Finite Group Databasebeta

← C22×C7×ΓL(1,𝔽8)×C11⋊C5 #44568615

Subgroups — C22×C7×ΓL(1,𝔽8)×C11⋊C5 #44568615

Order 32340.

200 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 90 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⋊C5C213yesmaximal, direct product factor
2C22×C72×C11⋊C5C311yesmaximal, characteristic, complemented, Hall
3C7×ΓL(1,𝔽8)×C11⋊C5C2211yescharacteristic, direct product factor, Hall
4C22×C7×C11×ΓL(1,𝔽8)C511yesmaximal, characteristic, complemented, Hall
5C2×C72×C11⋊C5C2×C313yescomplemented
6C22×C3×C7×C11⋊C5—71nomaximal
7C22×ΓL(1,𝔽8)×C11⋊C5C711yesmaximal, characteristic, direct product factor
8C2×C7×C11×ΓL(1,𝔽8)C2×C513yescomplemented
9C22×C5×C7×ΓL(1,𝔽8)—111nomaximal, Hall
10C72×C11⋊C5C22×C311yescharacteristic, complemented, Hall
11C2×ΓL(1,𝔽8)×C11⋊C5C2×C713yesdirect product factor
12C2×C3×C7×C11⋊C5—73no—
13C22×C72×C11C3×C511yescharacteristic, complemented, Hall
14C7×C11×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented, Hall
15C22×C7×C11⋊C5—32no—
16C22×C7×C11⋊C5ΓL(1,𝔽8)11yescharacteristic, direct product factor
17C22×C7×C11⋊C5C3×C711yescharacteristic, complemented
18C2×C5×C7×ΓL(1,𝔽8)—113no—
19C3×C7×C11⋊C5—71no—
20ΓL(1,𝔽8)×C11⋊C5C22×C711yescharacteristic, direct product factor
21C2×C72×C11C2×C3×C513yescomplemented
22C22×C5×C72—111noHall
23C22×C3×C7×C11—71no—
24C22×C11×ΓL(1,𝔽8)C5×C711yescharacteristic, complemented
25C2×C7×C11⋊C5—36no—
26C2×C7×C11⋊C5C2×ΓL(1,𝔽8)13yesdirect product factor
27C2×C7×C11⋊C5C2×C3×C713yescomplemented
28C5×C7×ΓL(1,𝔽8)—111noHall
29C22×C3×C11⋊C5—71noHall
30C22×C7×ΓL(1,𝔽8)C11⋊C511yescharacteristic, direct product factor, Hall
31C72×C11C22×C3×C511yescharacteristic, complemented, Hall
32C2×C5×C72—113no—
33C2×C3×C7×C11—73no—
34C2×C11×ΓL(1,𝔽8)C2×C5×C713yescomplemented
35C22×C3×C5×C7—771no—
36C22×C5×ΓL(1,𝔽8)—111no—
37C7×C11⋊C5—32no—
38C7×C11⋊C5C22×ΓL(1,𝔽8)11yescharacteristic, direct product factor
39C7×C11⋊C5C22×C3×C711yescharacteristic, complemented
40C2×C3×C11⋊C5—73no—
41C22×C7×C11—32no—
42C22×C7×C11C5×ΓL(1,𝔽8)11yescharacteristic, complemented
43C22×C7×C11C3×C5×C711yescharacteristic, complemented
44C2×C7×ΓL(1,𝔽8)C2×C11⋊C513yesdirect product factor
45C5×C72—111noHall
46C3×C7×C11—71no—
47C11×ΓL(1,𝔽8)C22×C5×C711yescharacteristic, complemented
48C22×C11⋊C5C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
49C2×C3×C5×C7—773no—
50C2×C5×ΓL(1,𝔽8)—113no—
51C22×C72C3×C11⋊C511yescharacteristic, complemented, Hall
52C3×C11⋊C5—71noHall
53C2×C7×C11—36no—
54C2×C7×C11C2×C5×ΓL(1,𝔽8)13yescomplemented
55C2×C7×C11C2×C3×C5×C713yescomplemented
56C7×ΓL(1,𝔽8)C22×C11⋊C511yescharacteristic, direct product factor, Hall
57C22×C5×C7—111no—
58C22×C5×C7—111no—
59C22×C5×C7—332no—
60C22×C3×C11—71noHall
61C2×C11⋊C5C2×C7×ΓL(1,𝔽8)13yesdirect product factor
62C3×C5×C7—771no—
63C5×ΓL(1,𝔽8)—111no—
64C2×C72C2×C3×C11⋊C513yescomplemented
65C22×C3×C7—71no—
66C22×ΓL(1,𝔽8)C7×C11⋊C511yescharacteristic, direct product factor
67C7×C11—32no—
68C7×C11C22×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
69C7×C11C22×C3×C5×C711yescharacteristic, complemented
70C2×C5×C7—113no—
71C2×C5×C7—113no—
72C2×C5×C7—336no—
73C2×C3×C11—73no—
74C22×C3×C5—771noHall
75C11⋊C5C22×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
76C72C22×C3×C11⋊C511yescharacteristic, complemented, Sylow
77C22×C11C5×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
78C2×C3×C7—73no—
79C2×ΓL(1,𝔽8)C2×C7×C11⋊C513yesdirect product factor
80C5×C7—111no—
81C5×C7—111no—
82C5×C7—332no—
83C3×C11—71noHall
84C2×C3×C5—773no—
85C22×C7—32no—
86C22×C7ΓL(1,𝔽8)×C11⋊C511yescharacteristic, direct product factor
87C22×C7C3×C7×C11⋊C511yescharacteristic, complemented
88C2×C11C2×C5×C7×ΓL(1,𝔽8)13yescomplemented
89C3×C7—71no—
90ΓL(1,𝔽8)C22×C7×C11⋊C511yescharacteristic, direct product factor
91C22×C5—111noHall
92C3×C5—771noHall
93C2×C7—36no—
94C2×C7C2×C3×C7×C11⋊C513yescomplemented
95C2×C7C2×ΓL(1,𝔽8)×C11⋊C513yesdirect product factor
96C22×C3—71noHall
97C11C22×C5×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
98C2×C5—113no—
99C7—32no—
100C7C22×ΓL(1,𝔽8)×C11⋊C511yescharacteristic, direct product factor
101C7C22×C3×C7×C11⋊C511yescharacteristic, complemented
102C2×C3—73no—
103C5—111noSylow
104C22C7×ΓL(1,𝔽8)×C11⋊C511yescharacteristic, direct product factor, Sylow
105C3—71noSylow
106C2C2×C7×ΓL(1,𝔽8)×C11⋊C513yesdirect product factor
subgroupG/H
center, hypercenterC22×C7ΓL(1,𝔽8)×C11⋊C5
commutator, cosocle, hypocenterC7×C11C22×C3×C5×C7
FrattiniC1G
Fitting, socleC22×C72×C11C3×C5
LCSG ▷ C7×C11
UCSC1 ◁ C22×C7

