Finite Group Databasebeta

← C22×C11×ΓL(1,𝔽8)×C13⋊C3 #49114816

Subgroups — C22×C11×ΓL(1,𝔽8)×C13⋊C3 #49114816

Order 36036.

240 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 110 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⋊C3C213yesmaximal, direct product factor
2C22×C7×C11×C13⋊C3C311yesmaximal, characteristic, complemented
3C22×C11×C13×ΓL(1,𝔽8)C311yesmaximal, characteristic, complemented
4C22×C11×#63271C311yesmaximal, characteristic, complemented
5C22×C11×#63270C311yesmaximal, characteristic, complemented
6C11×ΓL(1,𝔽8)×C13⋊C3C2211yescharacteristic, direct product factor, Hall
7C2×C11×#63271C2×C313yescomplemented
8C2×C11×#63270C2×C313yescomplemented
9C2×C11×C13×ΓL(1,𝔽8)C2×C313yescomplemented
10C2×C7×C11×C13⋊C3C2×C313yescomplemented
11C22×C3×C11×C13⋊C3—71nomaximal, Hall
12C22×C7×C11×C13C3211yescharacteristic, complemented, Hall
13C22×ΓL(1,𝔽8)×C13⋊C3C1111yesmaximal, characteristic, direct product factor, Hall
14C11×#63271C22×C311yescharacteristic, complemented
15C11×#63270C22×C311yescharacteristic, complemented
16C11×C13×ΓL(1,𝔽8)C22×C311yescharacteristic, complemented
17C7×C11×C13⋊C3C22×C311yescharacteristic, complemented
18C22×C3×C11×ΓL(1,𝔽8)—131nomaximal, Hall
19C2×C3×C11×C13⋊C3—73no—
20C2×C7×C11×C13C2×C3213yescomplemented
21C22×C3×C11×C13—71no—
22C22×C11×C13⋊C3—71no—
23C22×C11×C13⋊C3—71no—
24C22×C11×C13⋊C3ΓL(1,𝔽8)11yescharacteristic, direct product factor
25C2×ΓL(1,𝔽8)×C13⋊C3C2×C1113yesdirect product factor
26C2×C3×C11×ΓL(1,𝔽8)—133no—
27C3×C11×C13⋊C3—71noHall
28C22×#63270C3×C1111yescharacteristic, complemented
29C22×#63271C3×C1111yescharacteristic, complemented
30C22×C13×ΓL(1,𝔽8)C3×C1111yescharacteristic, complemented
31C22×C7×C13⋊C3C3×C1111yescharacteristic, complemented
32C7×C11×C13C22×C3211yescharacteristic, complemented, Hall
33C22×C3×C7×C11—131no—
34C22×C11×ΓL(1,𝔽8)—131no—
35C22×C11×ΓL(1,𝔽8)—131no—
36C22×C11×ΓL(1,𝔽8)C13⋊C311yescharacteristic, direct product factor
37C2×C3×C11×C13—73no—
38C2×C11×C13⋊C3—73no—
39C2×C11×C13⋊C3—73no—
40C2×C11×C13⋊C3C2×ΓL(1,𝔽8)13yesdirect product factor
41ΓL(1,𝔽8)×C13⋊C3C22×C1111yescharacteristic, direct product factor, Hall
42C3×C11×ΓL(1,𝔽8)—131noHall
43C22×C11×C13C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
44C2×#63271C2×C3×C1113yescomplemented
45C2×#63270C2×C3×C1113yescomplemented
46C2×C13×ΓL(1,𝔽8)C2×C3×C1113yescomplemented
47C2×C7×C13⋊C3C2×C3×C1113yescomplemented
48C22×C3×C13⋊C3—71noHall
49C2×C3×C7×C11—133no—
50C2×C11×ΓL(1,𝔽8)—133no—
51C2×C11×ΓL(1,𝔽8)—133no—
52C2×C11×ΓL(1,𝔽8)C2×C13⋊C313yesdirect product factor
53C3×C11×C13—71no—
54C11×C13⋊C3—71no—
55C11×C13⋊C3—71no—
56C11×C13⋊C3C22×ΓL(1,𝔽8)11yescharacteristic, direct product factor
57C22×C32×C11—911noHall
58C22×C7×C13C32×C1111yescharacteristic, complemented, Hall
59C22×C7×C11C3×C13⋊C311yescharacteristic, complemented, Hall
60C2×C11×C13C2×C3×ΓL(1,𝔽8)13yescomplemented
