Finite Group Databasebeta

← #31795599

Subgroups — #31795599

Order 17536.

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#22578239C2112yesmaximal, complemented
2#22578238C2116yesmaximal, complemented
3C2×#17984133C2118yesmaximal, complemented
4H(5,𝔽2)×D137C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C137C211yesmaximal, characteristic, complemented
6#17984136C2218yescomplemented
7#17984135C22148yescomplemented
8#17984134C22148yescomplemented
9#17984133C221144yescomplemented
10#17984133C22172yescomplemented
11C2×#14555510C22136yescomplemented
12C2×#14555509C22172yescomplemented
13#41×D137C22196yescomplemented
14C22×C22⋊D137C22124yescomplemented
15C2×D4×D137C22172yescomplemented
16H(5,𝔽2)×C137C22116yescomplemented
17C2×#41×C137C2216yescomplemented
18C22×D4×C137C2219yescomplemented
19C23×D137C23124yescomplemented
20#14555511C23132yescomplemented
21#14555510C231144yescomplemented
22#14555510C23148yescomplemented
23#14555509C231288yescomplemented
24C2×Dic274C23136yescomplemented
25Q8×D137C23132yescomplemented
26C2×D548C23136yescomplemented
27C2×C22⋊D137C231144yescomplemented
28C2×C22⋊D137C23172yescomplemented
29D4×D137C231288yescomplemented
30C22×Dic137C23124yescomplemented
31C2×C4×D137C23172yescomplemented
32C24×C137C2316yescomplemented
33#41×C137C23148yescomplemented
34C2×Q8×C137C2312yescomplemented
35C2×D4×C137C23172yescomplemented
36C2×D4×C137C23118yescomplemented
37C22×C4×C137C2319yescomplemented
38C22×D137—824no—
39C22×D137C24172yes—
40Dic274C24148yes—
41D548C24148yes—
42C22⋊D137C241144yes—
43C22⋊D137C24116yes—
44C2×Dic137C24172yes—
45C4×D137C24196yes—
46C23×C137—86no—
47C23×C137C2419yes—
48C23×C137C24124yes—
49Q8×C137C2418yes—
50D4×C137C24172yes—
51C2×C4×C137C2416yes—
52C2×C4×C137C24136yes—
53C22×C137—49no—
54C22×C137—424no—
55C22×C137C2511yescharacteristic
56C22×C137C25118yes—
57C2×D137—472no—
58C2×D137C25116yes—
59C4×C137C25112yes—
60Dic137C25116yes—
61C2×C137—218no—
62C2×C137—21no—
63C2×C137C2611yescharacteristic
64D137—216no—
65C137H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—1371nomaximal, Sylow
67#585—13716no—
68#585—13712no—
69C2×H(5,𝔽2)—13716no—
70C2×H(5,𝔽2)—13718no—
71C2×H(5,𝔽2)D13711yescharacteristic, complemented
722−1+4—13748no—
732−1+4—1378no—
74H(5,𝔽2)—13748no—
75H(5,𝔽2)—137144no—
76H(5,𝔽2)—13772no—
77H(5,𝔽2)C2×D137116yescomplemented
78C2×#41—13796no—
79C2×#41—13772no—
80C2×#41—13736no—
81C2×#41C2×D13716yescomplemented
82C22×D4—13724no—
83C22×D4—13772no—
84C22×D4C2×D13719yescomplemented
85C24—13724no—
86C24C22×D13716yescomplemented
87#41—13732no—
88#41—137288no—
89#41—137144no—
90#41—13748no—
91#41C22×D137148yescomplemented
92C2×Q8—13732no—
93C2×Q8—13736no—
94C2×Q8C22×D13712yescomplemented
95C2×D4—137144no—
96C2×D4—13772no—
97C2×D4—137288no—
98C2×D4—13736no—
99C2×D4C22×D137172yescomplemented
100C2×D4C22×D137118yescomplemented
101C22×C4—13724no—
102C22×C4—13772no—
103C22×C4C22×D13719yescomplemented
104C23—86no—
105C23—109624no—
106C23—13772no—
107C23C23×D137124yes—
108C23C23×D13719yes—
109Q8—13748no—
110Q8C23×D13718yes—
111D4—137144no—
112D4—13716no—
113D4—13748no—
114D4C23×D137172yes—
115C2×C4—13772no—
116C2×C4—13796no—
117C2×C4C23×D137136yes—
118C2×C4C23×D13716yes—
119C22—49no—
120C22—424no—
121C22—54872no—
122C22—13716no—
123C22C24×D137118yes—
124C22C24×D13711yescharacteristic
125C4—13716no—
126C4C24×D137112yes—
127C2—21no—
128C2—218no—
129C2—27416no—
130C2C25×D13711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D137
commutator, socle, cosocleC2×C137C26
FittingC2×H(5,𝔽2)×C137C2
hypercenterC2×H(5,𝔽2)D137
hypocenterC137H(7,𝔽2)
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22578239C2semidirect
2. #22578238C2semidirect
3. C2×#17984133C2semidirect
4. H(5,𝔽2)×D137C2semidirect
5. C2×H(5,𝔽2)×C137C2semidirect
6. #17984136C22semidirect
7. #17984135C22semidirect
8. #17984134C22semidirect
9. #17984133C22semidirect
10. #17984133C22semidirect
11. C2×#14555510C22semidirect
12. C2×#14555509C22semidirect
13. #41×D137C22semidirect
14. C22×C22⋊D137C22semidirect
15. C2×D4×D137C22semidirect
16. H(5,𝔽2)×C137C22semidirect
17. C2×#41×C137C22semidirect
18. C22×D4×C137C22semidirect
19. C23×D137C23semidirect
20. #14555511C23semidirect
21. #14555510C23semidirect
22. #14555510C23semidirect
23. #14555509C23semidirect
24. C2×Dic274C23semidirect
25. Q8×D137C23semidirect
26. C2×D548C23semidirect
27. C2×C22⋊D137C23semidirect
28. C2×C22⋊D137C23semidirect
29. D4×D137C23semidirect
30. C22×Dic137C23semidirect
31. C2×C4×D137C23semidirect
32. C24×C137C23semidirect
33. #41×C137C23semidirect
34. C2×Q8×C137C23semidirect
35. C2×D4×C137C23semidirect
36. C2×D4×C137C23semidirect
37. C22×C4×C137C23semidirect
65. C137H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D137semidirect
77. H(5,𝔽2)C2×D137semidirect
81. C2×#41C2×D137semidirect
84. C22×D4C2×D137semidirect
86. C24C22×D137semidirect
91. #41C22×D137semidirect
94. C2×Q8C22×D137semidirect
99. C2×D4C22×D137semidirect
100. C2×D4C22×D137semidirect
103. C22×C4C22×D137semidirect

Maximal subgroups

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

subgroupindex
#225782392
#225782382
C2×#179841332
H(5,𝔽2)×D1372
C2×H(5,𝔽2)×C1372
H(7,𝔽2)137