Finite Group Databasebeta

← #37545503

Subgroups — #37545503

Order 23168.

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

normal subgroups only

This group has 454 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#26042093C211yesmaximal, characteristic, complemented
2#26042090C212yesmaximal, complemented
3#26041662C211yesmaximal, characteristic, complemented
4#26041554C211yesmaximal, characteristic, complemented
5#26041546C211yesmaximal, characteristic, complemented
6#26041540C212yesmaximal, complemented
7#26041457C212yesmaximal, complemented
8#26041450C212yesmaximal, complemented
9#26041396C211yesmaximal, characteristic, complemented
10#26041394C211yesmaximal, characteristic, complemented
11#576×C181C211yesmaximal, characteristic, complemented
12#19153863C2212yescomplemented
13#19153846—22no—
14#19153846C2212yes—
15#19153846C2212yescomplemented
16#19153845—22no—
17#19153842—22no—
18#19153842C2212yescomplemented
19#19153836—22no—
20#19153836C2211yescharacteristic
21#19153835—21no—
22C2×Dic724C2211yescharacteristic, complemented
23Q16×D181—21no—
24C2×#15668437—22no—
25C2×#15668437C2212yescomplemented
26#19153823C2212yescomplemented
27#19153822C2211yescharacteristic, complemented
28#19153803C2212yescomplemented
29#19153752C2211yescharacteristic, complemented
30#19153751C2211yescharacteristic, complemented
31#19153743C2211yescharacteristic
32#19153743C2212yescomplemented
33#19153742C2211yescharacteristic, complemented
34#19153734C2212yescomplemented
35#19153733C2212yescomplemented
36C8×Dic181C2211yescharacteristic, complemented
37H(5,𝔽2)×C181C2212yes—
38AGL(1,ℤ/8)×C181—22no—
39AGL(1,ℤ/8)×C181C2212yes—
40#135×C181—21no—
41#135×C181C2211yescharacteristic
42C2×D8×C181—21no—
43C2×D8×C181C2211yescharacteristic
44#131×C181C2211yescharacteristic
45#15668449—22no—
46#15668449C2311yescharacteristic
47#15668448—22no—
48#15668448—42no—
49C2×Dic362—22no—
50C2×Dic362C2312yes—
51Q8×D181—22no—
52Dic724—21no—
53Dic724—21no—
54#15668439—22no—
55#15668439—22no—
56#15668439D411yescharacteristic
57#15668437—22no—
58#15668437—22no—
59#15668437—22no—
60#15668437—22no—
61#15668437—22no—
62#15668437D411yescharacteristic
63#15668436D411yescharacteristic
64#15668434—22no—
65#15668434D411yescharacteristic
66#15668433—22no—
67#15668433C2311yescharacteristic
68#15668433C2312yes—
69#15668432—21no—
70C2×#13089678—22no—
71C2×#13089678C2311yescharacteristic
72C8×D181—21no—
73C22⋊Dic181—42no—
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
84SD16×C181—22no—
85SD16×C181D412yes—
86D8×C181—22no—
87D8×C181—21no—
88D8×C181—21no—
89D8×C181D411yescharacteristic
90M16×C181—21no—
91M16×C181C2311yescharacteristic
92C2×C8×C181—21no—
93C2×C8×C181C2311yescharacteristic
94Dic362—22no—
95Dic362—22no—
96Dic362—42no—
97Dic362C2×D411yescharacteristic
98D724C2×D411yescharacteristic
99C22⋊D181—21no—
100C22⋊D181—42no—
101C2×Dic181—21no—
102C2×Dic181—22no—
103C2×Dic181—42no—
104C4×D181—21no—
105C4×D181—42no—
106#13089678—21no—
107#13089678—22no—
108#13089678C2×D411yescharacteristic
109#13089678C2×D411yescharacteristic
110C23×C181—22no—
111C23×C181—42no—
112Q8×C181C2×D411yescharacteristic
113Q8×C181C2×D412yes—
114D4×C181—21no—
115D4×C181—22no—
116D4×C181—22no—
117D4×C181—22no—
118D4×C181—42no—
119D4×C181—42no—
120D4×C181C2×D411yescharacteristic
121D4×C181C2×D412yes—
122C2×C4×C181—21no—
123C2×C4×C181—22no—
124C2×C4×C181—42no—
125C2×C4×C181C2411yescharacteristic
126C8×C181—21no—
127C8×C181C2×D411yescharacteristic
128C8×C181C2×D411yescharacteristic
129C22×C181—21no—
130C22×C181—22no—
131C22×C181—42no—
132C22×C181—82no—
133C22×C181—162no—
134C22×C181H(5,𝔽2)11yescharacteristic
135C2×D181—21no—
136C4×C181—21no—
137C4×C181—22no—
138C4×C181C22×D411yescharacteristic
139C4×C181C22×D411yescharacteristic
140Dic181—21no—
141Dic181—41no—
142Dic181—42no—
143C2×C181—21no—
144C2×C181—42no—
145C2×C181—41no—
146C2×C181—82no—
147C2×C181D4211yescharacteristic
