Finite Group Databasebeta

← #37545518

Subgroups — #37545518

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#26042082C212yesmaximal, complemented
2#26042081C211yesmaximal, characteristic, complemented
3#26041654C211yesmaximal, characteristic, complemented
4#26041549C211yesmaximal, characteristic, complemented
5#26041547C212yesmaximal, complemented
6#26041546C211yesmaximal, characteristic, complemented
7#26041447C212yesmaximal, complemented
8#26041445C212yesmaximal, complemented
9#26041396C211yesmaximal, characteristic, complemented
10#26041394C211yesmaximal, characteristic, complemented
11#576×C181C211yesmaximal, characteristic, complemented
12#19153862C2212yescomplemented
13#19153843—22no—
14#19153842—22no—
15#19153842C2212yescomplemented
16#19153841—22no—
17#19153841C2212yescomplemented
18#19153841C2212yescomplemented
19#19153839—21no—
20#19153836—22no—
21#19153836C2211yescharacteristic
22C2×D1448C2211yescharacteristic, complemented
23C2×#15668434—22no—
24C2×#15668434C2212yescomplemented
25D8×D181—21no—
26#19153823C2212yescomplemented
27#19153822C2211yescharacteristic, complemented
28#19153812C2212yescomplemented
29#19153752C2211yescharacteristic, complemented
30#19153751C2211yescharacteristic, complemented
31#19153743C2211yescharacteristic, complemented
32#19153743C2212yescomplemented
33#19153742C2211yescharacteristic, complemented
34#19153733C2212yescomplemented
35#19153732C2212yescomplemented
36C8×Dic181C2211yescharacteristic, complemented
37H(5,𝔽2)×C181C2212yescomplemented
38AGL(1,ℤ/8)×C181—22no—
39AGL(1,ℤ/8)×C181C2212yescomplemented
40#135×C181—21no—
41#135×C181C2211yescharacteristic
42C2×D8×C181—21no—
43C2×D8×C181C2211yescharacteristic, complemented
44#131×C181C2211yescharacteristic, complemented
45#15668450—22no—
46#15668449—22no—
47#15668449C2311yescharacteristic
48C2×D724—22no—
49C2×D724C2312yes—
50C2×C22⋊D181—42no—
51D4×D181—22no—
52D4×D181—42no—
53#15668439D411yescharacteristic, complemented
54#15668437—22no—
55#15668437D411yescharacteristic, complemented
56#15668436—22no—
57#15668436—22no—
58#15668436D411yescharacteristic, complemented
59D1448—21no—
60D1448—21no—
61#15668434—22no—
62#15668434—22no—
63#15668434—22no—
64#15668434—22no—
65#15668434—22no—
66#15668434D411yescharacteristic, complemented
67#15668433—22no—
68#15668433C2312yes—
69#15668433C2311yescharacteristic
70#15668432—21no—
71C2×#13089678—22no—
72C2×#13089678C2311yescharacteristic
73C8×D181—21no—
74C4×Dic181C2311yescharacteristic
75#41×C181—22no—
76#41×C181—22no—
77#41×C181C2311yescharacteristic
78#41×C181C2312yes—
79C2×D4×C181—22no—
80C2×D4×C181—22no—
81C2×D4×C181—42no—
82C2×D4×C181C2312yes—
83Q16×C181D411yescharacteristic, complemented
84SD16×C181—22no—
85SD16×C181D412yescomplemented
86D8×C181—22no—
87D8×C181—21no—
88D8×C181—21no—
89D8×C181D411yescharacteristic, complemented
90M16×C181—21no—
91M16×C181C2311yescharacteristic
92C2×C8×C181—21no—
93C2×C8×C181C2311yescharacteristic
94C22×D181—22no—
95C22×D181—42no—
96Dic362C2×D411yescharacteristic
97D724—22no—
98D724—22no—
99D724—42no—
100D724C2×D411yescharacteristic
101C22⋊D181—21no—
102C22⋊D181—42no—
103C22⋊D181—84no—
104C2×Dic181—21no—
105C4×D181—21no—
106C4×D181—42no—
107#13089678—21no—
108#13089678—22no—
109#13089678C2×D411yescharacteristic
110#13089678C2×D411yescharacteristic
111C23×C181—22no—
112C23×C181—42no—
113Q8×C181C2×D411yescharacteristic
114Q8×C181C2×D412yes—
115D4×C181—21no—
116D4×C181—22no—
117D4×C181—22no—
118D4×C181—22no—
119D4×C181—42no—
120D4×C181—42no—
121D4×C181C2×D411yescharacteristic
122D4×C181C2×D412yes—
123C2×C4×C181—21no—
124C2×C4×C181—22no—
125C2×C4×C181—42no—
126C2×C4×C181C2411yescharacteristic
127C8×C181—21no—
128C8×C181C2×D411yescharacteristic
129C8×C181C2×D411yescharacteristic
130C22×C181—21no—
131C22×C181—22no—
132C22×C181—42no—
133C22×C181—82no—
134C22×C181—162no—
135C22×C181H(5,𝔽2)11yescharacteristic
136C2×D181—21no—
137C2×D181—42no—
138C2×D181—82no—
139C2×D181—162no—
140C4×C181—21no—
141C4×C181—22no—
142C4×C181C22×D411yescharacteristic
143C4×C181C22×D411yescharacteristic
144Dic181—22no—
145Dic181—21no—
146C2×C181—21no—
147C2×C181—42no—
148C2×C181—41no—
149C2×C181—82no—
150C2×C181D4211yescharacteristic
151D181—41no—
152D181—82no—
153C181#329211yescharacteristic, complemented, Sylow
154#3292—1811nomaximal, Sylow
155#576—1812no—
156#576—1811no—
