,𝔽8)direct (N × H)
79. C2×ΓL(1,𝔽8)C2×C5×C7×C11direct (N × H)
81. C5×C7C22×C11×ΓL(1,𝔽8)direct (N × H)
82. C5×C7C22×C3×C7×C11semidirect
86. C22×C7C5×C11×ΓL(1,𝔽8)direct (N × H)
87. C22×C7C3×C5×C7×C11semidirect
88. C2×C11C2×C5×C7×ΓL(1,𝔽8)direct (N × H)
90. ΓL(1,𝔽8)C22×C5×C7×C11direct (N × H)
91. C22×C5C7×C11×ΓL(1,𝔽8)direct (N × H)
94. C2×C7C2×C3×C5×C7×C11semidirect
95. C2×C7C2×C5×C11×ΓL(1,𝔽8)direct (N × H)
97. C11C22×C5×C7×ΓL(1,𝔽8)direct (N × H)
98. C2×C5C2×C7×C11×ΓL(1,𝔽8)direct (N × H)
100. C7C22×C3×C5×C7×C11semidirect
101. C7C22×C5×C11×ΓL(1,𝔽8)direct (N × H)
103. C5C22×C7×C11×ΓL(1,𝔽8)direct (N × H)
104. C22C5×C7×C11×ΓL(1,𝔽8)direct (N × H)
106. C2C2×C5×C7×C11×ΓL(1,𝔽8)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C7×C11×ΓL(1,𝔽8)2
C22×C5×C72×C113
C22×C7×C11×ΓL(1,𝔽8)5
C22×C5×C11×ΓL(1,𝔽8)7
C22×C3×C5×C7×C117
C22×C5×C7×ΓL(1,𝔽8)11