Finite Group Databasebeta

← C2×ΓL(1,𝔽8)×D429 #49114872

Subgroups — C2×ΓL(1,𝔽8)×D429 #49114872

Order 36036.

176 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 57 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
2ΓL(1,𝔽8)×D429C212yesmaximal, direct product factor
3C2×C7×D429C311yesmaximal, characteristic, complemented
4C2×ΓL(1,𝔽8)×D143—31nomaximal
5C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6ΓL(1,𝔽8)×D143—32no—
7C7×D429C2×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×D429—71nomaximal, Hall
12C2×C7×D143—31noHall
13C2×ΓL(1,𝔽8)×D39—111nomaximal, Hall
14C11×C13×ΓL(1,𝔽8)—21no—
15C11×C13×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
16C3×C7×C11×C13C22×C311yescharacteristic, complemented
17C2×ΓL(1,𝔽8)×D33—131nomaximal, Hall
18C3×D429—72no—
19C2×C32×C11×C13—71no—
20C7×D143—32no—
21C2×C7×C11×C13C3×D311yescharacteristic, complemented
22C2×D429ΓL(1,𝔽8)11yescharacteristic, direct product factor
23C2×C3×D143—211no—
24C2×C3×C13×ΓL(1,𝔽8)D1111yescharacteristic, complemented
25ΓL(1,𝔽8)×D39—112no—
26C2×C3×C11×ΓL(1,𝔽8)D1311yescharacteristic, complemented
27ΓL(1,𝔽8)×D33—132no—
28C32×C11×C13—71noHall
29C2×C7×D39—111no—
30C2×ΓL(1,𝔽8)×D13—331no—
31C7×C11×C13C2×C3×D311yescharacteristic, complemented, Hall
32C2×C7×D33—131no—
33C2×ΓL(1,𝔽8)×D11—391no—
34C2×C3×C11×C13—71no—
35C2×C3×C11×C13—141no—
36C2×C3×C11×C13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
37D429C2×ΓL(1,𝔽8)12yesdirect product factor
38C3×D143—212no—
39C3×C13×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
40C3×C11×ΓL(1,𝔽8)C2×D1311yescharacteristic, complemented, Hall
41C2×D143—31noHall
42C2×C3×C7×C13C3×D1111yescharacteristic, complemented
43C7×D39—112no—
44C2×C13×ΓL(1,𝔽8)—21no—
45C2×C13×ΓL(1,𝔽8)D3311yescharacteristic, complemented
46ΓL(1,𝔽8)×D13—332no—
47C2×C3×D39—771noHall
48C2×C3×C7×C11C3×D1311yescharacteristic, complemented
49C7×D33—132no—
50C2×C11×ΓL(1,𝔽8)—21no—
51C2×C11×ΓL(1,𝔽8)D3911yescharacteristic, complemented
52ΓL(1,𝔽8)×D11—392no—
53C3×C11×C13—71no—
54C3×C11×C13—141no—
55C3×C11×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented
56C2×C3×D33—911noHall
57C2×C7×D13—331noHall
58C2×C7×D11—391noHall
59C2×C11×C13D3×ΓL(1,𝔽8)11yescharacteristic, complemented
60D143—32no—
61C3×C7×C13C2×C3×D1111yescharacteristic, complemented
62C13×ΓL(1,𝔽8)—21no—
63C13×ΓL(1,𝔽8)C2×D3311yescharacteristic, complemented
64C2×D3×ΓL(1,𝔽8)—1431noHall
65C2×C32×C13—71no—
66C3×D39—772no—
67C3×C7×C11C2×C3×D1311yescharacteristic, complemented
68C11×ΓL(1,𝔽8)—21no—
69C11×ΓL(1,𝔽8)C2×D3911yescharacteristic, complemented
70C2×C32×C11—71no—
71C3×D33—912no—
72C2×C7×C13C3×D3311yescharacteristic, complemented
73C7×D13—332no—
74C2×D39—111no—
75C2×C3×D13—2311no—
76C2×C7×C11C3×D3911yescharacteristic, complemented
77C7×D11—392no—
78C11×C13C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
79C2×D33—131no—
80C2×C3×D11—2731no—
81C2×C3×ΓL(1,𝔽8)D14311yescharacteristic, complemented
82D3×ΓL(1,𝔽8)—1432no—
83C32×C13—71noHall
84C32×C11—71noHall
85C7×C13C2×C3×D3311yescharacteristic, complemented, Hall
86C2×D3×C7—1431no—
87C22×ΓL(1,𝔽8)—4291no—
88C2×C3×C13—71no—
89C2×C3×C13—141no—
90C2×C3×C13ΓL(1,𝔽8)×D1111yescharacteristic, complemented
91D39—112no—
92C3×D13—2312no—
93C7×C11C2×C3×D3911yescharacteristic, complemented, Hall
94C2×C3×C11—71no—
95C2×C3×C11—141no—
96C2×C3×C11ΓL(1,𝔽8)×D1311yescharacteristic, complemented
97D33—132no—
98C3×D11—2732no—
99C3×ΓL(1,𝔽8)C2×D14311yescharacteristic, complemented, Hall
