Finite Group Databasebeta

← C2×C3×C5×ΓL(1,𝔽8)×D11 #28357635

Subgroups — C2×C3×C5×ΓL(1,𝔽8)×D11 #28357635

Order 13860.

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×C5×C11×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C3×C5×ΓL(1,𝔽8)×D11C212yesmaximal, direct product factor
3C2×C3×C5×C7×D11C311yesmaximal, characteristic, complemented
4C2×C5×ΓL(1,𝔽8)×D11C313yesmaximal, direct product factor
5C3×C5×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C2×C3×ΓL(1,𝔽8)×D11C511yesmaximal, characteristic, direct product factor, Hall
7C5×ΓL(1,𝔽8)×D11C2×C316yesdirect product factor
8C3×C5×C7×D11C2×C312yescomplemented
9C2×C5×C11×ΓL(1,𝔽8)C2×C313yescomplemented
10C2×C3×C5×C7×C11C2×C311yescharacteristic, complemented
11C2×C32×C5×D11—71nomaximal, Hall
12C2×C5×C7×D11C3211yescharacteristic, complemented, Hall
13C2×C3×C11×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
14C3×ΓL(1,𝔽8)×D11C2×C512yesdirect product factor
15C22×C3×C5×ΓL(1,𝔽8)—111nomaximal, Hall
16C3×C5×C7×C11C22×C311yescharacteristic, complemented
17C5×C11×ΓL(1,𝔽8)C22×C313yescomplemented
18C2×C32×C5×C11—71no—
19C32×C5×D11—72no—
20C2×C3×C7×D11C3×C511yescharacteristic, complemented
21C2×ΓL(1,𝔽8)×D11C3×C513yesdirect product factor
22C2×C5×C7×C11C2×C3211yescharacteristic, complemented
23C5×C7×D11C2×C3212yescomplemented
24C3×C11×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented, Hall
25C2×C3×C5×D11—73no—
26C2×C3×C5×D11ΓL(1,𝔽8)11yescharacteristic, direct product factor
27C2×C3×C5×ΓL(1,𝔽8)—112no—
28C2×C3×C5×ΓL(1,𝔽8)D1111yescharacteristic, direct product factor
29C32×C5×C11—71noHall
30C2×C3×C7×C11C2×C3×C511yescharacteristic, complemented
31C3×C7×D11C2×C3×C512yescomplemented
32C2×C11×ΓL(1,𝔽8)C2×C3×C513yescomplemented
33ΓL(1,𝔽8)×D11C2×C3×C516yesdirect product factor
34C22×C3×C5×C7—111no—
35C22×C5×ΓL(1,𝔽8)—113no—
36C2×C32×D11—71noHall
37C5×C7×C11C22×C3211yescharacteristic, complemented, Hall
38C2×C3×C5×C11—73no—
39C2×C3×C5×C11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
40C3×C5×D11—76no—
41C3×C5×D11C2×ΓL(1,𝔽8)12yesdirect product factor
42C3×C5×ΓL(1,𝔽8)C2×D1111yescharacteristic, direct product factor, Hall
43C2×C7×D11C32×C511yescharacteristic, complemented, Hall
44C22×C3×ΓL(1,𝔽8)—111noHall
45C3×C7×C11C22×C3×C511yescharacteristic, complemented
46C11×ΓL(1,𝔽8)C22×C3×C513yescomplemented
47C2×C5×D11C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
48C2×C3×C5×C7—112no—
49C2×C3×C5×C7C3×D1111yescharacteristic, complemented
50C2×C5×ΓL(1,𝔽8)—116no—
51C2×C5×ΓL(1,𝔽8)C3×D1113yesdirect product factor
52C2×C32×C11—71no—
53C32×D11—72no—
54C22×C32×C5—771noHall
55C3×C5×C11—73no—
56C3×C5×C11C22×ΓL(1,𝔽8)11yescharacteristic, complemented
57C2×C7×C11C2×C32×C511yescharacteristic, complemented
