Finite Group Databasebeta

← #496×D181 #37544151

Subgroups — #496×D181 #37544151

Order 23168.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D181C212yesmaximal, complemented
2C2×D8×D181C212yesmaximal, complemented
3#127×D181C211yesmaximal, characteristic, complemented
4#124×D181C211yesmaximal, characteristic, complemented
5#26041669C211yesmaximal, characteristic, complemented
6#26041668C211yesmaximal, characteristic, complemented
7#26041667C212yesmaximal, complemented
8#26041666C211yesmaximal, characteristic, complemented
9#26041665C212yesmaximal, complemented
10#97×D181C211yesmaximal, characteristic, complemented
11#496×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181C2211yescharacteristic
13C2×D4×D181—22no—
14C2×D4×D181C2211yescharacteristic
15C2×D4×D181C2212yes—
16C2×#15668438C2212yescomplemented
17C2×#15668437C2212yescomplemented
18C2×#15668436C2212yescomplemented
19SD16×D181—28no—
20C2×D1448C2212yescomplemented
21C2×#15668434C2212yescomplemented
22C2×#15668434C2212yescomplemented
23D8×D181—28no—
24C2×C8×D181C2212yes—
25#19153813C2211yescharacteristic, complemented
26#19153812—22no—
27#19153812C2212yescomplemented
28#19153805C2211yescharacteristic, complemented
29#19153804C2211yescharacteristic, complemented
30#19153803C2211yescharacteristic, complemented
31#19153802—22no—
32C22⋊C4×D181—22no—
33C42×D181C2211yescharacteristic
34#19153727C2211yescharacteristic, complemented
35#19153726C2212yescomplemented
36C2×SD16×C181C2212yescomplemented
37C2×D8×C181C2212yescomplemented
38#127×C181C2211yescharacteristic, complemented
39#124×C181C2211yescharacteristic, complemented
40#97×C181C2211yescharacteristic, complemented
41C23×D181—21no—
42C23×D181—22no—
43C2×Dic362—21no—
44C2×Dic362C2311yescharacteristic
45Q8×D181—24no—
46C2×D724—22no—
47C2×D724C2311yescharacteristic
48C2×D724C2312yes—
49C2×C22⋊D181—21no—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—24no—
52D4×D181—24no—
53D4×D181—24no—
54D4×D181—24no—
55D4×D181—48no—
56C2×C4×D181—21no—
57C2×C4×D181D412yes—
58C2×C4×D181C2311yescharacteristic
59#15668438—24no—
60#15668437—24no—
61#15668436—24no—
62D1448—24no—
63#15668434—24no—
64#15668434—24no—
65C2×#13089678C2312yes—
66C8×D181D418yescomplemented
67C22⋊Dic181—22no—
68#15668426—22no—
69#15668426—22no—
70C4×Dic181D412yes—
71C4×Dic181C2311yescharacteristic
72C2×Q8×C181C2311yescharacteristic
73C2×D4×C181—22no—
74C2×D4×C181C2311yescharacteristic
75C2×D4×C181C2312yes—
76SD16×C181—24no—
77D8×C181—24no—
78C2×C8×C181C2312yes—
79C22⋊C4×C181—22no—
80C42×C181C2311yescharacteristic
81C22×D181—21no—
82C22×D181—22no—
83C22×D181—42no—
84C22×D181—44no—
85C22×D181—48no—
86C22×D181—44no—
87C22×D181C2×D411yescharacteristic
88Dic362—22no—
89Dic362—42no—
90D724—22no—
91D724—22no—
92D724—22no—
93D724—44no—
94C22⋊D181—48no—
95C22⋊D181—42no—
96C22⋊D181—44no—
97C2×Dic181—21no—
98C2×Dic181C2×D412yes—
99C2×Dic181C2×D411yescharacteristic
100C4×D181—24no—
101C4×D181—42no—
102C4×D181C2×D414yes—
103#13089678C2×D414yescomplemented
104C23×C181—21no—
105C23×C181—22no—
106Q8×C181—22no—
107D4×C181—22no—
108D4×C181—22no—
109D4×C181—22no—
110D4×C181—44no—
111C2×C4×C181—21no—
112C2×C4×C181C2×D412yes—
113C2×C4×C181C2411yescharacteristic
114C8×C181C2×D414yescomplemented
115C22×C181—41no—
116C22×C181—42no—
117C22×C181—44no—
118C22×C181—42no—
119C22×C181C22×D411yescharacteristic
120C2×D181—41no—
121C2×D181—42no—
122C2×D181—44no—
123C2×D181—42no—
124C2×D181—88no—
125C2×D181—84no—
126C2×D181#12712yes—
127C2×D181AGL(1,ℤ/8)14yes—
128C4×C181—22no—
129C4×C181—41no—
130C4×C181C22×D412yes—
131Dic181—22no—
132Dic181—41no—
133Dic181#12712yes—
134C2×C181—82no—
135C2×C181—81no—
136C2×C181C2×#12711yescharacteristic
137C2×C181C2×AGL(1,ℤ/8)12yes—
138D181—82no—
139D181—81no—
140D181#49614yesdirect product factor
141C181C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1811nomaximal, Sylow
143C22×SD16—1812no—
144C22×D8—1812no—
145C2×#127—1811no—
146C2×#124—1811no—
