Finite Group Databasebeta

← D3×C11×ΓL(1,𝔽8)×D13 #49114765

Subgroups — D3×C11×ΓL(1,𝔽8)×D13 #49114765

Order 36036.

176 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C11×ΓL(1,𝔽8)×D39C211yesmaximal, characteristic, complemented
2D3×C11×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C3×C11×ΓL(1,𝔽8)×D13C211yesmaximal, characteristic, complemented
4D3×C7×C11×D13C311yesmaximal, characteristic, complemented
5C2×C11×ΓL(1,𝔽8)×D13—31nomaximal
6C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C11×ΓL(1,𝔽8)×D13—21no—
8C11×ΓL(1,𝔽8)×D13—31no—
9C11×ΓL(1,𝔽8)×D13D311yescharacteristic, direct product factor
10C7×C11×D39C2×C311yescharacteristic, complemented
11D3×C7×C11×C13C2×C311yescharacteristic, complemented
12C3×C7×C11×D13C2×C311yescharacteristic, complemented
13C2×C11×C13×ΓL(1,𝔽8)—31no—
14C3×D3×C11×D13—71nomaximal, Hall
15C2×C7×C11×D13—31noHall
16D3×ΓL(1,𝔽8)×D13C1111yesmaximal, characteristic, direct product factor, Hall
17C11×C13×ΓL(1,𝔽8)—21no—
18C11×C13×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
19C3×C7×C11×C13C22×C311yescharacteristic, complemented
20C2×D3×C11×ΓL(1,𝔽8)—131nomaximal, Hall
21C3×C11×D39—71no—
22C3×D3×C11×C13—71no—
23C32×C11×D13—71no—
24C7×C11×D13—31no—
25C7×C11×D13C3×D311yescharacteristic, complemented
26C2×C7×C11×C13—31no—
27C2×C3×C11×D13—211no—
28D3×C11×D13ΓL(1,𝔽8)11yescharacteristic, direct product factor
29ΓL(1,𝔽8)×D39C2×C1111yescharacteristic, complemented
30D3×C13×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
31C3×ΓL(1,𝔽8)×D13C2×C1111yescharacteristic, complemented
32C2×C3×C11×ΓL(1,𝔽8)—131no—
33D3×C11×ΓL(1,𝔽8)—131no—
34D3×C11×ΓL(1,𝔽8)D1311yescharacteristic, direct product factor
35C32×C11×C13—71noHall
36D3×C7×D13C3×C1111yescharacteristic, complemented
37C2×ΓL(1,𝔽8)×D13—31no—
38C7×C11×C13C2×C3×D311yescharacteristic, complemented, Hall
39C2×D3×C7×C11—131no—
40C22×C11×ΓL(1,𝔽8)—391no—
41C2×C3×C11×C13—211no—
42C11×D39C2×ΓL(1,𝔽8)11yescharacteristic, complemented
43D3×C11×C13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
44C3×C11×D13—71no—
45C3×C11×D13—141no—
46C3×C11×D13—211no—
47C3×C11×D13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
48C3×C13×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented, Hall
49C3×C11×ΓL(1,𝔽8)C2×D1311yescharacteristic, complemented, Hall
50C2×C11×D13—31noHall
51C7×D39C2×C3×C1111yescharacteristic, complemented
52D3×C7×C13C2×C3×C1111yescharacteristic, complemented
53C3×C7×D13C2×C3×C1111yescharacteristic, complemented
54C2×C13×ΓL(1,𝔽8)—31no—
55ΓL(1,𝔽8)×D13—21no—
56ΓL(1,𝔽8)×D13—31no—
57ΓL(1,𝔽8)×D13D3×C1111yescharacteristic, direct product factor
58C3×D3×D13—71noHall
59C2×C3×C7×C11—131no—
60D3×C7×C11—131no—
61D3×C7×C11C3×D1311yescharacteristic, complemented
62C2×C11×ΓL(1,𝔽8)—31no—
63C2×C11×ΓL(1,𝔽8)—131no—
64C2×C11×ΓL(1,𝔽8)—261no—
65C2×C11×ΓL(1,𝔽8)—391no—
66C3×C11×C13—71no—
67C3×C11×C13—141no—
68C3×C11×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented
69C2×C3×D3×C11—911noHall
70C2×C7×D13—31noHall
71C22×C7×C11—391noHall
72C2×C11×C13—31no—
73C11×D13—31no—
74C11×D13D3×ΓL(1,𝔽8)11yescharacteristic, direct product factor
