Finite Group Databasebeta

← C2×C3×C13×ΓL(1,𝔽8)×D11 #49114835

Subgroups — C2×C3×C13×ΓL(1,𝔽8)×D11 #49114835

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×C13×ΓL(1,𝔽8)×D11C212yesmaximal, direct product factor
3C2×C3×C7×C13×D11C311yesmaximal, characteristic, complemented
4C2×C13×ΓL(1,𝔽8)×D11C313yesmaximal, direct product factor
5C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C13×ΓL(1,𝔽8)×D11C2×C316yesdirect product factor
7C3×C7×C13×D11C2×C312yescomplemented
8C2×C11×C13×ΓL(1,𝔽8)C2×C313yescomplemented
9C2×C3×C7×C11×C13C2×C311yescharacteristic, complemented
10C2×C32×C13×D11—71nomaximal, Hall
11C2×C7×C13×D11C3211yescharacteristic, complemented, Hall
12C22×C3×C13×ΓL(1,𝔽8)—111nomaximal, Hall
13C11×C13×ΓL(1,𝔽8)C22×C313yescomplemented
14C3×C7×C11×C13C22×C311yescharacteristic, complemented
15C2×C3×ΓL(1,𝔽8)×D11C1311yesmaximal, characteristic, direct product factor, Hall
16C32×C13×D11—72no—
17C2×C32×C11×C13—71no—
18C7×C13×D11C2×C3212yescomplemented
19C2×C7×C11×C13C2×C3211yescharacteristic, complemented
20C2×C3×C13×D11—73no—
21C2×C3×C13×D11ΓL(1,𝔽8)11yescharacteristic, direct product factor
22C2×C3×C13×ΓL(1,𝔽8)—112no—
23C2×C3×C13×ΓL(1,𝔽8)D1111yescharacteristic, direct product factor
24C2×C3×C11×ΓL(1,𝔽8)C2×C1311yescharacteristic, complemented
25C3×ΓL(1,𝔽8)×D11C2×C1312yesdirect product factor
26C32×C11×C13—71noHall
27C22×C3×C7×C13—111no—
28C22×C13×ΓL(1,𝔽8)—113no—
29C7×C11×C13C22×C3211yescharacteristic, complemented, Hall
30C2×C3×C7×D11C3×C1311yescharacteristic, complemented
31C2×ΓL(1,𝔽8)×D11C3×C1313yesdirect product factor
32C2×C3×C11×C13—73no—
33C2×C3×C11×C13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
34C3×C13×D11—76no—
35C3×C13×D11C2×ΓL(1,𝔽8)12yesdirect product factor
36C3×C13×ΓL(1,𝔽8)C2×D1111yescharacteristic, direct product factor, Hall
37C3×C11×ΓL(1,𝔽8)C22×C1311yescharacteristic, complemented, Hall
38C2×C13×D11C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
39C2×C3×C7×C13—112no—
40C2×C3×C7×C13C3×D1111yescharacteristic, complemented
41C2×C13×ΓL(1,𝔽8)—116no—
42C2×C13×ΓL(1,𝔽8)C3×D1113yesdirect product factor
43C22×C32×C13—771noHall
44C2×C3×C7×C11C2×C3×C1311yescharacteristic, complemented
45C3×C7×D11C2×C3×C1312yescomplemented
46C2×C11×ΓL(1,𝔽8)C2×C3×C1313yescomplemented
47ΓL(1,𝔽8)×D11C2×C3×C1316yesdirect product factor
48C3×C11×C13—73no—
49C3×C11×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented
50C2×C32×D11—71noHall
51C22×C7×C13—111noHall
52C2×C7×D11C32×C1311yescharacteristic, complemented, Hall
53C2×C11×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
54C13×D11C2×C3×ΓL(1,𝔽8)12yesdirect product factor
55C3×C7×C13C2×C3×D1111yescharacteristic, complemented
56C13×ΓL(1,𝔽8)C2×C3×D1113yesdirect product factor
57C22×C3×ΓL(1,𝔽8)—111noHall
