Finite Group Databasebeta

← #37548502

Subgroups — #37548502

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
1#26042115C211yesmaximal, characteristic, complemented
2C2×#19153862C211yesmaximal, characteristic, complemented
3#26042091C218yesmaximal, complemented
4#26042084C218yesmaximal, complemented
5#26042079C216yesmaximal, complemented
6C2×#19153844C216yesmaximal, complemented
7#568×C181C211yesmaximal, characteristic, complemented
8#19153864—21no—
9#19153863C2214yescomplemented
10#19153862—24no—
11#19153862C2218yescomplemented
12#19153862C2214yescomplemented
13#19153861—23no—
14C2×#15668450—21no—
15C2×#15668449C2213yescomplemented
16C2×#15668449C2216yescomplemented
17#41×D181—24no—
18C22×D724C2213yescomplemented
19C2×D4×D181—23no—
20#19153849C2214yescomplemented
21#19153849C22112yescomplemented
22#19153844C22112yescomplemented
23#19153844C22124yescomplemented
24#19153844C22112yescomplemented
25#19153838C22124yescomplemented
26C2×#15668438C22112yescomplemented
27C2×D1448C22112yescomplemented
28C2×#41×C181C2211yescharacteristic
29#131×C181C2218yescomplemented
30C2×M16×C181C2216yescomplemented
31C23×D181—23no—
32#15668450—24no—
33#15668450—28no—
34#15668449—26no—
35#15668449—26no—
36#15668449—26no—
37#15668449C23112yes—
38#15668449C2314yes—
39#15668449C23112yescomplemented
40#15668448—212no—
41C2×Dic362C2311yescharacteristic
42C2×Dic362C2316yescomplemented
43Q8×D181—24no—
44C2×D724—23no—
45C2×D724—26no—
46C2×D724C23112yes—
47C2×D724C2313yes—
48C2×D724C2316yescomplemented
49C2×C22⋊D181—23no—
50C2×C22⋊D181—26no—
51D4×D181—224no—
52D4×D181—212no—
53C2×C4×D181—23no—
54C2×C4×D181—26no—
55Dic724C2318yescomplemented
56#15668438C23124yescomplemented
57#15668438C2318yescomplemented
58D1448C23124yescomplemented
59#41×C181D414yes—
60#41×C181C2314yes—
61C2×Q8×C181D411yescharacteristic
62C2×D4×C181D413yes—
63C22×C4×C181C2313yes—
64M16×C181C2314yes—
65M16×C181C23112yescomplemented
66C2×C8×C181C23112yescomplemented
67C22×D181—23no—
68C22×D181—26no—
69C22×D181—212no—
70C22×D181—86no—
71Dic362—26no—
72Dic362—21no—
73Dic362C2414yes—
74Dic362C2412yes—
75D724—212no—
76D724—26no—
77D724—23no—
78D724C2414yes—
79D724C2416yes—
80C22⋊D181—212no—
81C22⋊D181—212no—
82C22⋊D181—24no—
83C2×Dic181—21no—
84C2×Dic181—26no—
85C4×D181—24no—
86C4×D181—212no—
87C4×D181—212no—
88C23×C181C2×D413yes—
89Q8×C181C2×D414yes—
90D4×C181C2×D4112yes—
91C2×C4×C181C2×D413yes—
92C2×C4×C181C2×D416yes—
93C2×C4×C181C2416yes—
94C2×C4×C181C2411yescharacteristic
95C8×C181C2418yes—
96C22×C181—43no—
97C22×C181C22×D411yescharacteristic
98C22×C181C22×D416yes—
99C2×D181—24no—
100C2×D181—26no—
101C2×D181—46no—
102C2×D181—86no—
103C2×D181—812no—
104C4×C181C22×D411yescharacteristic
105C4×C181C22×D416yes—
106C4×C181C2511yescharacteristic
107Dic181—22no—
108Dic181—24no—
109C2×C181—21no—
110C2×C181—26no—
111C2×C181C23×D411yescharacteristic
112D181—46no—
113D181—44no—
114C181#358511yescharacteristic, complemented, Sylow
115#3585—1811nomaximal, Sylow
116#585—1811no—
117C2×H(5,𝔽2)—1811no—
118#577—1818no—
119#576—1818no—
120#575—1816no—
121C2×AGL(1,ℤ/8)—1816no—
122#568D18111yescharacteristic, complemented
1232−1+4—3621no—
1242−1+4—1814no—
125H(5,𝔽2)—3624no—
126H(5,𝔽2)—3623no—