58C7×D11C2×C32×C512yescomplemented
59C22×C5×C7—111noHall
60C2×C3×D11—73no—
61C2×C3×D11C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
62C2×C3×ΓL(1,𝔽8)—112no—
63C2×C3×ΓL(1,𝔽8)C5×D1111yescharacteristic, direct product factor
64C2×C5×C11C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
65C5×D11C2×C3×ΓL(1,𝔽8)12yesdirect product factor
66C3×C5×C7C2×C3×D1111yescharacteristic, complemented
67C5×ΓL(1,𝔽8)C2×C3×D1113yesdirect product factor
68C32×C11—71noHall
69C2×C32×C5—71no—
70C2×C32×C5—772no—
71C22×C3×C7—111no—
72C22×ΓL(1,𝔽8)—113no—
73C7×C11C22×C32×C511yescharacteristic, complemented, Hall
74C2×C5×C7—112no—
75C2×C5×C7C32×D1111yescharacteristic, complemented
76C2×C3×C11—73no—
77C2×C3×C11C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
78C3×D11—76no—
79C3×D11C2×C5×ΓL(1,𝔽8)12yesdirect product factor
80C3×ΓL(1,𝔽8)C2×C5×D1111yescharacteristic, direct product factor, Hall
81C22×C3×C5—111no—
82C22×C3×C5—773no—
83C5×C11C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
84C32×C5—71noHall
85C2×D11C3×C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
86C2×C3×C7—112no—
87C2×C3×C7C3×C5×D1111yescharacteristic, complemented
88C2×ΓL(1,𝔽8)—116no—
89C2×ΓL(1,𝔽8)C3×C5×D1113yesdirect product factor
90C22×C32—771noHall
91C5×C7C2×C32×D1111yescharacteristic, complemented, Hall
92C3×C11—73no—
93C3×C11C22×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
94C2×C3×C5—73no—
95C2×C3×C5—112no—
96C2×C3×C5—776no—
97C2×C3×C5ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
98C22×C7—111noHall
99C2×C11C2×C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
100D11C2×C3×C5×ΓL(1,𝔽8)12yesdirect product factor
101C3×C7C2×C3×C5×D1111yescharacteristic, complemented
102ΓL(1,𝔽8)C2×C3×C5×D1113yesdirect product factor
103C22×C5—111noHall
104C2×C32—71no—
105C2×C32—772no—
106C3×C5—73no—
107C3×C5C2×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
108C2×C7—112no—
109C2×C7C32×C5×D1111yescharacteristic, complemented
110C22×C3—111no—
111C22×C3—773no—
112C11C22×C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
113C2×C5—112no—
114C2×C5C3×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
115C32—71noSylow
116C7C2×C32×C5×D1111yescharacteristic, complemented, Sylow
117C2×C3—73no—
118C2×C3—112no—
119C2×C3—776no—
120C2×C3C5×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
121C5C2×C3×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor, Sylow
122C22—111noSylow
123C3—73no—
124C3C2×C5×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
125C2—112no—
126C2C3×C5×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5ΓL(1,𝔽8)×D11
commutator, cosocle, hypocenterC7×C11C22×C32×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C11C2×C3
LCSG ▷ C7×C11