147#496—1812no—
148#496—1812no—
149#496—1811no—
150#496—1811no—
151#496—1811no—
152#496D18111yescharacteristic, direct product factor
153C2×#97—1811no—
154C22×Q8—1811no—
155C22×D4—3622no—
156C22×D4—1811no—
157C22×D4—1812no—
158C2×SD16—3628no—
159C2×SD16—1812no—
160C2×SD16—1812no—
161C2×SD16—1812no—
162C2×SD16C2×D18112yescomplemented
163C2×D8—3628no—
164C2×D8—1812no—
165C2×D8—1812no—
166C2×D8—1812no—
167C2×D8C2×D18112yescomplemented
168C22×C8—1812no—
169#127—3622no—
170#127—1812no—
171#127—1811no—
172#127C2×D18111yescharacteristic, complemented
173#124—3622no—
174#124—1811no—
175#124—1811no—
176#124—1811no—
177#124C2×D18111yescharacteristic, complemented
178C2×C22⋊C4—3622no—
179C2×C42—1811no—
180#97—1812no—
181#97—1811no—
182#97C2×D18111yescharacteristic, complemented
183C24—3621no—
184C24—3622no—
185C2×Q8—3621no—
186C2×Q8—3624no—
187C2×Q8—1811no—
188C2×Q8C22×D18111yescharacteristic
189C2×D4—22no—
190C2×D4—3624no—
191C2×D4—3624no—
192C2×D4—3624no—
193C2×D4—3621no—
194C2×D4—3622no—
195C2×D4—3624no—
196C2×D4—3622no—
197C2×D4—1811no—
198C2×D4—1812no—
199C2×D4—7248no—
200C2×D4C22×D18111yescharacteristic
201C2×D4C22×D18112yes—
202C22×C4—3621no—
203C22×C4—1812no—
204C22×C4—1811no—
205SD16—24no—
206SD16—3624no—
207SD16—3624no—
208SD16—3624no—
209D8—24no—
210D8—3624no—
211D8—3624no—
212D8—3624no—
213C2×C8—1812no—
214C2×C8—1818no—
215C2×C8C22×D18112yes—
216C22⋊C4—22no—
217C22⋊C4—3622no—
218C22⋊C4—3622no—
219C22⋊C4—3622no—
220C42—1812no—
221C42—1811no—
222C42C22×D18111yescharacteristic
223C23—21no—
224C23—22no—
225C23—3621no—
226C23—3622no—
227C23—1811no—
228C23—7242no—
229C23—7244no—
230C23—7248no—
231C23—7244no—
232Q8—22no—
233Q8—3622no—
234Q8—7242no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—3622no—
240D4—3622no—
241D4—3622no—
242D4—7244no—
243D4—7248no—
244D4—7242no—
245D4—7244no—
246C2×C4—21no—
247C2×C4—3624no—
248C2×C4—3621no—
249C2×C4—1812no—
250C2×C4—1811no—
251C2×C4—1814no—
252C2×C4—7242no—
253C2×C4D4×D18112yes—
254C2×C4C23×D18111yescharacteristic
255C8—1814no—
256C8D4×D18114yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—14488no—
262C22—14484no—
263C22—1814no—
264C22—1812no—
265C22—7244no—
266C22—7242no—
267C22—7241no—
268C22—7242no—
269C22C2×D4×D18111yescharacteristic
270C4—22no—
271C4—41no—
272C4—3622no—
273C4—1812no—
274C4—7241no—
275C4C2×D4×D18112yes—
276C2—82no—
277C2—81no—
278C2—14482no—
279C2—14481no—
280C2—1814no—
281C2#127×D18111yescharacteristic
282C2AGL(1,ℤ/8)×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#496×C181C2
socleC22×C181C22×D4
hypercenter#496D181
hypocenterC181C2×#496
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D181C2semidirect
2. C2×D8×D181C2semidirect
3. #127×D181C2semidirect
4. #124×D181C2semidirect
5. #26041669C2semidirect
6. #26041668C2semidirect
7. #26041667C2semidirect
8. #26041666C2semidirect
9. #26041665C2semidirect
10. #97×D181C2semidirect
11. #496×C181C2semidirect
16. C2×#15668438C22semidirect
17. C2×#15668437C22semidirect
18. C2×#15668436C22semidirect
20. C2×D1448C22semidirect
21. C2×#15668434C22semidirect
22. C2×#15668434C22semidirect
25. #19153813C22semidirect
27. #19153812C22semidirect
28. #19153805C22semidirect
29. #19153804C22semidirect
30. #19153803C22semidirect
34. #19153727C22semidirect
35. #19153726C22semidirect
36. C2×SD16×C181C22semidirect
37. C2×D8×C181C22semidirect
38. #127×C181C22semidirect
39. #124×C181C22semidirect
40. #97×C181C22semidirect
66. C8×D181D4semidirect
103. #13089678C2×D4semidirect
114. C8×C181C2×D4semidirect
140. D181#496direct (N × H)
141. C181C2×#496semidirect
152. #496D181direct (N × H)
162. C2×SD16C2×D181semidirect
167. C2×D8C2×D181semidirect
172. #127C2×D181semidirect
177. #124C2×D181semidirect
182. #97C2×D181semidirect
256. C8D4×D181semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1812
C2×D8×D1812
#127×D1812
#124×D1812
#260416692
#260416682
#260416672
#260416662
#260416652
#97×D1812
#496×C1812
C2×#496181