Finite Group Databasebeta

← #37544689

Subgroups — #37544689

Order 23168.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#26041815C211yesmaximal, characteristic, complemented
2C2×#19153803C211yesmaximal, characteristic, complemented
3#26041660C212yesmaximal, complemented
4#26041632C212yesmaximal, complemented
5#26041499C212yesmaximal, complemented
6#26041493C212yesmaximal, complemented
7#26041434C211yesmaximal, characteristic, complemented
8#26041420C211yesmaximal, characteristic, complemented
9#26041413C211yesmaximal, characteristic
10#26041329C211yesmaximal, characteristic, complemented
11#450×C181C211yesmaximal, characteristic, complemented
12C22×Dic362C2211yescharacteristic
13C2×#15668439C2212yescomplemented
14C2×#15668438C2212yescomplemented
15C2×#15668434C2212yes—
16#19153811—21no—
17#19153807C2211yescharacteristic, complemented
18#19153805C2211yescharacteristic, complemented
19#19153803—21no—
20#19153803—21no—
21#19153803C2212yes—
22#19153803C2212yes—
23#19153800—21no—
24#19153790C2211yescharacteristic
25Q8×Dic181C2211yescharacteristic, complemented
26#19153778C2211yescharacteristic, complemented
27#19153776—21no—
28#19153774—21no—
29C2×C22⋊Dic181—21no—
30C2×C22⋊Dic181—21no—
31C2×C4×Dic181C2211yescharacteristic
32#19153745C2212yescomplemented
33#19153741C2212yescomplemented
34#19153739C2212yescomplemented
35#19153737C2212yes—
36C22⋊#13089678C2211yescharacteristic
37C8×Dic181C2212yescomplemented
38C22×D4×C181C2211yescharacteristic
39C2×SD16×C181C2212yes—
40C22⋊Q8×C181C2211yescharacteristic
41#102×C181C2212yes—
42C22⋊C8×C181C2211yescharacteristic
43C2×Dic362—21no—
44C2×Dic362—42no—
45C2×Dic362C2312yes—
46C2×D724C2311yescharacteristic
47C2×C22⋊D181—21no—
48C22×Dic181—21no—
49C22×Dic181D411yescharacteristic
50C22×Dic181D411yescharacteristic
51C2×C4×D181—21no—
52#15668439—42no—
53#15668438—42no—
54#15668434—42no—
55C2×#13089678C2312yes—
56C4⋊Dic181—21no—
57C4⋊Dic181—21no—
58C4⋊Dic181C2311yescharacteristic
59#15668428—21no—
60C22⋊Dic181—22no—
61C22⋊Dic181—22no—
62C22⋊Dic181—22no—
63C22⋊Dic181—22no—
64#15668426—21no—
65#15668426—21no—
66#15668426—21no—
67#15668426—21no—
68C4×Dic181—21no—
69C4×Dic181D411yescharacteristic
70C4×Dic181D411yescharacteristic
71C4×Dic181C2311yescharacteristic
72C4×Dic181C2311yescharacteristic
73C24×C181—21no—
74C2×Q8×C181C2311yescharacteristic
75C2×D4×C181—21no—
76C2×D4×C181—42no—
77C2×D4×C181C2312yes—
78C22×C4×C181C2311yescharacteristic
79SD16×C181—42no—
80C2×C8×C181C2312yes—
81C4⋊C4×C181—21no—
82C4⋊C4×C181C2311yescharacteristic
83C22⋊C4×C181—21no—
84C22×D181—21no—
85Dic362—22no—
86Dic362—22no—
87Dic362—42no—
88D724—22no—
89C22⋊D181—42no—
90C2×Dic181—22no—
91C2×Dic181—21no—
92C2×Dic181—22no—
93C2×Dic181—41no—
94C2×Dic181—42no—
95C2×Dic181C2×D411yescharacteristic
96C2×Dic181C2×D411yescharacteristic
97C2×Dic181C2×D411yescharacteristic
98C2×Dic181C2×D411yescharacteristic
99C4×D181—42no—
100#13089678—22no—
101C23×C181—22no—
102C23×C181—42no—
103C23×C181—42no—
104C23×C181—42no—
105C23×C181C2×D411yescharacteristic
106Q8×C181—22no—
107D4×C181—22no—
108D4×C181—22no—
109D4×C181—42no—
110C2×C4×C181—22no—
111C2×C4×C181—21no—
112C2×C4×C181—21no—
113C2×C4×C181C2×D411yescharacteristic
114C2×C4×C181C2411yescharacteristic
115C8×C181—22no—
116C22×C181—21no—
117C22×C181—21no—
118C22×C181—42no—
119C22×C181—42no—
120C22×C181—42no—
121C22×C181—84no—
122C22×C181#13512yes—
123C22×C181C22×D411yescharacteristic
124C2×D181—41no—
125C2×D181—41no—
126C2×D181—41no—
127C4×C181—21no—
128C4×C181—41no—
129C4×C181—41no—
130C4×C181H(5,𝔽2)12yes—
131Dic181—21no—
132Dic181—21no—
133Dic181—21no—
134Dic181—21no—
135Dic181—42no—
136Dic181—41no—
137C2×C181—22no—
138C2×C181—82no—
139C2×C181#53511yescharacteristic
