Finite Group Databasebeta

← #37545521

Subgroups — #37545521

Order 23168.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#26042084C212yesmaximal, complemented
2#26042081C212yesmaximal, complemented
3#26041656C211yesmaximal, characteristic, complemented
4#26041550C212yesmaximal, complemented
5#26041549C212yesmaximal, complemented
6#26041448C214yesmaximal, complemented
7#26041397C211yesmaximal, characteristic, complemented
8#443×C181C211yesmaximal, characteristic, complemented
9#19153862C2214yescomplemented
10#19153844—24no—
11#19153844C2214yescomplemented
12#19153843—24no—
13#19153841C2214yescomplemented
14#19153839—22no—
15#19153838—22no—
16#19153838C2211yescharacteristic
17C2×D1448—22no—
18C2×D1448C2212yescomplemented
19C2×D1448C2212yescomplemented
20D8×D181—22no—
21#19153822C2212yescomplemented
22#19153813C2212yescomplemented
23#19153752C2211yescharacteristic, complemented
24#19153744C2212yescomplemented
25#19153732C2214yescomplemented
26#135×C181C2211yescharacteristic
27#131×C181C2212yescomplemented
28#108×C181C2211yescharacteristic, complemented
29#104×C181C2212yescomplemented
30C4×C8×C181C2211yescharacteristic, complemented
31#15668450—24no—
32#15668449—24no—
33#15668449C2312yes—
34C2×D724—24no—
35C2×D724—42no—
36C2×D724C2314yes—
37D4×D181—24no—
38D4×D181—44no—
39Dic724D411yescharacteristic, complemented
40#15668438—24no—
41#15668438D412yescomplemented
42#15668436—24no—
43D1448—24no—
44D1448—22no—
45D1448—22no—
46D1448—22no—
47D1448—22no—
48D1448D411yescharacteristic, complemented
49#15668434—24no—
50#15668433C2312yes—
51#15668432—22no—
52C8×D181—22no—
53#41×C181C2312yes—
54Q16×C181D411yescharacteristic, complemented
55SD16×C181D412yescomplemented
56D8×C181D411yescharacteristic, complemented
57M16×C181—22no—
58M16×C181C2312yes—
59C2×C8×C181—22no—
60C2×C8×C181C2311yescharacteristic
61C2×C8×C181C2311yescharacteristic
62C42×C181C2311yescharacteristic
63C22×D181—24no—
64C22×D181—44no—
65Dic362C2×D412yes—
66D724—24no—
67D724—24no—
68D724—44no—
69D724—84no—
70D724C2×D412yes—
71C22⋊D181—22no—
72C22⋊D181—44no—
73C4×D181—22no—
74C4×D181—44no—
75#13089678—22no—
76Q8×C181C2×D412yes—
77D4×C181—22no—
78D4×C181C2×D412yes—
79C2×C4×C181—21no—
80C2×C4×C181—22no—
81C2×C4×C181C2411yescharacteristic
82C8×C181—22no—
83C8×C181C2×D411yescharacteristic
84C8×C181C2×D411yescharacteristic
85C8×C181C2×D411yescharacteristic
86C8×C181C2×D411yescharacteristic
87C22×C181—22no—
88C22×C181H(5,𝔽2)11yescharacteristic
89C2×D181—22no—
90C2×D181—44no—
91C2×D181—84no—
92C2×D181—164no—
93C4×C181—22no—
94C4×C181—22no—
95C4×C181C22×D411yescharacteristic
96C4×C181C22×D411yescharacteristic
97Dic181—22no—
98C2×C181—21no—
99C2×C181—42no—
100C2×C181D4211yescharacteristic
101D181—42no—
102D181—84no—
103C181#329211yescharacteristic, complemented, Sylow
104#3292—1811nomaximal, Sylow
105#576—1812no—
106#576—1812no—
107#493—1811no—
108#472—1812no—
109#472—1812no—
110#453—1814no—
111#443—1811no—
112#443D18111yescharacteristic, complemented
113H(5,𝔽2)—1814no—
114AGL(1,ℤ/8)—3624no—
115AGL(1,ℤ/8)—3624no—
116AGL(1,ℤ/8)—1814no—
117AGL(1,ℤ/8)—1814no—
118#135—3622no—
119#135—3622no—
