Finite Group Databasebeta

← C11×ΓL(1,𝔽8)×Dic39 #49114579

Subgroups — C11×ΓL(1,𝔽8)×Dic39 #49114579

Order 36036.

112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 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
2C11×ΓL(1,𝔽8)×Dic13—31nomaximal
3C7×C11×Dic39C311yesmaximal, characteristic, complemented
4C3×C11×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5C2×C11×C13×ΓL(1,𝔽8)—21no—
6C2×C11×C13×ΓL(1,𝔽8)D311yescharacteristic
7C2×C3×C7×C11×C13C2×C311yescharacteristic
8C3×C11×Dic39—71nomaximal, Hall
9C7×C11×Dic13—31noHall
10ΓL(1,𝔽8)×Dic39C1111yesmaximal, characteristic, direct product factor, Hall
11C11×C13×ΓL(1,𝔽8)—21no—
12C11×C13×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
13C3×C7×C11×C13C3×C411yescharacteristic, complemented
14C11×Dic3×ΓL(1,𝔽8)—131nomaximal, Hall
15C2×C32×C11×C13—71no—
16C2×C7×C11×C13C3×D311yescharacteristic
17C11×Dic39ΓL(1,𝔽8)11yescharacteristic, direct product factor
18C3×C11×Dic13—211no—
19C2×C3×C13×ΓL(1,𝔽8)C2×C1111yescharacteristic
20C2×C3×C11×ΓL(1,𝔽8)D1311yescharacteristic
21C32×C11×C13—71noHall
22C7×Dic39C3×C1111yescharacteristic, complemented
23ΓL(1,𝔽8)×Dic13—31no—
24C7×C11×C13C3×Dic311yescharacteristic, complemented, Hall
25C7×C11×Dic3—131no—
26C4×C11×ΓL(1,𝔽8)—391no—
27C2×C3×C11×C13—71no—
28C2×C3×C11×C13—141no—
29C2×C3×C11×C13C2×ΓL(1,𝔽8)11yescharacteristic
30C3×C13×ΓL(1,𝔽8)C4×C1111yescharacteristic, complemented, Hall
31C3×C11×ΓL(1,𝔽8)Dic1311yescharacteristic, complemented, Hall
32C11×Dic13—31noHall
33C2×C3×C7×C13C2×C3×C1111yescharacteristic
34C2×C13×ΓL(1,𝔽8)—21no—
35C2×C13×ΓL(1,𝔽8)D3×C1111yescharacteristic
36C3×Dic39—71noHall
37C2×C3×C7×C11C3×D1311yescharacteristic
38C2×C11×ΓL(1,𝔽8)—21no—
39C2×C11×ΓL(1,𝔽8)D3911yescharacteristic
40C3×C11×C13—71no—
41C3×C11×C13—141no—
42C3×C11×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
43C3×C11×Dic3—911noHall
44C7×Dic13—31noHall
45C4×C7×C11—391noHall
46C2×C11×C13D3×ΓL(1,𝔽8)11yescharacteristic
47C3×C7×C13C3×C4×C1111yescharacteristic, complemented
48C13×ΓL(1,𝔽8)—21no—
49C13×ΓL(1,𝔽8)C11×Dic311yescharacteristic, complemented
50Dic3×ΓL(1,𝔽8)—131noHall
51C2×C32×C13—71no—
52C3×C7×C11C3×Dic1311yescharacteristic, complemented
53C11×ΓL(1,𝔽8)—21no—
54C11×ΓL(1,𝔽8)Dic3911yescharacteristic, direct product factor
55C2×C32×C11—71no—
56C2×C7×C13C3×D3×C1111yescharacteristic
57Dic39C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
58C3×Dic13—211no—
59C2×C7×C11C3×D3911yescharacteristic
60C11×C13Dic3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
61C3×C4×C11—2731no—
62C11×Dic3—131no—
63C2×C3×ΓL(1,𝔽8)C11×D1311yescharacteristic
64C32×C13—71noHall
65C32×C11—71noHall
66C7×C13C3×C11×Dic311yescharacteristic, complemented, Hall
67C7×Dic3—131no—
68C4×ΓL(1,𝔽8)—391no—
69C2×C3×C13—71no—
70C2×C3×C13—141no—
71C2×C3×C13C2×C11×ΓL(1,𝔽8)11yescharacteristic
72C7×C11C3×Dic3911yescharacteristic, complemented, Hall
73C2×C3×C11—71no—
74C2×C3×C11—141no—
75C2×C3×C11ΓL(1,𝔽8)×D1311yescharacteristic
76C3×ΓL(1,𝔽8)C11×Dic1311yescharacteristic, complemented, Hall
77Dic13—31noHall
78C4×C11—391noHall
79C2×C3×C7C3×C11×D1311yescharacteristic
80C2×ΓL(1,𝔽8)—21no—
81C2×ΓL(1,𝔽8)C11×D3911yescharacteristic
82C3×C13—71no—
83C3×C13—141no—
84C3×C13C4×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
85C3×Dic3—911noHall
86C3×C11—71no—
87C3×C11—141no—
88C3×C11ΓL(1,𝔽8)×Dic1311yescharacteristic, complemented
89C4×C7—391noHall
90C2×C13D3×C11×ΓL(1,𝔽8)11yescharacteristic
91C2×C11ΓL(1,𝔽8)×D3911yescharacteristic
92C3×C7C3×C11×Dic1311yescharacteristic, complemented
93ΓL(1,𝔽8)—21no—
94ΓL(1,𝔽8)C11×Dic3911yescharacteristic, direct product factor
95C2×C32—71no—
96C2×C7C3×C11×D3911yescharacteristic
97C13C11×Dic3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
98C3×C4—2731no—
99Dic3—131no—
100C11ΓL(1,𝔽8)×Dic3911yescharacteristic, direct product factor, Sylow
101C32—71noSylow
102C7C3×C11×Dic3911yescharacteristic, complemented, Sylow
103C2×C3—71no—
104C2×C3—141no—
105C2×C3C11×ΓL(1,𝔽8)×D1311yescharacteristic
106C4—391noSylow
107C3—71no—
108C3—141no—
109C3C11×ΓL(1,𝔽8)×Dic1311yescharacteristic, complemented
110C2C11×ΓL(1,𝔽8)×D3911yescharacteristic
subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)×D39
commutator, hypocenterC3×C7×C13C3×C4×C11
FrattiniC2C11×ΓL(1,𝔽8)×D39
Fitting, socleC2×C3×C7×C11×C13C2×C3
prime radicalC2×C3×C11×C13×ΓL(1,𝔽8)C2
cosocleC2×C3×C7×C13C2×C3×C11
LCSG ▷ C3×C7×C13
UCSC1 ◁ C2×C11

