Finite Group Databasebeta

← #585×D97 #27156542

Subgroups — #585×D97 #27156542

Order 12416.

5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 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#20261572C216yesmaximal, complemented
2#20261571C2115yesmaximal, complemented
3#20261570C2110yesmaximal, complemented
42−1+4×D97C216yesmaximal, complemented
5H(5,𝔽2)×D97C2110yesmaximal, complemented
6C2×#41×D97C2115yesmaximal, complemented
7#585×C97C211yesmaximal, characteristic, complemented
8#15690858C22130yescomplemented
9#15690857C22160yescomplemented
10#15690856C22160yescomplemented
11#15690855C22190yescomplemented
12C2×#13095512C22115yescomplemented
13C2×#13095511C22145yescomplemented
14C2×#13095510C22145yescomplemented
15#41×D97C22120yescomplemented
16#41×D97C22160yescomplemented
17#41×D97C22160yescomplemented
18#41×D97C22160yescomplemented
19C2×Q8×D97C22115yescomplemented
20C2×D4×D97C22145yescomplemented
21C22×C4×D97C22115yescomplemented
222−1+4×C97C2216yescomplemented
23H(5,𝔽2)×C97C22110yescomplemented
24C2×#41×C97C22115yescomplemented
25C23×D97C23115yes—
26#13095512C23120yescomplemented
27#13095512C23160yescomplemented
28#13095511C23160yescomplemented
29#13095511C231180yescomplemented
30#13095510C23160yescomplemented
31#13095510C231180yescomplemented
32C2×Dic194C23145yescomplemented
33Q8×D97C23120yes—
34Q8×D97C23160yescomplemented
35C2×D388C23145yescomplemented
36C2×C22⋊D97C23190yescomplemented
37D4×D97C23160yes—
38D4×D97C231180yescomplemented
39C22×Dic97C23115yescomplemented
40C2×C4×D97C23115yes—
41C2×C4×D97C23145yes—
42C2×C4×D97C23145yescomplemented
43C2×C4×D97C23145yescomplemented
44#41×C97C23120yescomplemented
45#41×C97C23160yescomplemented
46C2×Q8×C97C23115yescomplemented
47C2×D4×C97C23145yescomplemented
48C22×C4×C97C23115yescomplemented
49C23×C97C24115yes—
50C22×D97—430no—
51C22×D97C24115yes—
52C22×D97C24145yes—
53Q8×C97C24120yes—
54D4×C97C24160yes—
55C2×C4×C97C24115yes—
56C2×C4×C97C24145yes—
57C22⋊D97C241120yes—
58C2×Dic97C24115yes—
59C2×Dic97C24145yes—
60D388C24160yes—
61C4×D97C2411yescharacteristic
62C4×D97C24115yes—
63C4×D97C24115yes—
64C4×D97C24115yes—
65C4×D97C24190yes—
66Dic194C24160yes—
67C22×C97—415no—
68C22×C97C25115yes—
69C2×D97—230no—
70C2×D97—445no—
71C2×D97C25115yes—
72C2×D97C2511yescharacteristic
73C4×C97C25115yes—
74C4×C97C2511yescharacteristic
75Dic97C25115yes—
76Dic97C2511yescharacteristic
77C2×C97—215no—
78C2×C97C2611yescharacteristic
79D97—215no—
80D97#58512yesdirect product factor
81C2×#585—971nomaximal, Sylow
82C97C2×#58511yescharacteristic, complemented, Sylow
83#585—976no—
84#585—9710no—
85#585—9715no—
86#585D9711yescharacteristic, direct product factor
87C2×2−1+4—976no—
88C2×H(5,𝔽2)—9710no—
89C22×#41—9715no—
902−1+4—9760no—
912−1+4—9730no—
922−1+4C2×D9716yescomplemented
93H(5,𝔽2)—9760no—
94H(5,𝔽2)—9790no—
95H(5,𝔽2)C2×D97110yescomplemented
96C2×#41—9720no—
97C2×#41—9760no—
98C2×#41—9760no—
99C2×#41—9760no—
