Finite Group Databasebeta

← #37548500

Subgroups — #37548500

Order 23168.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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
1C2×#19153862C211yesmaximal, characteristic, complemented
2#26042087C218yesmaximal, complemented
3#26042082C218yesmaximal, complemented
4#26042077C216yesmaximal, complemented
5C2×#19153841C216yesmaximal, complemented
6#26042046C211yesmaximal, characteristic, complemented
7#585×C181C211yesmaximal, characteristic, complemented
8#19153862—24no—
9#19153862C2218yescomplemented
10C2×#15668450—21no—
11C2×#15668449C2213yescomplemented
12C22×D724C2213yescomplemented
13C2×D4×D181—23no—
14#19153847C22112yescomplemented
15#19153846C2214yescomplemented
16#19153842C22112yescomplemented
17#19153841C22112yescomplemented
18#19153841C22124yescomplemented
19#19153836C22124yescomplemented
20C2×#15668436C22112yescomplemented
21C2×#15668434C22112yescomplemented
22#19153823C2218yescomplemented
23C2×#15668433C2216yescomplemented
242−1+4×C181—21no—
252−1+4×C181C2214yescomplemented
26H(5,𝔽2)×C181—23no—
27H(5,𝔽2)×C181C2214yescomplemented
28C2×#41×C181—24no—
29C2×#41×C181C2211yescharacteristic
30C2×#41×C181C2216yescomplemented
31C23×D181—23no—
32#15668450—28no—
33#15668449—26no—
34#15668449C23112yes—
35C2×Dic362C2311yescharacteristic
36C2×D724—23no—
37C2×D724—26no—
38C2×D724C23112yes—
39C2×D724C2313yes—
40C2×C22⋊D181—23no—
41D4×D181—224no—
42C2×C4×D181—23no—
43#15668439C2318yescomplemented
44#15668437C2318yescomplemented
45#15668436C23124yescomplemented
46#15668434C23124yescomplemented
47#15668433C2314yes—
48#15668433C23112yescomplemented
49C2×#13089678C23112yescomplemented
50#41×C181—24no—
51#41×C181—212no—
52#41×C181—26no—
53#41×C181—26no—
54#41×C181D414yes—
55#41×C181C2314yes—
56#41×C181C2314yes—
57#41×C181C23112yescomplemented
58C2×Q8×C181—24no—
59C2×Q8×C181D411yescharacteristic
60C2×Q8×C181C2316yescomplemented
61C2×D4×C181—26no—
62C2×D4×C181—212no—
63C2×D4×C181D413yes—
64C2×D4×C181C2316yescomplemented
65C22×C4×C181—26no—
66C22×C4×C181C2313yes—
67C22×D181—23no—
68C22×D181—212no—
69C22×D181—86no—
70Dic362—21no—
71Dic362C2412yes—
72D724—212no—
73D724—23no—
74D724C2416yes—
75C22⋊D181—212no—
76C2×Dic181—21no—
77C4×D181—212no—
78#13089678C2418yes—
79C23×C181—26no—
80C23×C181C2×D413yes—
81Q8×C181—26no—
82Q8×C181C2×D414yes—
83Q8×C181C2414yes—
84D4×C181—24no—
85D4×C181—212no—
86D4×C181—26no—
87D4×C181C2×D4112yes—
88D4×C181C2414yes—
89C2×C4×C181—26no—
90C2×C4×C181—24no—
91C2×C4×C181—212no—
92C2×C4×C181C2×D416yes—
93C2×C4×C181C2×D413yes—
94C2×C4×C181C2411yescharacteristic
95C2×C4×C181C2416yes—
96C22×C181—24no—
97C22×C181—43no—
98C22×C181—86no—
99C22×C181C22×D416yes—
100C22×C181C22×D411yescharacteristic
101C2×D181—26no—
102C2×D181—46no—
103C2×D181—812no—
104C4×C181—24no—
105C4×C181C22×D411yescharacteristic
106C4×C181C22×D416yes—
107C4×C181C2511yescharacteristic
108Dic181—22no—
109C2×C181—21no—
110C2×C181—26no—
111C2×C181—44no—
112C2×C181C23×D411yescharacteristic
113D181—46no—
114C181#358511yescharacteristic, complemented, Sylow
115#3585—1811nomaximal, Sylow
116#585D18111yescharacteristic, complemented
117C2×H(5,𝔽2)—1811no—
118#577—1818no—
119#576—1818no—
120#575—1816no—
121C2×AGL(1,ℤ/8)—1816no—
122#568—1811no—
1232−1+4—21no—
1242−1+4C2×D18114yescomplemented
125H(5,𝔽2)—23no—
126H(5,𝔽2)—3624no—
127H(5,𝔽2)—1818no—
128H(5,𝔽2)C2×D18114yescomplemented
129C2×#41—24no—
