Finite Group Databasebeta

← #49136071

Subgroups — #49136071

Order 36224.

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#31827301C2112yesmaximal, complemented
2#31827300C2116yesmaximal, complemented
3C2×#22583655C2118yesmaximal, complemented
4H(5,𝔽2)×D283C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C283C211yesmaximal, characteristic, complemented
6#22583658C2218yescomplemented
7#22583657C22148yescomplemented
8#22583656C22148yescomplemented
9#22583655C221144yescomplemented
10#22583655C22172yescomplemented
11C2×#17985392C22136yescomplemented
12C2×#17985391C22172yescomplemented
13#41×D283C22196yescomplemented
14C22×C22⋊D283C22124yescomplemented
15C2×D4×D283C22172yescomplemented
16H(5,𝔽2)×C283C22116yescomplemented
17C2×#41×C283C2216yescomplemented
18C22×D4×C283C2219yescomplemented
19C23×D283C23124yescomplemented
20#17985393C23132yescomplemented
21#17985392C231144yescomplemented
22#17985392C23148yescomplemented
23#17985391C231288yescomplemented
24C2×Dic566C23136yescomplemented
25Q8×D283C23132yescomplemented
26C2×D1132C23136yescomplemented
27C2×C22⋊D283C231144yescomplemented
28C2×C22⋊D283C23172yescomplemented
29D4×D283C231288yescomplemented
30C22×Dic283C23124yescomplemented
31C2×C4×D283C23172yescomplemented
32C24×C283C2316yescomplemented
33#41×C283C23148yescomplemented
34C2×Q8×C283C2312yescomplemented
35C2×D4×C283C23172yescomplemented
36C2×D4×C283C23118yescomplemented
37C22×C4×C283C2319yescomplemented
38C22×D283—824no—
39C22×D283C24172yes—
40Dic566C24148yes—
41D1132C24148yes—
42C22⋊D283C241144yes—
43C22⋊D283C24116yes—
44C2×Dic283C24172yes—
45C4×D283C24196yes—
46C23×C283—86no—
47C23×C283C2419yes—
48C23×C283C24124yes—
49Q8×C283C2418yes—
50D4×C283C24172yes—
51C2×C4×C283C2416yes—
52C2×C4×C283C24136yes—
53C22×C283—49no—
54C22×C283—424no—
55C22×C283C2511yescharacteristic
56C22×C283C25118yes—
57C2×D283—472no—
58C2×D283C25116yes—
59C4×C283C25112yes—
60Dic283C25116yes—
61C2×C283—218no—
62C2×C283—21no—
63C2×C283C2611yescharacteristic
64D283—216no—
65C283H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—2831nomaximal, Sylow
67#585—28316no—
68#585—28312no—
69C2×H(5,𝔽2)—28316no—
70C2×H(5,𝔽2)—28318no—
71C2×H(5,𝔽2)D28311yescharacteristic, complemented
722−1+4—28348no—
732−1+4—2838no—
74H(5,𝔽2)—28348no—
75H(5,𝔽2)—283144no—
76H(5,𝔽2)—28372no—
77H(5,𝔽2)C2×D283116yescomplemented
78C2×#41—28396no—
79C2×#41—28372no—
80C2×#41—28336no—
81C2×#41C2×D28316yescomplemented
82C22×D4—28324no—
83C22×D4—28372no—
84C22×D4C2×D28319yescomplemented
85C24—28324no—
86C24C22×D28316yescomplemented
87#41—28332no—
88#41—283288no—
89#41—283144no—
90#41—28348no—
91#41C22×D283148yescomplemented
92C2×Q8—28332no—
93C2×Q8—28336no—
94C2×Q8C22×D28312yescomplemented
95C2×D4—283144no—
96C2×D4—28372no—
97C2×D4—283288no—
98C2×D4—28336no—
99C2×D4C22×D283172yescomplemented
100C2×D4C22×D283118yescomplemented
101C22×C4—28324no—
102C22×C4—28372no—
103C22×C4C22×D28319yescomplemented
104C23—86no—
105C23—28372no—
106C23—226424no—
107C23C23×D283124yes—
108C23C23×D28319yes—
109Q8—28348no—
110Q8C23×D28318yes—
111D4—283144no—
112D4—28316no—
113D4—28348no—
114D4C23×D283172yes—
115C2×C4—28372no—
116C2×C4—28396no—
117C2×C4C23×D283136yes—
118C2×C4C23×D28316yes—
119C22—49no—
120C22—424no—
121C22—28316no—
122C22—113272no—
123C22C24×D283118yes—
124C22C24×D28311yescharacteristic
125C4—28316no—
126C4C24×D283112yes—
127C2—21no—
128C2—218no—
129C2—56616no—
130C2C25×D28311yescharacteristic
subgroupG/H
center, FrattiniC2C25×D283
commutator, socle, cosocleC2×C283C26
FittingC2×H(5,𝔽2)×C283C2
hypercenterC2×H(5,𝔽2)D283
hypocenterC283H(7,𝔽2)
LCSG ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #31827301C2semidirect
2. #31827300C2semidirect
3. C2×#22583655C2semidirect
4. H(5,𝔽2)×D283C2semidirect
5. C2×H(5,𝔽2)×C283C2semidirect
6. #22583658C22semidirect
7. #22583657C22semidirect
8. #22583656C22semidirect
9. #22583655C22semidirect
10. #22583655C22semidirect
11. C2×#17985392C22semidirect
12. C2×#17985391C22semidirect
13. #41×D283C22semidirect
14. C22×C22⋊D283C22semidirect
15. C2×D4×D283C22semidirect
16. H(5,𝔽2)×C283C22semidirect
17. C2×#41×C283C22semidirect
18. C22×D4×C283C22semidirect
19. C23×D283C23semidirect
20. #17985393C23semidirect
21. #17985392C23semidirect
22. #17985392C23semidirect
23. #17985391C23semidirect
24. C2×Dic566C23semidirect
25. Q8×D283C23semidirect
26. C2×D1132C23semidirect
27. C2×C22⋊D283C23semidirect
28. C2×C22⋊D283C23semidirect
29. D4×D283C23semidirect
30. C22×Dic283C23semidirect
31. C2×C4×D283C23semidirect
32. C24×C283C23semidirect
33. #41×C283C23semidirect
34. C2×Q8×C283C23semidirect
35. C2×D4×C283C23semidirect
36. C2×D4×C283C23semidirect
37. C22×C4×C283C23semidirect
65. C283H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D283semidirect
77. H(5,𝔽2)C2×D283semidirect
81. C2×#41C2×D283semidirect
84. C22×D4C2×D283semidirect
86. C24C22×D283semidirect
91. #41C22×D283semidirect
94. C2×Q8C22×D283semidirect
99. C2×D4C22×D283semidirect
100. C2×D4C22×D283semidirect
103. C22×C4C22×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
#318273002
C2×#225836552
H(5,𝔽2)×D2832
C2×H(5,𝔽2)×C2832
H(7,𝔽2)283