Finite Group Databasebeta

← #19149405

Subgroups — #19149405

Order 5504.

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#14583735C2112yesmaximal, complemented
2#14583734C2116yesmaximal, complemented
3C2×#13063370C2118yesmaximal, complemented
4H(5,𝔽2)×D43C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C43C211yesmaximal, characteristic, complemented
6#13063373C2218yescomplemented
7#13063372C22148yescomplemented
8#13063371C22148yescomplemented
9#13063370C221144yescomplemented
10#13063370C22172yescomplemented
11C2×#10621898C22136yescomplemented
12C2×#10621900C22172yescomplemented
13#41×D43C22196yescomplemented
14C22×C22⋊D43C22124yescomplemented
15C2×D4×D43C22172yescomplemented
16H(5,𝔽2)×C43C22116yescomplemented
17C2×#41×C43C2216yescomplemented
18C22×D4×C43C2219yescomplemented
19C24×C43C2316yescomplemented
20C23×D43C23124yescomplemented
21#41×C43C23148yescomplemented
22C2×Q8×C43C2312yescomplemented
23C2×D4×C43C23118yescomplemented
24C2×D4×C43C23172yescomplemented
25C22×C4×C43C2319yescomplemented
26C2×C22⋊D43C231144yescomplemented
27C2×C22⋊D43C23172yescomplemented
28C22×Dic43C23124yescomplemented
29#10621902C23132yescomplemented
30Q8×D43C23132yescomplemented
31#10621900C231288yescomplemented
32D4×D43C231288yescomplemented
33#10621898C231144yescomplemented
34#10621898C23148yescomplemented
35C2×D172C23136yescomplemented
36C2×C4×D43C23172yescomplemented
37C2×Dic86C23136yescomplemented
38C23×C43—86no—
39C23×C43C24124yes—
40C23×C43C2419yes—
41C22×D43—824no—
42C22×D43C24172yes—
43Q8×C43C2418yes—
44D4×C43C24172yes—
45C2×C4×C43C24136yes—
46C2×C4×C43C2416yes—
47C22⋊D43C24116yes—
48C22⋊D43C241144yes—
49C2×Dic43C24172yes—
50D172C24148yes—
51C4×D43C24196yes—
52Dic86C24148yes—
53C22×C43—424no—
54C22×C43—49no—
55C22×C43C25118yes—
56C22×C43C2511yescharacteristic
57C2×D43—472no—
58C2×D43C25116yes—
59C4×C43C25112yes—
60Dic43C25116yes—
61H(7,𝔽2)—431nomaximal, Sylow
62C2×C43—218no—
63C2×C43—21no—
64C2×C43C2611yescharacteristic
65D43—216no—
66#585—4316no—
67#585—4312no—
68C2×H(5,𝔽2)—4316no—
69C2×H(5,𝔽2)—4318no—
70C2×H(5,𝔽2)D4311yescharacteristic, complemented
71C43H(7,𝔽2)11yescharacteristic, complemented, Sylow
722−1+4—4348no—
732−1+4—438no—
74H(5,𝔽2)—4348no—
75H(5,𝔽2)—43144no—
76H(5,𝔽2)—4372no—
77H(5,𝔽2)C2×D43116yescomplemented
78C2×#41—4396no—
79C2×#41—4372no—
80C2×#41—4336no—
81C2×#41C2×D4316yescomplemented
82C22×D4—4324no—
83C22×D4—4372no—
84C22×D4C2×D4319yescomplemented
85C24—4324no—
86C24C22×D4316yescomplemented
87#41—4332no—
88#41—43288no—
89#41—43144no—
90#41—4348no—
91#41C22×D43148yescomplemented
92C2×Q8—4332no—
93C2×Q8—4336no—
94C2×Q8C22×D4312yescomplemented
95C2×D4—43144no—
96C2×D4—4372no—
97C2×D4—43288no—
98C2×D4—4336no—
99C2×D4C22×D43172yescomplemented
100C2×D4C22×D43118yescomplemented
101C22×C4—4324no—
102C22×C4—4372no—
103C22×C4C22×D4319yescomplemented
104C23—86no—
105C23—4372no—
106C23—34424no—
107C23C23×D43124yes—
108C23C23×D4319yes—
109Q8—4348no—
110Q8C23×D4318yes—
111D4—43144no—
112D4—4316no—
113D4—4348no—
114D4C23×D43172yes—
115C2×C4—4372no—
116C2×C4—4396no—
117C2×C4C23×D43136yes—
118C2×C4C23×D4316yes—
119C22—49no—
120C22—424no—
121C22—4316no—
122C22—17272no—
123C22C24×D43118yes—
124C22C24×D4311yescharacteristic
125C4—4316no—
126C4C24×D43112yes—
127C2—21no—
128C2—218no—
129C2—8616no—
130C2C25×D4311yescharacteristic
subgroupG/H
center, FrattiniC2C25×D43
commutator, socleC2×C43C26
FittingC2×H(5,𝔽2)×C43C2
hypercenterC2×H(5,𝔽2)D43
hypocenterC43H(7,𝔽2)
LCSG ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14583735C2semidirect
2. #14583734C2semidirect
3. C2×#13063370C2semidirect
4. H(5,𝔽2)×D43C2semidirect
5. C2×H(5,𝔽2)×C43C2semidirect
6. #13063373C22semidirect
7. #13063372C22semidirect
8. #13063371C22semidirect
9. #13063370C22semidirect
10. #13063370C22semidirect
11. C2×#10621898C22semidirect
12. C2×#10621900C22semidirect
13. #41×D43C22semidirect
14. C22×C22⋊D43C22semidirect
15. C2×D4×D43C22semidirect
16. H(5,𝔽2)×C43C22semidirect
17. C2×#41×C43C22semidirect
18. C22×D4×C43C22semidirect
19. C24×C43C23semidirect
20. C23×D43C23semidirect
21. #41×C43C23semidirect
22. C2×Q8×C43C23semidirect
23. C2×D4×C43C23semidirect
24. C2×D4×C43C23semidirect
25. C22×C4×C43C23semidirect
26. C2×C22⋊D43C23semidirect
27. C2×C22⋊D43C23semidirect
28. C22×Dic43C23semidirect
29. #10621902C23semidirect
30. Q8×D43C23semidirect
31. #10621900C23semidirect
32. D4×D43C23semidirect
33. #10621898C23semidirect
34. #10621898C23semidirect
35. C2×D172C23semidirect
36. C2×C4×D43C23semidirect
37. C2×Dic86C23semidirect
70. C2×H(5,𝔽2)D43semidirect
71. C43H(7,𝔽2)semidirect
77. H(5,𝔽2)C2×D43semidirect
81. C2×#41C2×D43semidirect
84. C22×D4C2×D43semidirect
86. C24C22×D43semidirect
91. #41C22×D43semidirect
94. C2×Q8C22×D43semidirect
99. C2×D4C22×D43semidirect
100. C2×D4C22×D43semidirect
103. C22×C4C22×D43semidirect