Finite Group Databasebeta

← #585×D167 #36338944

Subgroups — #585×D167 #36338944

Order 21376.

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#24924106C216yesmaximal, complemented
2#24924105C2115yesmaximal, complemented
3#24924104C2110yesmaximal, complemented
42−1+4×D167C216yesmaximal, complemented
5H(5,𝔽2)×D167C2110yesmaximal, complemented
6C2×#41×D167C2115yesmaximal, complemented
7#585×C167C211yesmaximal, characteristic, complemented
8#19128880C22130yescomplemented
9#19128879C22160yescomplemented
10#19128878C22160yescomplemented
11#19128877C22190yescomplemented
12C2×#14581951C22115yescomplemented
13C2×#14581950C22145yescomplemented
14C2×#14581949C22145yescomplemented
15#41×D167C22120yescomplemented
16#41×D167C22160yescomplemented
17#41×D167C22160yescomplemented
18#41×D167C22160yescomplemented
19C2×Q8×D167C22115yescomplemented
20C2×D4×D167C22145yescomplemented
21C22×C4×D167C22115yescomplemented
222−1+4×C167C2216yescomplemented
23H(5,𝔽2)×C167C22110yescomplemented
24C2×#41×C167C22115yescomplemented
25C23×D167C23115yes—
26#14581951C23120yescomplemented
27#14581951C23160yescomplemented
28#14581950C23160yescomplemented
29#14581950C231180yescomplemented
30#14581949C23160yescomplemented
31#14581949C231180yescomplemented
32C2×Dic334C23145yescomplemented
33Q8×D167C23120yes—
34Q8×D167C23160yescomplemented
35C2×D668C23145yescomplemented
36C2×C22⋊D167C23190yescomplemented
37D4×D167C23160yes—
38D4×D167C231180yescomplemented
39C22×Dic167C23115yescomplemented
40C2×C4×D167C23115yes—
41C2×C4×D167C23145yes—
42C2×C4×D167C23145yescomplemented
43C2×C4×D167C23145yescomplemented
44#41×C167C23120yescomplemented
45#41×C167C23160yescomplemented
46C2×Q8×C167C23115yescomplemented
47C2×D4×C167C23145yescomplemented
48C22×C4×C167C23115yescomplemented
49C22×D167—430no—
50C22×D167C24115yes—
51C22×D167C24145yes—
52Dic334C24160yes—
53D668C24160yes—
54C22⋊D167C241120yes—
55C2×Dic167C24115yes—
56C2×Dic167C24145yes—
57C4×D167C2411yescharacteristic
58C4×D167C24115yes—
59C4×D167C24115yes—
60C4×D167C24115yes—
61C4×D167C24190yes—
62C23×C167C24115yes—
63Q8×C167C24120yes—
64D4×C167C24160yes—
65C2×C4×C167C24115yes—
66C2×C4×C167C24145yes—
67C22×C167—415no—
68C22×C167C25115yes—
69C2×D167—230no—
70C2×D167—445no—
71C2×D167C25115yes—
72C2×D167C2511yescharacteristic
73C4×C167C25115yes—
74C4×C167C2511yescharacteristic
75Dic167C2511yescharacteristic
76Dic167C25115yes—
77C2×C167—215no—
78C2×C167C2611yescharacteristic
79D167—215no—
80D167#58512yesdirect product factor
81C167C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1671nomaximal, Sylow
83#585—1676no—
84#585—16710no—
85#585—16715no—
86#585D16711yescharacteristic, direct product factor
87C2×2−1+4—1676no—
88C2×H(5,𝔽2)—16710no—
89C22×#41—16715no—
902−1+4—16760no—
912−1+4—16730no—
922−1+4C2×D16716yescomplemented
93H(5,𝔽2)—16760no—
94H(5,𝔽2)—16790no—
95H(5,𝔽2)C2×D167110yescomplemented
96C2×#41—16720no—
97C2×#41—16760no—
98C2×#41—16760no—
99C2×#41—16760no—
100C2×#41—16715no—
101C2×#41—16745no—
102C2×#41—16745no—
103C2×#41C2×D167115yescomplemented
104C22×Q8—16715no—