58C2×C32×C13—71no—
59C2×C32×C13—772no—
60C3×C7×C11C22×C3×C1311yescharacteristic, complemented
61C11×ΓL(1,𝔽8)C22×C3×C1313yescomplemented
62C2×C32×C11—71no—
63C32×D11—72no—
64C2×C7×C13—112no—
65C2×C7×C13C32×D1111yescharacteristic, complemented
66C22×C3×C13—111no—
67C22×C3×C13—773no—
68C2×C7×C11C2×C32×C1311yescharacteristic, complemented
69C7×D11C2×C32×C1312yescomplemented
70C11×C13C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
71C2×C3×D11—73no—
72C2×C3×D11C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor
73C2×C3×ΓL(1,𝔽8)—112no—
74C2×C3×ΓL(1,𝔽8)C13×D1111yescharacteristic, direct product factor
75C32×C13—71noHall
76C32×C11—71noHall
77C7×C13C2×C32×D1111yescharacteristic, complemented, Hall
78C22×C3×C7—111no—
79C22×ΓL(1,𝔽8)—113no—
80C2×C3×C13—73no—
81C2×C3×C13—112no—
82C2×C3×C13—776no—
83C2×C3×C13ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
84C7×C11C22×C32×C1311yescharacteristic, complemented, Hall
85C2×C3×C11—73no—
86C2×C3×C11C2×C13×ΓL(1,𝔽8)11yescharacteristic, complemented
87C3×D11—76no—
88C3×D11C2×C13×ΓL(1,𝔽8)12yesdirect product factor
89C3×ΓL(1,𝔽8)C2×C13×D1111yescharacteristic, direct product factor, Hall
90C22×C13—111noHall
91C2×D11C3×C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
92C2×C3×C7—112no—
93C2×C3×C7C3×C13×D1111yescharacteristic, complemented
94C2×ΓL(1,𝔽8)—116no—
95C2×ΓL(1,𝔽8)C3×C13×D1113yesdirect product factor
96C3×C13—73no—
97C3×C13C2×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
98C22×C32—771noHall
99C3×C11—73no—
100C3×C11C22×C13×ΓL(1,𝔽8)11yescharacteristic, complemented
101C22×C7—111noHall
102C2×C13—112no—
103C2×C13C3×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
104C2×C11C2×C3×C13×ΓL(1,𝔽8)11yescharacteristic, complemented
105D11C2×C3×C13×ΓL(1,𝔽8)12yesdirect product factor
106C3×C7C2×C3×C13×D1111yescharacteristic, complemented
107ΓL(1,𝔽8)C2×C3×C13×D1113yesdirect product factor
108C2×C32—71no—
109C2×C32—772no—
110C2×C7—112no—
111C2×C7C32×C13×D1111yescharacteristic, complemented
112C13C2×C3×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor, Sylow
113C22×C3—111no—
114C22×C3—773no—
115C11C22×C3×C13×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
116C32—71noSylow
117C7C2×C32×C13×D1111yescharacteristic, complemented, Sylow
118C2×C3—73no—
119C2×C3—112no—
120C2×C3—776no—
121C2×C3C13×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
122C22—111noSylow
123C3—73no—
124C3C2×C13×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
125C2—112no—
126C2C3×C13×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C13ΓL(1,𝔽8)×D11
commutator, cosocle, hypocenterC7×C11C22×C32×C13
FrattiniC1G
Fitting, socleC2×C3×C7×C11×C13C2×C3
LCSG ▷ C7×C11
UCSC1 ◁ C2×C3×C13

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×C13×ΓL(1,𝔽8)×D11C2direct (N × H)