75C3×C7×C13C22×C3×C1111yescharacteristic, complemented
76C13×ΓL(1,𝔽8)—21no—
77C13×ΓL(1,𝔽8)C2×D3×C1111yescharacteristic, complemented
78C2×D3×ΓL(1,𝔽8)—131noHall
79C3×D39—71no—
80C3×D3×C13—71no—
81C32×D13—71no—
82C3×C7×C11C2×C3×D1311yescharacteristic, complemented
83C11×ΓL(1,𝔽8)—21no—
84C11×ΓL(1,𝔽8)D3×D1311yescharacteristic, direct product factor
85C2×C32×C11—911no—
86C3×D3×C11—71no—
87C3×D3×C11—911no—
88C2×C7×C13—31no—
89C7×D13—31no—
90C7×D13C3×D3×C1111yescharacteristic, complemented
91C2×C3×D13—211no—
92D3×D13C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
93C2×C7×C11—31no—
94C2×C7×C11—131no—
95C2×C7×C11—391no—
96C11×C13C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
97C22×C3×C11—2731no—
98C2×D3×C11—131no—
99C2×C3×ΓL(1,𝔽8)—131no—
100D3×ΓL(1,𝔽8)—131no—
101D3×ΓL(1,𝔽8)C11×D1311yescharacteristic, direct product factor
102C32×C13—71noHall
103C32×C11—71noHall
104C7×C13C2×C3×D3×C1111yescharacteristic, complemented, Hall
105C2×D3×C7—131no—
106C22×ΓL(1,𝔽8)—391no—
107C2×C3×C13—211no—
108D39C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
109C3×D13—71no—
110C3×D13—141no—
111C3×D13—211no—
112C3×D13C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
113D3×C13C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
114C7×C11C3×D3×D1311yescharacteristic, complemented, Hall
115C2×C3×C11—131no—
116C2×C3×C11—2731no—
117C2×C3×C11—211no—
118C2×C3×C11—911no—
119C2×C3×C11—1821no—
120D3×C11—131no—
121D3×C11ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
122C3×ΓL(1,𝔽8)C2×C11×D1311yescharacteristic, complemented, Hall
123C2×D13—31noHall
124C22×C11—391noHall
125C2×C3×C7—131no—
126D3×C7—131no—
127D3×C7C3×C11×D1311yescharacteristic, complemented
128C2×ΓL(1,𝔽8)—31no—
129C2×ΓL(1,𝔽8)—131no—
130C2×ΓL(1,𝔽8)—261no—
131C2×ΓL(1,𝔽8)—391no—
132C3×C13—71no—
133C3×C13—141no—
134C3×C13C22×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
135C2×C3×D3—911noHall
136C3×C11—71no—
137C3×C11—141no—
138C3×C11C2×ΓL(1,𝔽8)×D1311yescharacteristic, complemented
139C22×C7—391noHall
140C2×C13—31no—
141D13—31no—
142D13D3×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
143C2×C11—31no—
144C2×C11—131no—
145C2×C11—391no—
146C3×C7C2×C3×C11×D1311yescharacteristic, complemented
147ΓL(1,𝔽8)—21no—
148ΓL(1,𝔽8)D3×C11×D1311yescharacteristic, direct product factor
149C2×C32—911no—
150C3×D3—71no—
151C3×D3—911no—
152C2×C7—31no—
153C2×C7—131no—
154C2×C7—391no—
155C13C2×D3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
156C22×C3—2731no—
157C2×D3—131no—
158C11D3×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor, Sylow
159C32—71noSylow
160C7C3×D3×C11×D1311yescharacteristic, complemented, Sylow
161C2×C3—131no—
162C2×C3—2731no—
163C2×C3—211no—
164C2×C3—911no—
165C2×C3—1821no—
166D3—131no—
167D3C11×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
168C22—391noSylow
169C3—71no—
170C3—141no—
171C3C2×C11×ΓL(1,𝔽8)×D1311yescharacteristic, complemented
172C2—31no—
173C2—131no—
174C2—391no—
subgroupG/H
center, hypercenterC11D3×ΓL(1,𝔽8)×D13
commutator, cosocle, hypocenterC3×C7×C13C22×C3×C11
FrattiniC1G
Fitting, socleC3×C7×C11×C13C22×C3
LCSG ▷ C3×C7×C13
UCSC1 ◁ C11

