Finite Group Databasebeta

← C2×C11×ΓL(1,𝔽8)×D39 #49114866

Subgroups — C2×C11×ΓL(1,𝔽8)×D39 #49114866

Order 36036.

176 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 66 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
2C11×ΓL(1,𝔽8)×D39C212yesmaximal, direct product factor
3C2×C7×C11×D39C311yesmaximal, characteristic, complemented
4C2×C11×ΓL(1,𝔽8)×D13—31nomaximal
5C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C11×ΓL(1,𝔽8)×D13—32no—
7C7×C11×D39C2×C312yescomplemented
8C2×C11×C13×ΓL(1,𝔽8)—21no—
9C2×C11×C13×ΓL(1,𝔽8)D311yescharacteristic, complemented
10C2×C3×C7×C11×C13C2×C311yescharacteristic, complemented
11C2×C3×C11×D39—71nomaximal, Hall
12C2×C7×C11×D13—31noHall
13C2×ΓL(1,𝔽8)×D39C1111yesmaximal, characteristic, direct product factor, Hall
14C11×C13×ΓL(1,𝔽8)—21no—
15C11×C13×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
16C3×C7×C11×C13C22×C311yescharacteristic, complemented
17C2×D3×C11×ΓL(1,𝔽8)—131nomaximal, Hall
18C3×C11×D39—72no—
19C2×C32×C11×C13—71no—
20C7×C11×D13—32no—
21C2×C7×C11×C13C3×D311yescharacteristic, complemented
22C2×C11×D39ΓL(1,𝔽8)11yescharacteristic, direct product factor
23C2×C3×C11×D13—211no—
24C2×C3×C13×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
25ΓL(1,𝔽8)×D39C2×C1112yesdirect product factor
26C2×C3×C11×ΓL(1,𝔽8)D1311yescharacteristic, complemented
27D3×C11×ΓL(1,𝔽8)—132no—
28C32×C11×C13—71noHall
29C2×C7×D39C3×C1111yescharacteristic, complemented
30C2×ΓL(1,𝔽8)×D13—31no—
31C7×C11×C13C2×C3×D311yescharacteristic, complemented, Hall
32C2×D3×C7×C11—131no—
33C22×C11×ΓL(1,𝔽8)—391no—
34C2×C3×C11×C13—71no—
35C2×C3×C11×C13—141no—
36C2×C3×C11×C13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
37C11×D39C2×ΓL(1,𝔽8)12yesdirect product factor
38C3×C11×D13—212no—
39C3×C13×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented, Hall
40C3×C11×ΓL(1,𝔽8)C2×D1311yescharacteristic, complemented, Hall
41C2×C11×D13—31noHall
42C2×C3×C7×C13C2×C3×C1111yescharacteristic, complemented
43C7×D39C2×C3×C1112yescomplemented
44C2×C13×ΓL(1,𝔽8)—21no—
45C2×C13×ΓL(1,𝔽8)D3×C1111yescharacteristic, complemented
46ΓL(1,𝔽8)×D13—32no—
47C2×C3×D39—71noHall
48C2×C3×C7×C11C3×D1311yescharacteristic, complemented
49D3×C7×C11—132no—
50C2×C11×ΓL(1,𝔽8)—21no—
51C2×C11×ΓL(1,𝔽8)—392no—
52C2×C11×ΓL(1,𝔽8)D3911yescharacteristic, direct product factor
53C3×C11×C13—71no—
54C3×C11×C13—141no—
55C3×C11×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented
56C2×C3×D3×C11—911noHall
57C2×C7×D13—31noHall
58C22×C7×C11—391noHall
59C2×C11×C13D3×ΓL(1,𝔽8)11yescharacteristic, complemented
60C11×D13—32no—