100C2×#41—9715no—
101C2×#41—9745no—
102C2×#41—9745no—
103C2×#41C2×D97115yescomplemented
104C22×Q8—9715no—
105C22×D4—9745no—
106C23×C4—9715no—
107C24—9715no—
108#41—9720no—
109#41—9760no—
110#41—9760no—
111#41—97180no—
112#41—9760no—
113#41—97180no—
114#41C22×D97120yescomplemented
115#41C22×D97160yescomplemented
116C2×Q8—9720no—
117C2×Q8—9760no—
118C2×Q8—9745no—
119C2×Q8C22×D97115yescomplemented
120C2×D4—9790no—
121C2×D4—9760no—
122C2×D4—97180no—
123C2×D4—9745no—
124C2×D4C22×D97145yescomplemented
125C22×C4—9715no—
126C22×C4—9715no—
127C22×C4—9745no—
128C22×C4—9745no—
129C22×C4—9745no—
130C22×C4C22×D97115yescomplemented
131C23—9745no—
132C23—9715no—
133C23—38830no—
134C23C23×D97115yes—
135Q8—9760no—
136Q8C23×D97120yes—
137D4—97120no—
138D4—9760no—
139D4C23×D97160yes—
140C2×C4—9715no—
141C2×C4—9745no—
142C2×C4—9715no—
143C2×C4—9715no—
144C2×C4—9790no—
145C2×C4—971no—
146C2×C4—9715no—
147C2×C4C23×D97145yes—
148C2×C4C23×D97115yes—
149C22—415no—
150C22—971no—
151C22—9715no—
152C22—38845no—
153C22—19430no—
154C22C24×D97115yes—
155C4—971no—
156C4—9715no—
157C4C24×D97115yes—
158C4C24×D9711yescharacteristic
159C2—215no—
160C2—972no—
161C2—19415no—
162C2C25×D9711yescharacteristic
subgroupG/H
centerC4C24×D97
commutator, socle, cosocleC2×C97C26
FrattiniC2C25×D97
Fitting#585×C97C2
hypercenter#585D97
hypocenterC97C2×#585
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20261572C2semidirect
2. #20261571C2semidirect
3. #20261570C2semidirect
4. 2−1+4×D97C2semidirect
5. H(5,𝔽2)×D97C2semidirect
6. C2×#41×D97C2semidirect
7. #585×C97C2semidirect
8. #15690858C22semidirect
9. #15690857C22semidirect
10. #15690856C22semidirect
11. #15690855C22semidirect
12. C2×#13095512C22semidirect
13. C2×#13095511C22semidirect
14. C2×#13095510C22semidirect
15. #41×D97C22semidirect
16. #41×D97C22semidirect
17. #41×D97C22semidirect
18. #41×D97C22semidirect
19. C2×Q8×D97C22semidirect
20. C2×D4×D97C22semidirect
21. C22×C4×D97C22semidirect
22. 2−1+4×C97C22semidirect
23. H(5,𝔽2)×C97C22semidirect
24. C2×#41×C97C22semidirect
26. #13095512C23semidirect
27. #13095512C23semidirect
28. #13095511C23semidirect
29. #13095511C23semidirect
30. #13095510C23semidirect
31. #13095510C23semidirect
32. C2×Dic194C23semidirect
34. Q8×D97C23semidirect
35. C2×D388C23semidirect
36. C2×C22⋊D97C23semidirect
38. D4×D97C23semidirect
39. C22×Dic97C23semidirect
42. C2×C4×D97C23semidirect
43. C2×C4×D97C23semidirect
44. #41×C97C23semidirect
45. #41×C97C23semidirect
46. C2×Q8×C97C23semidirect
47. C2×D4×C97C23semidirect
48. C22×C4×C97C23semidirect
80. D97#585direct (N × H)
82. C97C2×#585semidirect
86. #585D97direct (N × H)
92. 2−1+4C2×D97semidirect
95. H(5,𝔽2)C2×D97semidirect
103. C2×#41C2×D97semidirect
114. #41C22×D97semidirect
115. #41C22×D97semidirect
119. C2×Q8C22×D97semidirect
124. C2×D4C22×D97semidirect
130. 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
#202615702
2−1+4×D972
H(5,𝔽2)×D972
C2×#41×D972
#585×C972
C2×#58597