Finite Group Databasebeta

← C22×C5×C7×C13×ΓL(1,𝔽8) #50304502

Subgroups — C22×C5×C7×C13×ΓL(1,𝔽8) #50304502

Order 38220.

200 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 120 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×C5×C7×C13×ΓL(1,𝔽8)C213yesmaximal, direct product factor
2C22×C5×C72×C13C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C13×ΓL(1,𝔽8)C2211yescharacteristic, direct product factor, Hall
4C22×C7×C13×ΓL(1,𝔽8)C511yesmaximal, characteristic, direct product factor, Hall
5C2×C5×C72×C13C2×C313yescomplemented
6C22×C5×C13×ΓL(1,𝔽8)C711yesmaximal, characteristic, direct product factor
7C22×C3×C5×C7×C13—71nomaximal
8C2×C7×C13×ΓL(1,𝔽8)C2×C513yesdirect product factor
9C5×C72×C13C22×C311yescharacteristic, complemented, Hall
10C22×C5×C7×ΓL(1,𝔽8)C1311yesmaximal, characteristic, direct product factor, Hall
11C2×C5×C13×ΓL(1,𝔽8)C2×C713yesdirect product factor
12C2×C3×C5×C7×C13—73no—
13C22×C72×C13C3×C511yescharacteristic, complemented, Hall
14C7×C13×ΓL(1,𝔽8)C22×C511yescharacteristic, direct product factor, Hall
15C22×C5×C7×C13—32no—
16C22×C5×C7×C13ΓL(1,𝔽8)11yescharacteristic, direct product factor
17C22×C5×C7×C13C3×C711yescharacteristic, complemented
18C2×C5×C7×ΓL(1,𝔽8)C2×C1313yesdirect product factor
19C3×C5×C7×C13—71no—
20C5×C13×ΓL(1,𝔽8)C22×C711yescharacteristic, direct product factor
21C2×C72×C13C2×C3×C513yescomplemented
22C22×C3×C7×C13—71no—
23C22×C13×ΓL(1,𝔽8)C5×C711yescharacteristic, direct product factor
24C22×C5×C72C3×C1311yescharacteristic, complemented, Hall
25C2×C5×C7×C13—36no—
26C2×C5×C7×C13C2×ΓL(1,𝔽8)13yesdirect product factor
27C2×C5×C7×C13C2×C3×C713yescomplemented
28C22×C3×C5×C13—71noHall
29C5×C7×ΓL(1,𝔽8)C22×C1311yescharacteristic, direct product factor, Hall
30C72×C13C22×C3×C511yescharacteristic, complemented, Hall
31C22×C7×ΓL(1,𝔽8)C5×C1311yescharacteristic, direct product factor, Hall
32C2×C3×C7×C13—73no—
33C2×C13×ΓL(1,𝔽8)C2×C5×C713yesdirect product factor
34C2×C5×C72C2×C3×C1313yescomplemented
35C5×C7×C13—32no—
36C5×C7×C13C22×ΓL(1,𝔽8)11yescharacteristic, direct product factor
37C5×C7×C13C22×C3×C711yescharacteristic, complemented
38C22×C3×C5×C7—71no—
39C22×C5×ΓL(1,𝔽8)C7×C1311yescharacteristic, direct product factor
40C2×C3×C5×C13—73no—
41C22×C7×C13—32no—
42C22×C7×C13C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
43C22×C7×C13C3×C5×C711yescharacteristic, complemented
44C2×C7×ΓL(1,𝔽8)C2×C5×C1313yesdirect product factor
45C3×C7×C13—71no—
46C13×ΓL(1,𝔽8)C22×C5×C711yescharacteristic, direct product factor
47C22×C5×C13C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
48C5×C72C22×C3×C1311yescharacteristic, complemented, Hall
