Finite Group Databasebeta

← #27185942

Subgroups — #27185942

Order 12928.

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#20266013C2112yesmaximal, complemented
2#20266012C2116yesmaximal, complemented
3C2×#15692300C2118yesmaximal, complemented
4H(5,𝔽2)×D101C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C101C211yesmaximal, characteristic, complemented
6#15692303C2218yescomplemented
7#15692302C22148yescomplemented
8#15692301C22148yescomplemented
9#15692300C221144yescomplemented
10#15692300C22172yescomplemented
11C2×#13108074C22136yescomplemented
12C2×#13108073C22172yescomplemented
13#41×D101C22196yescomplemented
14C22×C22⋊D101C22124yescomplemented
15C2×D4×D101C22172yescomplemented
16H(5,𝔽2)×C101C22116yescomplemented
17C2×#41×C101C2216yescomplemented
18C22×D4×C101C2219yescomplemented
19C23×D101C23124yescomplemented
20#13108075C23132yescomplemented
21#13108074C231144yescomplemented
22#13108074C23148yescomplemented
23#13108073C231288yescomplemented
24C2×Dic202C23136yescomplemented
25Q8×D101C23132yescomplemented
26C2×D404C23136yescomplemented
27C2×C22⋊D101C231144yescomplemented
28C2×C22⋊D101C23172yescomplemented
29D4×D101C231288yescomplemented
30C22×Dic101C23124yescomplemented
31C2×C4×D101C23172yescomplemented
32C24×C101C2316yescomplemented
33#41×C101C23148yescomplemented
34C2×Q8×C101C2312yescomplemented
35C2×D4×C101C23172yescomplemented
36C2×D4×C101C23118yescomplemented
37C22×C4×C101C2319yescomplemented
38C23×C101—86no—
39C23×C101C2419yes—
40C23×C101C24124yes—
41C22×D101—824no—
42C22×D101C24172yes—
43Q8×C101C2418yes—
44D4×C101C24172yes—
45C2×C4×C101C2416yes—
46C2×C4×C101C24136yes—
47C22⋊D101C241144yes—
48C22⋊D101C24116yes—
49C2×Dic101C24172yes—
50D404C24148yes—
51C4×D101C24196yes—
52Dic202C24148yes—
53C22×C101—424no—
54C22×C101—49no—
55C22×C101C25118yes—
56C22×C101C2511yescharacteristic
57C2×D101—472no—
58C2×D101C25116yes—
59C4×C101C25112yes—
60Dic101C25116yes—
61C2×C101—218no—
62C2×C101—21no—
63C2×C101C2611yescharacteristic
64D101—216no—
65H(7,𝔽2)—1011nomaximal, Sylow
66C101H(7,𝔽2)11yescharacteristic, complemented, Sylow
67#585—10116no—
68#585—10112no—
69C2×H(5,𝔽2)—10116no—
70C2×H(5,𝔽2)—10118no—
71C2×H(5,𝔽2)D10111yescharacteristic, complemented
722−1+4—10148no—
732−1+4—1018no—
74H(5,𝔽2)—10148no—
75H(5,𝔽2)—101144no—
76H(5,𝔽2)—10172no—
77H(5,𝔽2)C2×D101116yescomplemented
78C2×#41—10196no—
79C2×#41—10172no—
80C2×#41—10136no—
81C2×#41C2×D10116yescomplemented
82C22×D4—10124no—
83C22×D4—10172no—
84C22×D4C2×D10119yescomplemented
85C24—10124no—
86C24C22×D10116yescomplemented
87#41—10132no—
88#41—101288no—
89#41—101144no—
90#41—10148no—
91#41C22×D101148yescomplemented
92C2×Q8—10132no—
93C2×Q8—10136no—
94C2×Q8C22×D10112yescomplemented
95C2×D4—101144no—
96C2×D4—10172no—
97C2×D4—101288no—
98C2×D4—10136no—
99C2×D4C22×D101172yescomplemented
100C2×D4C22×D101118yescomplemented
101C22×C4—10124no—
102C22×C4—10172no—
103C22×C4C22×D10119yescomplemented
104C23—86no—
105C23—80824no—
106C23—10172no—
107C23C23×D101124yes—
108C23C23×D10119yes—
109Q8—10148no—
110Q8C23×D10118yes—
111D4—101144no—
112D4—10116no—
113D4—10148no—
114D4C23×D101172yes—
115C2×C4—10172no—
116C2×C4—10196no—
117C2×C4C23×D101136yes—
118C2×C4C23×D10116yes—
119C22—49no—
120C22—424no—
121C22—10116no—
122C22—40472no—
123C22C24×D101118yes—
124C22C24×D10111yescharacteristic
125C4—10116no—
126C4C24×D101112yes—
127C2—21no—
128C2—218no—
129C2—20216no—
130C2C25×D10111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D101
commutator, socle, cosocleC2×C101C26
FittingC2×H(5,𝔽2)×C101C2
hypercenterC2×H(5,𝔽2)D101
hypocenterC101H(7,𝔽2)
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20266013C2semidirect
2. #20266012C2semidirect
3. C2×#15692300C2semidirect
4. H(5,𝔽2)×D101C2semidirect
5. C2×H(5,𝔽2)×C101C2semidirect
6. #15692303C22semidirect
7. #15692302C22semidirect
8. #15692301C22semidirect
9. #15692300C22semidirect
10. #15692300C22semidirect
11. C2×#13108074C22semidirect
12. C2×#13108073C22semidirect
13. #41×D101C22semidirect
14. C22×C22⋊D101C22semidirect
15. C2×D4×D101C22semidirect
16. H(5,𝔽2)×C101C22semidirect
17. C2×#41×C101C22semidirect
18. C22×D4×C101C22semidirect
19. C23×D101C23semidirect
20. #13108075C23semidirect
21. #13108074C23semidirect
22. #13108074C23semidirect
23. #13108073C23semidirect
24. C2×Dic202C23semidirect
25. Q8×D101C23semidirect
26. C2×D404C23semidirect
27. C2×C22⋊D101C23semidirect
28. C2×C22⋊D101C23semidirect
29. D4×D101C23semidirect
30. C22×Dic101C23semidirect
31. C2×C4×D101C23semidirect
32. C24×C101C23semidirect
33. #41×C101C23semidirect
34. C2×Q8×C101C23semidirect
35. C2×D4×C101C23semidirect
36. C2×D4×C101C23semidirect
37. C22×C4×C101C23semidirect
66. C101H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D101semidirect
77. H(5,𝔽2)C2×D101semidirect
81. C2×#41C2×D101semidirect
84. C22×D4C2×D101semidirect
86. C24C22×D101semidirect
91. #41C22×D101semidirect
94. C2×Q8C22×D101semidirect
99. C2×D4C22×D101semidirect
100. C2×D4C22×D101semidirect
103. C22×C4C22×D101semidirect

Maximal subgroups

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

subgroupindex
#202660132
#202660122
C2×#156923002
H(5,𝔽2)×D1012
C2×H(5,𝔽2)×C1012
H(7,𝔽2)101