Finite Group Databasebeta

← #26035062

Subgroups — #26035062

Order 11392.

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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
1#19152434C2112yesmaximal, complemented
2#19152433C2116yesmaximal, complemented
3C2×#15668057C2118yesmaximal, complemented
4H(5,𝔽2)×D89C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C89C211yesmaximal, characteristic, complemented
6#15668060C2218yescomplemented
7#15668059C22148yescomplemented
8#15668058C22148yescomplemented
9#15668057C221144yescomplemented
10#15668057C22172yescomplemented
11C2×#13083574C22136yescomplemented
12C2×#13083573C22172yescomplemented
13#41×D89C22196yescomplemented
14C22×C22⋊D89C22124yescomplemented
15C2×D4×D89C22172yescomplemented
16H(5,𝔽2)×C89C22116yescomplemented
17C2×#41×C89C2216yescomplemented
18C22×D4×C89C2219yescomplemented
19C23×D89C23124yescomplemented
20#13083575C23132yescomplemented
21#13083574C231144yescomplemented
22#13083574C23148yescomplemented
23#13083573C231288yescomplemented
24C2×Dic178C23136yescomplemented
25Q8×D89C23132yescomplemented
26C2×D356C23136yescomplemented
27C2×C22⋊D89C231144yescomplemented
28C2×C22⋊D89C23172yescomplemented
29D4×D89C231288yescomplemented
30C22×Dic89C23124yescomplemented
31C2×C4×D89C23172yescomplemented
32C24×C89C2316yescomplemented
33#41×C89C23148yescomplemented
34C2×Q8×C89C2312yescomplemented
35C2×D4×C89C23172yescomplemented
36C2×D4×C89C23118yescomplemented
37C22×C4×C89C2319yescomplemented
38C23×C89—86no—
39C23×C89C2419yes—
40C23×C89C24124yes—
41C22×D89—824no—
42C22×D89C24172yes—
43Q8×C89C2418yes—
44D4×C89C24172yes—
45C2×C4×C89C2416yes—
46C2×C4×C89C24136yes—
47C22⋊D89C241144yes—
48C22⋊D89C24116yes—
49C2×Dic89C24172yes—
50D356C24148yes—
51C4×D89C24196yes—
52Dic178C24148yes—
53C22×C89—424no—
54C22×C89—49no—
55C22×C89C25118yes—
56C22×C89C2511yescharacteristic
57C2×D89—472no—
58C2×D89C25116yes—
59C4×C89C25112yes—
60Dic89C25116yes—
61C2×C89—218no—
62C2×C89—21no—
63C2×C89C2611yescharacteristic
64D89—216no—
65H(7,𝔽2)—891nomaximal, Sylow
66C89H(7,𝔽2)11yescharacteristic, complemented, Sylow
67#585—8916no—
68#585—8912no—
69C2×H(5,𝔽2)—8916no—
70C2×H(5,𝔽2)—8918no—
71C2×H(5,𝔽2)D8911yescharacteristic, complemented
722−1+4—8948no—
732−1+4—898no—
74H(5,𝔽2)—8948no—
75H(5,𝔽2)—89144no—
76H(5,𝔽2)—8972no—
77H(5,𝔽2)C2×D89116yescomplemented
78C2×#41—8996no—
79C2×#41—8972no—
80C2×#41—8936no—
81C2×#41C2×D8916yescomplemented
82C22×D4—8924no—
83C22×D4—8972no—
84C22×D4C2×D8919yescomplemented
85C24—8924no—
86C24C22×D8916yescomplemented
87#41—8932no—
88#41—89288no—
89#41—89144no—
90#41—8948no—
91#41C22×D89148yescomplemented
92C2×Q8—8932no—
93C2×Q8—8936no—
94C2×Q8C22×D8912yescomplemented
95C2×D4—89144no—
96C2×D4—8972no—
97C2×D4—89288no—
98C2×D4—8936no—
99C2×D4C22×D89172yescomplemented
100C2×D4C22×D89118yescomplemented
101C22×C4—8924no—
102C22×C4—8972no—
103C22×C4C22×D8919yescomplemented
104C23—86no—
105C23—8972no—
106C23—71224no—
107C23C23×D89124yes—
108C23C23×D8919yes—
109Q8—8948no—
110Q8C23×D8918yes—
111D4—89144no—
112D4—8916no—
113D4—8948no—
114D4C23×D89172yes—
115C2×C4—8972no—
116C2×C4—8996no—
117C2×C4C23×D89136yes—
118C2×C4C23×D8916yes—
119C22—49no—
120C22—424no—
121C22—8916no—
122C22—35672no—
123C22C24×D89118yes—
124C22C24×D8911yescharacteristic
125C4—8916no—
126C4C24×D89112yes—
127C2—21no—
128C2—218no—
129C2—17816no—
130C2C25×D8911yescharacteristic
subgroupG/H
center, FrattiniC2C25×D89
commutator, socle, cosocleC2×C89C26
FittingC2×H(5,𝔽2)×C89C2
hypercenterC2×H(5,𝔽2)D89
hypocenterC89H(7,𝔽2)
LCSG ▷ C2×C89 ▷ C89
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #19152434C2semidirect
2. #19152433C2semidirect
3. C2×#15668057C2semidirect
4. H(5,𝔽2)×D89C2semidirect
5. C2×H(5,𝔽2)×C89C2semidirect
6. #15668060C22semidirect
7. #15668059C22semidirect
8. #15668058C22semidirect
9. #15668057C22semidirect
10. #15668057C22semidirect
11. C2×#13083574C22semidirect
12. C2×#13083573C22semidirect
13. #41×D89C22semidirect
14. C22×C22⋊D89C22semidirect
15. C2×D4×D89C22semidirect
16. H(5,𝔽2)×C89C22semidirect
17. C2×#41×C89C22semidirect
18. C22×D4×C89C22semidirect
19. C23×D89C23semidirect
20. #13083575C23semidirect
21. #13083574C23semidirect
22. #13083574C23semidirect
23. #13083573C23semidirect
24. C2×Dic178C23semidirect
25. Q8×D89C23semidirect
26. C2×D356C23semidirect
27. C2×C22⋊D89C23semidirect
28. C2×C22⋊D89C23semidirect
29. D4×D89C23semidirect
30. C22×Dic89C23semidirect
31. C2×C4×D89C23semidirect
32. C24×C89C23semidirect
33. #41×C89C23semidirect
34. C2×Q8×C89C23semidirect
35. C2×D4×C89C23semidirect
36. C2×D4×C89C23semidirect
37. C22×C4×C89C23semidirect
66. C89H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D89semidirect
77. H(5,𝔽2)C2×D89semidirect
81. C2×#41C2×D89semidirect
84. C22×D4C2×D89semidirect
86. C24C22×D89semidirect
91. #41C22×D89semidirect
94. C2×Q8C22×D89semidirect
99. C2×D4C22×D89semidirect
100. C2×D4C22×D89semidirect
103. C22×C4C22×D89semidirect

Maximal subgroups

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

subgroupindex
#191524342
#191524332
C2×#156680572
H(5,𝔽2)×D892
C2×H(5,𝔽2)×C892
H(7,𝔽2)89