Finite Group Databasebeta

← #27156547

Subgroups — #27156547

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#20261572C2112yesmaximal, complemented
2#20261571C2115yesmaximal, complemented
3C2×#15690856C2115yesmaximal, complemented
4H(5,𝔽2)×D97C2120yesmaximal, complemented
5#585×C97C211yesmaximal, characteristic, complemented
6#15690858C22130yescomplemented
7#15690857C22120yescomplemented
8#15690856C221120yescomplemented
9#15690856C22160yescomplemented
10#15690855C22190yescomplemented
11C2×#13095512C22130yescomplemented
12C2×#13095511C22145yescomplemented
13#41×D97C221120yescomplemented
14C22×D388C22115yescomplemented
15C2×D4×D97C22190yescomplemented
162−1+4×C97C2216yescomplemented
17H(5,𝔽2)×C97C22110yescomplemented
18C2×#41×C97C22115yescomplemented
19C23×D97C23130yescomplemented
20#13095512C231120yescomplemented
21#13095511C231180yescomplemented
22#13095511C23160yescomplemented
23#13095510C231120yescomplemented
24C2×Dic194C23115yescomplemented
25Q8×D97C23140yescomplemented
26C2×D388C23190yescomplemented
27C2×D388C23145yescomplemented
28C2×C22⋊D97C23190yescomplemented
29D4×D97C231360yescomplemented
30C2×C4×D97C23190yescomplemented
31#41×C97C23160yescomplemented
32#41×C97C23120yescomplemented
33C2×Q8×C97C23115yescomplemented
34C2×D4×C97C23145yescomplemented
35C22×C4×C97C23115yescomplemented
36C23×C97C24115yes—
37C22×D97—830no—
38C22×D97C24190yes—
39Q8×C97C24120yes—
40D4×C97C24160yes—
41C2×C4×C97C24115yes—
42C2×C4×C97C24145yes—
43C22⋊D97C241120yes—
44C2×Dic97C24130yes—
45D388C24190yes—
46D388C24110yes—
47C4×D97C241120yes—
48Dic194C24130yes—
49Dic194C2416yes—
50C22×C97—415no—
51C22×C97C25115yes—
52C2×D97—490no—
53C2×D97C25120yes—
54C4×C97C2511yescharacteristic
55C4×C97C25115yes—
56Dic97C25112yes—
57C2×C97—215no—
58C2×C97C2611yescharacteristic
59D97—220no—
60H(7,𝔽2)—971nomaximal, Sylow
61C97H(7,𝔽2)11yescharacteristic, complemented, Sylow
62#585—9712no—
63#585—9715no—
64#585D9711yescharacteristic, complemented
65C2×H(5,𝔽2)—9720no—
66C2×H(5,𝔽2)—9715no—
672−1+4—9720no—
682−1+4—9730no—
692−1+4C2×D9716yescomplemented
70H(5,𝔽2)—97120no—
71H(5,𝔽2)—9760no—
72H(5,𝔽2)—9790no—
73H(5,𝔽2)C2×D97110yescomplemented
74C2×#41—97120no—
75C2×#41—9730no—
76C2×#41—9745no—
77C2×#41C2×D97115yescomplemented
78C22×D4—9790no—
79C22×D4—9715no—
80C24—9730no—
81#41—97120no—
82#41—97120no—
83#41—97180no—
84#41—9760no—
85#41C22×D97160yescomplemented
86#41C22×D97120yescomplemented
87C2×Q8—9740no—
88C2×Q8—9715no—
89C2×Q8C22×D97115yescomplemented
90C2×D4—9790no—
91C2×D4—97360no—
92C2×D4—9790no—
93C2×D4—9745no—
94C2×D4C22×D97145yescomplemented
95C22×C4—9790no—
96C22×C4C22×D97115yescomplemented
97C23—77630no—
98C23—9790no—
99C23C23×D97115yes—
100Q8—9730no—
101Q8—976no—
102Q8C23×D97120yes—
103D4—97120no—
104D4—9790no—
105D4—9710no—
106D4C23×D97160yes—
107C2×C4—9730no—
108C2×C4—97120no—
109C2×C4C23×D97145yes—
110C2×C4C23×D97115yes—
111C22—415no—
112C22—9720no—
113C22—38890no—
114C22C24×D97115yes—
115C4—9712no—
116C4C24×D97115yes—
117C4C24×D9711yescharacteristic
118C2—215no—
119C2—19420no—
120C2C25×D9711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D97
commutator, socle, cosocleC2×C97C26
Fitting#585×C97C2
hypercenter#585D97
hypocenterC97H(7,𝔽2)
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20261572C2semidirect
2. #20261571C2semidirect
3. C2×#15690856C2semidirect
4. H(5,𝔽2)×D97C2semidirect
5. #585×C97C2semidirect
6. #15690858C22semidirect
7. #15690857C22semidirect
8. #15690856C22semidirect
9. #15690856C22semidirect
10. #15690855C22semidirect
11. C2×#13095512C22semidirect
12. C2×#13095511C22semidirect
13. #41×D97C22semidirect
14. C22×D388C22semidirect
15. C2×D4×D97C22semidirect
16. 2−1+4×C97C22semidirect
17. H(5,𝔽2)×C97C22semidirect
18. C2×#41×C97C22semidirect
19. C23×D97C23semidirect
20. #13095512C23semidirect
21. #13095511C23semidirect
22. #13095511C23semidirect
23. #13095510C23semidirect
24. C2×Dic194C23semidirect
25. Q8×D97C23semidirect
26. C2×D388C23semidirect
27. C2×D388C23semidirect
28. C2×C22⋊D97C23semidirect
29. D4×D97C23semidirect
30. C2×C4×D97C23semidirect
31. #41×C97C23semidirect
32. #41×C97C23semidirect
33. C2×Q8×C97C23semidirect
34. C2×D4×C97C23semidirect
35. C22×C4×C97C23semidirect
61. C97H(7,𝔽2)semidirect
64. #585D97semidirect
69. 2−1+4C2×D97semidirect
73. H(5,𝔽2)C2×D97semidirect
77. C2×#41C2×D97semidirect
85. #41C22×D97semidirect
86. #41C22×D97semidirect
89. C2×Q8C22×D97semidirect
94. C2×D4C22×D97semidirect
96. C22×C4C22×D97semidirect

Maximal subgroups

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

subgroupindex
#202615722
#202615712
C2×#156908562
H(5,𝔽2)×D972
#585×C972
H(7,𝔽2)97