Finite Group Databasebeta

← C2×D7×ΓL(1,𝔽8)×C11⋊C5 #44568573

Subgroups — C2×D7×ΓL(1,𝔽8)×C11⋊C5 #44568573

Order 32340.

168 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 63 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×C7×ΓL(1,𝔽8)×C11⋊C5C211yesmaximal, characteristic, complemented
2D7×ΓL(1,𝔽8)×C11⋊C5C212yesmaximal, direct product factor
3C2×C7×D7×C11⋊C5C311yesmaximal, characteristic, complemented, Hall
4C7×ΓL(1,𝔽8)×C11⋊C5C2211yescharacteristic, complemented, Hall
5C2×C11×D7×ΓL(1,𝔽8)C511yesmaximal, characteristic, complemented, Hall
6C2×C72×C11⋊C5C2×C311yescharacteristic, complemented
7C7×D7×C11⋊C5C2×C312yescomplemented
8C2×C3×D7×C11⋊C5—71nomaximal
9C22×ΓL(1,𝔽8)×C11⋊C5—71nomaximal
10C2×C7×C11×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
11C11×D7×ΓL(1,𝔽8)C2×C512yescomplemented
12C2×C5×D7×ΓL(1,𝔽8)—111nomaximal, Hall
13C72×C11⋊C5C22×C311yescharacteristic, complemented, Hall
14C3×D7×C11⋊C5—72no—
15C2×ΓL(1,𝔽8)×C11⋊C5—72no—
16C2×ΓL(1,𝔽8)×C11⋊C5D711yescharacteristic, direct product factor
17C2×C3×C7×C11⋊C5—71no—
18C2×C7×C11×D7C3×C511yescharacteristic, complemented, Hall
19C7×C11×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented, Hall
20C22×C7×C11⋊C5—71no—
21C2×D7×C11⋊C5ΓL(1,𝔽8)11yescharacteristic, direct product factor
22C2×C5×C7×ΓL(1,𝔽8)—111no—
23C5×D7×ΓL(1,𝔽8)—112no—
24C3×C7×C11⋊C5—71no—
25ΓL(1,𝔽8)×C11⋊C5C2×D711yescharacteristic, direct product factor
26C2×C72×C11C2×C3×C511yescharacteristic, complemented
27C7×C11×D7C2×C3×C512yescomplemented
28C2×C5×C7×D7—111noHall
29C2×C3×C11×D7—71no—
30C22×C11×ΓL(1,𝔽8)—71no—
31C2×C7×C11⋊C5—61no—
32C2×C7×C11⋊C5—72no—
33C2×C7×C11⋊C5C2×ΓL(1,𝔽8)11yescharacteristic, complemented
34C2×C7×C11⋊C5C3×D711yescharacteristic, complemented
35D7×C11⋊C5C2×ΓL(1,𝔽8)12yesdirect product factor
36C5×C7×ΓL(1,𝔽8)—111noHall
37C22×C3×C11⋊C5—491noHall
38C2×D7×ΓL(1,𝔽8)C11⋊C511yescharacteristic, direct product factor, Hall
39C72×C11C22×C3×C511yescharacteristic, complemented, Hall
40C2×C5×C72—111no—
41C5×C7×D7—112no—
42C2×C3×C7×C11—71no—
43C3×C11×D7—72no—
44C2×C11×ΓL(1,𝔽8)—72no—
45C2×C11×ΓL(1,𝔽8)C5×D711yescharacteristic, complemented
46C2×C3×C5×D7—771no—
47C22×C5×ΓL(1,𝔽8)—771no—
48C7×C11⋊C5—61no—
49C7×C11⋊C5C22×ΓL(1,𝔽8)11yescharacteristic, complemented
50C7×C11⋊C5C2×C3×D711yescharacteristic, complemented
51C2×C3×C11⋊C5—71no—
52C2×C3×C11⋊C5—492no—
53C22×C7×C11—71no—
