Finite Group Databasebeta

← C2×C3×C11×ΓL(1,𝔽8)×D13 #49114834

Subgroups — C2×C3×C11×ΓL(1,𝔽8)×D13 #49114834

Order 36036.

240 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 112 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×C3×C11×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C3×C11×ΓL(1,𝔽8)×D13C212yesmaximal, direct product factor
3C2×C3×C7×C11×D13C311yesmaximal, characteristic, complemented
4C2×C11×ΓL(1,𝔽8)×D13C313yesmaximal, direct product factor
5C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C11×ΓL(1,𝔽8)×D13C2×C316yesdirect product factor
7C3×C7×C11×D13C2×C312yescomplemented
8C2×C11×C13×ΓL(1,𝔽8)C2×C313yescomplemented
9C2×C3×C7×C11×C13C2×C311yescharacteristic, complemented
10C2×C32×C11×D13—71nomaximal, Hall
11C2×C7×C11×D13C3211yescharacteristic, complemented, Hall
12C2×C3×ΓL(1,𝔽8)×D13C1111yesmaximal, characteristic, direct product factor, Hall
13C11×C13×ΓL(1,𝔽8)C22×C313yescomplemented
14C3×C7×C11×C13C22×C311yescharacteristic, complemented
15C22×C3×C11×ΓL(1,𝔽8)—131nomaximal, Hall
16C32×C11×D13—72no—
17C2×C32×C11×C13—71no—
18C7×C11×D13C2×C3212yescomplemented
19C2×C7×C11×C13C2×C3211yescharacteristic, complemented
20C2×C3×C11×D13—73no—
21C2×C3×C11×D13ΓL(1,𝔽8)11yescharacteristic, direct product factor
22C2×C3×C13×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
23C3×ΓL(1,𝔽8)×D13C2×C1112yesdirect product factor
24C2×C3×C11×ΓL(1,𝔽8)—132no—
25C2×C3×C11×ΓL(1,𝔽8)D1311yescharacteristic, direct product factor
26C32×C11×C13—71noHall
27C2×C3×C7×D13C3×C1111yescharacteristic, complemented
28C2×ΓL(1,𝔽8)×D13C3×C1113yesdirect product factor
29C7×C11×C13C22×C3211yescharacteristic, complemented, Hall
30C22×C3×C7×C11—131no—
31C22×C11×ΓL(1,𝔽8)—133no—
32C2×C3×C11×C13—73no—
33C2×C3×C11×C13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
34C3×C11×D13—76no—
35C3×C11×D13C2×ΓL(1,𝔽8)12yesdirect product factor
36C3×C13×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented, Hall
37C3×C11×ΓL(1,𝔽8)C2×D1311yescharacteristic, direct product factor, Hall
38C2×C11×D13C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
39C2×C3×C7×C13C2×C3×C1111yescharacteristic, complemented
40C3×C7×D13C2×C3×C1112yescomplemented
41C2×C13×ΓL(1,𝔽8)C2×C3×C1113yescomplemented
42ΓL(1,𝔽8)×D13C2×C3×C1116yesdirect product factor
43C2×C32×D13—71noHall
44C2×C3×C7×C11—132no—
45C2×C3×C7×C11C3×D1311yescharacteristic, complemented
46C2×C11×ΓL(1,𝔽8)—136no—
47C2×C11×ΓL(1,𝔽8)C3×D1313yesdirect product factor
48C3×C11×C13—73no—
49C3×C11×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented
50C22×C32×C11—911noHall
51C2×C7×D13C32×C1111yescharacteristic, complemented, Hall
52C22×C7×C11—131noHall
53C2×C11×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
54C11×D13C2×C3×ΓL(1,𝔽8)12yesdirect product factor
55C3×C7×C13C22×C3×C1111yescharacteristic, complemented