105C22×D4—16745no—
106C23×C4—16715no—
107C24—16715no—
108#41—16720no—
109#41—16760no—
110#41—16760no—
111#41—167180no—
112#41—16760no—
113#41—167180no—
114#41C22×D167120yescomplemented
115#41C22×D167160yescomplemented
116C2×Q8—16720no—
117C2×Q8—16760no—
118C2×Q8—16745no—
119C2×Q8C22×D167115yescomplemented
120C2×D4—16790no—
121C2×D4—16760no—
122C2×D4—167180no—
123C2×D4—16745no—
124C2×D4C22×D167145yescomplemented
125C22×C4—16715no—
126C22×C4—16715no—
127C22×C4—16745no—
128C22×C4—16745no—
129C22×C4—16745no—
130C22×C4C22×D167115yescomplemented
131C23—66830no—
132C23—16715no—
133C23—16745no—
134C23C23×D167115yes—
135Q8—16760no—
136Q8C23×D167120yes—
137D4—167120no—
138D4—16760no—
139D4C23×D167160yes—
140C2×C4—16715no—
141C2×C4—16745no—
142C2×C4—16715no—
143C2×C4—16715no—
144C2×C4—16790no—
145C2×C4—1671no—
146C2×C4—16715no—
147C2×C4C23×D167145yes—
148C2×C4C23×D167115yes—
149C22—415no—
150C22—33430no—
151C22—66845no—
152C22—16715no—
153C22—1671no—
154C22C24×D167115yes—
155C4—1671no—
156C4—16715no—
157C4C24×D167115yes—
158C4C24×D16711yescharacteristic
159C2—215no—
160C2—33415no—
161C2—1672no—
162C2C25×D16711yescharacteristic
subgroupG/H
centerC4C24×D167
commutator, socle, cosocleC2×C167C26
FrattiniC2C25×D167
Fitting#585×C167C2
hypercenter#585D167
hypocenterC167C2×#585
LCSG ▷ C2×C167 ▷ C167
UCSC1 ◁ C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #24924106C2semidirect
2. #24924105C2semidirect
3. #24924104C2semidirect
4. 2−1+4×D167C2semidirect
5. H(5,𝔽2)×D167C2semidirect
6. C2×#41×D167C2semidirect
7. #585×C167C2semidirect
8. #19128880C22semidirect
9. #19128879C22semidirect
10. #19128878C22semidirect
11. #19128877C22semidirect
12. C2×#14581951C22semidirect
13. C2×#14581950C22semidirect
14. C2×#14581949C22semidirect
15. #41×D167C22semidirect
16. #41×D167C22semidirect
17. #41×D167C22semidirect
18. #41×D167C22semidirect
19. C2×Q8×D167C22semidirect
20. C2×D4×D167C22semidirect
21. C22×C4×D167C22semidirect
22. 2−1+4×C167C22semidirect
23. H(5,𝔽2)×C167C22semidirect
24. C2×#41×C167C22semidirect
26. #14581951C23semidirect
27. #14581951C23semidirect
28. #14581950C23semidirect
29. #14581950C23semidirect
30. #14581949C23semidirect
31. #14581949C23semidirect
32. C2×Dic334C23semidirect
34. Q8×D167C23semidirect
35. C2×D668C23semidirect
36. C2×C22⋊D167C23semidirect
38. D4×D167C23semidirect
39. C22×Dic167C23semidirect
42. C2×C4×D167C23semidirect
43. C2×C4×D167C23semidirect
44. #41×C167C23semidirect
45. #41×C167C23semidirect
46. C2×Q8×C167C23semidirect
47. C2×D4×C167C23semidirect
48. C22×C4×C167C23semidirect
80. D167#585direct (N × H)
81. C167C2×#585semidirect
86. #585D167direct (N × H)
92. 2−1+4C2×D167semidirect
95. H(5,𝔽2)C2×D167semidirect
103. C2×#41C2×D167semidirect
114. #41C22×D167semidirect
115. #41C22×D167semidirect
119. C2×Q8C22×D167semidirect
124. C2×D4C22×D167semidirect
130. C22×C4C22×D167semidirect

Maximal subgroups

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

subgroupindex
#249241062
#249241052
#249241042
2−1+4×D1672
H(5,𝔽2)×D1672
C2×#41×D1672
#585×C1672
C2×#585167