54C2×C11×D7C5×ΓL(1,𝔽8)11yescharacteristic, complemented
55C2×C7×ΓL(1,𝔽8)C2×C11⋊C511yescharacteristic, complemented
56D7×ΓL(1,𝔽8)C2×C11⋊C512yesdirect product factor
57C5×C72—111noHall
58C3×C7×C11—71no—
59C11×ΓL(1,𝔽8)C2×C5×D711yescharacteristic, complemented
60C22×C11⋊C5—71noHall
61C2×C3×C5×C7—771no—
62C3×C5×D7—772no—
63C2×C5×ΓL(1,𝔽8)—111no—
64C2×C5×ΓL(1,𝔽8)—772no—
65C2×C7×D7C3×C11⋊C511yescharacteristic, complemented, Hall
66C3×C11⋊C5—71noHall
67C2×C7×C11—61no—
68C2×C7×C11—72no—
69C2×C7×C11C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
70C2×C7×C11C3×C5×D711yescharacteristic, complemented
71C11×D7C2×C5×ΓL(1,𝔽8)12yescomplemented
72C7×ΓL(1,𝔽8)C22×C11⋊C511yescharacteristic, complemented, Hall
73C22×C5×C7—771no—
74C2×C5×D7—111no—
75C22×C3×C11—491noHall
76C2×C11⋊C5—72no—
77C2×C11⋊C5D7×ΓL(1,𝔽8)11yescharacteristic, direct product factor
78C3×C5×C7—771no—
79C5×ΓL(1,𝔽8)—111no—
80C2×C72C2×C3×C11⋊C511yescharacteristic, complemented
81C7×D7C2×C3×C11⋊C512yescomplemented
82C2×C3×D7—71no—
83C22×ΓL(1,𝔽8)—71no—
84C7×C11—61no—
85C7×C11C22×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
86C7×C11C2×C3×C5×D711yescharacteristic, complemented
87C2×C5×C7—111no—
88C2×C5×C7—111no—
89C2×C5×C7—661no—
90C2×C5×C7—772no—
91C5×D7—112no—
92C2×C3×C11—71no—
93C2×C3×C11—492no—
94C22×C3×C5—5391noHall
95C11⋊C5C2×D7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
96C72C22×C3×C11⋊C511yescharacteristic, complemented, Sylow
97C22×C11—71noHall
98C2×C3×C7—71no—
99C3×D7—72no—
100C2×ΓL(1,𝔽8)—72no—
101C2×ΓL(1,𝔽8)D7×C11⋊C511yescharacteristic, direct product factor
102C5×C7—111no—
103C5×C7—111no—
104C5×C7—661no—
105C3×C11—71noHall
106C2×C3×C5—5392no—
107C2×C3×C5—771no—
108C22×C7—71no—
109C2×D7ΓL(1,𝔽8)×C11⋊C511yescharacteristic, direct product factor
110C2×C11—72no—
111C2×C11C5×D7×ΓL(1,𝔽8)11yescharacteristic, complemented
112C3×C7—71no—
113ΓL(1,𝔽8)C2×D7×C11⋊C511yescharacteristic, direct product factor
114C22×C5—771noHall
115C3×C5—771noHall
116C2×C7—61no—
117C2×C7—72no—
118C2×C7C2×ΓL(1,𝔽8)×C11⋊C511yescharacteristic, complemented
119C2×C7C3×D7×C11⋊C511yescharacteristic, complemented
120D7C2×ΓL(1,𝔽8)×C11⋊C512yesdirect product factor
121C22×C3—491noHall
122C11C2×C5×D7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
123C2×C5—111no—
124C2×C5—772no—
125C7—61no—
126C7C22×ΓL(1,𝔽8)×C11⋊C511yescharacteristic, complemented
127C7C2×C3×D7×C11⋊C511yescharacteristic, complemented
