Finite Group Databasebeta

← #16860522

Subgroups — #16860522

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#14521886C2112yesmaximal, complemented
2#14521885C2115yesmaximal, complemented
3C2×#11758566C2115yesmaximal, complemented
4H(5,𝔽2)×D31C2120yesmaximal, complemented
5C31×#585C211yesmaximal, characteristic, complemented
6C31×2−1+4C2216yescomplemented
7C31×H(5,𝔽2)C22110yescomplemented
8C2×#41×C31C22115yescomplemented
9#11758567C22120yescomplemented
10#11758566C221120yescomplemented
11#11758566C22160yescomplemented
12#41×D31C221120yescomplemented
13#11758564C22130yescomplemented
14C2×#92499C22130yescomplemented
15#11758561C22190yescomplemented
16C2×D4×D31C22190yescomplemented
17C2×#92495C22145yescomplemented
18C22×D124C22115yescomplemented
19C23×D31C23130yescomplemented
20#41×C31C23160yescomplemented
21#41×C31C23120yescomplemented
22C2×Q8×C31C23115yescomplemented
23C2×D4×C31C23145yescomplemented
24C22×C4×C31C23115yescomplemented
25C2×C22⋊D31C23190yescomplemented
26#92499C231120yescomplemented
27Q8×D31C23140yescomplemented
28#92497C231120yescomplemented
29D4×D31C231360yescomplemented
30#92495C23160yescomplemented
31#92495C231180yescomplemented
32C2×D124C23145yescomplemented
33C2×D124C23190yescomplemented
34C2×C4×D31C23190yescomplemented
35C2×Dic62C23115yescomplemented
36C23×C31C24115yes—
37C22×D31—830no—
38C22×D31C24190yes—
39Q8×C31C24120yes—
40D4×C31C24160yes—
41C2×C4×C31C24115yes—
42C2×C4×C31C24145yes—
43C22⋊D31C241120yes—
44C2×Dic31C24130yes—
45D124C24110yes—
46D124C24190yes—
47C4×D31C241120yes—
48Dic62C2416yes—
49Dic62C24130yes—
50H(7,𝔽2)—311nomaximal, Sylow
51C22×C31—415no—
52C22×C31C25115yes—
53C2×D31—490no—
54C2×D31C25120yes—
55C4×C31C2511yescharacteristic
56C4×C31C25115yes—
57Dic31C25112yes—
58#585—3112no—
59#585—3115no—
60#585D3111yescharacteristic, complemented
61C2×H(5,𝔽2)—3120no—
62C2×H(5,𝔽2)—3115no—
63C2×C31—215no—
64C2×C31C2611yescharacteristic
65D31—220no—
662−1+4—3120no—
672−1+4—3130no—
682−1+4C2×D3116yescomplemented
69H(5,𝔽2)—31120no—
70H(5,𝔽2)—3160no—
71H(5,𝔽2)—3190no—
72H(5,𝔽2)C2×D31110yescomplemented
73C2×#41—31120no—
74C2×#41—3130no—
75C2×#41—3145no—
76C2×#41C2×D31115yescomplemented
77C22×D4—3190no—
78C22×D4—3115no—
79C31H(7,𝔽2)11yescharacteristic, complemented, Sylow
80C24—3130no—
81#41—31120no—
82#41—31120no—
83#41—31180no—
84#41—3160no—
85#41C22×D31160yescomplemented
86#41C22×D31120yescomplemented
87C2×Q8—3140no—
88C2×Q8—3115no—
89C2×Q8C22×D31115yescomplemented
90C2×D4—3190no—
91C2×D4—31360no—
92C2×D4—3190no—
93C2×D4—3145no—
94C2×D4C22×D31145yescomplemented
95C22×C4—3190no—
96C22×C4C22×D31115yescomplemented
97C23—3190no—
98C23—24830no—
99C23C23×D31115yes—
100Q8—3130no—
101Q8—316no—
102Q8C23×D31120yes—
103D4—31120no—
104D4—3190no—
105D4—3110no—
106D4C23×D31160yes—
107C2×C4—3130no—
108C2×C4—31120no—
109C2×C4C23×D31145yes—
110C2×C4C23×D31115yes—
111C22—415no—
112C22—3120no—
113C22—12490no—
114C22C24×D31115yes—
115C4—3112no—
116C4C24×D31115yes—
117C4C24×D3111yescharacteristic
118C2—215no—
119C2—6220no—
120C2C25×D3111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D31
commutator, socle, cosocleC2×C31C26
FittingC31×#585C2
hypercenter#585D31
hypocenterC31H(7,𝔽2)
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #14521886C2semidirect
2. #14521885C2semidirect
3. C2×#11758566C2semidirect
4. H(5,𝔽2)×D31C2semidirect
5. C31×#585C2semidirect
6. C31×2−1+4C22semidirect
7. C31×H(5,𝔽2)C22semidirect
8. C2×#41×C31C22semidirect
9. #11758567C22semidirect
10. #11758566C22semidirect
11. #11758566C22semidirect
12. #41×D31C22semidirect
13. #11758564C22semidirect
14. C2×#92499C22semidirect
15. #11758561C22semidirect
16. C2×D4×D31C22semidirect
17. C2×#92495C22semidirect
18. C22×D124C22semidirect
19. C23×D31C23semidirect
20. #41×C31C23semidirect
21. #41×C31C23semidirect
22. C2×Q8×C31C23semidirect
23. C2×D4×C31C23semidirect
24. C22×C4×C31C23semidirect
25. C2×C22⋊D31C23semidirect
26. #92499C23semidirect
27. Q8×D31C23semidirect
28. #92497C23semidirect
29. D4×D31C23semidirect
30. #92495C23semidirect
31. #92495C23semidirect
32. C2×D124C23semidirect
33. C2×D124C23semidirect
34. C2×C4×D31C23semidirect
35. C2×Dic62C23semidirect
60. #585D31semidirect
68. 2−1+4C2×D31semidirect
72. H(5,𝔽2)C2×D31semidirect
76. C2×#41C2×D31semidirect
79. C31H(7,𝔽2)semidirect
85. #41C22×D31semidirect
86. #41C22×D31semidirect
89. C2×Q8C22×D31semidirect
94. C2×D4C22×D31semidirect
96. C22×C4C22×D31semidirect

Maximal subgroups

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

subgroupindex
#145218862
#145218852
C2×#117585662
H(5,𝔽2)×D312
C31×#5852
H(7,𝔽2)31