Finite Group Databasebeta

← #36338949

Subgroups — #36338949

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#24924106C2112yesmaximal, complemented
2#24924105C2115yesmaximal, complemented
3C2×#19128878C2115yesmaximal, complemented
4H(5,𝔽2)×D167C2120yesmaximal, complemented
5#585×C167C211yesmaximal, characteristic, complemented
6#19128880C22130yescomplemented
7#19128879C22120yescomplemented
8#19128878C221120yescomplemented
9#19128878C22160yescomplemented
10#19128877C22190yescomplemented
11C2×#14581951C22130yescomplemented
12C2×#14581950C22145yescomplemented
13#41×D167C221120yescomplemented
14C22×D668C22115yescomplemented
15C2×D4×D167C22190yescomplemented
162−1+4×C167C2216yescomplemented
17H(5,𝔽2)×C167C22110yescomplemented
18C2×#41×C167C22115yescomplemented
19C23×D167C23130yescomplemented
20#14581951C231120yescomplemented
21#14581950C231180yescomplemented
22#14581950C23160yescomplemented
23#14581949C231120yescomplemented
24C2×Dic334C23115yescomplemented
25Q8×D167C23140yescomplemented
26C2×D668C23190yescomplemented
27C2×D668C23145yescomplemented
28C2×C22⋊D167C23190yescomplemented
29D4×D167C231360yescomplemented
30C2×C4×D167C23190yescomplemented
31#41×C167C23160yescomplemented
32#41×C167C23120yescomplemented
33C2×Q8×C167C23115yescomplemented
34C2×D4×C167C23145yescomplemented
35C22×C4×C167C23115yescomplemented
36C22×D167—830no—
37C22×D167C24190yes—
38Dic334C24130yes—
39Dic334C2416yes—
40D668C24190yes—
41D668C24110yes—
42C22⋊D167C241120yes—
43C2×Dic167C24130yes—
44C4×D167C241120yes—
45C23×C167C24115yes—
46Q8×C167C24120yes—
47D4×C167C24160yes—
48C2×C4×C167C24115yes—
49C2×C4×C167C24145yes—
50C22×C167—415no—
51C22×C167C25115yes—
52C2×D167—490no—
53C2×D167C25120yes—
54C4×C167C2511yescharacteristic
55C4×C167C25115yes—
56Dic167C25112yes—
57C2×C167—215no—
58C2×C167C2611yescharacteristic
59D167—220no—
60C167H(7,𝔽2)11yescharacteristic, complemented, Sylow
61H(7,𝔽2)—1671nomaximal, Sylow
62#585—16712no—
63#585—16715no—
64#585D16711yescharacteristic, complemented
65C2×H(5,𝔽2)—16720no—
66C2×H(5,𝔽2)—16715no—
672−1+4—16720no—
682−1+4—16730no—
692−1+4C2×D16716yescomplemented
70H(5,𝔽2)—167120no—
71H(5,𝔽2)—16760no—
72H(5,𝔽2)—16790no—
73H(5,𝔽2)C2×D167110yescomplemented
74C2×#41—167120no—
75C2×#41—16730no—
76C2×#41—16745no—
77C2×#41C2×D167115yescomplemented
78C22×D4—16790no—
79C22×D4—16715no—
80C24—16730no—
81#41—167120no—
82#41—167120no—
83#41—167180no—
84#41—16760no—
85#41C22×D167160yescomplemented
86#41C22×D167120yescomplemented
87C2×Q8—16740no—
88C2×Q8—16715no—
89C2×Q8C22×D167115yescomplemented
90C2×D4—16790no—
91C2×D4—167360no—
92C2×D4—16790no—
93C2×D4—16745no—
94C2×D4C22×D167145yescomplemented
95C22×C4—16790no—
96C22×C4C22×D167115yescomplemented
97C23—133630no—
98C23—16790no—
99C23C23×D167115yes—
100Q8—16730no—
101Q8—1676no—
102Q8C23×D167120yes—
103D4—167120no—
104D4—16790no—
105D4—16710no—
106D4C23×D167160yes—
107C2×C4—16730no—
108C2×C4—167120no—
109C2×C4C23×D167145yes—
110C2×C4C23×D167115yes—
111C22—415no—
112C22—66890no—
113C22—16720no—
114C22C24×D167115yes—
115C4—16712no—
116C4C24×D167115yes—
117C4C24×D16711yescharacteristic
118C2—215no—
119C2—33420no—
120C2C25×D16711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D167
commutator, socle, cosocleC2×C167C26
Fitting#585×C167C2
hypercenter#585D167
hypocenterC167H(7,𝔽2)
LCSG ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #24924106C2semidirect
2. #24924105C2semidirect
3. C2×#19128878C2semidirect
4. H(5,𝔽2)×D167C2semidirect
5. #585×C167C2semidirect
6. #19128880C22semidirect
7. #19128879C22semidirect
8. #19128878C22semidirect
9. #19128878C22semidirect
10. #19128877C22semidirect
11. C2×#14581951C22semidirect
12. C2×#14581950C22semidirect
13. #41×D167C22semidirect
14. C22×D668C22semidirect
15. C2×D4×D167C22semidirect
16. 2−1+4×C167C22semidirect
17. H(5,𝔽2)×C167C22semidirect
18. C2×#41×C167C22semidirect
19. C23×D167C23semidirect
20. #14581951C23semidirect
21. #14581950C23semidirect
22. #14581950C23semidirect
23. #14581949C23semidirect
24. C2×Dic334C23semidirect
25. Q8×D167C23semidirect
26. C2×D668C23semidirect
27. C2×D668C23semidirect
28. C2×C22⋊D167C23semidirect
29. D4×D167C23semidirect
30. C2×C4×D167C23semidirect
31. #41×C167C23semidirect
32. #41×C167C23semidirect
33. C2×Q8×C167C23semidirect
34. C2×D4×C167C23semidirect
35. C22×C4×C167C23semidirect
60. C167H(7,𝔽2)semidirect
64. #585D167semidirect
69. 2−1+4C2×D167semidirect
73. H(5,𝔽2)C2×D167semidirect
77. C2×#41C2×D167semidirect
85. #41C22×D167semidirect
86. #41C22×D167semidirect
89. C2×Q8C22×D167semidirect
94. C2×D4C22×D167semidirect
96. C22×C4C22×D167semidirect

Maximal subgroups

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

subgroupindex
#249241062
#249241052
C2×#191288782
H(5,𝔽2)×D1672
#585×C1672
H(7,𝔽2)167