148D181—41no—
149C181#329011yescharacteristic, complemented, Sylow
150#3290—1811nomaximal, Sylow
151#578—1811no—
152#577—1812no—
153#576D18111yescharacteristic, complemented
154#495—1811no—
155#473—1811no—
156#472—1811no—
157#471—1812no—
158#455—1812no—
159#454—1812no—
160#443—1811no—
161#443—1811no—
1622−1+4—1812no—
163H(5,𝔽2)C2×D18112yes—
164#137—3622no—
165#137—3622no—
166#137—1812no—
167#137—1812no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—3622no—
170AGL(1,ℤ/8)—1812no—
171AGL(1,ℤ/8)C2×D18112yes—
172#135—21no—
173#135—3622no—
174#135—3621no—
175#135—1811no—
176#135C2×D18111yescharacteristic
177C2×Q16—3621no—
178C2×Q16—1811no—
179C2×SD16—3622no—
180C2×SD16—1812no—
181C2×D8—21no—
182C2×D8C2×D18111yescharacteristic
183#131—1812no—
184#131—1811no—
185#131C2×D18111yescharacteristic
186#124—1812no—
187#108—1811no—
188#108—1811no—
189#104—1811no—
190#104—1812no—
191#104—1811no—
192#101—1812no—
193#100—1812no—
194C4×C8—1811no—
195#41—22no—
196#41—22no—
197#41—3622no—
198#41—3622no—
199#41—1811no—
200#41—7242no—
201#41C22×D18111yescharacteristic
202#41C22×D18112yes—
203C2×Q8—3622no—
204C2×Q8—3622no—
205C2×Q8—1812no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4C22×D18112yes—
210Q16—3622no—
211Q16—3622no—
212Q16—3621no—
213Q16—3621no—
214Q16—1811no—
215Q16C22⋊D18111yescharacteristic
216SD16—22no—
217SD16—3622no—
218SD16—3622no—
219SD16—3622no—
220SD16—3622no—
221SD16—3622no—
222SD16—1811no—
223SD16—1811no—
224SD16C22⋊D18112yes—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—3622no—
229D8—1811no—
230D8C22⋊D18111yescharacteristic
231M16—21no—
232M16—3622no—
233M16—3621no—
234M16—1811no—
235M16—1812no—
236M16C22×D18111yescharacteristic
237C2×C8—21no—
238C2×C8—3622no—
239C2×C8—3621no—
240C2×C8—1811no—
241C2×C8C22×D18111yescharacteristic
242C22⋊C4—7242no—
243C42—1811no—
244C23—22no—
245C23—42no—
246Q8—3622no—
247Q8—3622no—
248Q8—1811no—
249Q8—7242no—
250Q8D4×D18111yescharacteristic
251Q8C2×C22⋊D18112yes—
252D4—21no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3621no—
259D4—1811no—
260D4—7242no—
261D4D4×D18111yescharacteristic
262D4C2×C22⋊D18112yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—3621no—
267C2×C4—3621no—
268C2×C4—3622no—
269C2×C4—7242no—
270C2×C4—7242no—
271C2×C4C23×D18111yescharacteristic
272C8—21no—
273C8—3622no—
274C8—3621no—
275C8—1811no—
276C8—1811no—
277C8C2×C22⋊D18111yescharacteristic
278C8C2×C22⋊D18111yescharacteristic
279C22—22no—
280C22—21no—
281C22—42no—
282C22—82no—
283C22—162no—
284C22—3621no—
285C22#1915386111yescharacteristic
286C4—21no—
287C4—22no—
288C4—3621no—
289C4—7241no—
290C4—7242no—
291C4C2×D4×D18111yescharacteristic
292C4C22×C22⋊D18111yescharacteristic
293C2—21no—
294C2—41no—
295C2—42no—
296C2—82no—
297C2—7241no—
298C2D4×C22⋊D18111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#576×C181C2
socleC2×C181D42
hypercenter#576D181
hypocenterC181#3290
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. #26042093C2semidirect
2. #26042090C2semidirect
3. #26041662C2semidirect
4. #26041554C2semidirect
5. #26041546C2semidirect
6. #26041540C2semidirect
7. #26041457C2semidirect
8. #26041450C2semidirect
9. #26041396C2semidirect
10. #26041394C2semidirect
11. #576×C181C2semidirect
12. #19153863C22semidirect
15. #19153846C22semidirect
18. #19153842C22semidirect
22. C2×Dic724C22semidirect
25. C2×#15668437C22semidirect
26. #19153823C22semidirect
27. #19153822C22semidirect
28. #19153803C22semidirect
29. #19153752C22semidirect
30. #19153751C22semidirect
32. #19153743C22semidirect
33. #19153742C22semidirect
34. #19153734C22semidirect
35. #19153733C22semidirect
36. C8×Dic181C22semidirect
149. C181#3290semidirect
153. #576D181semidirect

Maximal subgroups

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

subgroupindex
#260420932
#260420902
#260416622
#260415542
#260415462
#260415402
#260414572
#260414502
#260413962
#260413942
#576×C1812
#3290181