Finite Group Databasebeta

← #585×D137 #31795595

Subgroups — #585×D137 #31795595

Order 17536.

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#22578240C216yesmaximal, complemented
2#22578239C2115yesmaximal, complemented
3#22578238C2110yesmaximal, complemented
42−1+4×D137C216yesmaximal, complemented
5H(5,𝔽2)×D137C2110yesmaximal, complemented
6C2×#41×D137C2115yesmaximal, complemented
7#585×C137C211yesmaximal, characteristic, complemented
8#17984136C22130yescomplemented
9#17984135C22160yescomplemented
10#17984134C22160yescomplemented
11#17984133C22190yescomplemented
12C2×#14555511C22115yescomplemented
13C2×#14555510C22145yescomplemented
14C2×#14555509C22145yescomplemented
15#41×D137C22120yescomplemented
16#41×D137C22160yescomplemented
17#41×D137C22160yescomplemented
18#41×D137C22160yescomplemented
19C2×Q8×D137C22115yescomplemented
20C2×D4×D137C22145yescomplemented
21C22×C4×D137C22115yescomplemented
222−1+4×C137C2216yescomplemented
23H(5,𝔽2)×C137C22110yescomplemented
24C2×#41×C137C22115yescomplemented
25C23×D137C23115yes—
26#14555511C23120yescomplemented
27#14555511C23160yescomplemented
28#14555510C23160yescomplemented
29#14555510C231180yescomplemented
30#14555509C23160yescomplemented
31#14555509C231180yescomplemented
32C2×Dic274C23145yescomplemented
33Q8×D137C23120yes—
34Q8×D137C23160yescomplemented
35C2×D548C23145yescomplemented
36C2×C22⋊D137C23190yescomplemented
37D4×D137C23160yes—
38D4×D137C231180yescomplemented
39C22×Dic137C23115yescomplemented
40C2×C4×D137C23115yes—
41C2×C4×D137C23145yes—
42C2×C4×D137C23145yescomplemented
43C2×C4×D137C23145yescomplemented
44#41×C137C23120yescomplemented
45#41×C137C23160yescomplemented
46C2×Q8×C137C23115yescomplemented
47C2×D4×C137C23145yescomplemented
48C22×C4×C137C23115yescomplemented
49C22×D137—430no—
50C22×D137C24115yes—
51C22×D137C24145yes—
52Dic274C24160yes—
53D548C24160yes—
54C22⋊D137C241120yes—
55C2×Dic137C24115yes—
56C2×Dic137C24145yes—
57C4×D137C2411yescharacteristic
58C4×D137C24115yes—
59C4×D137C24115yes—
60C4×D137C24115yes—
61C4×D137C24190yes—
62C23×C137C24115yes—
63Q8×C137C24120yes—
64D4×C137C24160yes—
65C2×C4×C137C24115yes—
66C2×C4×C137C24145yes—
67C22×C137—415no—
68C22×C137C25115yes—
69C2×D137—230no—
70C2×D137—445no—
71C2×D137C25115yes—
72C2×D137C2511yescharacteristic
73C4×C137C25115yes—
74C4×C137C2511yescharacteristic
75Dic137C2511yescharacteristic
76Dic137C25115yes—
77C2×C137—215no—
78C2×C137C2611yescharacteristic
79D137—215no—
80D137#58512yesdirect product factor
81C137C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1371nomaximal, Sylow
83#585—1376no—
84#585—13710no—
85#585—13715no—
86#585D13711yescharacteristic, direct product factor
87C2×2−1+4—1376no—
88C2×H(5,𝔽2)—13710no—
89C22×#41—13715no—
902−1+4—13760no—
912−1+4—13730no—
922−1+4C2×D13716yescomplemented
93H(5,𝔽2)—13760no—
94H(5,𝔽2)—13790no—
95H(5,𝔽2)C2×D137110yescomplemented
96C2×#41—13720no—
97C2×#41—13760no—
98C2×#41—13760no—
99C2×#41—13760no—
100C2×#41—13715no—
101C2×#41—13745no—
102C2×#41—13745no—
103C2×#41C2×D137115yescomplemented