61C3×C7×C13C22×C3×C1111yescharacteristic, complemented
62C13×ΓL(1,𝔽8)—21no—
63C13×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
64C2×D3×ΓL(1,𝔽8)—131noHall
65C2×C32×C13—71no—
66C3×D39—72no—
67C3×C7×C11C2×C3×D1311yescharacteristic, complemented
68C11×ΓL(1,𝔽8)—21no—
69C11×ΓL(1,𝔽8)C2×D3911yescharacteristic, direct product factor
70C2×C32×C11—71no—
71C3×D3×C11—912no—
72C2×C7×C13C3×D3×C1111yescharacteristic, complemented
73C7×D13—32no—
74C2×D39C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
75C2×C3×D13—211no—
76C2×C7×C11—392no—
77C2×C7×C11C3×D3911yescharacteristic, complemented
78C11×C13C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
79C22×C3×C11—2731no—
80C2×D3×C11—131no—
81C2×C3×ΓL(1,𝔽8)C11×D1311yescharacteristic, complemented
82D3×ΓL(1,𝔽8)—132no—
83C32×C13—71noHall
84C32×C11—71noHall
85C7×C13C2×C3×D3×C1111yescharacteristic, complemented, Hall
86C2×D3×C7—131no—
87C22×ΓL(1,𝔽8)—391no—
88C2×C3×C13—71no—
89C2×C3×C13—141no—
90C2×C3×C13C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
91D39C2×C11×ΓL(1,𝔽8)12yesdirect product factor
92C3×D13—212no—
93C7×C11C2×C3×D3911yescharacteristic, complemented, Hall
94C2×C3×C11—71no—
95C2×C3×C11—141no—
96C2×C3×C11—2732no—
97C2×C3×C11ΓL(1,𝔽8)×D1311yescharacteristic, complemented
98D3×C11—132no—
99C3×ΓL(1,𝔽8)C2×C11×D1311yescharacteristic, complemented, Hall
100C2×D13—31noHall
101C22×C11—391noHall
102C2×C3×C7C3×C11×D1311yescharacteristic, complemented
103D3×C7—132no—
104C2×ΓL(1,𝔽8)—21no—
105C2×ΓL(1,𝔽8)—392no—
106C2×ΓL(1,𝔽8)C11×D3911yescharacteristic, direct product factor
107C3×C13—71no—
108C3×C13—141no—
109C3×C13C22×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
110C2×C3×D3—911noHall
111C3×C11—71no—
112C3×C11—141no—
113C3×C11C2×ΓL(1,𝔽8)×D1311yescharacteristic, complemented
114C22×C7—391noHall
115C2×C13D3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
116D13—32no—
117C2×C11—392no—
118C2×C11ΓL(1,𝔽8)×D3911yescharacteristic, direct product factor
119C3×C7C2×C3×C11×D1311yescharacteristic, complemented
120ΓL(1,𝔽8)—21no—
121ΓL(1,𝔽8)C2×C11×D3911yescharacteristic, direct product factor
122C2×C32—71no—
123C3×D3—912no—
124C2×C7—392no—
125C2×C7C3×C11×D3911yescharacteristic, complemented
126C13C2×D3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
127C22×C3—2731no—
128C2×D3—131no—
129C11C2×ΓL(1,𝔽8)×D3911yescharacteristic, direct product factor, Sylow
130C32—71noSylow
131C7C2×C3×C11×D3911yescharacteristic, complemented, Sylow
132C2×C3—71no—
133C2×C3—141no—
134C2×C3—2732no—
135C2×C3C11×ΓL(1,𝔽8)×D1311yescharacteristic, complemented
136D3—132no—
137C22—391noSylow
