Finite Group Databasebeta

← #36338948

Subgroups — #36338948

Order 21376.

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#24924105C2112yesmaximal, complemented
2#24924104C2116yesmaximal, complemented
3C2×#19128877C2118yesmaximal, complemented
4H(5,𝔽2)×D167C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C167C211yesmaximal, characteristic, complemented
6#19128880C2218yescomplemented
7#19128879C22148yescomplemented
8#19128878C22148yescomplemented
9#19128877C221144yescomplemented
10#19128877C22172yescomplemented
11C2×#14581950C22136yescomplemented
12C2×#14581949C22172yescomplemented
13#41×D167C22196yescomplemented
14C22×C22⋊D167C22124yescomplemented
15C2×D4×D167C22172yescomplemented
16H(5,𝔽2)×C167C22116yescomplemented
17C2×#41×C167C2216yescomplemented
18C22×D4×C167C2219yescomplemented
19C23×D167C23124yescomplemented
20#14581951C23132yescomplemented
21#14581950C231144yescomplemented
22#14581950C23148yescomplemented
23#14581949C231288yescomplemented
24C2×Dic334C23136yescomplemented
25Q8×D167C23132yescomplemented
26C2×D668C23136yescomplemented
27C2×C22⋊D167C231144yescomplemented
28C2×C22⋊D167C23172yescomplemented
29D4×D167C231288yescomplemented
30C22×Dic167C23124yescomplemented
31C2×C4×D167C23172yescomplemented
32C24×C167C2316yescomplemented
33#41×C167C23148yescomplemented
34C2×Q8×C167C2312yescomplemented
35C2×D4×C167C23172yescomplemented
36C2×D4×C167C23118yescomplemented
37C22×C4×C167C2319yescomplemented
38C22×D167—824no—
39C22×D167C24172yes—
40Dic334C24148yes—
41D668C24148yes—
42C22⋊D167C241144yes—
43C22⋊D167C24116yes—
44C2×Dic167C24172yes—
45C4×D167C24196yes—
46C23×C167—86no—
47C23×C167C2419yes—
48C23×C167C24124yes—
49Q8×C167C2418yes—
50D4×C167C24172yes—
51C2×C4×C167C2416yes—
52C2×C4×C167C24136yes—
53C22×C167—49no—
54C22×C167—424no—
55C22×C167C2511yescharacteristic
56C22×C167C25118yes—
57C2×D167—472no—
58C2×D167C25116yes—
59C4×C167C25112yes—
60Dic167C25116yes—
61C2×C167—218no—
62C2×C167—21no—
63C2×C167C2611yescharacteristic
64D167—216no—
65C167H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—1671nomaximal, Sylow
67#585—16716no—
68#585—16712no—
69C2×H(5,𝔽2)—16716no—
70C2×H(5,𝔽2)—16718no—
71C2×H(5,𝔽2)D16711yescharacteristic, complemented
722−1+4—16748no—
732−1+4—1678no—
74H(5,𝔽2)—16748no—
75H(5,𝔽2)—167144no—
76H(5,𝔽2)—16772no—
77H(5,𝔽2)C2×D167116yescomplemented
78C2×#41—16796no—
79C2×#41—16772no—
80C2×#41—16736no—
81C2×#41C2×D16716yescomplemented
82C22×D4—16724no—
83C22×D4—16772no—
84C22×D4C2×D16719yescomplemented
85C24—16724no—
86C24C22×D16716yescomplemented
87#41—16732no—
88#41—167288no—
89#41—167144no—
90#41—16748no—
91#41C22×D167148yescomplemented
92C2×Q8—16732no—
93C2×Q8—16736no—
94C2×Q8C22×D16712yescomplemented
95C2×D4—167144no—
96C2×D4—16772no—
97C2×D4—167288no—
98C2×D4—16736no—
99C2×D4C22×D167172yescomplemented
100C2×D4C22×D167118yescomplemented
101C22×C4—16724no—
102C22×C4—16772no—
103C22×C4C22×D16719yescomplemented
104C23—86no—
105C23—133624no—
106C23—16772no—
107C23C23×D167124yes—
108C23C23×D16719yes—
109Q8—16748no—
110Q8C23×D16718yes—
111D4—167144no—
112D4—16716no—
113D4—16748no—
114D4C23×D167172yes—
115C2×C4—16772no—
116C2×C4—16796no—
117C2×C4C23×D167136yes—
118C2×C4C23×D16716yes—
119C22—49no—
120C22—424no—
121C22—66872no—
122C22—16716no—
123C22C24×D167118yes—
124C22C24×D16711yescharacteristic
125C4—16716no—
126C4C24×D167112yes—
127C2—21no—
128C2—218no—
129C2—33416no—
130C2C25×D16711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D167
commutator, socle, cosocleC2×C167C26
FittingC2×H(5,𝔽2)×C167C2
hypercenterC2×H(5,𝔽2)D167
hypocenterC167H(7,𝔽2)
LCSG ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #24924105C2semidirect
2. #24924104C2semidirect
3. C2×#19128877C2semidirect
4. H(5,𝔽2)×D167C2semidirect
5. C2×H(5,𝔽2)×C167C2semidirect
6. #19128880C22semidirect
7. #19128879C22semidirect
8. #19128878C22semidirect
9. #19128877C22semidirect
10. #19128877C22semidirect
11. C2×#14581950C22semidirect
12. C2×#14581949C22semidirect
13. #41×D167C22semidirect
14. C22×C22⋊D167C22semidirect
15. C2×D4×D167C22semidirect
16. H(5,𝔽2)×C167C22semidirect
17. C2×#41×C167C22semidirect
18. C22×D4×C167C22semidirect
19. C23×D167C23semidirect
20. #14581951C23semidirect
21. #14581950C23semidirect
22. #14581950C23semidirect
23. #14581949C23semidirect
24. C2×Dic334C23semidirect
25. Q8×D167C23semidirect
26. C2×D668C23semidirect
27. C2×C22⋊D167C23semidirect
28. C2×C22⋊D167C23semidirect
29. D4×D167C23semidirect
30. C22×Dic167C23semidirect
31. C2×C4×D167C23semidirect
32. C24×C167C23semidirect
33. #41×C167C23semidirect
34. C2×Q8×C167C23semidirect
35. C2×D4×C167C23semidirect
36. C2×D4×C167C23semidirect
37. C22×C4×C167C23semidirect
65. C167H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D167semidirect
77. H(5,𝔽2)C2×D167semidirect
81. C2×#41C2×D167semidirect
84. C22×D4C2×D167semidirect
86. C24C22×D167semidirect
91. #41C22×D167semidirect
94. C2×Q8C22×D167semidirect
99. C2×D4C22×D167semidirect
100. C2×D4C22×D167semidirect
103. C22×C4C22×D167semidirect

Maximal subgroups

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

subgroupindex
#249241052
#249241042
C2×#191288772
H(5,𝔽2)×D1672
C2×H(5,𝔽2)×C1672
H(7,𝔽2)167