Finite Group Databasebeta

← #44565175

Subgroups — #44565175

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#30632523C2112yesmaximal, complemented
2#30632522C2116yesmaximal, complemented
3C2×#22548760C2118yesmaximal, complemented
4H(5,𝔽2)×D251C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C251C211yesmaximal, characteristic, complemented
6#22548763C2218yescomplemented
7#22548762C22148yescomplemented
8#22548761C22148yescomplemented
9#22548760C221144yescomplemented
10#22548760C22172yescomplemented
11C2×#16860810C22136yescomplemented
12C2×#16860809C22172yescomplemented
13#41×D251C22196yescomplemented
14C22×C22⋊D251C22124yescomplemented
15C2×D4×D251C22172yescomplemented
16H(5,𝔽2)×C251C22116yescomplemented
17C2×#41×C251C2216yescomplemented
18C22×D4×C251C2219yescomplemented
19C23×D251C23124yescomplemented
20#16860811C23132yescomplemented
21#16860810C231144yescomplemented
22#16860810C23148yescomplemented
23#16860809C231288yescomplemented
24C2×Dic502C23136yescomplemented
25Q8×D251C23132yescomplemented
26C2×D1004C23136yescomplemented
27C2×C22⋊D251C231144yescomplemented
28C2×C22⋊D251C23172yescomplemented
29D4×D251C231288yescomplemented
30C22×Dic251C23124yescomplemented
31C2×C4×D251C23172yescomplemented
32C24×C251C2316yescomplemented
33#41×C251C23148yescomplemented
34C2×Q8×C251C2312yescomplemented
35C2×D4×C251C23172yescomplemented
36C2×D4×C251C23118yescomplemented
37C22×C4×C251C2319yescomplemented
38C22×D251—824no—
39C22×D251C24172yes—
40Dic502C24148yes—
41D1004C24148yes—
42C22⋊D251C241144yes—
43C22⋊D251C24116yes—
44C2×Dic251C24172yes—
45C4×D251C24196yes—
46C23×C251—86no—
47C23×C251C2419yes—
48C23×C251C24124yes—
49Q8×C251C2418yes—
50D4×C251C24172yes—
51C2×C4×C251C2416yes—
52C2×C4×C251C24136yes—
53C22×C251—49no—
54C22×C251—424no—
55C22×C251C2511yescharacteristic
56C22×C251C25118yes—
57C2×D251—472no—
58C2×D251C25116yes—
59C4×C251C25112yes—
60Dic251C25116yes—
61C2×C251—218no—
62C2×C251—21no—
63C2×C251C2611yescharacteristic
64D251—216no—
65C251H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—2511nomaximal, Sylow
67#585—25116no—
68#585—25112no—
69C2×H(5,𝔽2)—25116no—
70C2×H(5,𝔽2)—25118no—
71C2×H(5,𝔽2)D25111yescharacteristic, complemented
722−1+4—25148no—
732−1+4—2518no—
74H(5,𝔽2)—25148no—
75H(5,𝔽2)—251144no—
76H(5,𝔽2)—25172no—
77H(5,𝔽2)C2×D251116yescomplemented
78C2×#41—25196no—
79C2×#41—25172no—
80C2×#41—25136no—
81C2×#41C2×D25116yescomplemented
82C22×D4—25124no—
83C22×D4—25172no—
84C22×D4C2×D25119yescomplemented
85C24—25124no—
86C24C22×D25116yescomplemented
87#41—25132no—
88#41—251288no—
89#41—251144no—
90#41—25148no—
91#41C22×D251148yescomplemented
92C2×Q8—25132no—
93C2×Q8—25136no—
94C2×Q8C22×D25112yescomplemented
95C2×D4—251144no—
96C2×D4—25172no—
97C2×D4—251288no—
98C2×D4—25136no—
99C2×D4C22×D251172yescomplemented
100C2×D4C22×D251118yescomplemented
101C22×C4—25124no—
102C22×C4—25172no—
103C22×C4C22×D25119yescomplemented
104C23—86no—
105C23—200824no—
106C23—25172no—
107C23C23×D251124yes—
108C23C23×D25119yes—
109Q8—25148no—
110Q8C23×D25118yes—
111D4—251144no—
112D4—25116no—
113D4—25148no—
114D4C23×D251172yes—
115C2×C4—25172no—
116C2×C4—25196no—
117C2×C4C23×D251136yes—
118C2×C4C23×D25116yes—
119C22—49no—
120C22—424no—
121C22—100472no—
122C22—25116no—
123C22C24×D251118yes—
124C22C24×D25111yescharacteristic
125C4—25116no—
126C4C24×D251112yes—
127C2—21no—
128C2—218no—
129C2—50216no—
130C2C25×D25111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D251
commutator, socle, cosocleC2×C251C26
FittingC2×H(5,𝔽2)×C251C2
hypercenterC2×H(5,𝔽2)D251
hypocenterC251H(7,𝔽2)
LCSG ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #30632523C2semidirect
2. #30632522C2semidirect
3. C2×#22548760C2semidirect
4. H(5,𝔽2)×D251C2semidirect
5. C2×H(5,𝔽2)×C251C2semidirect
6. #22548763C22semidirect
7. #22548762C22semidirect
8. #22548761C22semidirect
9. #22548760C22semidirect
10. #22548760C22semidirect
11. C2×#16860810C22semidirect
12. C2×#16860809C22semidirect
13. #41×D251C22semidirect
14. C22×C22⋊D251C22semidirect
15. C2×D4×D251C22semidirect
16. H(5,𝔽2)×C251C22semidirect
17. C2×#41×C251C22semidirect
18. C22×D4×C251C22semidirect
19. C23×D251C23semidirect
20. #16860811C23semidirect
21. #16860810C23semidirect
22. #16860810C23semidirect
23. #16860809C23semidirect
24. C2×Dic502C23semidirect
25. Q8×D251C23semidirect
26. C2×D1004C23semidirect
27. C2×C22⋊D251C23semidirect
28. C2×C22⋊D251C23semidirect
29. D4×D251C23semidirect
30. C22×Dic251C23semidirect
31. C2×C4×D251C23semidirect
32. C24×C251C23semidirect
33. #41×C251C23semidirect
34. C2×Q8×C251C23semidirect
35. C2×D4×C251C23semidirect
36. C2×D4×C251C23semidirect
37. C22×C4×C251C23semidirect
65. C251H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D251semidirect
77. H(5,𝔽2)C2×D251semidirect
81. C2×#41C2×D251semidirect
84. C22×D4C2×D251semidirect
86. C24C22×D251semidirect
91. #41C22×D251semidirect
94. C2×Q8C22×D251semidirect
99. C2×D4C22×D251semidirect
100. C2×D4C22×D251semidirect
103. C22×C4C22×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306325222
C2×#225487602
H(5,𝔽2)×D2512
C2×H(5,𝔽2)×C2512
H(7,𝔽2)251