Finite Group Databasebeta

← C2×ΓL(1,𝔽8)×D11×C13⋊C3 #49114635

Subgroups — C2×ΓL(1,𝔽8)×D11×C13⋊C3 #49114635

Order 36036.

240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 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×C11×ΓL(1,𝔽8)×C13⋊C3C211yesmaximal, characteristic, complemented
2ΓL(1,𝔽8)×D11×C13⋊C3C212yesmaximal, direct product factor
3C2×D11×#63271C311yesmaximal, characteristic, complemented
4C2×D11×#63270C311yesmaximal, characteristic, complemented
5C2×C13×ΓL(1,𝔽8)×D11C311yesmaximal, characteristic, complemented
6C2×C7×D11×C13⋊C3C311yesmaximal, characteristic, complemented
7C11×ΓL(1,𝔽8)×C13⋊C3C2211yescharacteristic, complemented, Hall
8D11×#63271C2×C312yescomplemented
9D11×#63270C2×C312yescomplemented
10C13×ΓL(1,𝔽8)×D11C2×C312yescomplemented
11C7×D11×C13⋊C3C2×C312yescomplemented
12C2×C11×#63271C2×C311yescharacteristic, complemented
13C2×C11×#63270C2×C311yescharacteristic, complemented
14C2×C11×C13×ΓL(1,𝔽8)C2×C311yescharacteristic, complemented
15C2×C7×C11×C13⋊C3C2×C311yescharacteristic, complemented
16C2×C3×D11×C13⋊C3—71nomaximal, Hall
17C2×C7×C13×D11C3211yescharacteristic, complemented, Hall
18C22×ΓL(1,𝔽8)×C13⋊C3—111nomaximal, Hall
19C11×#63271C22×C311yescharacteristic, complemented
20C11×#63270C22×C311yescharacteristic, complemented
21C11×C13×ΓL(1,𝔽8)C22×C311yescharacteristic, complemented
22C7×C11×C13⋊C3C22×C311yescharacteristic, complemented
23C2×C3×ΓL(1,𝔽8)×D11—131nomaximal, Hall
24C2×C3×C11×C13⋊C3—71no—
25C3×D11×C13⋊C3—72no—
26C7×C13×D11C2×C3212yescomplemented
27C2×C7×C11×C13C2×C3211yescharacteristic, complemented
28C2×C3×C13×D11—71no—
29C2×D11×C13⋊C3—71no—
30C2×D11×C13⋊C3—71no—
31C2×D11×C13⋊C3ΓL(1,𝔽8)11yescharacteristic, direct product factor
32C2×ΓL(1,𝔽8)×C13⋊C3—112no—
33C2×ΓL(1,𝔽8)×C13⋊C3D1111yescharacteristic, direct product factor
34C2×C3×C11×ΓL(1,𝔽8)—131no—
35C3×ΓL(1,𝔽8)×D11—132no—
36C3×C11×C13⋊C3—71noHall
37C22×#63270—111no—
38C22×#63271—111no—
39C22×C13×ΓL(1,𝔽8)—111no—
40C22×C7×C13⋊C3—111no—
41C7×C11×C13C22×C3211yescharacteristic, complemented, Hall
42C2×C3×C7×D11—131no—
43C2×ΓL(1,𝔽8)×D11—131no—
44C2×ΓL(1,𝔽8)×D11—131no—
45C2×ΓL(1,𝔽8)×D11C13⋊C311yescharacteristic, direct product factor
46C2×C3×C11×C13—71no—
47C3×C13×D11—72no—
48C2×C11×C13⋊C3—71no—
49C2×C11×C13⋊C3—71no—
50C2×C11×C13⋊C3C2×ΓL(1,𝔽8)11yescharacteristic, complemented