128C2×C3—71no—
129C2×C3—492no—
130C5—111noSylow
131C22—71noSylow
132C3—71noSylow
133C2—72no—
134C2D7×ΓL(1,𝔽8)×C11⋊C511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2D7×ΓL(1,𝔽8)×C11⋊C5
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C72×C11C2×C3×C5
LCSG ▷ C72×C11
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C7×ΓL(1,𝔽8)×C11⋊C5C2semidirect
2. D7×ΓL(1,𝔽8)×C11⋊C5C2direct (N × H)
3. C2×C7×D7×C11⋊C5C3semidirect
4. C7×ΓL(1,𝔽8)×C11⋊C5C22semidirect
5. C2×C11×D7×ΓL(1,𝔽8)C5semidirect
6. C2×C72×C11⋊C5C2×C3semidirect
7. C7×D7×C11⋊C5C2×C3semidirect
10. C2×C7×C11×ΓL(1,𝔽8)C2×C5semidirect
11. C11×D7×ΓL(1,𝔽8)C2×C5semidirect
13. C72×C11⋊C5C22×C3semidirect
16. C2×ΓL(1,𝔽8)×C11⋊C5D7direct (N × H)
18. C2×C7×C11×D7C3×C5semidirect
19. C7×C11×ΓL(1,𝔽8)C22×C5semidirect
21. C2×D7×C11⋊C5ΓL(1,𝔽8)direct (N × H)
25. ΓL(1,𝔽8)×C11⋊C5C2×D7direct (N × H)
26. C2×C72×C11C2×C3×C5semidirect
27. C7×C11×D7C2×C3×C5semidirect
33. C2×C7×C11⋊C5C2×ΓL(1,𝔽8)semidirect
34. C2×C7×C11⋊C5C3×D7semidirect
35. D7×C11⋊C5C2×ΓL(1,𝔽8)direct (N × H)
38. C2×D7×ΓL(1,𝔽8)C11⋊C5direct (N × H)
39. C72×C11C22×C3×C5semidirect
45. C2×C11×ΓL(1,𝔽8)C5×D7semidirect
49. C7×C11⋊C5C22×ΓL(1,𝔽8)semidirect
50. C7×C11⋊C5C2×C3×D7semidirect
54. C2×C11×D7C5×ΓL(1,𝔽8)semidirect
55. C2×C7×ΓL(1,𝔽8)C2×C11⋊C5semidirect
56. D7×ΓL(1,𝔽8)C2×C11⋊C5direct (N × H)
59. C11×ΓL(1,𝔽8)C2×C5×D7semidirect
65. C2×C7×D7C3×C11⋊C5semidirect
69. C2×C7×C11C2×C5×ΓL(1,𝔽8)semidirect
70. C2×C7×C11C3×C5×D7semidirect
71. C11×D7C2×C5×ΓL(1,𝔽8)semidirect
72. C7×ΓL(1,𝔽8)C22×C11⋊C5semidirect
77. C2×C11⋊C5D7×ΓL(1,𝔽8)direct (N × H)
80. C2×C72C2×C3×C11⋊C5semidirect
81. C7×D7C2×C3×C11⋊C5semidirect
85. C7×C11C22×C5×ΓL(1,𝔽8)semidirect
86. C7×C11C2×C3×C5×D7semidirect
95. C11⋊C5C2×D7×ΓL(1,𝔽8)direct (N × H)
96. C72C22×C3×C11⋊C5semidirect
101. C2×ΓL(1,𝔽8)D7×C11⋊C5direct (N × H)
109. C2×D7ΓL(1,𝔽8)×C11⋊C5direct (N × H)
111. C2×C11C5×D7×ΓL(1,𝔽8)semidirect
113. ΓL(1,𝔽8)C2×D7×C11⋊C5direct (N × H)
118. C2×C7C2×ΓL(1,𝔽8)×C11⋊C5semidirect
119. C2×C7C3×D7×C11⋊C5semidirect
120. D7C2×ΓL(1,𝔽8)×C11⋊C5direct (N × H)
122. C11C2×C5×D7×ΓL(1,𝔽8)semidirect
126. C7C22×ΓL(1,𝔽8)×C11⋊C5semidirect
127. C7C2×C3×D7×C11⋊C5semidirect
134. C2D7×ΓL(1,𝔽8)×C11⋊C5direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×ΓL(1,𝔽8)×C11⋊C52
D7×ΓL(1,𝔽8)×C11⋊C52
C2×C7×D7×C11⋊C53
C2×C11×D7×ΓL(1,𝔽8)5
C2×C3×D7×C11⋊C57
C22×ΓL(1,𝔽8)×C11⋊C57
C2×C5×D7×ΓL(1,𝔽8)11