Finite Group Databasebeta

← #37543005

Subgroups — #37543005

Order 23168.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153863C211yesmaximal, characteristic, complemented
2C2×#19153842C211yesmaximal, characteristic, complemented
3C22×#15668437C211yesmaximal, characteristic, complemented
4#26041422C214yesmaximal, complemented
5#26041417C214yesmaximal, complemented
6#26041287C211yesmaximal, characteristic, complemented
7#26041273C211yesmaximal, characteristic, complemented
8#26041246C211yesmaximal, characteristic, complemented
9#535×C181C211yesmaximal, characteristic, complemented
10#19153863—24no—
11#19153863—24no—
12C2×#15668449C2211yescharacteristic
13C2×#15668449C2212yescomplemented
14C2×#15668448—22no—
15C2×#15668448—21no—
16C22×Dic362C2211yescharacteristic
17C22×Dic362C2212yescomplemented
18C2×Q8×D181—21no—
19#19153842—24no—
20C2×#15668437—24no—
21C2×#15668437C2212yes—
22C2×#15668437C2214yes—
23C2×#15668434C2212yes—
24C2×#15668433C2211yescharacteristic
25C22×#13089678C2211yescharacteristic
26#19153739C2212yescomplemented
27#19153739C2214yescomplemented
28#19153736C2212yescomplemented
29C22⋊#13089678C2214yescomplemented
30C2×#41×C181C2211yescharacteristic
31C22×D4×C181C2211yescharacteristic
32#127×C181—21no—
33#124×C181—21no—
34#121×C181C2214yes—
35#120×C181—22no—
36#117×C181C2211yescharacteristic
37#15668449—24no—
38#15668449—24no—
39#15668449—22no—
40#15668449D414yescomplemented
41#15668448—216no—
42#15668448—28no—
43C2×Dic362—24no—
44C2×Dic362—24no—
45C2×Dic362—21no—
46C2×Dic362—22no—
47C2×Dic362—22no—
48C2×Dic362D414yescomplemented
49C2×Dic362C2312yes—
50C2×Dic362C2311yescharacteristic
51Q8×D181—28no—
52C2×D724C2311yescharacteristic
53C2×C22⋊D181—21no—
54C2×C22⋊D181—22no—
55C22×Dic181—21no—
56C22×Dic181—22no—
57C2×C4×D181—21no—
58C2×C4×D181—22no—
59#15668437—28no—
60#15668437—24no—
61#15668434—24no—
62#15668433—22no—
63C2×#13089678—22no—
64C2×#13089678C2312yes—
65C2×#13089678C2312yes—
66C24×C181—21no—
67#41×C181—22no—
68#41×C181—22no—
69C2×Q8×C181D411yescharacteristic
70C2×D4×C181—24no—
71C2×D4×C181—22no—
72C2×D4×C181—22no—
73C2×D4×C181—21no—
74C2×D4×C181D412yes—
75C2×D4×C181D411yescharacteristic
76C2×D4×C181C2312yes—
77C22×C4×C181C2312yes—
78C22×C4×C181C2311yescharacteristic
79C4⋊C4×C181C2312yes—
80C22⋊C4×C181—21no—
81C22⋊C4×C181—24no—
82C42×C181—21no—
83C22×D181—21no—
84Dic362—28no—
85Dic362—24no—
86Dic362—21no—
87Dic362—22no—
88Dic362C2×D412yes—
89Dic362C2×D414yes—
90D724—21no—
91D724C2×D412yes—
92C22⋊D181—28no—
93C22⋊D181—24no—
94C2×Dic181—28no—
95C2×Dic181—24no—
96C2×Dic181—21no—
97C2×Dic181—22no—
98C4×D181—28no—
99C4×D181—24no—
100#13089678—22no—
101#13089678—22no—
102C23×C181—22no—
103C23×C181—42no—
104C23×C181—82no—
105C23×C181C2×D412yes—
106C23×C181C2×D411yescharacteristic
107Q8×C181—22no—
108D4×C181—24no—
109D4×C181—22no—
110D4×C181—22no—
111D4×C181—22no—
112D4×C181—42no—
113D4×C181—84no—
114C2×C4×C181—22no—
115C2×C4×C181—22no—
116C2×C4×C181—22no—
117C2×C4×C181C2×D412yes—
118C2×C4×C181C2×D412yes—
119C2×C4×C181C2×D412yes—
120C2×C4×C181C2×D411yescharacteristic
121C2×C4×C181C2411yescharacteristic
122C22×C181—22no—
123C22×C181—22no—
124C22×C181—24no—
125C22×C181—42no—
126C22×C181—44no—
127C22×C181—84no—
128C22×C181#12012yes—
129C22×C181C22×D411yescharacteristic
130C2×D181—22no—
131C2×D181—42no—
132C4×C181—22no—
133C4×C181—42no—
134C4×C181#12012yes—
135C4×C181C22×D411yescharacteristic
136C4×C181C22×D411yescharacteristic
137Dic181—24no—
138Dic181—22no—
139C2×C181—22no—
140C2×C181—42no—