49C2×C3×C5×C7—73no—
50C2×C5×ΓL(1,𝔽8)C2×C7×C1313yesdirect product factor
51C22×C72C3×C5×C1311yescharacteristic, complemented, Hall
52C3×C5×C13—71noHall
53C2×C7×C13—36no—
54C2×C7×C13C2×C5×ΓL(1,𝔽8)13yesdirect product factor
55C2×C7×C13C2×C3×C5×C713yescomplemented
56C22×C3×C13—71noHall
57C7×ΓL(1,𝔽8)C22×C5×C1311yescharacteristic, direct product factor, Hall
58C22×C5×C7—32no—
59C22×C5×C7C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor
60C22×C5×C7C3×C7×C1311yescharacteristic, complemented
61C2×C5×C13C2×C7×ΓL(1,𝔽8)13yesdirect product factor
62C3×C5×C7—71no—
63C5×ΓL(1,𝔽8)C22×C7×C1311yescharacteristic, direct product factor
64C2×C72C2×C3×C5×C1313yescomplemented
65C7×C13—32no—
66C7×C13C22×C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
67C7×C13C22×C3×C5×C711yescharacteristic, complemented
68C22×C3×C7—71no—
69C22×ΓL(1,𝔽8)C5×C7×C1311yescharacteristic, direct product factor
70C2×C3×C13—73no—
71C2×C5×C7—36no—
72C2×C5×C7C2×C13×ΓL(1,𝔽8)13yesdirect product factor
73C2×C5×C7C2×C3×C7×C1313yescomplemented
74C5×C13C22×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
75C22×C3×C5—71noHall
76C22×C13C5×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
77C72C22×C3×C5×C1311yescharacteristic, complemented, Sylow
78C2×C3×C7—73no—
79C2×ΓL(1,𝔽8)C2×C5×C7×C1313yesdirect product factor
80C3×C13—71noHall
81C5×C7—32no—
82C5×C7C22×C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor
83C5×C7C22×C3×C7×C1311yescharacteristic, complemented
84C2×C3×C5—73no—
85C22×C7—32no—
86C22×C7C5×C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor
87C22×C7C3×C5×C7×C1311yescharacteristic, complemented
88C2×C13C2×C5×C7×ΓL(1,𝔽8)13yesdirect product factor
89C3×C7—71no—
90ΓL(1,𝔽8)C22×C5×C7×C1311yescharacteristic, direct product factor
91C22×C5C7×C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
92C3×C5—71noHall
93C2×C7—36no—
94C2×C7C2×C3×C5×C7×C1313yescomplemented
95C2×C7C2×C5×C13×ΓL(1,𝔽8)13yesdirect product factor
96C13C22×C5×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
97C22×C3—71noHall
98C2×C5C2×C7×C13×ΓL(1,𝔽8)13yesdirect product factor
99C7—32no—
100C7C22×C3×C5×C7×C1311yescharacteristic, complemented
101C7C22×C5×C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor
102C2×C3—73no—
103C5C22×C7×C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
104C22C5×C7×C13×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
105C3—71noSylow
106C2C2×C5×C7×C13×ΓL(1,𝔽8)13yesdirect product factor
subgroupG/H
center, hypercenterC22×C5×C7×C13ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C5×C7×C13
FrattiniC1G
Fitting, socleC22×C5×C72×C13C3
LCSG ▷ C7
UCSC1 ◁ C22×C5×C7×C13