61#63271C22×C3×C1111yescharacteristic, complemented
62#63270C22×C3×C1111yescharacteristic, complemented
63C7×C13⋊C3C22×C3×C1111yescharacteristic, complemented
64C13×ΓL(1,𝔽8)C22×C3×C1111yescharacteristic, complemented
65C22×C3×ΓL(1,𝔽8)—131noHall
66C2×C3×C13⋊C3—73no—
67C3×C7×C11—131no—
68C11×ΓL(1,𝔽8)—131no—
69C11×ΓL(1,𝔽8)—131no—
70C11×ΓL(1,𝔽8)C22×C13⋊C311yescharacteristic, direct product factor
71C2×C32×C11—913no—
72C2×C7×C13C2×C32×C1113yescomplemented
73C22×C3×C13—71no—
74C22×C13⋊C3—71no—
75C22×C13⋊C3—71no—
76C22×C13⋊C3C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
77C2×C7×C11C2×C3×C13⋊C313yescomplemented
78C11×C13C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
79C22×C3×C11—71no—
80C22×C3×C11—131no—
81C22×C3×C11—911no—
82C22×C3×C11—911no—
83C2×C3×ΓL(1,𝔽8)—133no—
84C3×C13⋊C3—71noHall
85C32×C11—911noHall
86C7×C13C22×C32×C1111yescharacteristic, complemented, Hall
87C22×C3×C7—131no—
88C22×ΓL(1,𝔽8)—131no—
89C22×ΓL(1,𝔽8)—131no—
90C22×ΓL(1,𝔽8)C11×C13⋊C311yescharacteristic, direct product factor
91C2×C3×C13—73no—
92C2×C13⋊C3—73no—
93C2×C13⋊C3—73no—
94C2×C13⋊C3C2×C11×ΓL(1,𝔽8)13yesdirect product factor
95C7×C11C22×C3×C13⋊C311yescharacteristic, complemented, Hall
96C2×C3×C11—73no—
97C2×C3×C11—133no—
98C2×C3×C11—913no—
99C2×C3×C11—913no—
100C3×ΓL(1,𝔽8)—131noHall
101C22×C13C3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
102C22×C11ΓL(1,𝔽8)×C13⋊C311yescharacteristic, direct product factor, Hall
103C2×C3×C7—133no—
104C2×ΓL(1,𝔽8)—133no—
105C2×ΓL(1,𝔽8)—133no—
106C2×ΓL(1,𝔽8)C2×C11×C13⋊C313yesdirect product factor
107C3×C13—71no—
108C13⋊C3—71no—
109C13⋊C3—71no—
110C13⋊C3C22×C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
111C22×C32—911noHall
112C3×C11—71no—
113C3×C11—131no—
114C3×C11—911no—
115C3×C11—911no—
116C22×C7C3×C11×C13⋊C311yescharacteristic, complemented, Hall
117C2×C13C2×C3×C11×ΓL(1,𝔽8)13yescomplemented
118C2×C11C2×ΓL(1,𝔽8)×C13⋊C313yesdirect product factor
119C3×C7—131no—
120ΓL(1,𝔽8)—131no—
121ΓL(1,𝔽8)—131no—
122ΓL(1,𝔽8)C22×C11×C13⋊C311yescharacteristic, direct product factor
123C2×C32—913no—
124C2×C7C2×C3×C11×C13⋊C313yescomplemented
125C13C22×C3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
126C22×C3—71no—
127C22×C3—131no—
128C22×C3—911no—
129C22×C3—911no—
130C11C22×ΓL(1,𝔽8)×C13⋊C311yescharacteristic, direct product factor, Sylow
131C32—911noSylow
132C7C22×C3×C11×C13⋊C311yescharacteristic, complemented, Sylow
133C2×C3—73no—
134C2×C3—133no—
135C2×C3—913no—
136C2×C3—913no—
137C22C11×ΓL(1,𝔽8)×C13⋊C311yescharacteristic, direct product factor, Sylow
138C3—71no—
139C3—131no—
140C3—911no—
141C3—911no—
142C2C2×C11×ΓL(1,𝔽8)×C13⋊C313yesdirect product factor
subgroupG/H
center, hypercenterC22×C11ΓL(1,𝔽8)×C13⋊C3
commutator, cosocle, hypocenterC7×C13C22×C32×C11
FrattiniC1G
Fitting, socleC22×C7×C11×C13C32
LCSG ▷ C7×C13
UCSC1 ◁ C22×C11

