Finite Group Databasebeta

← ΓL(1,𝔽8)×D11×D39 #49114781

Subgroups — ΓL(1,𝔽8)×D11×D39 #49114781

Order 36036.

176 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 48 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
1ΓL(1,𝔽8)×D429C211yesmaximal, characteristic, complemented
2C11×ΓL(1,𝔽8)×D39C211yesmaximal, characteristic, complemented
3C3×C13×ΓL(1,𝔽8)×D11C211yesmaximal, characteristic, complemented
4C7×D11×D39C311yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×D11×D13—31nomaximal
6C3×C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7ΓL(1,𝔽8)×D143—31no—
8C13×ΓL(1,𝔽8)×D11—21no—
9C13×ΓL(1,𝔽8)×D11D311yescharacteristic, complemented
10C11×ΓL(1,𝔽8)×D13—31no—
11C7×D429C2×C311yescharacteristic, complemented
12C7×C11×D39C2×C311yescharacteristic, complemented
13C3×C7×C13×D11C2×C311yescharacteristic, complemented
14C3×D11×D39—71nomaximal, Hall
15C7×D11×D13—31noHall
16C2×ΓL(1,𝔽8)×D39—111nomaximal, Hall
17C11×C13×ΓL(1,𝔽8)—21no—
18C11×C13×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
19C3×C7×C11×C13C22×C311yescharacteristic, complemented
20D3×ΓL(1,𝔽8)×D11—131nomaximal, Hall
21C3×D429—71no—
22C3×C11×D39—71no—
23C32×C13×D11—71no—
24C7×D143—31no—
25C7×C13×D11C3×D311yescharacteristic, complemented
26C7×C11×D13—31no—
27D11×D39ΓL(1,𝔽8)11yescharacteristic, direct product factor
28C3×D11×D13—211no—
29C2×C3×C13×ΓL(1,𝔽8)—111no—
30ΓL(1,𝔽8)×D39—111no—
31ΓL(1,𝔽8)×D39D1111yescharacteristic, direct product factor
32ΓL(1,𝔽8)×D33—131no—
33D3×C11×ΓL(1,𝔽8)—131no—
34C3×ΓL(1,𝔽8)×D11D1311yescharacteristic, complemented
35C32×C11×C13—71noHall
36C2×C7×D39—111no—
37C2×ΓL(1,𝔽8)×D13—331no—
38C7×C11×C13C2×C3×D311yescharacteristic, complemented, Hall
39D3×C7×D11—131no—
40C2×ΓL(1,𝔽8)×D11—391no—
41D429C2×ΓL(1,𝔽8)11yescharacteristic, complemented
42C11×D39C2×ΓL(1,𝔽8)11yescharacteristic, complemented
43C3×D143—211no—
44C3×C13×D11—71no—
45C3×C13×D11—141no—
46C3×C13×D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
47C3×C11×D13—211no—
48C3×C13×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
49C3×C11×ΓL(1,𝔽8)C2×D1311yescharacteristic, complemented, Hall
50D11×D13—31noHall
51C2×C3×C7×C13—111no—
52C7×D39—111no—
53C7×D39C3×D1111yescharacteristic, complemented
54C2×C13×ΓL(1,𝔽8)—111no—
55C2×C13×ΓL(1,𝔽8)—221no—
56ΓL(1,𝔽8)×D13—31no—
57ΓL(1,𝔽8)×D13—331no—
58C2×C3×D39—771noHall
59C7×D33—131no—
60D3×C7×C11—131no—
61C3×C7×D11C3×D1311yescharacteristic, complemented