51D11×C13⋊C3—72no—
52D11×C13⋊C3—72no—
53D11×C13⋊C3C2×ΓL(1,𝔽8)12yesdirect product factor
54ΓL(1,𝔽8)×C13⋊C3C2×D1111yescharacteristic, direct product factor, Hall
55C3×C11×ΓL(1,𝔽8)—131noHall
56C2×C13×D11C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
57C2×#63271—112no—
58C2×#63271C3×D1111yescharacteristic, complemented
59C2×#63270—112no—
60C2×#63270C3×D1111yescharacteristic, complemented
61C2×C13×ΓL(1,𝔽8)—112no—
62C2×C13×ΓL(1,𝔽8)C3×D1111yescharacteristic, complemented
63C2×C7×C13⋊C3—112no—
64C2×C7×C13⋊C3C3×D1111yescharacteristic, complemented
65C22×C3×C13⋊C3—771noHall
66C2×C3×C7×C11—131no—
67C3×C7×D11—132no—
68C2×C11×ΓL(1,𝔽8)—131no—
69C2×C11×ΓL(1,𝔽8)—131no—
70C2×C11×ΓL(1,𝔽8)C2×C13⋊C311yescharacteristic, complemented
71ΓL(1,𝔽8)×D11—132no—
72ΓL(1,𝔽8)×D11—132no—
73ΓL(1,𝔽8)×D11C2×C13⋊C312yesdirect product factor
74C3×C11×C13—71no—
75C11×C13⋊C3—71no—
76C11×C13⋊C3—71no—
77C11×C13⋊C3C22×ΓL(1,𝔽8)11yescharacteristic, complemented
78C2×C32×D11—911noHall
79C22×C7×C13—111noHall
80C2×C7×D11C3×C13⋊C311yescharacteristic, complemented, Hall
81C2×C11×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
82C13×D11C2×C3×ΓL(1,𝔽8)12yescomplemented
83#63271C2×C3×D1111yescharacteristic, complemented
84#63270C2×C3×D1111yescharacteristic, complemented
85C7×C13⋊C3C2×C3×D1111yescharacteristic, complemented
86C13×ΓL(1,𝔽8)C2×C3×D1111yescharacteristic, complemented
87C22×C3×ΓL(1,𝔽8)—1431noHall
88C2×C3×C13⋊C3—71no—
89C2×C3×C13⋊C3—772no—
90C3×C7×C11—131no—
91C11×ΓL(1,𝔽8)—131no—
92C11×ΓL(1,𝔽8)—131no—
93C11×ΓL(1,𝔽8)C22×C13⋊C311yescharacteristic, complemented
94C2×C32×C11—911no—
95C32×D11—912no—
96C2×C7×C13—112no—
97C2×C7×C13C32×D1111yescharacteristic, complemented
98C22×C3×C13—771no—
99C22×C13⋊C3—111no—
100C22×C13⋊C3—771no—
101C22×C13⋊C3—771no—
102C2×C7×C11C2×C3×C13⋊C311yescharacteristic, complemented
103C7×D11C2×C3×C13⋊C312yescomplemented
104C11×C13C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
105C2×C3×D11—71no—
106C2×C3×D11—131no—
107C2×C3×D11—911no—
108C2×C3×D11—911no—
109C2×C3×ΓL(1,𝔽8)—131no—
110C2×C3×ΓL(1,𝔽8)—1432no—
111C3×C13⋊C3—71noHall
112C32×C11—911noHall
113C7×C13C2×C32×D1111yescharacteristic, complemented, Hall
114C22×C3×C7—1431no—
115C22×ΓL(1,𝔽8)—111no—
116C22×ΓL(1,𝔽8)—1431no—
117C22×ΓL(1,𝔽8)—1431no—
118C2×C3×C13—71no—
119C2×C3×C13—772no—