Complements

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

NH ≅ G/Ntype
3. C7×C11×Dic39C3semidirect
4. C3×C11×C13×ΓL(1,𝔽8)C4semidirect
10. ΓL(1,𝔽8)×Dic39C11direct (N × H)
12. C11×C13×ΓL(1,𝔽8)Dic3semidirect
13. C3×C7×C11×C13C3×C4semidirect
17. C11×Dic39ΓL(1,𝔽8)direct (N × H)
22. C7×Dic39C3×C11semidirect
24. C7×C11×C13C3×Dic3semidirect
30. C3×C13×ΓL(1,𝔽8)C4×C11semidirect
31. C3×C11×ΓL(1,𝔽8)Dic13semidirect
42. C3×C11×C13C4×ΓL(1,𝔽8)semidirect
47. C3×C7×C13C3×C4×C11semidirect
49. C13×ΓL(1,𝔽8)C11×Dic3semidirect
52. C3×C7×C11C3×Dic13semidirect
54. C11×ΓL(1,𝔽8)Dic39direct (N × H)
57. Dic39C11×ΓL(1,𝔽8)direct (N × H)
60. C11×C13Dic3×ΓL(1,𝔽8)semidirect
66. C7×C13C3×C11×Dic3semidirect
72. C7×C11C3×Dic39semidirect
76. C3×ΓL(1,𝔽8)C11×Dic13semidirect
84. C3×C13C4×C11×ΓL(1,𝔽8)semidirect
88. C3×C11ΓL(1,𝔽8)×Dic13semidirect
92. C3×C7C3×C11×Dic13semidirect
94. ΓL(1,𝔽8)C11×Dic39direct (N × H)
97. C13C11×Dic3×ΓL(1,𝔽8)semidirect
100. C11ΓL(1,𝔽8)×Dic39direct (N × H)
102. C7C3×C11×Dic39semidirect
109. C3C11×ΓL(1,𝔽8)×Dic13semidirect

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
C11×ΓL(1,𝔽8)×Dic133
C7×C11×Dic393
C3×C11×Dic397
ΓL(1,𝔽8)×Dic3911
C11×Dic3×ΓL(1,𝔽8)13