130C2×#41—3621no—
131C2×#41—1813no—
132C2×#41C2×D18111yescharacteristic
133C2×#41C2×D18116yescomplemented
134C22×D4—3623no—
135C22×D4—1813no—
136#137—1814no—
137#137—18112no—
138AGL(1,ℤ/8)—18112no—
139AGL(1,ℤ/8)—18124no—
140AGL(1,ℤ/8)—18112no—
141#135—18124no—
142C2×SD16—18112no—
143C2×D8—18112no—
144#131—1818no—
145C2×M16—1816no—
146C24—3623no—
147#41—24no—
148#41—212no—
149#41—26no—
150#41—26no—
151#41—3628no—
152#41—3626no—
153#41—18112no—
154#41C22⋊D18114yes—
155#41C22×D18114yes—
156#41C22×D18114yes—
157#41C22×D181112yescomplemented
158C2×Q8—24no—
159C2×Q8—1811no—
160C2×Q8C22⋊D18111yescharacteristic
161C2×Q8C22×D18116yescomplemented
162C2×D4—26no—
163C2×D4—212no—
164C2×D4—3623no—
165C2×D4—36224no—
166C2×D4—3623no—
167C2×D4—3626no—
168C2×D4—18112no—
169C2×D4—1813no—
170C2×D4C22⋊D18113yes—
171C2×D4C22×D18116yescomplemented
172C22×C4—26no—
173C22×C4—3623no—
174C22×C4C22×D18113yes—
175Q16—1818no—
176SD16—18124no—
177SD16—1818no—
178D8—18124no—
179M16—1814no—
180M16—18112no—
181C2×C8—18112no—
182C23—26no—
183C23—36212no—
184C23—3623no—
185C23—14486no—
186C23C2×C22⋊D18113yes—
187Q8—26no—
188Q8—3621no—
189Q8—1812no—
190Q8C2×C22⋊D18114yes—
191Q8C23×D18114yes—
192D4—24no—
193D4—212no—
194D4—26no—
195D4—36212no—
196D4—3623no—
197D4—36212no—
198D4—1816no—
199D4C2×C22⋊D181112yes—
200D4C23×D18114yes—
201C2×C4—26no—
202C2×C4—24no—
203C2×C4—212no—
204C2×C4—3621no—
205C2×C4—36212no—
206C2×C4C2×C22⋊D18116yes—
207C2×C4C2×C22⋊D18113yes—
208C2×C4C23×D18116yes—
209C2×C4C23×D18111yescharacteristic
210C8—1818no—
211C22—24no—
212C22—43no—
213C22—86no—
214C22—3626no—
215C22—144812no—
216C22—7246no—
217C22C22×C22⋊D18116yes—
218C22C22×C22⋊D18111yescharacteristic
219C4—24no—
220C4—3622no—
221C4C22×C22⋊D18111yescharacteristic
222C4C22×C22⋊D18116yes—
223C4C24×D18111yescharacteristic
224C2—21no—
225C2—26no—
226C2—44no—
227C2—7246no—
228C2C23×C22⋊D18111yescharacteristic
subgroupG/H
centerC2C23×C22⋊D181
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
Fitting#585×C181C2
socleC2×C181C23×D4
hypercenter#585D181
hypocenterC181#3585
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×#41 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2semidirect
2. #26042087C2semidirect
3. #26042082C2semidirect
4. #26042077C2semidirect
5. C2×#19153841C2semidirect
6. #26042046C2semidirect
7. #585×C181C2semidirect
9. #19153862C22semidirect
11. C2×#15668449C22semidirect
12. C22×D724C22semidirect
14. #19153847C22semidirect
15. #19153846C22semidirect
16. #19153842C22semidirect
17. #19153841C22semidirect
18. #19153841C22semidirect
19. #19153836C22semidirect
20. C2×#15668436C22semidirect
21. C2×#15668434C22semidirect
22. #19153823C22semidirect
23. C2×#15668433C22semidirect
25. 2−1+4×C181C22semidirect
27. H(5,𝔽2)×C181C22semidirect
30. C2×#41×C181C22semidirect
43. #15668439C23semidirect
44. #15668437C23semidirect
45. #15668436C23semidirect
46. #15668434C23semidirect
48. #15668433C23semidirect
49. C2×#13089678C23semidirect
57. #41×C181C23semidirect
60. C2×Q8×C181C23semidirect
64. C2×D4×C181C23semidirect
114. C181#3585semidirect
116. #585D181semidirect
124. 2−1+4C2×D181semidirect
128. H(5,𝔽2)C2×D181semidirect
133. C2×#41C2×D181semidirect
157. #41C22×D181semidirect
161. C2×Q8C22×D181semidirect
171. C2×D4C22×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538622
#260420872
#260420822
#260420772
C2×#191538412
#260420462
#585×C1812
#3585181