120C2×C13⋊C3—71no—
121C2×C13⋊C3—71no—
122C2×C13⋊C3—112no—
123C2×C13⋊C3—772no—
124C2×C13⋊C3—772no—
125C2×C13⋊C3ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
126C7×C11C22×C3×C13⋊C311yescharacteristic, complemented, Hall
127C2×C3×C11—71no—
128C2×C3×C11—131no—
129C2×C3×C11—911no—
130C2×C3×C11—911no—
131C3×D11—72no—
132C3×D11—132no—
133C3×D11—912no—
134C3×D11—912no—
135C3×ΓL(1,𝔽8)—131noHall
136C22×C13—111noHall
137C2×D11ΓL(1,𝔽8)×C13⋊C311yescharacteristic, direct product factor, Hall
138C2×C3×C7—131no—
139C2×C3×C7—1432no—
140C2×ΓL(1,𝔽8)—112no—
141C2×ΓL(1,𝔽8)—131no—
142C2×ΓL(1,𝔽8)—131no—
143C2×ΓL(1,𝔽8)—1432no—
144C2×ΓL(1,𝔽8)—1432no—
145C2×ΓL(1,𝔽8)D11×C13⋊C311yescharacteristic, direct product factor
146C3×C13—71no—
147C13⋊C3—71no—
148C13⋊C3—71no—
149C13⋊C3C2×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
150C22×C32—10011noHall
151C3×C11—71no—
152C3×C11—131no—
153C3×C11—911no—
154C3×C11—911no—
155C22×C7—111noHall
156C2×C13—112no—
157C2×C13C3×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
158C2×C11C2×ΓL(1,𝔽8)×C13⋊C311yescharacteristic, complemented
159D11C2×ΓL(1,𝔽8)×C13⋊C312yesdirect product factor
160C3×C7—131no—
161ΓL(1,𝔽8)—131no—
162ΓL(1,𝔽8)—131no—
163ΓL(1,𝔽8)C2×D11×C13⋊C311yescharacteristic, direct product factor
164C2×C32—911no—
165C2×C32—10012no—
166C2×C7—112no—
167C2×C7C3×D11×C13⋊C311yescharacteristic, complemented
168C13C2×C3×ΓL(1,𝔽8)×D1111yescharacteristic, complemented, Sylow
169C22×C3—771no—
170C22×C3—1431no—
171C22×C3—10011no—
172C22×C3—10011no—
173C11C22×ΓL(1,𝔽8)×C13⋊C311yescharacteristic, complemented, Sylow
174C32—911noSylow
175C7C2×C3×D11×C13⋊C311yescharacteristic, complemented, Sylow
176C2×C3—71no—
177C2×C3—131no—
178C2×C3—772no—
179C2×C3—911no—
180C2×C3—911no—
181C2×C3—1432no—
182C2×C3—10012no—
183C2×C3—10012no—
184C22—111noSylow
185C3—71no—
186C3—131no—
187C3—911no—
188C3—911no—
189C2—112no—
190C2ΓL(1,𝔽8)×D11×C13⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D11×C13⋊C3
commutator, cosocle, hypocenterC7×C11×C13C22×C32
FrattiniC1G
Fitting, socleC2×C7×C11×C13C2×C32
LCSG ▷ C7×C11×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C11×ΓL(1,𝔽8)×C13⋊C3C2semidirect
2. ΓL(1,𝔽8)×D11×C13⋊C3C2direct (N × H)
3. C2×D11×#63271C3semidirect
4. C2×D11×#63270C3semidirect
5. C2×C13×ΓL(1,𝔽8)×D11C3semidirect
6. C2×C7×D11×C13⋊C3C3semidirect