104C22×Q8—13715no—
105C22×D4—13745no—
106C23×C4—13715no—
107C24—13715no—
108#41—13720no—
109#41—13760no—
110#41—13760no—
111#41—137180no—
112#41—13760no—
113#41—137180no—
114#41C22×D137120yescomplemented
115#41C22×D137160yescomplemented
116C2×Q8—13720no—
117C2×Q8—13760no—
118C2×Q8—13745no—
119C2×Q8C22×D137115yescomplemented
120C2×D4—13790no—
121C2×D4—13760no—
122C2×D4—137180no—
123C2×D4—13745no—
124C2×D4C22×D137145yescomplemented
125C22×C4—13715no—
126C22×C4—13715no—
127C22×C4—13745no—
128C22×C4—13745no—
129C22×C4—13745no—
130C22×C4C22×D137115yescomplemented
131C23—54830no—
132C23—13715no—
133C23—13745no—
134C23C23×D137115yes—
135Q8—13760no—
136Q8C23×D137120yes—
137D4—137120no—
138D4—13760no—
139D4C23×D137160yes—
140C2×C4—13715no—
141C2×C4—13745no—
142C2×C4—13715no—
143C2×C4—13715no—
144C2×C4—13790no—
145C2×C4—1371no—
146C2×C4—13715no—
147C2×C4C23×D137145yes—
148C2×C4C23×D137115yes—
149C22—415no—
150C22—27430no—
151C22—54845no—
152C22—13715no—
153C22—1371no—
154C22C24×D137115yes—
155C4—1371no—
156C4—13715no—
157C4C24×D137115yes—
158C4C24×D13711yescharacteristic
159C2—215no—
160C2—27415no—
161C2—1372no—
162C2C25×D13711yescharacteristic
subgroupG/H
centerC4C24×D137
commutator, socle, cosocleC2×C137C26
FrattiniC2C25×D137
Fitting#585×C137C2
hypercenter#585D137
hypocenterC137C2×#585
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22578240C2semidirect
2. #22578239C2semidirect
3. #22578238C2semidirect
4. 2−1+4×D137C2semidirect
5. H(5,𝔽2)×D137C2semidirect
6. C2×#41×D137C2semidirect
7. #585×C137C2semidirect
8. #17984136C22semidirect
9. #17984135C22semidirect
10. #17984134C22semidirect
11. #17984133C22semidirect
12. C2×#14555511C22semidirect
13. C2×#14555510C22semidirect
14. C2×#14555509C22semidirect
15. #41×D137C22semidirect
16. #41×D137C22semidirect
17. #41×D137C22semidirect
18. #41×D137C22semidirect
19. C2×Q8×D137C22semidirect
20. C2×D4×D137C22semidirect
21. C22×C4×D137C22semidirect
22. 2−1+4×C137C22semidirect
23. H(5,𝔽2)×C137C22semidirect
24. C2×#41×C137C22semidirect
26. #14555511C23semidirect
27. #14555511C23semidirect
28. #14555510C23semidirect
29. #14555510C23semidirect
30. #14555509C23semidirect
31. #14555509C23semidirect
32. C2×Dic274C23semidirect
34. Q8×D137C23semidirect
35. C2×D548C23semidirect
36. C2×C22⋊D137C23semidirect
38. D4×D137C23semidirect
39. C22×Dic137C23semidirect
42. C2×C4×D137C23semidirect
43. C2×C4×D137C23semidirect
44. #41×C137C23semidirect
45. #41×C137C23semidirect
46. C2×Q8×C137C23semidirect
47. C2×D4×C137C23semidirect
48. C22×C4×C137C23semidirect
80. D137#585direct (N × H)
81. C137C2×#585semidirect
86. #585D137direct (N × H)
92. 2−1+4C2×D137semidirect
95. H(5,𝔽2)C2×D137semidirect
103. C2×#41C2×D137semidirect
114. #41C22×D137semidirect
115. #41C22×D137semidirect
119. C2×Q8C22×D137semidirect
124. C2×D4C22×D137semidirect
130. C22×C4C22×D137semidirect

Maximal subgroups

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

subgroupindex
#225782402
#225782392
#225782382
2−1+4×D1372
H(5,𝔽2)×D1372
C2×#41×D1372
#585×C1372
C2×#585137