141C2×C181—82no—
142C2×C181C2×#12011yescharacteristic
143C2×C181#57712yes—
144D181—42no—
145C181#301011yescharacteristic, complemented, Sylow
146#3010—1811nomaximal, Sylow
147C2×2−1+4—1811no—
148C2×AGL(1,ℤ/8)—1811no—
149C22×SD16—1811no—
150#535D18111yescharacteristic, complemented
151#449—1814no—
152#448—1814no—
153#419—1811no—
154#416—1811no—
155#408—1811no—
1562−1+4—3624no—
1572−1+4—3624no—
158C2×#41—3622no—
159C2×#41—3621no—
160C2×#41—1812no—
161C2×#41—1811no—
162C2×#41C2×D18111yescharacteristic
163C22×Q8—3621no—
164C22×Q8—1812no—
165C22×Q8—1811no—
166C22×D4C2×D18111yescharacteristic
167AGL(1,ℤ/8)—3624no—
168C2×SD16—3624no—
169C2×SD16—1814no—
170C2×SD16—1812no—
171C2×D8—1812no—
172C2×M16—1811no—
173C22×C8—1811no—
174#127—21no—
175#124—21no—
176#121C2×D18114yes—
177#120—22no—
178#117C2×D18111yescharacteristic
179#103—1812no—
180#103—1814no—
181#102—1812no—
182C22⋊C8—1814no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—3624no—
187#41—3624no—
188#41—36216no—
189#41—3628no—
190#41—3622no—
191#41—1814no—
192C2×Q8—3624no—
193C2×Q8—3624no—
194C2×Q8—3628no—
195C2×Q8—3622no—
196C2×Q8—3622no—
197C2×Q8—3621no—
198C2×Q8—1814no—
199C2×Q8—1811no—
200C2×Q8—1812no—
201C2×Q8C22⋊D18111yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—3622no—
207C2×D4—3621no—
208C2×D4—1811no—
209C2×D4C22⋊D18112yes—
210C2×D4C22⋊D18111yescharacteristic
211C2×D4C22×D18112yes—
212C22×C4—3622no—
213C22×C4—3621no—
214C22×C4—3622no—
215C22×C4—3621no—
216C22×C4C22×D18112yes—
217C22×C4C22×D18111yescharacteristic
218SD16—3628no—
219SD16—3624no—
220D8—3624no—
221M16—3622no—
222C2×C8—3622no—
223C2×C8—1812no—
224C2×C8—1812no—
225C4⋊C4C22×D18112yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—3621no—
233C23C2×C22⋊D18112yes—
234C23C2×C22⋊D18111yescharacteristic
235Q8—22no—
236Q8—3628no—
237Q8—3624no—
238Q8—3621no—
239Q8—3622no—
240Q8—1812no—
241Q8—1814no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—3628no—
249D4—3624no—
250D4—3621no—
251D4—1812no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3628no—
256C2×C4—3628no—
257C2×C4—3624no—
258C2×C4—3622no—
259C2×C4—3621no—
260C2×C4—3624no—
261C2×C4D4×D18112yes—
262C2×C4D4×D18112yes—
263C2×C4C2×C22⋊D18112yes—
264C2×C4C2×C22⋊D18111yescharacteristic
265C2×C4C23×D18111yescharacteristic
266C8—3622no—
267C8—3622no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—3622no—
275C22—7242no—
276C22#1915378412yes—
277C22C22×C22⋊D18111yescharacteristic
278C4—22no—
279C4—42no—
280C4—3624no—
281C4—3622no—
282C4#1915378412yes—
283C4C2×D4×D18111yescharacteristic
284C4C2×D4×D18111yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—7242no—
289C2C2×#1915378411yescharacteristic
290C2#2604209012yes—
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#535×C181C2
socleC22×C181C22×D4
hypercenter#535D181
hypocenterC181#3010
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19153863C2semidirect
2. C2×#19153842C2semidirect
3. C22×#15668437C2semidirect
4. #26041422C2semidirect
5. #26041417C2semidirect
6. #26041287C2semidirect
7. #26041273C2semidirect
8. #26041246C2semidirect
9. #535×C181C2semidirect
13. C2×#15668449C22semidirect
17. C22×Dic362C22semidirect
26. #19153739C22semidirect
27. #19153739C22semidirect
28. #19153736C22semidirect
29. C22⋊#13089678C22semidirect
40. #15668449D4semidirect
48. C2×Dic362D4semidirect
145. C181#3010semidirect
150. #535D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538632
C2×#191538422
C22×#156684372
#260414222
#260414172
#260412872
#260412732
#260412462
#535×C1812
#3010181