Finite Group Databasebeta

← #27156546

Subgroups — #27156546

Order 12416.

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#20261571C2112yesmaximal, complemented
2#20261570C2116yesmaximal, complemented
3C2×#15690855C2118yesmaximal, complemented
4H(5,𝔽2)×D97C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
6#15690858C2218yescomplemented
7#15690857C22148yescomplemented
8#15690856C22148yescomplemented
9#15690855C221144yescomplemented
10#15690855C22172yescomplemented
11C2×#13095511C22136yescomplemented
12C2×#13095510C22172yescomplemented
13#41×D97C22196yescomplemented
14C22×C22⋊D97C22124yescomplemented
15C2×D4×D97C22172yescomplemented
16H(5,𝔽2)×C97C22116yescomplemented
17C2×#41×C97C2216yescomplemented
18C22×D4×C97C2219yescomplemented
19C23×D97C23124yescomplemented
20#13095512C23132yescomplemented
21#13095511C231144yescomplemented
22#13095511C23148yescomplemented
23#13095510C231288yescomplemented
24C2×Dic194C23136yescomplemented
25Q8×D97C23132yescomplemented
26C2×D388C23136yescomplemented
27C2×C22⋊D97C231144yescomplemented
28C2×C22⋊D97C23172yescomplemented
29D4×D97C231288yescomplemented
30C22×Dic97C23124yescomplemented
31C2×C4×D97C23172yescomplemented
32C24×C97C2316yescomplemented
33#41×C97C23148yescomplemented
34C2×Q8×C97C2312yescomplemented
35C2×D4×C97C23172yescomplemented
36C2×D4×C97C23118yescomplemented
37C22×C4×C97C2319yescomplemented
38C23×C97—86no—
39C23×C97C2419yes—
40C23×C97C24124yes—
41C22×D97—824no—
42C22×D97C24172yes—
43Q8×C97C2418yes—
44D4×C97C24172yes—
45C2×C4×C97C2416yes—
46C2×C4×C97C24136yes—
47C22⋊D97C241144yes—
48C22⋊D97C24116yes—
49C2×Dic97C24172yes—
50D388C24148yes—
51C4×D97C24196yes—
52Dic194C24148yes—
53C22×C97—424no—
54C22×C97—49no—
55C22×C97C25118yes—
56C22×C97C2511yescharacteristic
57C2×D97—472no—
58C2×D97C25116yes—
59C4×C97C25112yes—
60Dic97C25116yes—
61C2×C97—218no—
62C2×C97—21no—
63C2×C97C2611yescharacteristic
64D97—216no—
65H(7,𝔽2)—971nomaximal, Sylow
66C97H(7,𝔽2)11yescharacteristic, complemented, Sylow
67#585—9716no—
68#585—9712no—
69C2×H(5,𝔽2)—9716no—
70C2×H(5,𝔽2)—9718no—
71C2×H(5,𝔽2)D9711yescharacteristic, complemented
722−1+4—9748no—
732−1+4—978no—
74H(5,𝔽2)—9748no—
75H(5,𝔽2)—97144no—
76H(5,𝔽2)—9772no—
77H(5,𝔽2)C2×D97116yescomplemented
78C2×#41—9796no—
79C2×#41—9772no—
80C2×#41—9736no—
81C2×#41C2×D9716yescomplemented
82C22×D4—9724no—
83C22×D4—9772no—
84C22×D4C2×D9719yescomplemented
85C24—9724no—
86C24C22×D9716yescomplemented
87#41—9732no—
88#41—97288no—
89#41—97144no—
90#41—9748no—
91#41C22×D97148yescomplemented
92C2×Q8—9732no—
93C2×Q8—9736no—
94C2×Q8C22×D9712yescomplemented
95C2×D4—97144no—
96C2×D4—9772no—
97C2×D4—97288no—
98C2×D4—9736no—
99C2×D4C22×D97172yescomplemented
100C2×D4C22×D97118yescomplemented
101C22×C4—9724no—
102C22×C4—9772no—
103C22×C4C22×D9719yescomplemented
104C23—86no—
105C23—77624no—
106C23—9772no—
107C23C23×D97124yes—
108C23C23×D9719yes—
109Q8—9748no—
110Q8C23×D9718yes—
111D4—97144no—
112D4—9716no—
113D4—9748no—
114D4C23×D97172yes—
115C2×C4—9772no—
116C2×C4—9796no—
117C2×C4C23×D97136yes—
118C2×C4C23×D9716yes—
119C22—49no—
120C22—424no—
121C22—9716no—
122C22—38872no—
123C22C24×D97118yes—
124C22C24×D9711yescharacteristic
125C4—9716no—
126C4C24×D97112yes—
127C2—21no—
128C2—218no—
129C2—19416no—
130C2C25×D9711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D97
commutator, socle, cosocleC2×C97C26
FittingC2×H(5,𝔽2)×C97C2
hypercenterC2×H(5,𝔽2)D97
hypocenterC97H(7,𝔽2)
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20261571C2semidirect
2. #20261570C2semidirect
3. C2×#15690855C2semidirect
4. H(5,𝔽2)×D97C2semidirect
5. C2×H(5,𝔽2)×C97C2semidirect
6. #15690858C22semidirect
7. #15690857C22semidirect
8. #15690856C22semidirect
9. #15690855C22semidirect
10. #15690855C22semidirect
11. C2×#13095511C22semidirect
12. C2×#13095510C22semidirect
13. #41×D97C22semidirect
14. C22×C22⋊D97C22semidirect
15. C2×D4×D97C22semidirect
16. H(5,𝔽2)×C97C22semidirect
17. C2×#41×C97C22semidirect
18. C22×D4×C97C22semidirect
19. C23×D97C23semidirect
20. #13095512C23semidirect
21. #13095511C23semidirect
22. #13095511C23semidirect
23. #13095510C23semidirect
24. C2×Dic194C23semidirect
25. Q8×D97C23semidirect
26. C2×D388C23semidirect
27. C2×C22⋊D97C23semidirect
28. C2×C22⋊D97C23semidirect
29. D4×D97C23semidirect
30. C22×Dic97C23semidirect
31. C2×C4×D97C23semidirect
32. C24×C97C23semidirect
33. #41×C97C23semidirect
34. C2×Q8×C97C23semidirect
35. C2×D4×C97C23semidirect
36. C2×D4×C97C23semidirect
37. C22×C4×C97C23semidirect
66. C97H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D97semidirect
77. H(5,𝔽2)C2×D97semidirect
81. C2×#41C2×D97semidirect
84. C22×D4C2×D97semidirect
86. C24C22×D97semidirect
91. #41C22×D97semidirect
94. C2×Q8C22×D97semidirect
99. C2×D4C22×D97semidirect
100. C2×D4C22×D97semidirect
103. C22×C4C22×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
#202615702
C2×#156908552
H(5,𝔽2)×D972
C2×H(5,𝔽2)×C972
H(7,𝔽2)97