Finite Group Databasebeta

← #585×D131 #30680767

Subgroups — #585×D131 #30680767

Order 16768.

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#22554149C216yesmaximal, complemented
2#22554148C2115yesmaximal, complemented
3#22554147C2110yesmaximal, complemented
42−1+4×D131C216yesmaximal, complemented
5H(5,𝔽2)×D131C2110yesmaximal, complemented
6C2×#41×D131C2115yesmaximal, complemented
7#585×C131C211yesmaximal, characteristic, complemented
8#16862294C22130yescomplemented
9#16862293C22160yescomplemented
10#16862292C22160yescomplemented
11#16862291C22190yescomplemented
12C2×#14523514C22115yescomplemented
13C2×#14523513C22145yescomplemented
14C2×#14523512C22145yescomplemented
15#41×D131C22120yescomplemented
16#41×D131C22160yescomplemented
17#41×D131C22160yescomplemented
18#41×D131C22160yescomplemented
19C2×Q8×D131C22115yescomplemented
20C2×D4×D131C22145yescomplemented
21C22×C4×D131C22115yescomplemented
222−1+4×C131C2216yescomplemented
23H(5,𝔽2)×C131C22110yescomplemented
24C2×#41×C131C22115yescomplemented
25C23×D131C23115yes—
26#14523514C23120yescomplemented
27#14523514C23160yescomplemented
28#14523513C23160yescomplemented
29#14523513C231180yescomplemented
30#14523512C23160yescomplemented
31#14523512C231180yescomplemented
32C2×Dic262C23145yescomplemented
33Q8×D131C23120yes—
34Q8×D131C23160yescomplemented
35C2×D524C23145yescomplemented
36C2×C22⋊D131C23190yescomplemented
37D4×D131C23160yes—
38D4×D131C231180yescomplemented
39C22×Dic131C23115yescomplemented
40C2×C4×D131C23115yes—
41C2×C4×D131C23145yes—
42C2×C4×D131C23145yescomplemented
43C2×C4×D131C23145yescomplemented
44#41×C131C23120yescomplemented
45#41×C131C23160yescomplemented
46C2×Q8×C131C23115yescomplemented
47C2×D4×C131C23145yescomplemented
48C22×C4×C131C23115yescomplemented
49C22×D131—430no—
50C22×D131C24115yes—
51C22×D131C24145yes—
52Dic262C24160yes—
53D524C24160yes—
54C22⋊D131C241120yes—
55C2×Dic131C24115yes—
56C2×Dic131C24145yes—
57C4×D131C2411yescharacteristic
58C4×D131C24115yes—
59C4×D131C24115yes—
60C4×D131C24115yes—
61C4×D131C24190yes—
62C23×C131C24115yes—
63Q8×C131C24120yes—
64D4×C131C24160yes—
65C2×C4×C131C24115yes—
66C2×C4×C131C24145yes—
67C22×C131—415no—
68C22×C131C25115yes—
69C2×D131—230no—
70C2×D131—445no—
71C2×D131C25115yes—
72C2×D131C2511yescharacteristic
73C4×C131C25115yes—
74C4×C131C2511yescharacteristic
75Dic131C2511yescharacteristic
76Dic131C25115yes—
77C2×C131—215no—
78C2×C131C2611yescharacteristic
79D131—215no—
80D131#58512yesdirect product factor
81C131C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1311nomaximal, Sylow
83#585—1316no—
84#585—13110no—
85#585—13115no—
86#585D13111yescharacteristic, direct product factor
87C2×2−1+4—1316no—
88C2×H(5,𝔽2)—13110no—
89C22×#41—13115no—
902−1+4—13160no—
912−1+4—13130no—
922−1+4C2×D13116yescomplemented
93H(5,𝔽2)—13160no—
94H(5,𝔽2)—13190no—
95H(5,𝔽2)C2×D131110yescomplemented
96C2×#41—13120no—
97C2×#41—13160no—
98C2×#41—13160no—
99C2×#41—13160no—
100C2×#41—13115no—
101C2×#41—13145no—
102C2×#41—13145no—
103C2×#41C2×D131115yescomplemented
104C22×Q8—13115no—