Complements

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

NH ≅ G/Ntype
1. C2×C5×C7×C13×ΓL(1,𝔽8)C2direct (N × H)
2. C22×C5×C72×C13C3semidirect
3. C5×C7×C13×ΓL(1,𝔽8)C22direct (N × H)
4. C22×C7×C13×ΓL(1,𝔽8)C5direct (N × H)
5. C2×C5×C72×C13C2×C3semidirect
6. C22×C5×C13×ΓL(1,𝔽8)C7direct (N × H)
8. C2×C7×C13×ΓL(1,𝔽8)C2×C5direct (N × H)
9. C5×C72×C13C22×C3semidirect
10. C22×C5×C7×ΓL(1,𝔽8)C13direct (N × H)
11. C2×C5×C13×ΓL(1,𝔽8)C2×C7direct (N × H)
13. C22×C72×C13C3×C5semidirect
14. C7×C13×ΓL(1,𝔽8)C22×C5direct (N × H)
16. C22×C5×C7×C13ΓL(1,𝔽8)direct (N × H)
17. C22×C5×C7×C13C3×C7semidirect
18. C2×C5×C7×ΓL(1,𝔽8)C2×C13direct (N × H)
20. C5×C13×ΓL(1,𝔽8)C22×C7direct (N × H)
21. C2×C72×C13C2×C3×C5semidirect
23. C22×C13×ΓL(1,𝔽8)C5×C7direct (N × H)
24. C22×C5×C72C3×C13semidirect
26. C2×C5×C7×C13C2×ΓL(1,𝔽8)direct (N × H)
27. C2×C5×C7×C13C2×C3×C7semidirect
29. C5×C7×ΓL(1,𝔽8)C22×C13direct (N × H)
30. C72×C13C22×C3×C5semidirect
31. C22×C7×ΓL(1,𝔽8)C5×C13direct (N × H)
33. C2×C13×ΓL(1,𝔽8)C2×C5×C7direct (N × H)
34. C2×C5×C72C2×C3×C13semidirect
36. C5×C7×C13C22×ΓL(1,𝔽8)direct (N × H)
37. C5×C7×C13C22×C3×C7semidirect
39. C22×C5×ΓL(1,𝔽8)C7×C13direct (N × H)
42. C22×C7×C13C5×ΓL(1,𝔽8)direct (N × H)
43. C22×C7×C13C3×C5×C7semidirect
44. C2×C7×ΓL(1,𝔽8)C2×C5×C13direct (N × H)
46. C13×ΓL(1,𝔽8)C22×C5×C7direct (N × H)
47. C22×C5×C13C7×ΓL(1,𝔽8)direct (N × H)
48. C5×C72C22×C3×C13semidirect
50. C2×C5×ΓL(1,𝔽8)C2×C7×C13direct (N × H)
51. C22×C72C3×C5×C13semidirect
54. C2×C7×C13C2×C5×ΓL(1,𝔽8)direct (N × H)
55. C2×C7×C13C2×C3×C5×C7semidirect
57. C7×ΓL(1,𝔽8)C22×C5×C13direct (N × H)
59. C22×C5×C7C13×ΓL(1,𝔽8)direct (N × H)
60. C22×C5×C7C3×C7×C13semidirect
61. C2×C5×C13C2×C7×ΓL(1,𝔽8)direct (N × H)
63. C5×ΓL(1,𝔽8)C22×C7×C13direct (N × H)
64. C2×C72C2×C3×C5×C13semidirect
66. C7×C13C22×C5×ΓL(1,𝔽8)direct (N × H)
67. C7×C13C22×C3×C5×C7semidirect
69. C22×ΓL(1,𝔽8)C5×C7×C13direct (N × H)
72. C2×C5×C7C2×C13×ΓL(1,𝔽8)direct (N × H)
73. C2×C5×C7C2×C3×C7×C13semidirect
74. C5×C13C22×C7×ΓL(1,𝔽8)direct (N × H)
76. C22×C13C5×C7×ΓL(1,𝔽8)direct (N × H)
77. C72C22×C3×C5×C13semidirect
79. C2×ΓL(1,𝔽8)C2×C5×C7×C13direct (N × H)
82. C5×C7C22×C13×ΓL(1,𝔽8)direct (N × H)
83. C5×C7C22×C3×C7×C13semidirect
86. C22×C7C5×C13×ΓL(1,𝔽8)direct (N × H)
87. C22×C7C3×C5×C7×C13semidirect
88. C2×C13C2×C5×C7×ΓL(1,𝔽8)direct (N × H)
90. ΓL(1,𝔽8)C22×C5×C7×C13direct (N × H)
91. C22×C5C7×C13×ΓL(1,𝔽8)direct (N × H)
94. C2×C7C2×C3×C5×C7×C13semidirect
95. C2×C7C2×C5×C13×ΓL(1,𝔽8)direct (N × H)
96. C13C22×C5×C7×ΓL(1,𝔽8)direct (N × H)
98. C2×C5C2×C7×C13×ΓL(1,𝔽8)direct (N × H)
100. C7C22×C3×C5×C7×C13semidirect
101. C7C22×C5×C13×ΓL(1,𝔽8)direct (N × H)
103. C5C22×C7×C13×ΓL(1,𝔽8)direct (N × H)
104. C22C5×C7×C13×ΓL(1,𝔽8)direct (N × H)
106. C2C2×C5×C7×C13×ΓL(1,𝔽8)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C7×C13×ΓL(1,𝔽8)2
C22×C5×C72×C133
C22×C7×C13×ΓL(1,𝔽8)5
C22×C5×C13×ΓL(1,𝔽8)7
C22×C3×C5×C7×C137
C22×C5×C7×ΓL(1,𝔽8)13