62C2×C11×ΓL(1,𝔽8)—391no—
63ΓL(1,𝔽8)×D11—21no—
64ΓL(1,𝔽8)×D11—391no—
65ΓL(1,𝔽8)×D11D3911yescharacteristic, direct product factor
66C3×C11×C13—71no—
67C3×C11×C13—141no—
68C3×C11×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented
69C3×D3×D11—911noHall
70C2×C7×D13—331noHall
71C2×C7×D11—391noHall
72D143—31no—
73C11×D13—31no—
74C13×D11D3×ΓL(1,𝔽8)11yescharacteristic, complemented
75C3×C7×C13C2×C3×D1111yescharacteristic, complemented
76C13×ΓL(1,𝔽8)—21no—
77C13×ΓL(1,𝔽8)D3×D1111yescharacteristic, complemented
78C2×D3×ΓL(1,𝔽8)—1431noHall
79C2×C32×C13—771no—
80C3×D39—71no—
81C3×D39—771no—
82C3×C7×C11C2×C3×D1311yescharacteristic, complemented
83C11×ΓL(1,𝔽8)—21no—
84C11×ΓL(1,𝔽8)C2×D3911yescharacteristic, complemented
85C3×D33—911no—
86C3×D3×C11—911no—
87C32×D11—71no—
88C2×C7×C13—111no—
89C7×D13—31no—
90C7×D13—331no—
91C2×D39—111no—
92C2×C3×D13—2311no—
93C2×C7×C11—391no—
94C7×D11—391no—
95C7×D11C3×D3911yescharacteristic, complemented
96C11×C13C2×D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
97C2×C3×D11—2731no—
98D3×D11—131no—
99C2×C3×ΓL(1,𝔽8)—111no—
100D3×ΓL(1,𝔽8)—131no—
101D3×ΓL(1,𝔽8)—1431no—
102C32×C13—71noHall
103C32×C11—71noHall
104C7×C13C3×D3×D1111yescharacteristic, complemented, Hall
105C2×D3×C7—1431no—
106C22×ΓL(1,𝔽8)—4291no—
107C2×C3×C13—111no—
108C2×C3×C13—771no—
109C2×C3×C13—1541no—
110D39—111no—
111D39ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
112C3×D13—211no—
113C3×D13—2311no—
114C7×C11C2×C3×D3911yescharacteristic, complemented, Hall
115C2×C3×C11—2731no—
116D33—131no—
117C3×D11—71no—
118C3×D11—141no—
119C3×D11—2731no—
120C3×D11ΓL(1,𝔽8)×D1311yescharacteristic, complemented
121D3×C11—131no—
122C3×ΓL(1,𝔽8)D11×D1311yescharacteristic, complemented, Hall
123C2×D13—331noHall
124C2×D11—391noHall
125C2×C3×C7—111no—
126D3×C7—131no—
127D3×C7—1431no—
128C2×ΓL(1,𝔽8)—111no—
129C2×ΓL(1,𝔽8)—221no—
130C2×ΓL(1,𝔽8)—391no—
131C2×ΓL(1,𝔽8)—4291no—
132C3×C13—71no—
133C3×C13—141no—
134C3×C13C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
135C2×C3×D3—10011noHall
136C3×C11—71no—
137C3×C11—141no—
138C3×C11C2×ΓL(1,𝔽8)×D1311yescharacteristic, complemented
139C22×C7—4291noHall
140C2×C13—111no—
141D13—31no—
142D13—331no—
143C2×C11—391no—
144D11—391no—
145D11ΓL(1,𝔽8)×D3911yescharacteristic, direct product factor
146C3×C7C3×D11×D1311yescharacteristic, complemented
147ΓL(1,𝔽8)—21no—
148ΓL(1,𝔽8)D11×D3911yescharacteristic, direct product factor