Complements

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

NH ≅ G/Ntype
1. C11×ΓL(1,𝔽8)×D39C2semidirect
2. D3×C11×C13×ΓL(1,𝔽8)C2semidirect
3. C3×C11×ΓL(1,𝔽8)×D13C2semidirect
4. D3×C7×C11×D13C3semidirect
6. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
9. C11×ΓL(1,𝔽8)×D13D3direct (N × H)
10. C7×C11×D39C2×C3semidirect
11. D3×C7×C11×C13C2×C3semidirect
12. C3×C7×C11×D13C2×C3semidirect
16. D3×ΓL(1,𝔽8)×D13C11direct (N × H)
18. C11×C13×ΓL(1,𝔽8)C2×D3semidirect
19. C3×C7×C11×C13C22×C3semidirect
25. C7×C11×D13C3×D3semidirect
28. D3×C11×D13ΓL(1,𝔽8)direct (N × H)
29. ΓL(1,𝔽8)×D39C2×C11semidirect
30. D3×C13×ΓL(1,𝔽8)C2×C11semidirect
31. C3×ΓL(1,𝔽8)×D13C2×C11semidirect
34. D3×C11×ΓL(1,𝔽8)D13direct (N × H)
36. D3×C7×D13C3×C11semidirect
38. C7×C11×C13C2×C3×D3semidirect
42. C11×D39C2×ΓL(1,𝔽8)semidirect
43. D3×C11×C13C2×ΓL(1,𝔽8)semidirect
47. C3×C11×D13C2×ΓL(1,𝔽8)semidirect
48. C3×C13×ΓL(1,𝔽8)C22×C11semidirect
49. C3×C11×ΓL(1,𝔽8)C2×D13semidirect
51. C7×D39C2×C3×C11semidirect
52. D3×C7×C13C2×C3×C11semidirect
53. C3×C7×D13C2×C3×C11semidirect
57. ΓL(1,𝔽8)×D13D3×C11direct (N × H)
61. D3×C7×C11C3×D13semidirect
68. C3×C11×C13C22×ΓL(1,𝔽8)semidirect
74. C11×D13D3×ΓL(1,𝔽8)direct (N × H)
75. C3×C7×C13C22×C3×C11semidirect
77. C13×ΓL(1,𝔽8)C2×D3×C11semidirect
82. C3×C7×C11C2×C3×D13semidirect
84. C11×ΓL(1,𝔽8)D3×D13direct (N × H)
90. C7×D13C3×D3×C11semidirect
92. D3×D13C11×ΓL(1,𝔽8)direct (N × H)
96. C11×C13C2×D3×ΓL(1,𝔽8)semidirect
101. D3×ΓL(1,𝔽8)C11×D13direct (N × H)
104. C7×C13C2×C3×D3×C11semidirect
108. D39C2×C11×ΓL(1,𝔽8)semidirect
112. C3×D13C2×C11×ΓL(1,𝔽8)semidirect
113. D3×C13C2×C11×ΓL(1,𝔽8)semidirect
114. C7×C11C3×D3×D13semidirect
121. D3×C11ΓL(1,𝔽8)×D13direct (N × H)
122. C3×ΓL(1,𝔽8)C2×C11×D13semidirect
127. D3×C7C3×C11×D13semidirect
134. C3×C13C22×C11×ΓL(1,𝔽8)semidirect
138. C3×C11C2×ΓL(1,𝔽8)×D13semidirect
142. D13D3×C11×ΓL(1,𝔽8)direct (N × H)
146. C3×C7C2×C3×C11×D13semidirect
148. ΓL(1,𝔽8)D3×C11×D13direct (N × H)
155. C13C2×D3×C11×ΓL(1,𝔽8)semidirect
158. C11D3×ΓL(1,𝔽8)×D13direct (N × H)
160. C7C3×D3×C11×D13semidirect
167. D3C11×ΓL(1,𝔽8)×D13direct (N × H)
171. C3C2×C11×ΓL(1,𝔽8)×D13semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C11×ΓL(1,𝔽8)×D392
D3×C11×C13×ΓL(1,𝔽8)2
C3×C11×ΓL(1,𝔽8)×D132
D3×C7×C11×D133
C2×C11×ΓL(1,𝔽8)×D133
C3×D3×C11×D137
D3×ΓL(1,𝔽8)×D1311
C2×D3×C11×ΓL(1,𝔽8)13