Finite Group Databasebeta

← #37545519

Subgroups — #37545519

Order 23168.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#26042084C211yesmaximal, characteristic, complemented
2#26042083C212yesmaximal, complemented
3#26042080C211yesmaximal, characteristic, complemented
4#26041654C211yesmaximal, characteristic, complemented
5#26041550C211yesmaximal, characteristic, complemented
6#26041548C212yesmaximal, complemented
7#26041449C212yesmaximal, complemented
8#26041445C212yesmaximal, complemented
9#26041395C211yesmaximal, characteristic, complemented
10#26041394C211yesmaximal, characteristic, complemented
11#472×C181C211yesmaximal, characteristic, complemented
12#19153862C2212yescomplemented
13#19153861C2212yescomplemented
14#19153844—22no—
15#19153844C2212yescomplemented
16#19153844C2212yescomplemented
17#19153843—22no—
18#19153842C2212yescomplemented
19#19153840—22no—
20#19153840—22no—
21#19153838—21no—
22#19153838C2211yescharacteristic
23#19153837—22no—
24#19153836C2211yescharacteristic
25#19153835—21no—
26C2×D1448—21no—
27C2×D1448C2211yescharacteristic, complemented
28C2×#15668434C2212yescomplemented
29D8×D181—21no—
30D8×D181—22no—
31#19153824C2212yescomplemented
32#19153822C2211yescharacteristic, complemented
33#19153812C2212yescomplemented
34#19153751C2211yescharacteristic, complemented
35#19153743C2211yescharacteristic, complemented
36#19153742C2211yescharacteristic, complemented
37#19153742C2212yescomplemented
38#19153732C2212yescomplemented
39C8×Dic181C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C181C2212yescomplemented
41C2×D8×C181C2211yescharacteristic, complemented
42#131×C181C2211yescharacteristic, complemented
43#108×C181C2211yescharacteristic, complemented
44#100×C181C2212yescomplemented
45#15668450—22no—
46#15668449—22no—
47#15668449C2311yescharacteristic
48#15668449C2312yes—
49#15668448—22no—
50#15668448—22no—
51C2×D724—22no—
52C2×D724C2312yes—
53C2×C22⋊D181—42no—
54C2×C22⋊D181—42no—
55D4×D181—22no—
56D4×D181—22no—
57D4×D181—22no—
58D4×D181—42no—
59Dic724D411yescharacteristic, complemented
60#15668439D411yescharacteristic, complemented
61#15668438—22no—
62#15668438—22no—
63#15668438D412yescomplemented
64#15668437—22no—
65#15668437D411yescharacteristic, complemented
66#15668436D411yescharacteristic, complemented
67D1448—22no—
68D1448—22no—
69D1448—21no—
70D1448—21no—
71D1448D411yescharacteristic, complemented
72#15668434—22no—
73#15668434—22no—
74#15668434—22no—
75#15668434D411yescharacteristic, complemented
76#15668433C2311yescharacteristic
77#15668432—22no—
78#15668432—21no—
79C2×#13089678C2311yescharacteristic
80C8×D181—22no—
81C8×D181—21no—
82C4×Dic181C2311yescharacteristic
83#41×C181C2311yescharacteristic
84C2×D4×C181C2312yes—
85SD16×C181—22no—
86D8×C181—22no—
87D8×C181—21no—
88D8×C181—21no—
89M16×C181—21no—
90M16×C181C2311yescharacteristic
91M16×C181C2312yes—
92C2×C8×C181—21no—
93C2×C8×C181C2311yescharacteristic
94C22×D181—22no—
95C22×D181—42no—
96C22×D181—42no—
97Dic362C2×D412yes—
98Dic362C2×D411yescharacteristic
99D724—22no—
100D724—22no—
101D724—42no—
102D724C2×D412yes—
103D724C2×D411yescharacteristic
104C22⋊D181—22no—
105C22⋊D181—21no—
106C22⋊D181—42no—
107C22⋊D181—42no—
108C22⋊D181—42no—
109C22⋊D181—84no—
110C2×Dic181—21no—
111C2×Dic181—42no—
112C4×D181—21no—
113C4×D181—22no—
114C4×D181—42no—
115#13089678—21no—
116#13089678C2×D411yescharacteristic
117#13089678C2×D411yescharacteristic
118C23×C181—22no—
119Q8×C181C2×D411yescharacteristic
120D4×C181—21no—
121D4×C181—22no—
122D4×C181—22no—
123D4×C181C2×D411yescharacteristic
124C2×C4×C181—21no—
125C2×C4×C181C2411yescharacteristic
126C8×C181—21no—
127C8×C181—22no—
128C8×C181C2×D411yescharacteristic
129C8×C181C2×D411yescharacteristic
130C22×C181—21no—
131C22×C181—42no—
132C22×C181—82no—
133C22×C181H(5,𝔽2)11yescharacteristic
134C2×D181—21no—
135C2×D181—22no—
136C2×D181—42no—
137C2×D181—82no—
138C2×D181—162no—
139C2×D181—162no—
140C4×C181—21no—
141C4×C181C22×D411yescharacteristic
142C4×C181C22×D411yescharacteristic
143Dic181—22no—
144Dic181—22no—
145Dic181—21no—
146C2×C181—21no—
147C2×C181—41no—
148C2×C181—82no—
149C2×C181D4211yescharacteristic
150D181—41no—
151D181—42no—