56C13×ΓL(1,𝔽8)C22×C3×C1113yescomplemented
57C22×C3×ΓL(1,𝔽8)—131noHall
58C2×C32×C13—71no—
59C32×D13—72no—
60C3×C7×C11C2×C3×D1311yescharacteristic, complemented
61C11×ΓL(1,𝔽8)C2×C3×D1313yesdirect product factor
62C2×C32×C11—71no—
63C2×C32×C11—912no—
64C2×C7×C13C2×C32×C1111yescharacteristic, complemented
65C7×D13C2×C32×C1112yescomplemented
66C2×C3×D13—73no—
67C2×C3×D13C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
68C2×C7×C11—132no—
69C2×C7×C11C32×D1311yescharacteristic, complemented
70C11×C13C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
71C22×C3×C11—131no—
72C22×C3×C11—913no—
73C2×C3×ΓL(1,𝔽8)—132no—
74C2×C3×ΓL(1,𝔽8)C11×D1311yescharacteristic, direct product factor
75C32×C13—71noHall
76C32×C11—71noHall
77C7×C13C22×C32×C1111yescharacteristic, complemented, Hall
78C22×C3×C7—131no—
79C22×ΓL(1,𝔽8)—133no—
80C2×C3×C13—73no—
81C2×C3×C13C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
82C3×D13—76no—
83C3×D13C2×C11×ΓL(1,𝔽8)12yesdirect product factor
84C7×C11C2×C32×D1311yescharacteristic, complemented, Hall
85C2×C3×C11—73no—
86C2×C3×C11—132no—
87C2×C3×C11—916no—
88C2×C3×C11ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
89C3×ΓL(1,𝔽8)C2×C11×D1311yescharacteristic, direct product factor, Hall
90C2×D13C3×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
91C22×C11—131noHall
92C2×C3×C7—132no—
93C2×C3×C7C3×C11×D1311yescharacteristic, complemented
94C2×ΓL(1,𝔽8)—136no—
95C2×ΓL(1,𝔽8)C3×C11×D1313yesdirect product factor
96C3×C13—73no—
97C3×C13C22×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
98C22×C32—911noHall
99C3×C11—73no—
100C3×C11C2×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
101C22×C7—131noHall
102C2×C13C2×C3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
103D13C2×C3×C11×ΓL(1,𝔽8)12yesdirect product factor
104C2×C11—132no—
105C2×C11C3×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
106C3×C7C2×C3×C11×D1311yescharacteristic, complemented
107ΓL(1,𝔽8)C2×C3×C11×D1313yesdirect product factor
108C2×C32—71no—
109C2×C32—912no—
110C2×C7—132no—
111C2×C7C32×C11×D1311yescharacteristic, complemented
112C13C22×C3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
113C22×C3—131no—
114C22×C3—913no—
115C11C2×C3×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor, Sylow
116C32—71noSylow
117C7C2×C32×C11×D1311yescharacteristic, complemented, Sylow
118C2×C3—73no—
119C2×C3—132no—
120C2×C3—916no—
121C2×C3C11×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
122C22—131noSylow
123C3—73no—
124C3C2×C11×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
125C2—132no—
126C2C3×C11×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C11ΓL(1,𝔽8)×D13
commutator, cosocle, hypocenterC7×C13C22×C32×C11
FrattiniC1G
Fitting, socleC2×C3×C7×C11×C13C2×C3
LCSG ▷ C7×C13
UCSC1 ◁ C2×C3×C11