120#135—1811no—
121#135C2×D18111yescharacteristic
122C2×D8—3622no—
123C2×D8—3622no—
124C2×D8—1812no—
125C2×D8—1812no—
126#131—1812no—
127#131C2×D18112yescomplemented
128#127—1812no—
129#108—1811no—
130#108C2×D18111yescharacteristic, complemented
131#104—1812no—
132#104C2×D18112yescomplemented
133#100—1814no—
134C4×C8C2×D18111yescharacteristic, complemented
135#41—3624no—
136#41—3624no—
137#41—1812no—
138#41C22×D18112yes—
139C2×D4—3624no—
140C2×D4—3624no—
141C2×D4—1814no—
142C2×D4—7242no—
143C2×D4—7244no—
144Q16—1811no—
145Q16D72411yescharacteristic, complemented
146SD16—3624no—
147SD16—3624no—
148SD16—1812no—
149SD16D72412yescomplemented
150D8—3624no—
151D8—3624no—
152D8—3622no—
153D8—3622no—
154D8—3622no—
155D8—3622no—
156D8—1811no—
157D8D72411yescharacteristic, complemented
158M16—22no—
159M16—3622no—
160M16—1812no—
161M16C22×D18112yes—
162C2×C8—22no—
163C2×C8—3622no—
164C2×C8C22×D18111yescharacteristic
165C2×C8C22×D18111yescharacteristic
166C42C22×D18111yescharacteristic
167C23—3624no—
168C23—7244no—
169Q8—1812no—
170Q8C2×D72412yes—
171D4—22no—
172D4—3622no—
173D4—3624no—
174D4—3624no—
175D4—14484no—
176D4—1812no—
177D4—7244no—
178D4—7244no—
179D4C2×D72412yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—3622no—
183C2×C4—7244no—
184C2×C4C23×D18111yescharacteristic
185C8—22no—
186C8—3622no—
187C8D4×D18111yescharacteristic
188C8D4×D18111yescharacteristic
189C8C2×D72411yescharacteristic
190C8C2×D72411yescharacteristic
191C22—22no—
192C22—28964no—
193C22—3622no—
194C22—14484no—
195C22—7244no—
196C22#1915386211yescharacteristic
197C4—22no—
198C4—22no—
199C4—3622no—
200C4C2×D4×D18111yescharacteristic
201C4C22×D72411yescharacteristic
202C2—21no—
203C2—42no—
204C2—14484no—
205C2—7242no—
206C2D4×D72411yescharacteristic
subgroupG/H
centerC2D4×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#443×C181C2
socleC2×C181D42
hypercenter#443D181
hypocenterC181#3292
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #26042084C2semidirect
2. #26042081C2semidirect
3. #26041656C2semidirect
4. #26041550C2semidirect
5. #26041549C2semidirect
6. #26041448C2semidirect
7. #26041397C2semidirect
8. #443×C181C2semidirect
9. #19153862C22semidirect
11. #19153844C22semidirect
13. #19153841C22semidirect
18. C2×D1448C22semidirect
19. C2×D1448C22semidirect
21. #19153822C22semidirect
22. #19153813C22semidirect
23. #19153752C22semidirect
24. #19153744C22semidirect
25. #19153732C22semidirect
27. #131×C181C22semidirect
28. #108×C181C22semidirect
29. #104×C181C22semidirect
30. C4×C8×C181C22semidirect
39. Dic724D4semidirect
41. #15668438D4semidirect
48. D1448D4semidirect
54. Q16×C181D4semidirect
55. SD16×C181D4semidirect
56. D8×C181D4semidirect
103. C181#3292semidirect
112. #443D181semidirect
127. #131C2×D181semidirect
130. #108C2×D181semidirect
132. #104C2×D181semidirect
134. C4×C8C2×D181semidirect
145. Q16D724semidirect
149. SD16D724semidirect
157. D8D724semidirect

Maximal subgroups

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

subgroupindex
#260420842
#260420812
#260416562
#260415502
#260415492
#260414482
#260413972
#443×C1812
#3292181