152D181—82no—
153C181#329211yescharacteristic, complemented, Sylow
154#3292—1811nomaximal, Sylow
155#576—1812no—
156#576—1811no—
157#576—1811no—
158#493—1811no—
159#472—1811no—
160#472—1812no—
161#472D18111yescharacteristic, complemented
162#453—1812no—
163#453—1812no—
164#443—1811no—
165#443—1811no—
166H(5,𝔽2)—1812no—
167H(5,𝔽2)—1812no—
168AGL(1,ℤ/8)—3622no—
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—3621no—
177#135—3621no—
178#135—3622no—
179#135—1811no—
180#135—1811no—
181C2×D8—3621no—
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#108C2×D18111yescharacteristic, complemented
193#104—1811no—
194#104—1812no—
195#104—1811no—
196#100—1812no—
197#100C2×D18112yescomplemented
198C4×C8—1811no—
199#41—3622no—
200#41—3622no—
201#41—3622no—
202#41—3622no—
203#41—1812no—
204#41—1811no—
205#41C22×D18111yescharacteristic
206C2×D4—3622no—
207C2×D4—3622no—
208C2×D4—3622no—
209C2×D4—3622no—
210C2×D4—1812no—
211C2×D4—7242no—
212C2×D4—7242no—
213C2×D4—7242no—
214C2×D4C22×D18112yes—
215Q16—1811no—
216Q16—1811no—
217SD16—22no—
218SD16—3622no—
219SD16—3622no—
220SD16—3622no—
221SD16—1811no—
222SD16—1811no—
223SD16—1812no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3622no—
228D8—3622no—
229D8—3622no—
230D8—3622no—
231D8—3622no—
232D8—3621no—
233D8—3621no—
234D8—1811no—
235D8—1811no—
236M16—21no—
237M16—3622no—
238M16—3621no—
239M16—1811no—
240M16C22×D18112yes—
241M16C22×D18111yescharacteristic
242C2×C8—21no—
243C2×C8—3622no—
244C2×C8—3621no—
245C2×C8—1811no—
246C2×C8C22×D18111yescharacteristic
247C42—1811no—
248C23—22no—
249C23—3622no—
250C23—7242no—
251C23—7242no—
252Q8—1811no—
253Q8—1812no—
254Q8D4×D18111yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—3622no—
259D4—3621no—
260D4—3622no—
261D4—3622no—
262D4—14484no—
263D4—1811no—
264D4—1812no—
265D4—7242no—
266D4—7242no—
267D4—7242no—
268D4—7242no—
269D4D4×D18111yescharacteristic
270C2×C4—21no—
271C2×C4—3621no—
272C2×C4—3622no—
273C2×C4—3621no—
274C2×C4—7242no—
275C2×C4—7242no—
276C2×C4C23×D18111yescharacteristic
277C8—21no—
278C8—22no—
279C8—3621no—
280C8—1811no—
281C8—1811no—
282C8D4×D18111yescharacteristic
283C8D4×D18111yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—28962no—
288C22—28962no—
289C22—3622no—
290C22—3621no—
291C22—14482no—
292C22—7242no—
293C22#1915386211yescharacteristic
294C4—21no—
295C4—3622no—
296C4—3622no—
297C4—3621no—
298C4C2×D4×D18111yescharacteristic
299C4C2×D4×D18111yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—14482no—
304C2—7242no—
305C2—7241no—
306C2#2604191311yescharacteristic
subgroupG/H
centerC2#26041913
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#472×C181C2
socleC2×C181D42
hypercenter#472D181
hypocenterC181#3292
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #26042084C2semidirect
2. #26042083C2semidirect
3. #26042080C2semidirect
4. #26041654C2semidirect
5. #26041550C2semidirect
6. #26041548C2semidirect
7. #26041449C2semidirect
8. #26041445C2semidirect
9. #26041395C2semidirect
10. #26041394C2semidirect
11. #472×C181C2semidirect
12. #19153862C22semidirect
13. #19153861C22semidirect
15. #19153844C22semidirect
16. #19153844C22semidirect
18. #19153842C22semidirect
27. C2×D1448C22semidirect
28. C2×#15668434C22semidirect
31. #19153824C22semidirect
32. #19153822C22semidirect
33. #19153812C22semidirect
34. #19153751C22semidirect
35. #19153743C22semidirect
36. #19153742C22semidirect
37. #19153742C22semidirect
38. #19153732C22semidirect
39. C8×Dic181C22semidirect
40. AGL(1,ℤ/8)×C181C22semidirect
41. C2×D8×C181C22semidirect
42. #131×C181C22semidirect
43. #108×C181C22semidirect
44. #100×C181C22semidirect
59. Dic724D4semidirect
60. #15668439D4semidirect
63. #15668438D4semidirect
65. #15668437D4semidirect
66. #15668436D4semidirect
71. D1448D4semidirect
75. #15668434D4semidirect
153. C181#3292semidirect
161. #472D181semidirect
175. AGL(1,ℤ/8)C2×D181semidirect
186. C2×D8C2×D181semidirect
189. #131C2×D181semidirect
192. #108C2×D181semidirect
197. #100C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420842
#260420832
#260420802
#260416542
#260415502
#260415482
#260414492
#260414452
#260413952
#260413942
#472×C1812
#3292181