Finite Group Databasebeta

← #585×D181 #37548527

Subgroups — #585×D181 #37548527

Order 23168.

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#26042116C216yesmaximal, complemented
2#26042115C2115yesmaximal, complemented
3#26042114C2110yesmaximal, complemented
42−1+4×D181C216yesmaximal, complemented
5H(5,𝔽2)×D181C2110yesmaximal, complemented
6C2×#41×D181C2115yesmaximal, complemented
7#585×C181C211yesmaximal, characteristic, complemented
8#19153864C22130yescomplemented
9#19153863C22160yescomplemented
10#19153862C22160yescomplemented
11#19153861C22190yescomplemented
12C2×#15668450C22115yescomplemented
13C2×#15668449C22145yescomplemented
14C2×#15668448C22145yescomplemented
15#41×D181C22120yescomplemented
16#41×D181C22160yescomplemented
17#41×D181C22160yescomplemented
18#41×D181C22160yescomplemented
19C2×Q8×D181C22115yescomplemented
20C2×D4×D181C22145yescomplemented
21C22×C4×D181C22115yescomplemented
222−1+4×C181C2216yescomplemented
23H(5,𝔽2)×C181C22110yescomplemented
24C2×#41×C181C22115yescomplemented
25C23×D181C23115yes—
26#15668450C23120yescomplemented
27#15668450C23160yescomplemented
28#15668449C23160yescomplemented
29#15668449C231180yescomplemented
30#15668448C23160yescomplemented
31#15668448C231180yescomplemented
32C2×Dic362C23145yescomplemented
33Q8×D181C23120yes—
34Q8×D181C23160yescomplemented
35C2×D724C23145yescomplemented
36C2×C22⋊D181C23190yescomplemented
37D4×D181C23160yes—
38D4×D181C231180yescomplemented
39C22×Dic181C23115yescomplemented
40C2×C4×D181C23115yes—
41C2×C4×D181C23145yes—
42C2×C4×D181C23145yescomplemented
43C2×C4×D181C23145yescomplemented
44#41×C181C23120yescomplemented
45#41×C181C23160yescomplemented
46C2×Q8×C181C23115yescomplemented
47C2×D4×C181C23145yescomplemented
48C22×C4×C181C23115yescomplemented
49C22×D181—430no—
50C22×D181C24115yes—
51C22×D181C24145yes—
52Dic362C24160yes—
53D724C24160yes—
54C22⋊D181C241120yes—
55C2×Dic181C24115yes—
56C2×Dic181C24145yes—
57C4×D181C2411yescharacteristic
58C4×D181C24115yes—
59C4×D181C24115yes—
60C4×D181C24115yes—
61C4×D181C24190yes—
62C23×C181C24115yes—
63Q8×C181C24120yes—
64D4×C181C24160yes—
65C2×C4×C181C24115yes—
66C2×C4×C181C24145yes—
67C22×C181—415no—
68C22×C181C25115yes—
69C2×D181—230no—
70C2×D181—445no—
71C2×D181C25115yes—
72C2×D181C2511yescharacteristic
73C4×C181C25115yes—
74C4×C181C2511yescharacteristic
75Dic181C2511yescharacteristic
76Dic181C25115yes—
77C2×C181—215no—
78C2×C181C2611yescharacteristic
79D181—215no—
80D181#58512yesdirect product factor
81C181C2×#58511yescharacteristic, complemented, Sylow
82C2×#585—1811nomaximal, Sylow
83#585—1816no—
84#585—18110no—
85#585—18115no—
86#585D18111yescharacteristic, direct product factor
87C2×2−1+4—1816no—
88C2×H(5,𝔽2)—18110no—
89C22×#41—18115no—
902−1+4—18160no—
912−1+4—18130no—
922−1+4C2×D18116yescomplemented
93H(5,𝔽2)—18160no—
94H(5,𝔽2)—18190no—
95H(5,𝔽2)C2×D181110yescomplemented
96C2×#41—18120no—
97C2×#41—18160no—
98C2×#41—18160no—
99C2×#41—18160no—
100C2×#41—18115no—
101C2×#41—18145no—
102C2×#41—18145no—
103C2×#41C2×D181115yescomplemented