127H(5,𝔽2)—1814no—
128H(5,𝔽2)—1818no—
129C2×#41—3624no—
130C2×#41—3621no—
131C2×#41—1816no—
132C2×#41—1813no—
133C2×#41C2×D18111yescharacteristic
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#131C2×D18118yescomplemented
145C2×M16C2×D18116yescomplemented
146C24—3623no—
147#41—3628no—
148#41—3624no—
149#41—36212no—
150#41—3626no—
151#41—3626no—
152#41—3626no—
153#41—18112no—
154#41—18112no—
155#41—1814no—
156#41D72414yes—
157#41C22×D18114yes—
158C2×Q8—3624no—
159C2×Q8—1816no—
160C2×Q8—1811no—
161C2×Q8D72411yescharacteristic
162C2×D4—3623no—
163C2×D4—3626no—
164C2×D4—36224no—
165C2×D4—36212no—
166C2×D4—3623no—
167C2×D4—3626no—
168C2×D4—1816no—
169C2×D4—18112no—
170C2×D4—1813no—
171C2×D4D72413yes—
172C22×C4—3623no—
173C22×C4—3626no—
174C22×C4C22×D18113yes—
175Q16—1818no—
176SD16—18124no—
177SD16—1818no—
178D8—18124no—
179M16C22×D18114yes—
180M16C22×D181112yescomplemented
181C2×C8C22×D181112yescomplemented
182C23—3626no—
183C23—36212no—
184C23—3623no—
185C23—14486no—
186C23C2×D72413yes—
187Q8—3626no—
188Q8—3621no—
189Q8—1814no—
190Q8—1812no—
191Q8C2×D72414yes—
192D4—36212no—
193D4—3624no—
194D4—36212no—
195D4—3623no—
196D4—36212no—
197D4—3626no—
198D4—1814no—
199D4—1816no—
200D4C2×D724112yes—
201C2×C4—3621no—
202C2×C4—3626no—
203C2×C4—3624no—
204C2×C4—36212no—
205C2×C4—36212no—
206C2×C4C2×D72413yes—
207C2×C4C2×D72416yes—
208C2×C4C23×D18116yes—
209C2×C4C23×D18111yescharacteristic
210C8C23×D18118yes—
211C22—43no—
212C22—3624no—
213C22—3626no—
214C22—144812no—
215C22—14486no—
216C22—7246no—
217C22C22×D72411yescharacteristic
218C22C22×D72416yes—
219C4—3622no—
220C4—3624no—
221C4C22×D72411yescharacteristic
222C4C22×D72416yes—
223C4C24×D18111yescharacteristic
224C2—21no—
225C2—26no—
226C2—7246no—
227C2—7244no—
228C2C23×D72411yescharacteristic
subgroupG/H
centerC2C23×D724
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
Fitting#568×C181C2
socleC2×C181C23×D4
hypercenter#568D181
hypocenterC181#3585
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #26042115C2semidirect
2. C2×#19153862C2semidirect
3. #26042091C2semidirect
4. #26042084C2semidirect
5. #26042079C2semidirect
6. C2×#19153844C2semidirect
7. #568×C181C2semidirect
9. #19153863C22semidirect
11. #19153862C22semidirect
12. #19153862C22semidirect
15. C2×#15668449C22semidirect
16. C2×#15668449C22semidirect
18. C22×D724C22semidirect
20. #19153849C22semidirect
21. #19153849C22semidirect
22. #19153844C22semidirect
23. #19153844C22semidirect
24. #19153844C22semidirect
25. #19153838C22semidirect
26. C2×#15668438C22semidirect
27. C2×D1448C22semidirect
29. #131×C181C22semidirect
30. C2×M16×C181C22semidirect
39. #15668449C23semidirect
42. C2×Dic362C23semidirect
48. C2×D724C23semidirect
55. Dic724C23semidirect
56. #15668438C23semidirect
57. #15668438C23semidirect
58. D1448C23semidirect
65. M16×C181C23semidirect
66. C2×C8×C181C23semidirect
114. C181#3585semidirect
122. #568D181semidirect
144. #131C2×D181semidirect
145. C2×M16C2×D181semidirect
180. M16C22×D181semidirect
181. C2×C8C22×D181semidirect

Maximal subgroups

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

subgroupindex
#260421152
C2×#191538622
#260420912
#260420842
#260420792
C2×#191538442
#568×C1812
#3585181