Complements

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

NH ≅ G/Ntype
1. C2×C11×ΓL(1,𝔽8)×C13⋊C3C2direct (N × H)
2. C22×C7×C11×C13⋊C3C3semidirect
3. C22×C11×C13×ΓL(1,𝔽8)C3semidirect
4. C22×C11×#63271C3semidirect
5. C22×C11×#63270C3semidirect
6. C11×ΓL(1,𝔽8)×C13⋊C3C22direct (N × H)
7. C2×C11×#63271C2×C3semidirect
8. C2×C11×#63270C2×C3semidirect
9. C2×C11×C13×ΓL(1,𝔽8)C2×C3semidirect
10. C2×C7×C11×C13⋊C3C2×C3semidirect
12. C22×C7×C11×C13C32semidirect
13. C22×ΓL(1,𝔽8)×C13⋊C3C11direct (N × H)
14. C11×#63271C22×C3semidirect
15. C11×#63270C22×C3semidirect
16. C11×C13×ΓL(1,𝔽8)C22×C3semidirect
17. C7×C11×C13⋊C3C22×C3semidirect
20. C2×C7×C11×C13C2×C32semidirect
24. C22×C11×C13⋊C3ΓL(1,𝔽8)direct (N × H)
25. C2×ΓL(1,𝔽8)×C13⋊C3C2×C11direct (N × H)
28. C22×#63270C3×C11semidirect
29. C22×#63271C3×C11semidirect
30. C22×C13×ΓL(1,𝔽8)C3×C11semidirect
31. C22×C7×C13⋊C3C3×C11semidirect
32. C7×C11×C13C22×C32semidirect
36. C22×C11×ΓL(1,𝔽8)C13⋊C3direct (N × H)
40. C2×C11×C13⋊C3C2×ΓL(1,𝔽8)direct (N × H)
41. ΓL(1,𝔽8)×C13⋊C3C22×C11direct (N × H)
43. C22×C11×C13C3×ΓL(1,𝔽8)semidirect
44. C2×#63271C2×C3×C11semidirect
45. C2×#63270C2×C3×C11semidirect
46. C2×C13×ΓL(1,𝔽8)C2×C3×C11semidirect
47. C2×C7×C13⋊C3C2×C3×C11semidirect
52. C2×C11×ΓL(1,𝔽8)C2×C13⋊C3direct (N × H)
56. C11×C13⋊C3C22×ΓL(1,𝔽8)direct (N × H)
58. C22×C7×C13C32×C11semidirect
59. C22×C7×C11C3×C13⋊C3semidirect
60. C2×C11×C13C2×C3×ΓL(1,𝔽8)semidirect
61. #63271C22×C3×C11semidirect
62. #63270C22×C3×C11semidirect
63. C7×C13⋊C3C22×C3×C11semidirect
64. C13×ΓL(1,𝔽8)C22×C3×C11semidirect
70. C11×ΓL(1,𝔽8)C22×C13⋊C3direct (N × H)
72. C2×C7×C13C2×C32×C11semidirect
76. C22×C13⋊C3C11×ΓL(1,𝔽8)direct (N × H)
77. C2×C7×C11C2×C3×C13⋊C3semidirect
78. C11×C13C22×C3×ΓL(1,𝔽8)semidirect
86. C7×C13C22×C32×C11semidirect
90. C22×ΓL(1,𝔽8)C11×C13⋊C3direct (N × H)
94. C2×C13⋊C3C2×C11×ΓL(1,𝔽8)direct (N × H)
95. C7×C11C22×C3×C13⋊C3semidirect
101. C22×C13C3×C11×ΓL(1,𝔽8)semidirect
102. C22×C11ΓL(1,𝔽8)×C13⋊C3direct (N × H)
106. C2×ΓL(1,𝔽8)C2×C11×C13⋊C3direct (N × H)
110. C13⋊C3C22×C11×ΓL(1,𝔽8)direct (N × H)
116. C22×C7C3×C11×C13⋊C3semidirect
117. C2×C13C2×C3×C11×ΓL(1,𝔽8)semidirect
118. C2×C11C2×ΓL(1,𝔽8)×C13⋊C3direct (N × H)
122. ΓL(1,𝔽8)C22×C11×C13⋊C3direct (N × H)
124. C2×C7C2×C3×C11×C13⋊C3semidirect
125. C13C22×C3×C11×ΓL(1,𝔽8)semidirect
130. C11C22×ΓL(1,𝔽8)×C13⋊C3direct (N × H)
132. C7C22×C3×C11×C13⋊C3semidirect
137. C22C11×ΓL(1,𝔽8)×C13⋊C3direct (N × H)
142. C2C2×C11×ΓL(1,𝔽8)×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
C22×C7×C11×C13⋊C33
C22×C11×C13×ΓL(1,𝔽8)3
C22×C11×#632713
C22×C11×#632703
C22×C3×C11×C13⋊C37
C22×ΓL(1,𝔽8)×C13⋊C311
C22×C3×C11×ΓL(1,𝔽8)13