Complements

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

NH ≅ G/Ntype
1. C2×C3×C11×C13×ΓL(1,𝔽8)C2semidirect
2. C3×C11×ΓL(1,𝔽8)×D13C2direct (N × H)
3. C2×C3×C7×C11×D13C3semidirect
4. C2×C11×ΓL(1,𝔽8)×D13C3direct (N × H)
5. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
6. C11×ΓL(1,𝔽8)×D13C2×C3direct (N × H)
7. C3×C7×C11×D13C2×C3semidirect
8. C2×C11×C13×ΓL(1,𝔽8)C2×C3semidirect
9. C2×C3×C7×C11×C13C2×C3semidirect
11. C2×C7×C11×D13C32semidirect
12. C2×C3×ΓL(1,𝔽8)×D13C11direct (N × H)
13. C11×C13×ΓL(1,𝔽8)C22×C3semidirect
14. C3×C7×C11×C13C22×C3semidirect
18. C7×C11×D13C2×C32semidirect
19. C2×C7×C11×C13C2×C32semidirect
21. C2×C3×C11×D13ΓL(1,𝔽8)direct (N × H)
22. C2×C3×C13×ΓL(1,𝔽8)C2×C11semidirect
23. C3×ΓL(1,𝔽8)×D13C2×C11direct (N × H)
25. C2×C3×C11×ΓL(1,𝔽8)D13direct (N × H)
27. C2×C3×C7×D13C3×C11semidirect
28. C2×ΓL(1,𝔽8)×D13C3×C11direct (N × H)
29. C7×C11×C13C22×C32semidirect
33. C2×C3×C11×C13C2×ΓL(1,𝔽8)semidirect
35. C3×C11×D13C2×ΓL(1,𝔽8)direct (N × H)
36. C3×C13×ΓL(1,𝔽8)C22×C11semidirect
37. C3×C11×ΓL(1,𝔽8)C2×D13direct (N × H)
38. C2×C11×D13C3×ΓL(1,𝔽8)direct (N × H)
39. C2×C3×C7×C13C2×C3×C11semidirect
40. C3×C7×D13C2×C3×C11semidirect
41. C2×C13×ΓL(1,𝔽8)C2×C3×C11semidirect
42. ΓL(1,𝔽8)×D13C2×C3×C11direct (N × H)
45. C2×C3×C7×C11C3×D13semidirect
47. C2×C11×ΓL(1,𝔽8)C3×D13direct (N × H)
49. C3×C11×C13C22×ΓL(1,𝔽8)semidirect
51. C2×C7×D13C32×C11semidirect
53. C2×C11×C13C2×C3×ΓL(1,𝔽8)semidirect
54. C11×D13C2×C3×ΓL(1,𝔽8)direct (N × H)
55. C3×C7×C13C22×C3×C11semidirect
56. C13×ΓL(1,𝔽8)C22×C3×C11semidirect
60. C3×C7×C11C2×C3×D13semidirect
61. C11×ΓL(1,𝔽8)C2×C3×D13direct (N × H)
64. C2×C7×C13C2×C32×C11semidirect
65. C7×D13C2×C32×C11semidirect
67. C2×C3×D13C11×ΓL(1,𝔽8)direct (N × H)
69. C2×C7×C11C32×D13semidirect
70. C11×C13C22×C3×ΓL(1,𝔽8)semidirect
74. C2×C3×ΓL(1,𝔽8)C11×D13direct (N × H)
77. C7×C13C22×C32×C11semidirect
81. C2×C3×C13C2×C11×ΓL(1,𝔽8)semidirect
83. C3×D13C2×C11×ΓL(1,𝔽8)direct (N × H)
84. C7×C11C2×C32×D13semidirect
88. C2×C3×C11ΓL(1,𝔽8)×D13direct (N × H)
89. C3×ΓL(1,𝔽8)C2×C11×D13direct (N × H)
90. C2×D13C3×C11×ΓL(1,𝔽8)direct (N × H)
93. C2×C3×C7C3×C11×D13semidirect
95. C2×ΓL(1,𝔽8)C3×C11×D13direct (N × H)
97. C3×C13C22×C11×ΓL(1,𝔽8)semidirect
100. C3×C11C2×ΓL(1,𝔽8)×D13direct (N × H)
102. C2×C13C2×C3×C11×ΓL(1,𝔽8)semidirect
103. D13C2×C3×C11×ΓL(1,𝔽8)direct (N × H)
105. C2×C11C3×ΓL(1,𝔽8)×D13direct (N × H)
106. C3×C7C2×C3×C11×D13semidirect
107. ΓL(1,𝔽8)C2×C3×C11×D13direct (N × H)
111. C2×C7C32×C11×D13semidirect
112. C13C22×C3×C11×ΓL(1,𝔽8)semidirect
115. C11C2×C3×ΓL(1,𝔽8)×D13direct (N × H)
117. C7C2×C32×C11×D13semidirect
121. C2×C3C11×ΓL(1,𝔽8)×D13direct (N × H)
124. C3C2×C11×ΓL(1,𝔽8)×D13direct (N × H)
126. C2C3×C11×ΓL(1,𝔽8)×D13direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C11×C13×ΓL(1,𝔽8)2
C3×C11×ΓL(1,𝔽8)×D132
C2×C3×C7×C11×D133
C2×C11×ΓL(1,𝔽8)×D133
C2×C32×C11×D137
C2×C3×ΓL(1,𝔽8)×D1311
C22×C3×C11×ΓL(1,𝔽8)13