Finite Group Databasebeta

← #44565176

Subgroups — #44565176

Order 32128.

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#30632524C2112yesmaximal, complemented
2#30632523C2115yesmaximal, complemented
3C2×#22548761C2115yesmaximal, complemented
4H(5,𝔽2)×D251C2120yesmaximal, complemented
5#585×C251C211yesmaximal, characteristic, complemented
6#22548763C22130yescomplemented
7#22548762C22120yescomplemented
8#22548761C221120yescomplemented
9#22548761C22160yescomplemented
10#22548760C22190yescomplemented
11C2×#16860811C22130yescomplemented
12C2×#16860810C22145yescomplemented
13#41×D251C221120yescomplemented
14C22×D1004C22115yescomplemented
15C2×D4×D251C22190yescomplemented
162−1+4×C251C2216yescomplemented
17H(5,𝔽2)×C251C22110yescomplemented
18C2×#41×C251C22115yescomplemented
19C23×D251C23130yescomplemented
20#16860811C231120yescomplemented
21#16860810C231180yescomplemented
22#16860810C23160yescomplemented
23#16860809C231120yescomplemented
24C2×Dic502C23115yescomplemented
25Q8×D251C23140yescomplemented
26C2×D1004C23190yescomplemented
27C2×D1004C23145yescomplemented
28C2×C22⋊D251C23190yescomplemented
29D4×D251C231360yescomplemented
30C2×C4×D251C23190yescomplemented
31#41×C251C23160yescomplemented
32#41×C251C23120yescomplemented
33C2×Q8×C251C23115yescomplemented
34C2×D4×C251C23145yescomplemented
35C22×C4×C251C23115yescomplemented
36C22×D251—830no—
37C22×D251C24190yes—
38Dic502C24130yes—
39Dic502C2416yes—
40D1004C24190yes—
41D1004C24110yes—
42C22⋊D251C241120yes—
43C2×Dic251C24130yes—
44C4×D251C241120yes—
45C23×C251C24115yes—
46Q8×C251C24120yes—
47D4×C251C24160yes—
48C2×C4×C251C24115yes—
49C2×C4×C251C24145yes—
50C22×C251—415no—
51C22×C251C25115yes—
52C2×D251—490no—
53C2×D251C25120yes—
54C4×C251C2511yescharacteristic
55C4×C251C25115yes—
56Dic251C25112yes—
57C2×C251—215no—
58C2×C251C2611yescharacteristic
59D251—220no—
60C251H(7,𝔽2)11yescharacteristic, complemented, Sylow
61H(7,𝔽2)—2511nomaximal, Sylow
62#585—25112no—
63#585—25115no—
64#585D25111yescharacteristic, complemented
65C2×H(5,𝔽2)—25120no—
66C2×H(5,𝔽2)—25115no—
672−1+4—25120no—
682−1+4—25130no—
692−1+4C2×D25116yescomplemented
70H(5,𝔽2)—251120no—
71H(5,𝔽2)—25160no—
72H(5,𝔽2)—25190no—
73H(5,𝔽2)C2×D251110yescomplemented
74C2×#41—251120no—
75C2×#41—25130no—
76C2×#41—25145no—
77C2×#41C2×D251115yescomplemented
78C22×D4—25190no—
79C22×D4—25115no—
80C24—25130no—
81#41—251120no—
82#41—251120no—
83#41—251180no—
84#41—25160no—
85#41C22×D251160yescomplemented
86#41C22×D251120yescomplemented
87C2×Q8—25140no—
88C2×Q8—25115no—
89C2×Q8C22×D251115yescomplemented
90C2×D4—25190no—
91C2×D4—251360no—
92C2×D4—25190no—
93C2×D4—25145no—
94C2×D4C22×D251145yescomplemented
95C22×C4—25190no—
96C22×C4C22×D251115yescomplemented
97C23—200830no—
98C23—25190no—
99C23C23×D251115yes—
100Q8—25130no—
101Q8—2516no—
102Q8C23×D251120yes—
103D4—251120no—
104D4—25190no—
105D4—25110no—
106D4C23×D251160yes—
107C2×C4—25130no—
108C2×C4—251120no—
109C2×C4C23×D251145yes—
110C2×C4C23×D251115yes—
111C22—415no—
112C22—100490no—
113C22—25120no—
114C22C24×D251115yes—
115C4—25112no—
116C4C24×D251115yes—
117C4C24×D25111yescharacteristic
118C2—215no—
119C2—50220no—
120C2C25×D25111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D251
commutator, socle, cosocleC2×C251C26
Fitting#585×C251C2
hypercenter#585D251
hypocenterC251H(7,𝔽2)
LCSG ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #30632524C2semidirect
2. #30632523C2semidirect
3. C2×#22548761C2semidirect
4. H(5,𝔽2)×D251C2semidirect
5. #585×C251C2semidirect
6. #22548763C22semidirect
7. #22548762C22semidirect
8. #22548761C22semidirect
9. #22548761C22semidirect
10. #22548760C22semidirect
11. C2×#16860811C22semidirect
12. C2×#16860810C22semidirect
13. #41×D251C22semidirect
14. C22×D1004C22semidirect
15. C2×D4×D251C22semidirect
16. 2−1+4×C251C22semidirect
17. H(5,𝔽2)×C251C22semidirect
18. C2×#41×C251C22semidirect
19. C23×D251C23semidirect
20. #16860811C23semidirect
21. #16860810C23semidirect
22. #16860810C23semidirect
23. #16860809C23semidirect
24. C2×Dic502C23semidirect
25. Q8×D251C23semidirect
26. C2×D1004C23semidirect
27. C2×D1004C23semidirect
28. C2×C22⋊D251C23semidirect
29. D4×D251C23semidirect
30. C2×C4×D251C23semidirect
31. #41×C251C23semidirect
32. #41×C251C23semidirect
33. C2×Q8×C251C23semidirect
34. C2×D4×C251C23semidirect
35. C22×C4×C251C23semidirect
60. C251H(7,𝔽2)semidirect
64. #585D251semidirect
69. 2−1+4C2×D251semidirect
73. H(5,𝔽2)C2×D251semidirect
77. C2×#41C2×D251semidirect
85. #41C22×D251semidirect
86. #41C22×D251semidirect
89. C2×Q8C22×D251semidirect
94. C2×D4C22×D251semidirect
96. C22×C4C22×D251semidirect

Maximal subgroups

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

subgroupindex
#306325242
#306325232
C2×#225487612
H(5,𝔽2)×D2512
#585×C2512
H(7,𝔽2)251