Finite Group Databasebeta

← #16860521

Subgroups — #16860521

Order 3968.

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#14521885C2112yesmaximal, complemented
2#14521884C2116yesmaximal, complemented
3C2×#11758561C2118yesmaximal, complemented
4H(5,𝔽2)×D31C2116yesmaximal, complemented
5C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
6C31×H(5,𝔽2)C22116yescomplemented
7C2×#41×C31C2216yescomplemented
8C22×D4×C31C2219yescomplemented
9C22×C22⋊D31C22124yescomplemented
10#11758567C22148yescomplemented
11#11758566C22148yescomplemented
12#41×D31C22196yescomplemented
13#11758564C2218yescomplemented
14#11758561C221144yescomplemented
15#11758561C22172yescomplemented
16C2×#92497C22172yescomplemented
17C2×D4×D31C22172yescomplemented
18C2×#92495C22136yescomplemented
19C24×C31C2316yescomplemented
20C23×D31C23124yescomplemented
21#41×C31C23148yescomplemented
22C2×Q8×C31C2312yescomplemented
23C2×D4×C31C23172yescomplemented
24C2×D4×C31C23118yescomplemented
25C22×C4×C31C2319yescomplemented
26C2×C22⋊D31C23172yescomplemented
27C2×C22⋊D31C231144yescomplemented
28C22×Dic31C23124yescomplemented
29#92499C23132yescomplemented
30Q8×D31C23132yescomplemented
31#92497C231288yescomplemented
32D4×D31C231288yescomplemented
33#92495C23148yescomplemented
34#92495C231144yescomplemented
35C2×D124C23136yescomplemented
36C2×C4×D31C23172yescomplemented
37C2×Dic62C23136yescomplemented
38C23×C31—86no—
39C23×C31C24124yes—
40C23×C31C2419yes—
41C22×D31—824no—
42C22×D31C24172yes—
43Q8×C31C2418yes—
44D4×C31C24172yes—
45C2×C4×C31C24136yes—
46C2×C4×C31C2416yes—
47C22⋊D31C24116yes—
48C22⋊D31C241144yes—
49C2×Dic31C24172yes—
50D124C24148yes—
51C4×D31C24196yes—
52Dic62C24148yes—
53H(7,𝔽2)—311nomaximal, Sylow
54C22×C31—424no—
55C22×C31—49no—
56C22×C31C25118yes—
57C22×C31C2511yescharacteristic
58C2×D31—472no—
59C2×D31C25116yes—
60C4×C31C25112yes—
61Dic31C25116yes—
62#585—3116no—
63#585—3112no—
64C2×H(5,𝔽2)—3116no—
65C2×H(5,𝔽2)—3118no—
66C2×H(5,𝔽2)D3111yescharacteristic, complemented
67C2×C31—218no—
68C2×C31—21no—
69C2×C31C2611yescharacteristic
70D31—216no—
712−1+4—3148no—
722−1+4—318no—
73H(5,𝔽2)—3148no—
74H(5,𝔽2)—31144no—
75H(5,𝔽2)—3172no—
76H(5,𝔽2)C2×D31116yescomplemented
77C2×#41—3196no—
78C2×#41—3172no—
79C2×#41—3136no—
80C2×#41C2×D3116yescomplemented
81C22×D4—3124no—
82C22×D4—3172no—
83C22×D4C2×D3119yescomplemented
84C31H(7,𝔽2)11yescharacteristic, complemented, Sylow
85C24—3124no—
86C24C22×D3116yescomplemented
87#41—3132no—
88#41—31288no—
89#41—31144no—
90#41—3148no—
91#41C22×D31148yescomplemented
92C2×Q8—3132no—
93C2×Q8—3136no—
94C2×Q8C22×D3112yescomplemented
95C2×D4—31144no—
96C2×D4—3172no—
97C2×D4—31288no—
98C2×D4—3136no—
99C2×D4C22×D31172yescomplemented
100C2×D4C22×D31118yescomplemented
101C22×C4—3124no—
102C22×C4—3172no—
103C22×C4C22×D3119yescomplemented
104C23—86no—
105C23—3172no—
106C23—24824no—
107C23C23×D31124yes—
108C23C23×D3119yes—
109Q8—3148no—
110Q8C23×D3118yes—
111D4—31144no—
112D4—3116no—
113D4—3148no—
114D4C23×D31172yes—
115C2×C4—3172no—
116C2×C4—3196no—
117C2×C4C23×D31136yes—
118C2×C4C23×D3116yes—
119C22—49no—
120C22—424no—
121C22—3116no—
122C22—12472no—
123C22C24×D31118yes—
124C22C24×D3111yescharacteristic
125C4—3116no—
126C4C24×D31112yes—
127C2—21no—
128C2—218no—
129C2—6216no—
130C2C25×D3111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D31
commutator, socle, cosocleC2×C31C26
FittingC2×C31×H(5,𝔽2)C2
hypercenterC2×H(5,𝔽2)D31
hypocenterC31H(7,𝔽2)
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. #14521884C2semidirect
3. C2×#11758561C2semidirect
4. H(5,𝔽2)×D31C2semidirect
5. C2×C31×H(5,𝔽2)C2semidirect
6. C31×H(5,𝔽2)C22semidirect
7. C2×#41×C31C22semidirect
8. C22×D4×C31C22semidirect
9. C22×C22⋊D31C22semidirect
10. #11758567C22semidirect
11. #11758566C22semidirect
12. #41×D31C22semidirect
13. #11758564C22semidirect
14. #11758561C22semidirect
15. #11758561C22semidirect
16. C2×#92497C22semidirect
17. C2×D4×D31C22semidirect
18. C2×#92495C22semidirect
19. C24×C31C23semidirect
20. C23×D31C23semidirect
21. #41×C31C23semidirect
22. C2×Q8×C31C23semidirect
23. C2×D4×C31C23semidirect
24. C2×D4×C31C23semidirect
25. C22×C4×C31C23semidirect
26. C2×C22⋊D31C23semidirect
27. C2×C22⋊D31C23semidirect
28. C22×Dic31C23semidirect
29. #92499C23semidirect
30. Q8×D31C23semidirect
31. #92497C23semidirect
32. D4×D31C23semidirect
33. #92495C23semidirect
34. #92495C23semidirect
35. C2×D124C23semidirect
36. C2×C4×D31C23semidirect
37. C2×Dic62C23semidirect
66. C2×H(5,𝔽2)D31semidirect
76. H(5,𝔽2)C2×D31semidirect
80. C2×#41C2×D31semidirect
83. C22×D4C2×D31semidirect
84. C31H(7,𝔽2)semidirect
86. C24C22×D31semidirect
91. #41C22×D31semidirect
94. C2×Q8C22×D31semidirect
99. C2×D4C22×D31semidirect
100. C2×D4C22×D31semidirect
103. C22×C4C22×D31semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145218842
C2×#117585612
H(5,𝔽2)×D312
C2×C31×H(5,𝔽2)2
H(7,𝔽2)31