157#576D18111yescharacteristic, complemented
158#493—1811no—
159#472—1812no—
160#472—1811no—
161#472—1811no—
162#453—1812no—
163#453—1812no—
164#443—1811no—
165#443—1811no—
166H(5,𝔽2)—1812no—
167H(5,𝔽2)C2×D18112yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—3622no—
170AGL(1,ℤ/8)—3622no—
171AGL(1,ℤ/8)—3622no—
172AGL(1,ℤ/8)—1812no—
173AGL(1,ℤ/8)—1812no—
174AGL(1,ℤ/8)—1812no—
175AGL(1,ℤ/8)C2×D18112yescomplemented
176#135—21no—
177#135—3622no—
178#135—3621no—
179#135—1811no—
180#135C2×D18111yescharacteristic
181C2×D8—21no—
182C2×D8—3622no—
183C2×D8—3621no—
184C2×D8—1812no—
185C2×D8—1811no—
186C2×D8C2×D18111yescharacteristic, complemented
187#131—1812no—
188#131—1811no—
189#131C2×D18111yescharacteristic, complemented
190#127—1812no—
191#108—1811no—
192#108—1811no—
193#104—1811no—
194#104—1812no—
195#104—1811no—
196#100—1812no—
197#100—1812no—
198C4×C8—1811no—
199#41—22no—
200#41—22no—
201#41—3622no—
202#41—3622no—
203#41—1811no—
204#41C22×D18111yescharacteristic
205#41C22×D18112yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—3622no—
210C2×D4—3622no—
211C2×D4—1812no—
212C2×D4—7242no—
213C2×D4—7242no—
214C2×D4C22×D18112yes—
215Q16—1811no—
216Q16C22⋊D18111yescharacteristic, complemented
217SD16—22no—
218SD16—3622no—
219SD16—3622no—
220SD16—3622no—
221SD16—1811no—
222SD16—1811no—
223SD16C22⋊D18112yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3622no—
228D8—3622no—
229D8—3622no—
230D8—3622no—
231D8—3622no—
232D8—3621no—
233D8—3621no—
234D8—1811no—
235D8C22⋊D18111yescharacteristic, complemented
236M16—21no—
237M16—3622no—
238M16—3621no—
239M16—1812no—
240M16—1811no—
241M16C22×D18111yescharacteristic
242C2×C8—21no—
243C2×C8—3622no—
244C2×C8—3621no—
245C2×C8—1811no—
246C2×C8C22×D18111yescharacteristic
247C42—1811no—
248C23—22no—
249C23—42no—
250C23—3622no—
251C23—7242no—
252Q8—1811no—
253Q8D4×D18111yescharacteristic
254Q8C2×C22⋊D18112yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—3621no—
262D4—3622no—
263D4—3622no—
264D4—14484no—
265D4—1811no—
266D4—7242no—
267D4—7242no—
268D4D4×D18111yescharacteristic
269D4C2×C22⋊D18112yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—3621no—
274C2×C4—3621no—
275C2×C4—7242no—
276C2×C4C23×D18111yescharacteristic
277C8—21no—
278C8—3622no—
279C8—3621no—
280C8—1811no—
281C8—1811no—
282C8C2×C22⋊D18111yescharacteristic
283C8C2×C22⋊D18111yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—28962no—
290C22—3621no—
291C22—14482no—
292C22—7242no—
293C22#1915386111yescharacteristic
294C4—21no—
295C4—22no—
296C4—3622no—
297C4—3621no—
298C4C2×D4×D18111yescharacteristic
299C4C22×C22⋊D18111yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—14482no—
305C2—7241no—
306C2D4×C22⋊D18111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#576×C181C2
socleC2×C181D42
hypercenter#576D181
hypocenterC181#3292
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #26042082C2semidirect
2. #26042081C2semidirect
3. #26041654C2semidirect
4. #26041549C2semidirect
5. #26041547C2semidirect
6. #26041546C2semidirect
7. #26041447C2semidirect
8. #26041445C2semidirect
9. #26041396C2semidirect
10. #26041394C2semidirect
11. #576×C181C2semidirect
12. #19153862C22semidirect
15. #19153842C22semidirect
17. #19153841C22semidirect
18. #19153841C22semidirect
22. C2×D1448C22semidirect
24. C2×#15668434C22semidirect
26. #19153823C22semidirect
27. #19153822C22semidirect
28. #19153812C22semidirect
29. #19153752C22semidirect
30. #19153751C22semidirect
31. #19153743C22semidirect
32. #19153743C22semidirect
33. #19153742C22semidirect
34. #19153733C22semidirect
35. #19153732C22semidirect
36. C8×Dic181C22semidirect
37. H(5,𝔽2)×C181C22semidirect
39. AGL(1,ℤ/8)×C181C22semidirect
43. C2×D8×C181C22semidirect
44. #131×C181C22semidirect
53. #15668439D4semidirect
55. #15668437D4semidirect
58. #15668436D4semidirect
66. #15668434D4semidirect
83. Q16×C181D4semidirect
85. SD16×C181D4semidirect
89. D8×C181D4semidirect
153. C181#3292semidirect
157. #576D181semidirect
167. H(5,𝔽2)C2×D181semidirect
175. AGL(1,ℤ/8)C2×D181semidirect
186. C2×D8C2×D181semidirect
189. #131C2×D181semidirect
216. Q16C22⋊D181semidirect
223. SD16C22⋊D181semidirect
235. D8C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
#260420822
#260420812
#260416542
#260415492
#260415472
#260415462
#260414472
#260414452
#260413962
#260413942
#576×C1812
#3292181