100C2×D13—331noHall
101C2×D11—391noHall
102C2×C3×C7C3×D14311yescharacteristic, complemented
103D3×C7—1432no—
104C2×ΓL(1,𝔽8)—21no—
105C2×ΓL(1,𝔽8)—4292no—
106C2×ΓL(1,𝔽8)D42911yescharacteristic, direct product factor
107C3×C13—71no—
108C3×C13—141no—
109C3×C13C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
110C2×C3×D3—10011noHall
111C3×C11—71no—
112C3×C11—141no—
113C3×C11C2×ΓL(1,𝔽8)×D1311yescharacteristic, complemented
114C22×C7—4291noHall
115C2×C13ΓL(1,𝔽8)×D3311yescharacteristic, complemented
116D13—332no—
117C2×C11ΓL(1,𝔽8)×D3911yescharacteristic, complemented
118D11—392no—
119C3×C7C2×C3×D14311yescharacteristic, complemented
120ΓL(1,𝔽8)—21no—
121ΓL(1,𝔽8)C2×D42911yescharacteristic, direct product factor
122C2×C32—71no—
123C3×D3—10012no—
124C2×C7—4292no—
125C2×C7C3×D42911yescharacteristic, complemented
126C13C2×ΓL(1,𝔽8)×D3311yescharacteristic, complemented, Sylow
127C22×C3—30031no—
128C2×D3—1431no—
129C11C2×ΓL(1,𝔽8)×D3911yescharacteristic, complemented, Sylow
130C32—71noSylow
131C7C2×C3×D42911yescharacteristic, complemented, Sylow
132C2×C3—71no—
133C2×C3—141no—
134C2×C3—30032no—
135C2×C3ΓL(1,𝔽8)×D14311yescharacteristic, complemented
136D3—1432no—
137C22—4291noSylow
138C3—71no—
139C3—141no—
140C3C2×ΓL(1,𝔽8)×D14311yescharacteristic, complemented
141C2—4292no—
142C2ΓL(1,𝔽8)×D42911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D429
commutator, cosocle, hypocenterC3×C7×C11×C13C22×C3
FrattiniC1G
Fitting, socleC2×C3×C7×C11×C13C2×C3
LCSG ▷ C3×C7×C11×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×C11×C13×ΓL(1,𝔽8)C2semidirect
2. ΓL(1,𝔽8)×D429C2direct (N × H)
3. C2×C7×D429C3semidirect
5. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
7. C7×D429C2×C3semidirect
9. C2×C11×C13×ΓL(1,𝔽8)D3semidirect
10. C2×C3×C7×C11×C13C2×C3semidirect
15. C11×C13×ΓL(1,𝔽8)C2×D3semidirect
16. C3×C7×C11×C13C22×C3semidirect
21. C2×C7×C11×C13C3×D3semidirect
22. C2×D429ΓL(1,𝔽8)direct (N × H)
24. C2×C3×C13×ΓL(1,𝔽8)D11semidirect
26. C2×C3×C11×ΓL(1,𝔽8)D13semidirect
31. C7×C11×C13C2×C3×D3semidirect
36. C2×C3×C11×C13C2×ΓL(1,𝔽8)semidirect
37. D429C2×ΓL(1,𝔽8)direct (N × H)
39. C3×C13×ΓL(1,𝔽8)C2×D11semidirect
40. C3×C11×ΓL(1,𝔽8)C2×D13semidirect
42. C2×C3×C7×C13C3×D11semidirect
45. C2×C13×ΓL(1,𝔽8)D33semidirect
48. C2×C3×C7×C11C3×D13semidirect
51. C2×C11×ΓL(1,𝔽8)D39semidirect
55. C3×C11×C13C22×ΓL(1,𝔽8)semidirect
59. C2×C11×C13D3×ΓL(1,𝔽8)semidirect
61. C3×C7×C13C2×C3×D11semidirect
63. C13×ΓL(1,𝔽8)C2×D33semidirect
67. C3×C7×C11C2×C3×D13semidirect
69. C11×ΓL(1,𝔽8)C2×D39semidirect
72. C2×C7×C13C3×D33semidirect
76. C2×C7×C11C3×D39semidirect
78. C11×C13C2×D3×ΓL(1,𝔽8)semidirect
81. C2×C3×ΓL(1,𝔽8)D143semidirect
85. C7×C13C2×C3×D33semidirect
90. C2×C3×C13ΓL(1,𝔽8)×D11semidirect
93. C7×C11C2×C3×D39semidirect
96. C2×C3×C11ΓL(1,𝔽8)×D13semidirect
99. C3×ΓL(1,𝔽8)C2×D143semidirect
102. C2×C3×C7C3×D143semidirect
106. C2×ΓL(1,𝔽8)D429direct (N × H)
109. C3×C13C2×ΓL(1,𝔽8)×D11semidirect
113. C3×C11C2×ΓL(1,𝔽8)×D13semidirect
115. C2×C13ΓL(1,𝔽8)×D33semidirect
117. C2×C11ΓL(1,𝔽8)×D39semidirect
119. C3×C7C2×C3×D143semidirect
121. ΓL(1,𝔽8)C2×D429direct (N × H)
125. C2×C7C3×D429semidirect
126. C13C2×ΓL(1,𝔽8)×D33semidirect
129. C11C2×ΓL(1,𝔽8)×D39semidirect
131. C7C2×C3×D429semidirect
135. C2×C3ΓL(1,𝔽8)×D143semidirect
140. C3C2×ΓL(1,𝔽8)×D143semidirect
142. C2ΓL(1,𝔽8)×D429direct (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
ΓL(1,𝔽8)×D4292
C2×C7×D4293
C2×ΓL(1,𝔽8)×D1433
C2×C3×D4297
C2×ΓL(1,𝔽8)×D3911
C2×ΓL(1,𝔽8)×D3313