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×C11×ΓL(1,𝔽8)C2semidirect
2. C3×C5×ΓL(1,𝔽8)×D11C2direct (N × H)
3. C2×C3×C5×C7×D11C3semidirect
4. C2×C5×ΓL(1,𝔽8)×D11C3direct (N × H)
5. C3×C5×C11×ΓL(1,𝔽8)C22semidirect
6. C2×C3×ΓL(1,𝔽8)×D11C5direct (N × H)
7. C5×ΓL(1,𝔽8)×D11C2×C3direct (N × H)
8. C3×C5×C7×D11C2×C3semidirect
9. C2×C5×C11×ΓL(1,𝔽8)C2×C3semidirect
10. C2×C3×C5×C7×C11C2×C3semidirect
12. C2×C5×C7×D11C32semidirect
13. C2×C3×C11×ΓL(1,𝔽8)C2×C5semidirect
14. C3×ΓL(1,𝔽8)×D11C2×C5direct (N × H)
16. C3×C5×C7×C11C22×C3semidirect
17. C5×C11×ΓL(1,𝔽8)C22×C3semidirect
20. C2×C3×C7×D11C3×C5semidirect
21. C2×ΓL(1,𝔽8)×D11C3×C5direct (N × H)
22. C2×C5×C7×C11C2×C32semidirect
23. C5×C7×D11C2×C32semidirect
24. C3×C11×ΓL(1,𝔽8)C22×C5semidirect
26. C2×C3×C5×D11ΓL(1,𝔽8)direct (N × H)
28. C2×C3×C5×ΓL(1,𝔽8)D11direct (N × H)
30. C2×C3×C7×C11C2×C3×C5semidirect
31. C3×C7×D11C2×C3×C5semidirect
32. C2×C11×ΓL(1,𝔽8)C2×C3×C5semidirect
33. ΓL(1,𝔽8)×D11C2×C3×C5direct (N × H)
37. C5×C7×C11C22×C32semidirect
39. C2×C3×C5×C11C2×ΓL(1,𝔽8)semidirect
41. C3×C5×D11C2×ΓL(1,𝔽8)direct (N × H)
42. C3×C5×ΓL(1,𝔽8)C2×D11direct (N × H)
43. C2×C7×D11C32×C5semidirect
45. C3×C7×C11C22×C3×C5semidirect
46. C11×ΓL(1,𝔽8)C22×C3×C5semidirect
47. C2×C5×D11C3×ΓL(1,𝔽8)direct (N × H)
49. C2×C3×C5×C7C3×D11semidirect
51. C2×C5×ΓL(1,𝔽8)C3×D11direct (N × H)
56. C3×C5×C11C22×ΓL(1,𝔽8)semidirect
57. C2×C7×C11C2×C32×C5semidirect
58. C7×D11C2×C32×C5semidirect
61. C2×C3×D11C5×ΓL(1,𝔽8)direct (N × H)
63. C2×C3×ΓL(1,𝔽8)C5×D11direct (N × H)
64. C2×C5×C11C2×C3×ΓL(1,𝔽8)semidirect
65. C5×D11C2×C3×ΓL(1,𝔽8)direct (N × H)
66. C3×C5×C7C2×C3×D11semidirect
67. C5×ΓL(1,𝔽8)C2×C3×D11direct (N × H)
73. C7×C11C22×C32×C5semidirect
75. C2×C5×C7C32×D11semidirect
77. C2×C3×C11C2×C5×ΓL(1,𝔽8)semidirect
79. C3×D11C2×C5×ΓL(1,𝔽8)direct (N × H)
80. C3×ΓL(1,𝔽8)C2×C5×D11direct (N × H)
83. C5×C11C22×C3×ΓL(1,𝔽8)semidirect
85. C2×D11C3×C5×ΓL(1,𝔽8)direct (N × H)
87. C2×C3×C7C3×C5×D11semidirect
89. C2×ΓL(1,𝔽8)C3×C5×D11direct (N × H)
91. C5×C7C2×C32×D11semidirect
93. C3×C11C22×C5×ΓL(1,𝔽8)semidirect
97. C2×C3×C5ΓL(1,𝔽8)×D11direct (N × H)
99. C2×C11C2×C3×C5×ΓL(1,𝔽8)semidirect
100. D11C2×C3×C5×ΓL(1,𝔽8)direct (N × H)
101. C3×C7C2×C3×C5×D11semidirect
102. ΓL(1,𝔽8)C2×C3×C5×D11direct (N × H)
107. C3×C5C2×ΓL(1,𝔽8)×D11direct (N × H)
109. C2×C7C32×C5×D11semidirect
112. C11C22×C3×C5×ΓL(1,𝔽8)semidirect
114. C2×C5C3×ΓL(1,𝔽8)×D11direct (N × H)
116. C7C2×C32×C5×D11semidirect
120. C2×C3C5×ΓL(1,𝔽8)×D11direct (N × H)
121. C5C2×C3×ΓL(1,𝔽8)×D11direct (N × H)
124. C3C2×C5×ΓL(1,𝔽8)×D11direct (N × H)
126. C2C3×C5×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×C5×C11×ΓL(1,𝔽8)2
C3×C5×ΓL(1,𝔽8)×D112
C2×C3×C5×C7×D113
C2×C5×ΓL(1,𝔽8)×D113
C2×C3×ΓL(1,𝔽8)×D115
C2×C32×C5×D117
C22×C3×C5×ΓL(1,𝔽8)11