Complements

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

NH ≅ G/Ntype
1. C2×C7×ΓL(1,𝔽8)×C11⋊C5C2direct (N × H)
2. C22×C72×C11⋊C5C3semidirect
3. C7×ΓL(1,𝔽8)×C11⋊C5C22direct (N × H)
4. C22×C7×C11×ΓL(1,𝔽8)C5semidirect
5. C2×C72×C11⋊C5C2×C3semidirect
7. C22×ΓL(1,𝔽8)×C11⋊C5C7direct (N × H)
8. C2×C7×C11×ΓL(1,𝔽8)C2×C5semidirect
10. C72×C11⋊C5C22×C3semidirect
11. C2×ΓL(1,𝔽8)×C11⋊C5C2×C7direct (N × H)
13. C22×C72×C11C3×C5semidirect
14. C7×C11×ΓL(1,𝔽8)C22×C5semidirect
16. C22×C7×C11⋊C5ΓL(1,𝔽8)direct (N × H)
17. C22×C7×C11⋊C5C3×C7semidirect
20. ΓL(1,𝔽8)×C11⋊C5C22×C7direct (N × H)
21. C2×C72×C11C2×C3×C5semidirect
24. C22×C11×ΓL(1,𝔽8)C5×C7semidirect
26. C2×C7×C11⋊C5C2×ΓL(1,𝔽8)direct (N × H)
27. C2×C7×C11⋊C5C2×C3×C7semidirect
30. C22×C7×ΓL(1,𝔽8)C11⋊C5direct (N × H)
31. C72×C11C22×C3×C5semidirect
34. C2×C11×ΓL(1,𝔽8)C2×C5×C7semidirect
38. C7×C11⋊C5C22×ΓL(1,𝔽8)direct (N × H)
39. C7×C11⋊C5C22×C3×C7semidirect
42. C22×C7×C11C5×ΓL(1,𝔽8)semidirect
43. C22×C7×C11C3×C5×C7semidirect
44. C2×C7×ΓL(1,𝔽8)C2×C11⋊C5direct (N × H)
47. C11×ΓL(1,𝔽8)C22×C5×C7semidirect
48. C22×C11⋊C5C7×ΓL(1,𝔽8)direct (N × H)
51. C22×C72C3×C11⋊C5semidirect
54. C2×C7×C11C2×C5×ΓL(1,𝔽8)semidirect
55. C2×C7×C11C2×C3×C5×C7semidirect
56. C7×ΓL(1,𝔽8)C22×C11⋊C5direct (N × H)
61. C2×C11⋊C5C2×C7×ΓL(1,𝔽8)direct (N × H)
64. C2×C72C2×C3×C11⋊C5semidirect
66. C22×ΓL(1,𝔽8)C7×C11⋊C5direct (N × H)
68. C7×C11C22×C5×ΓL(1,𝔽8)semidirect
69. C7×C11C22×C3×C5×C7semidirect
75. C11⋊C5C22×C7×ΓL(1,𝔽8)direct (N × H)
76. C72C22×C3×C11⋊C5semidirect
77. C22×C11C5×C7×ΓL(1,𝔽8)semidirect
79. C2×ΓL(1,𝔽8)C2×C7×C11⋊C5direct (N × H)
86. C22×C7ΓL(1,𝔽8)×C11⋊C5direct (N × H)
87. C22×C7C3×C7×C11⋊C5semidirect
88. C2×C11C2×C5×C7×ΓL(1,𝔽8)semidirect
90. ΓL(1,𝔽8)C22×C7×C11⋊C5direct (N × H)
94. C2×C7C2×C3×C7×C11⋊C5semidirect
95. C2×C7C2×ΓL(1,𝔽8)×C11⋊C5direct (N × H)
97. C11C22×C5×C7×ΓL(1,𝔽8)semidirect
100. C7C22×ΓL(1,𝔽8)×C11⋊C5direct (N × H)
101. C7C22×C3×C7×C11⋊C5semidirect
104. C22C7×ΓL(1,𝔽8)×C11⋊C5direct (N × H)
106. C2C2×C7×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
C22×C72×C11⋊C53
C22×C7×C11×ΓL(1,𝔽8)5
C22×C3×C7×C11⋊C57
C22×ΓL(1,𝔽8)×C11⋊C57
C22×C5×C7×ΓL(1,𝔽8)11