140C2×C181C2×#13511yescharacteristic
141C2×C181#57511yescharacteristic
142D181—81no—
143C181#321111yescharacteristic, complemented, Sylow
144#3211—1811nomaximal, Sylow
145#529—1811no—
146C2×#124—1811no—
147#495—1812no—
148#488—1812no—
149#464—1812no—
150#463—1812no—
151#451—1811no—
152#450D18111yescharacteristic, complemented
153#448—1811no—
154#447—1811no—
155#431—1811no—
156C22×Q8—1811no—
157C22×D4C2×D18111yescharacteristic
158C2×Q16—1812no—
159C2×SD16—1812no—
160C2×SD16C2×D18112yes—
161C2×D8—1812no—
162#126—3621no—
163#125—1811no—
164#124—3621no—
165#124—3621no—
166#124—1811no—
167#124—1812no—
168#124—1812no—
169#123—3621no—
170C22⋊Q8C2×D18111yescharacteristic
171#121—1811no—
172C4×Q8—1811no—
173C4×D4—3621no—
174C4×D4—1811no—
175#117—3621no—
176C2×C22⋊C4—3621no—
177C2×C22⋊C4—3621no—
178C2×C42—1811no—
179C4⋊C8—1812no—
180#103—1812no—
181#103—1812no—
182#102—1812no—
183#102C2×D18112yes—
184C22⋊C8—1811no—
185C22⋊C8C2×D18111yescharacteristic
186C4×C8—1812no—
187C24—21no—
188C2×Q8—3621no—
189C2×Q8—1812no—
190C2×Q8—7242no—
191C2×Q8C22×D18111yescharacteristic
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—3621no—
195C2×D4—1811no—
196C2×D4C22×D18112yes—
197C22×C4—3621no—
198C22×C4—3621no—
199C22×C4—1811no—
200C22×C4—1811no—
201C22×C4C22×D18111yescharacteristic
202Q16—7242no—
203SD16—42no—
204SD16—7242no—
205D8—7242no—
206C2×C8—1812no—
207C2×C8C22×D18112yes—
208C4⋊C4—21no—
209C4⋊C4—3621no—
210C4⋊C4—3621no—
211C4⋊C4—3621no—
212C4⋊C4—1811no—
213C4⋊C4C22×D18111yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—3621no—
216C22⋊C4—3621no—
217C22⋊C4—3621no—
218C22⋊C4—3622no—
219C22⋊C4—3622no—
220C22⋊C4—3622no—
221C22⋊C4—3622no—
222C22⋊C4—3621no—
223C42—3621no—
224C42—1811no—
225C42—1811no—
226C42—1811no—
227C42—1811no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—3621no—
233C23D4×D18111yescharacteristic
234Q8—22no—
235Q8—3622no—
236Q8—3622no—
237Q8—7242no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—3622no—
242D4—7242no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—3622no—
247C2×C4—3621no—
248C2×C4—3622no—
249C2×C4—1811no—
250C2×C4—1811no—
251C2×C4—1811no—
252C2×C4—1811no—
253C2×C4—7242no—
254C2×C4—7241no—
255C2×C4—7242no—
256C2×C4D4×D18111yescharacteristic
257C2×C4C23×D18111yescharacteristic
258C8—22no—
259C8—3622no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—7241no—
267C22—7241no—
268C22—7241no—
269C22#1915383712yes—
270C22C2×D4×D18111yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—3621no—
275C4—3621no—
276C4—3621no—
277C4—3621no—
278C4—7241no—
279C4—7242no—
280C4#1915386112yes—
281C2—22no—
282C2—82no—
283C2—14481no—
284C2#2604184411yescharacteristic
285C2C2×#1915383711yescharacteristic
286C2#2604207511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
prime radical#26041413C2
hypercenter#450D181
hypocenterC181#3211
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26041815C2semidirect
2. C2×#19153803C2semidirect
3. #26041660C2semidirect
4. #26041632C2semidirect
5. #26041499C2semidirect
6. #26041493C2semidirect
7. #26041434C2semidirect
8. #26041420C2semidirect
10. #26041329C2semidirect
11. #450×C181C2semidirect
13. C2×#15668439C22semidirect
14. C2×#15668438C22semidirect
17. #19153807C22semidirect
18. #19153805C22semidirect
25. Q8×Dic181C22semidirect
26. #19153778C22semidirect
32. #19153745C22semidirect
33. #19153741C22semidirect
34. #19153739C22semidirect
37. C8×Dic181C22semidirect
143. C181#3211semidirect
152. #450D181semidirect

Maximal subgroups

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

subgroupindex
#260418152
C2×#191538032
#260416602
#260416322
#260414992
#260414932
#260414342
#260414202
#260414132
#260413292
#450×C1812
#3211181