105C22×D4—13145no—
106C23×C4—13115no—
107C24—13115no—
108#41—13120no—
109#41—13160no—
110#41—13160no—
111#41—131180no—
112#41—13160no—
113#41—131180no—
114#41C22×D131120yescomplemented
115#41C22×D131160yescomplemented
116C2×Q8—13120no—
117C2×Q8—13160no—
118C2×Q8—13145no—
119C2×Q8C22×D131115yescomplemented
120C2×D4—13190no—
121C2×D4—13160no—
122C2×D4—131180no—
123C2×D4—13145no—
124C2×D4C22×D131145yescomplemented
125C22×C4—13115no—
126C22×C4—13115no—
127C22×C4—13145no—
128C22×C4—13145no—
129C22×C4—13145no—
130C22×C4C22×D131115yescomplemented
131C23—52430no—
132C23—13115no—
133C23—13145no—
134C23C23×D131115yes—
135Q8—13160no—
136Q8C23×D131120yes—
137D4—131120no—
138D4—13160no—
139D4C23×D131160yes—
140C2×C4—13115no—
141C2×C4—13145no—
142C2×C4—13115no—
143C2×C4—13115no—
144C2×C4—13190no—
145C2×C4—1311no—
146C2×C4—13115no—
147C2×C4C23×D131145yes—
148C2×C4C23×D131115yes—
149C22—415no—
150C22—26230no—
151C22—52445no—
152C22—13115no—
153C22—1311no—
154C22C24×D131115yes—
155C4—1311no—
156C4—13115no—
157C4C24×D131115yes—
158C4C24×D13111yescharacteristic
159C2—215no—
160C2—26215no—
161C2—1312no—
162C2C25×D13111yescharacteristic
subgroupG/H
centerC4C24×D131
commutator, socle, cosocleC2×C131C26
FrattiniC2C25×D131
Fitting#585×C131C2
hypercenter#585D131
hypocenterC131C2×#585
LCSG ▷ C2×C131 ▷ C131
UCSC1 ◁ C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22554149C2semidirect
2. #22554148C2semidirect
3. #22554147C2semidirect
4. 2−1+4×D131C2semidirect
5. H(5,𝔽2)×D131C2semidirect
6. C2×#41×D131C2semidirect
7. #585×C131C2semidirect
8. #16862294C22semidirect
9. #16862293C22semidirect
10. #16862292C22semidirect
11. #16862291C22semidirect
12. C2×#14523514C22semidirect
13. C2×#14523513C22semidirect
14. C2×#14523512C22semidirect
15. #41×D131C22semidirect
16. #41×D131C22semidirect
17. #41×D131C22semidirect
18. #41×D131C22semidirect
19. C2×Q8×D131C22semidirect
20. C2×D4×D131C22semidirect
21. C22×C4×D131C22semidirect
22. 2−1+4×C131C22semidirect
23. H(5,𝔽2)×C131C22semidirect
24. C2×#41×C131C22semidirect
26. #14523514C23semidirect
27. #14523514C23semidirect
28. #14523513C23semidirect
29. #14523513C23semidirect
30. #14523512C23semidirect
31. #14523512C23semidirect
32. C2×Dic262C23semidirect
34. Q8×D131C23semidirect
35. C2×D524C23semidirect
36. C2×C22⋊D131C23semidirect
38. D4×D131C23semidirect
39. C22×Dic131C23semidirect
42. C2×C4×D131C23semidirect
43. C2×C4×D131C23semidirect
44. #41×C131C23semidirect
45. #41×C131C23semidirect
46. C2×Q8×C131C23semidirect
47. C2×D4×C131C23semidirect
48. C22×C4×C131C23semidirect
80. D131#585direct (N × H)
81. C131C2×#585semidirect
86. #585D131direct (N × H)
92. 2−1+4C2×D131semidirect
95. H(5,𝔽2)C2×D131semidirect
103. C2×#41C2×D131semidirect
114. #41C22×D131semidirect
115. #41C22×D131semidirect
119. C2×Q8C22×D131semidirect
124. C2×D4C22×D131semidirect
130. C22×C4C22×D131semidirect

Maximal subgroups

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

subgroupindex
#225541492
#225541482
#225541472
2−1+4×D1312
H(5,𝔽2)×D1312
C2×#41×D1312
#585×C1312
C2×#585131