138C3—71no—
139C3—141no—
140C3C2×C11×ΓL(1,𝔽8)×D1311yescharacteristic, complemented
141C2—392no—
142C2C11×ΓL(1,𝔽8)×D3911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)×D39
commutator, cosocle, hypocenterC3×C7×C13C22×C3×C11
FrattiniC1G
Fitting, socleC2×C3×C7×C11×C13C2×C3
LCSG ▷ C3×C7×C13
UCSC1 ◁ C2×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. C11×ΓL(1,𝔽8)×D39C2direct (N × H)
3. C2×C7×C11×D39C3semidirect
5. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
7. C7×C11×D39C2×C3semidirect
9. C2×C11×C13×ΓL(1,𝔽8)D3semidirect
10. C2×C3×C7×C11×C13C2×C3semidirect
13. C2×ΓL(1,𝔽8)×D39C11direct (N × H)
15. C11×C13×ΓL(1,𝔽8)C2×D3semidirect
16. C3×C7×C11×C13C22×C3semidirect
21. C2×C7×C11×C13C3×D3semidirect
22. C2×C11×D39ΓL(1,𝔽8)direct (N × H)
24. C2×C3×C13×ΓL(1,𝔽8)C2×C11semidirect
25. ΓL(1,𝔽8)×D39C2×C11direct (N × H)
26. C2×C3×C11×ΓL(1,𝔽8)D13semidirect
29. C2×C7×D39C3×C11semidirect
31. C7×C11×C13C2×C3×D3semidirect
36. C2×C3×C11×C13C2×ΓL(1,𝔽8)semidirect
37. C11×D39C2×ΓL(1,𝔽8)direct (N × H)
39. C3×C13×ΓL(1,𝔽8)C22×C11semidirect
40. C3×C11×ΓL(1,𝔽8)C2×D13semidirect
42. C2×C3×C7×C13C2×C3×C11semidirect
43. C7×D39C2×C3×C11semidirect
45. C2×C13×ΓL(1,𝔽8)D3×C11semidirect
48. C2×C3×C7×C11C3×D13semidirect
52. C2×C11×ΓL(1,𝔽8)D39direct (N × H)
55. C3×C11×C13C22×ΓL(1,𝔽8)semidirect
59. C2×C11×C13D3×ΓL(1,𝔽8)semidirect
61. C3×C7×C13C22×C3×C11semidirect
63. C13×ΓL(1,𝔽8)C2×D3×C11semidirect
67. C3×C7×C11C2×C3×D13semidirect
69. C11×ΓL(1,𝔽8)C2×D39direct (N × H)
72. C2×C7×C13C3×D3×C11semidirect
74. C2×D39C11×ΓL(1,𝔽8)direct (N × H)
77. C2×C7×C11C3×D39semidirect
78. C11×C13C2×D3×ΓL(1,𝔽8)semidirect
81. C2×C3×ΓL(1,𝔽8)C11×D13semidirect
85. C7×C13C2×C3×D3×C11semidirect
90. C2×C3×C13C2×C11×ΓL(1,𝔽8)semidirect
91. D39C2×C11×ΓL(1,𝔽8)direct (N × H)
93. C7×C11C2×C3×D39semidirect
97. C2×C3×C11ΓL(1,𝔽8)×D13semidirect
99. C3×ΓL(1,𝔽8)C2×C11×D13semidirect
102. C2×C3×C7C3×C11×D13semidirect
106. C2×ΓL(1,𝔽8)C11×D39direct (N × H)
109. C3×C13C22×C11×ΓL(1,𝔽8)semidirect
113. C3×C11C2×ΓL(1,𝔽8)×D13semidirect
115. C2×C13D3×C11×ΓL(1,𝔽8)semidirect
118. C2×C11ΓL(1,𝔽8)×D39direct (N × H)
119. C3×C7C2×C3×C11×D13semidirect
121. ΓL(1,𝔽8)C2×C11×D39direct (N × H)
125. C2×C7C3×C11×D39semidirect
126. C13C2×D3×C11×ΓL(1,𝔽8)semidirect
129. C11C2×ΓL(1,𝔽8)×D39direct (N × H)
131. C7C2×C3×C11×D39semidirect
135. C2×C3C11×ΓL(1,𝔽8)×D13semidirect
140. C3C2×C11×ΓL(1,𝔽8)×D13semidirect
142. C2C11×ΓL(1,𝔽8)×D39direct (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
C11×ΓL(1,𝔽8)×D392
C2×C7×C11×D393
C2×C11×ΓL(1,𝔽8)×D133
C2×C3×C11×D397
C2×ΓL(1,𝔽8)×D3911
C2×D3×C11×ΓL(1,𝔽8)13