104C22×Q8—18115no—
105C22×D4—18145no—
106C23×C4—18115no—
107C24—18115no—
108#41—18120no—
109#41—18160no—
110#41—18160no—
111#41—181180no—
112#41—18160no—
113#41—181180no—
114#41C22×D181120yescomplemented
115#41C22×D181160yescomplemented
116C2×Q8—18120no—
117C2×Q8—18160no—
118C2×Q8—18145no—
119C2×Q8C22×D181115yescomplemented
120C2×D4—18190no—
121C2×D4—18160no—
122C2×D4—181180no—
123C2×D4—18145no—
124C2×D4C22×D181145yescomplemented
125C22×C4—18115no—
126C22×C4—18115no—
127C22×C4—18145no—
128C22×C4—18145no—
129C22×C4—18145no—
130C22×C4C22×D181115yescomplemented
131C23—18115no—
132C23—18145no—
133C23—72430no—
134C23C23×D181115yes—
135Q8—18160no—
136Q8C23×D181120yes—
137D4—181120no—
138D4—18160no—
139D4C23×D181160yes—
140C2×C4—18115no—
141C2×C4—18145no—
142C2×C4—18115no—
143C2×C4—18115no—
144C2×C4—18190no—
145C2×C4—1811no—
146C2×C4—18115no—
147C2×C4C23×D181145yes—
148C2×C4C23×D181115yes—
149C22—415no—
150C22—36230no—
151C22—18115no—
152C22—1811no—
153C22—72445no—
154C22C24×D181115yes—
155C4—1811no—
156C4—18115no—
157C4C24×D181115yes—
158C4C24×D18111yescharacteristic
159C2—215no—
160C2—36215no—
161C2—1812no—
162C2C25×D18111yescharacteristic
subgroupG/H
centerC4C24×D181
commutator, socle, cosocleC2×C181C26
FrattiniC2C25×D181
Fitting#585×C181C2
hypercenter#585D181
hypocenterC181C2×#585
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #26042116C2semidirect
2. #26042115C2semidirect
3. #26042114C2semidirect
4. 2−1+4×D181C2semidirect
5. H(5,𝔽2)×D181C2semidirect
6. C2×#41×D181C2semidirect
7. #585×C181C2semidirect
8. #19153864C22semidirect
9. #19153863C22semidirect
10. #19153862C22semidirect
11. #19153861C22semidirect
12. C2×#15668450C22semidirect
13. C2×#15668449C22semidirect
14. C2×#15668448C22semidirect
15. #41×D181C22semidirect
16. #41×D181C22semidirect
17. #41×D181C22semidirect
18. #41×D181C22semidirect
19. C2×Q8×D181C22semidirect
20. C2×D4×D181C22semidirect
21. C22×C4×D181C22semidirect
22. 2−1+4×C181C22semidirect
23. H(5,𝔽2)×C181C22semidirect
24. C2×#41×C181C22semidirect
26. #15668450C23semidirect
27. #15668450C23semidirect
28. #15668449C23semidirect
29. #15668449C23semidirect
30. #15668448C23semidirect
31. #15668448C23semidirect
32. C2×Dic362C23semidirect
34. Q8×D181C23semidirect
35. C2×D724C23semidirect
36. C2×C22⋊D181C23semidirect
38. D4×D181C23semidirect
39. C22×Dic181C23semidirect
42. C2×C4×D181C23semidirect
43. C2×C4×D181C23semidirect
44. #41×C181C23semidirect
45. #41×C181C23semidirect
46. C2×Q8×C181C23semidirect
47. C2×D4×C181C23semidirect
48. C22×C4×C181C23semidirect
80. D181#585direct (N × H)
81. C181C2×#585semidirect
86. #585D181direct (N × H)
92. 2−1+4C2×D181semidirect
95. H(5,𝔽2)C2×D181semidirect
103. C2×#41C2×D181semidirect
114. #41C22×D181semidirect
115. #41C22×D181semidirect
119. C2×Q8C22×D181semidirect
124. C2×D4C22×D181semidirect
130. C22×C4C22×D181semidirect

Maximal subgroups

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

subgroupindex
#260421162
#260421152
#260421142
2−1+4×D1812
H(5,𝔽2)×D1812
C2×#41×D1812
#585×C1812
C2×#585181