3. C2×C3×C7×C13×D11C3semidirect
4. C2×C13×ΓL(1,𝔽8)×D11C3direct (N × H)
5. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
6. C13×ΓL(1,𝔽8)×D11C2×C3direct (N × H)
7. C3×C7×C13×D11C2×C3semidirect
8. C2×C11×C13×ΓL(1,𝔽8)C2×C3semidirect
9. C2×C3×C7×C11×C13C2×C3semidirect
11. C2×C7×C13×D11C32semidirect
13. C11×C13×ΓL(1,𝔽8)C22×C3semidirect
14. C3×C7×C11×C13C22×C3semidirect
15. C2×C3×ΓL(1,𝔽8)×D11C13direct (N × H)
18. C7×C13×D11C2×C32semidirect
19. C2×C7×C11×C13C2×C32semidirect
21. C2×C3×C13×D11ΓL(1,𝔽8)direct (N × H)
23. C2×C3×C13×ΓL(1,𝔽8)D11direct (N × H)
24. C2×C3×C11×ΓL(1,𝔽8)C2×C13semidirect
25. C3×ΓL(1,𝔽8)×D11C2×C13direct (N × H)
29. C7×C11×C13C22×C32semidirect
30. C2×C3×C7×D11C3×C13semidirect
31. C2×ΓL(1,𝔽8)×D11C3×C13direct (N × H)
33. C2×C3×C11×C13C2×ΓL(1,𝔽8)semidirect
35. C3×C13×D11C2×ΓL(1,𝔽8)direct (N × H)
36. C3×C13×ΓL(1,𝔽8)C2×D11direct (N × H)
37. C3×C11×ΓL(1,𝔽8)C22×C13semidirect
38. C2×C13×D11C3×ΓL(1,𝔽8)direct (N × H)
40. C2×C3×C7×C13C3×D11semidirect
42. C2×C13×ΓL(1,𝔽8)C3×D11direct (N × H)
44. C2×C3×C7×C11C2×C3×C13semidirect
45. C3×C7×D11C2×C3×C13semidirect
46. C2×C11×ΓL(1,𝔽8)C2×C3×C13semidirect
47. ΓL(1,𝔽8)×D11C2×C3×C13direct (N × H)
49. C3×C11×C13C22×ΓL(1,𝔽8)semidirect
52. C2×C7×D11C32×C13semidirect
53. C2×C11×C13C2×C3×ΓL(1,𝔽8)semidirect
54. C13×D11C2×C3×ΓL(1,𝔽8)direct (N × H)
55. C3×C7×C13C2×C3×D11semidirect
56. C13×ΓL(1,𝔽8)C2×C3×D11direct (N × H)
60. C3×C7×C11C22×C3×C13semidirect
61. C11×ΓL(1,𝔽8)C22×C3×C13semidirect
65. C2×C7×C13C32×D11semidirect
68. C2×C7×C11C2×C32×C13semidirect
69. C7×D11C2×C32×C13semidirect
70. C11×C13C22×C3×ΓL(1,𝔽8)semidirect
72. C2×C3×D11C13×ΓL(1,𝔽8)direct (N × H)
74. C2×C3×ΓL(1,𝔽8)C13×D11direct (N × H)
77. C7×C13C2×C32×D11semidirect
83. C2×C3×C13ΓL(1,𝔽8)×D11direct (N × H)
84. C7×C11C22×C32×C13semidirect
86. C2×C3×C11C2×C13×ΓL(1,𝔽8)semidirect
88. C3×D11C2×C13×ΓL(1,𝔽8)direct (N × H)
89. C3×ΓL(1,𝔽8)C2×C13×D11direct (N × H)
91. C2×D11C3×C13×ΓL(1,𝔽8)direct (N × H)
93. C2×C3×C7C3×C13×D11semidirect
95. C2×ΓL(1,𝔽8)C3×C13×D11direct (N × H)
97. C3×C13C2×ΓL(1,𝔽8)×D11direct (N × H)
100. C3×C11C22×C13×ΓL(1,𝔽8)semidirect
103. C2×C13C3×ΓL(1,𝔽8)×D11direct (N × H)
104. C2×C11C2×C3×C13×ΓL(1,𝔽8)semidirect
105. D11C2×C3×C13×ΓL(1,𝔽8)direct (N × H)
106. C3×C7C2×C3×C13×D11semidirect
107. ΓL(1,𝔽8)C2×C3×C13×D11direct (N × H)
111. C2×C7C32×C13×D11semidirect
112. C13C2×C3×ΓL(1,𝔽8)×D11direct (N × H)
115. C11C22×C3×C13×ΓL(1,𝔽8)semidirect
117. C7C2×C32×C13×D11semidirect
121. C2×C3C13×ΓL(1,𝔽8)×D11direct (N × H)
124. C3C2×C13×ΓL(1,𝔽8)×D11direct (N × H)
126. C2C3×C13×ΓL(1,𝔽8)×D11direct (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×C13×ΓL(1,𝔽8)×D112
C2×C3×C7×C13×D113
C2×C13×ΓL(1,𝔽8)×D113
C2×C32×C13×D117
C22×C3×C13×ΓL(1,𝔽8)11
C2×C3×ΓL(1,𝔽8)×D1113