149C2×C32—771no—
150C3×D3—911no—
151C3×D3—10011no—
152C2×C7—111no—
153C2×C7—391no—
154C2×C7—4291no—
155C13D3×ΓL(1,𝔽8)×D1111yescharacteristic, complemented, Sylow
156C22×C3—30031no—
157C2×D3—1431no—
158C11C2×ΓL(1,𝔽8)×D3911yescharacteristic, complemented, Sylow
159C32—71noSylow
160C7C3×D11×D3911yescharacteristic, complemented, Sylow
161C2×C3—111no—
162C2×C3—2731no—
163C2×C3—771no—
164C2×C3—1541no—
165C2×C3—30031no—
166D3—131no—
167D3—1431no—
168C22—4291noSylow
169C3—71no—
170C3—141no—
171C3ΓL(1,𝔽8)×D11×D1311yescharacteristic, complemented
172C2—111no—
173C2—391no—
174C2—4291no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C11×C13C22×C3
LCSG ▷ C3×C7×C11×C13
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×D429C2semidirect
2. C11×ΓL(1,𝔽8)×D39C2semidirect
3. C3×C13×ΓL(1,𝔽8)×D11C2semidirect
4. C7×D11×D39C3semidirect
6. C3×C11×C13×ΓL(1,𝔽8)C22semidirect
9. C13×ΓL(1,𝔽8)×D11D3semidirect
11. C7×D429C2×C3semidirect
12. C7×C11×D39C2×C3semidirect
13. C3×C7×C13×D11C2×C3semidirect
18. C11×C13×ΓL(1,𝔽8)C2×D3semidirect
19. C3×C7×C11×C13C22×C3semidirect
25. C7×C13×D11C3×D3semidirect
27. D11×D39ΓL(1,𝔽8)direct (N × H)
31. ΓL(1,𝔽8)×D39D11direct (N × H)
34. C3×ΓL(1,𝔽8)×D11D13semidirect
38. C7×C11×C13C2×C3×D3semidirect
41. D429C2×ΓL(1,𝔽8)semidirect
42. C11×D39C2×ΓL(1,𝔽8)semidirect
46. C3×C13×D11C2×ΓL(1,𝔽8)semidirect
48. C3×C13×ΓL(1,𝔽8)C2×D11semidirect
49. C3×C11×ΓL(1,𝔽8)C2×D13semidirect
53. C7×D39C3×D11semidirect
61. C3×C7×D11C3×D13semidirect
65. ΓL(1,𝔽8)×D11D39direct (N × H)
68. C3×C11×C13C22×ΓL(1,𝔽8)semidirect
74. C13×D11D3×ΓL(1,𝔽8)semidirect
75. C3×C7×C13C2×C3×D11semidirect
77. C13×ΓL(1,𝔽8)D3×D11semidirect
82. C3×C7×C11C2×C3×D13semidirect
84. C11×ΓL(1,𝔽8)C2×D39semidirect
95. C7×D11C3×D39semidirect
96. C11×C13C2×D3×ΓL(1,𝔽8)semidirect
104. C7×C13C3×D3×D11semidirect
111. D39ΓL(1,𝔽8)×D11direct (N × H)
114. C7×C11C2×C3×D39semidirect
120. C3×D11ΓL(1,𝔽8)×D13semidirect
122. C3×ΓL(1,𝔽8)D11×D13semidirect
134. C3×C13C2×ΓL(1,𝔽8)×D11semidirect
138. C3×C11C2×ΓL(1,𝔽8)×D13semidirect
145. D11ΓL(1,𝔽8)×D39direct (N × H)
146. C3×C7C3×D11×D13semidirect
148. ΓL(1,𝔽8)D11×D39direct (N × H)
155. C13D3×ΓL(1,𝔽8)×D11semidirect
158. C11C2×ΓL(1,𝔽8)×D39semidirect
160. C7C3×D11×D39semidirect
171. C3ΓL(1,𝔽8)×D11×D13semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×D4292
C11×ΓL(1,𝔽8)×D392
C3×C13×ΓL(1,𝔽8)×D112
C7×D11×D393
ΓL(1,𝔽8)×D11×D133
C3×D11×D397
C2×ΓL(1,𝔽8)×D3911
D3×ΓL(1,𝔽8)×D1113