Finite Group Databasebeta

← #37548532

Subgroups — #37548532

Order 23168.

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#26042116C2112yesmaximal, complemented
2#26042115C2115yesmaximal, complemented
3C2×#19153862C2115yesmaximal, complemented
4H(5,𝔽2)×D181C2120yesmaximal, complemented
5#585×C181C211yesmaximal, characteristic, complemented
6#19153864C22130yescomplemented
7#19153863C22120yescomplemented
8#19153862C22160yescomplemented
9#19153862C221120yescomplemented
10#19153861C22190yescomplemented
11C2×#15668450C22130yescomplemented
12C2×#15668449C22145yescomplemented
13#41×D181C221120yescomplemented
14C22×D724C22115yescomplemented
15C2×D4×D181C22190yescomplemented
162−1+4×C181C2216yescomplemented
17H(5,𝔽2)×C181C22110yescomplemented
18C2×#41×C181C22115yescomplemented
19C23×D181C23130yescomplemented
20#15668450C231120yescomplemented
21#15668449C231180yescomplemented
22#15668449C23160yescomplemented
23#15668448C231120yescomplemented
24C2×Dic362C23115yescomplemented
25Q8×D181C23140yescomplemented
26C2×D724C23190yescomplemented
27C2×D724C23145yescomplemented
28C2×C22⋊D181C23190yescomplemented
29D4×D181C231360yescomplemented
30C2×C4×D181C23190yescomplemented
31#41×C181C23120yescomplemented
32#41×C181C23160yescomplemented
33C2×Q8×C181C23115yescomplemented
34C2×D4×C181C23145yescomplemented
35C22×C4×C181C23115yescomplemented
36C22×D181—830no—
37C22×D181C24190yes—
38Dic362C24130yes—
39Dic362C2416yes—
40D724C24190yes—
41D724C24110yes—
42C22⋊D181C241120yes—
43C2×Dic181C24130yes—
44C4×D181C241120yes—
45C23×C181C24115yes—
46Q8×C181C24120yes—
47D4×C181C24160yes—
48C2×C4×C181C24115yes—
49C2×C4×C181C24145yes—
50C22×C181—415no—
51C22×C181C25115yes—
52C2×D181—490no—
53C2×D181C25120yes—
54C4×C181C2511yescharacteristic
55C4×C181C25115yes—
56Dic181C25112yes—
57C2×C181—215no—
58C2×C181C2611yescharacteristic
59D181—220no—
60C181H(7,𝔽2)11yescharacteristic, complemented, Sylow
61H(7,𝔽2)—1811nomaximal, Sylow
62#585—18112no—
63#585—18115no—
64#585D18111yescharacteristic, complemented
65C2×H(5,𝔽2)—18120no—
66C2×H(5,𝔽2)—18115no—
672−1+4—18120no—
682−1+4—18130no—
692−1+4C2×D18116yescomplemented
70H(5,𝔽2)—181120no—
71H(5,𝔽2)—18160no—
72H(5,𝔽2)—18190no—
73H(5,𝔽2)C2×D181110yescomplemented
74C2×#41—181120no—
75C2×#41—18130no—
76C2×#41—18145no—
77C2×#41C2×D181115yescomplemented
78C22×D4—18190no—
79C22×D4—18115no—
80C24—18130no—
81#41—181120no—
82#41—181120no—
83#41—181180no—
84#41—18160no—
85#41C22×D181160yescomplemented
86#41C22×D181120yescomplemented
87C2×Q8—18140no—
88C2×Q8—18115no—
89C2×Q8C22×D181115yescomplemented
90C2×D4—18190no—
91C2×D4—181360no—
92C2×D4—18190no—
93C2×D4—18145no—
94C2×D4C22×D181145yescomplemented
95C22×C4—18190no—
96C22×C4C22×D181115yescomplemented
97C23—144830no—
98C23—18190no—
99C23C23×D181115yes—
100Q8—18130no—
101Q8—1816no—
102Q8C23×D181120yes—
103D4—181120no—
104D4—18190no—
105D4—18110no—
106D4C23×D181160yes—
107C2×C4—18130no—
108C2×C4—181120no—
109C2×C4C23×D181145yes—
110C2×C4C23×D181115yes—
111C22—415no—
112C22—18120no—
113C22—72490no—
114C22C24×D181115yes—
115C4—18112no—
116C4C24×D181115yes—
117C4C24×D18111yescharacteristic
118C2—215no—
119C2—36220no—
120C2C25×D18111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D181
commutator, socle, cosocleC2×C181C26
Fitting#585×C181C2
hypercenter#585D181
hypocenterC181H(7,𝔽2)
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #26042116C2semidirect
2. #26042115C2semidirect
3. C2×#19153862C2semidirect
4. H(5,𝔽2)×D181C2semidirect
5. #585×C181C2semidirect
6. #19153864C22semidirect
7. #19153863C22semidirect
8. #19153862C22semidirect
9. #19153862C22semidirect
10. #19153861C22semidirect
11. C2×#15668450C22semidirect
12. C2×#15668449C22semidirect
13. #41×D181C22semidirect
14. C22×D724C22semidirect
15. C2×D4×D181C22semidirect
16. 2−1+4×C181C22semidirect
17. H(5,𝔽2)×C181C22semidirect
18. C2×#41×C181C22semidirect
19. C23×D181C23semidirect
20. #15668450C23semidirect
21. #15668449C23semidirect
22. #15668449C23semidirect
23. #15668448C23semidirect
24. C2×Dic362C23semidirect
25. Q8×D181C23semidirect
26. C2×D724C23semidirect
27. C2×D724C23semidirect
28. C2×C22⋊D181C23semidirect
29. D4×D181C23semidirect
30. C2×C4×D181C23semidirect
31. #41×C181C23semidirect
32. #41×C181C23semidirect
33. C2×Q8×C181C23semidirect
34. C2×D4×C181C23semidirect
35. C22×C4×C181C23semidirect
60. C181H(7,𝔽2)semidirect
64. #585D181semidirect
69. 2−1+4C2×D181semidirect
73. H(5,𝔽2)C2×D181semidirect
77. C2×#41C2×D181semidirect
85. #41C22×D181semidirect
86. #41C22×D181semidirect
89. C2×Q8C22×D181semidirect
94. C2×D4C22×D181semidirect
96. C22×C4C22×D181semidirect

Maximal subgroups

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

subgroupindex
#260421162
#260421152
C2×#191538622
H(5,𝔽2)×D1812
#585×C1812
H(7,𝔽2)181