7. C11×ΓL(1,𝔽8)×C13⋊C3C22semidirect
8. D11×#63271C2×C3semidirect
9. D11×#63270C2×C3semidirect
10. C13×ΓL(1,𝔽8)×D11C2×C3semidirect
11. C7×D11×C13⋊C3C2×C3semidirect
12. C2×C11×#63271C2×C3semidirect
13. C2×C11×#63270C2×C3semidirect
14. C2×C11×C13×ΓL(1,𝔽8)C2×C3semidirect
15. C2×C7×C11×C13⋊C3C2×C3semidirect
17. C2×C7×C13×D11C32semidirect
19. C11×#63271C22×C3semidirect
20. C11×#63270C22×C3semidirect
21. C11×C13×ΓL(1,𝔽8)C22×C3semidirect
22. C7×C11×C13⋊C3C22×C3semidirect
26. C7×C13×D11C2×C32semidirect
27. C2×C7×C11×C13C2×C32semidirect
31. C2×D11×C13⋊C3ΓL(1,𝔽8)direct (N × H)
33. C2×ΓL(1,𝔽8)×C13⋊C3D11direct (N × H)
41. C7×C11×C13C22×C32semidirect
45. C2×ΓL(1,𝔽8)×D11C13⋊C3direct (N × H)
50. C2×C11×C13⋊C3C2×ΓL(1,𝔽8)semidirect
53. D11×C13⋊C3C2×ΓL(1,𝔽8)direct (N × H)
54. ΓL(1,𝔽8)×C13⋊C3C2×D11direct (N × H)
56. C2×C13×D11C3×ΓL(1,𝔽8)semidirect
58. C2×#63271C3×D11semidirect
60. C2×#63270C3×D11semidirect
62. C2×C13×ΓL(1,𝔽8)C3×D11semidirect
64. C2×C7×C13⋊C3C3×D11semidirect
70. C2×C11×ΓL(1,𝔽8)C2×C13⋊C3semidirect
73. ΓL(1,𝔽8)×D11C2×C13⋊C3direct (N × H)
77. C11×C13⋊C3C22×ΓL(1,𝔽8)semidirect
80. C2×C7×D11C3×C13⋊C3semidirect
81. C2×C11×C13C2×C3×ΓL(1,𝔽8)semidirect
82. C13×D11C2×C3×ΓL(1,𝔽8)semidirect
83. #63271C2×C3×D11semidirect
84. #63270C2×C3×D11semidirect
85. C7×C13⋊C3C2×C3×D11semidirect
86. C13×ΓL(1,𝔽8)C2×C3×D11semidirect
93. C11×ΓL(1,𝔽8)C22×C13⋊C3semidirect
97. C2×C7×C13C32×D11semidirect
102. C2×C7×C11C2×C3×C13⋊C3semidirect
103. C7×D11C2×C3×C13⋊C3semidirect
104. C11×C13C22×C3×ΓL(1,𝔽8)semidirect
113. C7×C13C2×C32×D11semidirect
125. C2×C13⋊C3ΓL(1,𝔽8)×D11direct (N × H)
126. C7×C11C22×C3×C13⋊C3semidirect
137. C2×D11ΓL(1,𝔽8)×C13⋊C3direct (N × H)
145. C2×ΓL(1,𝔽8)D11×C13⋊C3direct (N × H)
149. C13⋊C3C2×ΓL(1,𝔽8)×D11direct (N × H)
157. C2×C13C3×ΓL(1,𝔽8)×D11semidirect
158. C2×C11C2×ΓL(1,𝔽8)×C13⋊C3semidirect
159. D11C2×ΓL(1,𝔽8)×C13⋊C3direct (N × H)
163. ΓL(1,𝔽8)C2×D11×C13⋊C3direct (N × H)
167. C2×C7C3×D11×C13⋊C3semidirect
168. C13C2×C3×ΓL(1,𝔽8)×D11semidirect
173. C11C22×ΓL(1,𝔽8)×C13⋊C3semidirect
175. C7C2×C3×D11×C13⋊C3semidirect
190. C2ΓL(1,𝔽8)×D11×C13⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C11×ΓL(1,𝔽8)×C13⋊C32
ΓL(1,𝔽8)×D11×C13⋊C32
C2×D11×#632713
C2×D11×#632703
C2×C13×ΓL(1,𝔽8)×D113
C2×C7×D11×C13⋊C33
C2×C3×D11×C13⋊C37
C22×ΓL(1,𝔽8)×C13⋊C311
C2×C3×ΓL(1,𝔽8)×D1113