Finite Group Databasebeta

← #37544890

Subgroups — #37544890

Order 23168.

334 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041814C211yesmaximal, characteristic
2#125×D181C211yesmaximal, characteristic, complemented
3#26041678C211yesmaximal, characteristic, complemented
4#26041636C211yesmaximal, characteristic, complemented
5#26041634C211yesmaximal, characteristic, complemented
6#26041616C211yesmaximal, characteristic, complemented
7#26041611C211yesmaximal, characteristic, complemented
8#26041599C211yesmaximal, characteristic
9#26041594C211yesmaximal, characteristic
10#26041589C211yesmaximal, characteristic, complemented
11#26041585C211yesmaximal, characteristic, complemented
12#26041582C211yesmaximal, characteristic, complemented
13#26041578C211yesmaximal, characteristic, complemented
14#26041330C211yesmaximal, characteristic, complemented
15#499×C181C211yesmaximal, characteristic, complemented
16#19153811—21no—
17#19153808C2211yescharacteristic
18#19153807C2211yescharacteristic
19#19153807C2211yescharacteristic
20#19153806—21no—
21#19153806—21no—
22#19153804C2211yescharacteristic
23#19153801—21no—
24#19153796C2211yescharacteristic
25#19153791C2211yescharacteristic
26#19153781C2211yescharacteristic
27#19153778—21no—
28#19153778C2211yescharacteristic
29#19153774—21no—
30C4⋊C4×D181—21no—
31C4⋊C4×D181—21no—
32C4⋊C4×D181C2211yescharacteristic
33C4⋊C4×D181C2211yescharacteristic
34C42×D181C2211yescharacteristic
35#19153749C2211yescharacteristic
36#19153749C2211yescharacteristic
37#19153747C2211yescharacteristic
38#19153747C2211yescharacteristic
39#19153745C2211yescharacteristic
40#19153745C2211yescharacteristic
41#19153741C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51C4×#13089678C2211yescharacteristic
52#125×C181C2211yescharacteristic
53#125×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#106×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic
58C4×C8×C181C2211yescharacteristic
59C2×Dic362C2311yescharacteristic
60C2×D724—21no—
61C2×D724C2311yescharacteristic
62C2×C4×D181—21no—
63C2×C4×D181—21no—
64C2×C4×D181—21no—
65C2×C4×D181D411yescharacteristic
66C2×C4×D181D411yescharacteristic
67C2×C4×D181C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C4⋊Dic181—21no—
71C4⋊Dic181—21no—
72C4⋊Dic181—21no—
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668426—21no—
83#15668426—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87C4×Dic181—21no—
88C4×Dic181D411yescharacteristic
89C4×Dic181D411yescharacteristic
90C4×Dic181C2311yescharacteristic
91C2×C8×C181C2311yescharacteristic
92C2×C8×C181C2311yescharacteristic
93C4⋊C4×C181—21no—
94C4⋊C4×C181—21no—
95C4⋊C4×C181—21no—
96C4⋊C4×C181—21no—
97C4⋊C4×C181C2311yescharacteristic
98C4⋊C4×C181C2311yescharacteristic
99C4⋊C4×C181C2311yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C42×C181C2311yescharacteristic
102C22×D181—21no—
103C22×D181C2×D411yescharacteristic
104Dic362—21no—
105Dic362—21no—
106D724—21no—
107D724—21no—
108D724—42no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C2×Dic181C2×D411yescharacteristic
113C2×Dic181C2×D411yescharacteristic
114C2×Dic181C2×D411yescharacteristic
115C4×D181—22no—
116C4×D181—21no—
117C4×D181—21no—
118C4×D181—41no—
119C4×D181—42no—
120C4×D181—42no—
121C4×D181—42no—
122#13089678—21no—
123#13089678—21no—
124C2×C4×C181—21no—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181C2×D411yescharacteristic
129C2×C4×C181C2×D411yescharacteristic
130C2×C4×C181C2411yescharacteristic
131C8×C181—21no—
132C8×C181—21no—
133C22×C181C22×D411yescharacteristic
134C2×D181—21no—
135C2×D181—21no—
136C2×D181—41no—
137C2×D181—41no—
138C2×D181—41no—
139C2×D181#13512yes—
140C4×C181—21no—
141C4×C181—21no—
142C4×C181—41no—
143C4×C181—41no—
144C4×C181—41no—
145C4×C181—41no—
146C4×C1812−1+411yescharacteristic
147C4×C1812−1+411yescharacteristic
148Dic181—21no—
149Dic181—21no—
150Dic181—21no—
151Dic181—41no—
152Dic181—41no—
153Dic181—41no—
154C2×C181#53611yescharacteristic
155C2×C181C2×#13511yescharacteristic
156C2×C181#57511yescharacteristic
157D181—22no—
158D181—81no—
159C181#323211yescharacteristic, complemented, Sylow
160#3232—1811nomaximal, Sylow
161#529—1811no—
162C2×#125—1811no—
163#499—1811no—
164#499D18111yescharacteristic, complemented
165#488—1811no—
166#488—1811no—
167#484—1811no—
168#483—1811no—
169#481—1811no—
170#480—1811no—
171#479—1811no—
172#479—1811no—
173#478—1811no—
174#478—1811no—
175#431—1811no—
176#126—3621no—
177#125—3621no—
178#125—3621no—
179#125—1811no—
180#125—1811no—
181#125—1811no—
182#125C2×D18111yescharacteristic
183#125C2×D18111yescharacteristic
184#124—1811no—
185#123—3621no—
186C22⋊Q8—1811no—
187#121—1811no—
188C4×Q8—1811no—
189C4×D4—3621no—
190C4×D4—1811no—
191#117—3621no—
192C2×C4⋊C4—3621no—
193C2×C4⋊C4—3621no—
194C2×C4⋊C4—1811no—
195C2×C4⋊C4—1811no—
196C2×C42—1811no—
197#107—1811no—
198#107—1811no—
199#107C2×D18111yescharacteristic
200#107C2×D18111yescharacteristic
201#106—1811no—
202#106—1811no—
203#106C2×D18111yescharacteristic
204#106C2×D18111yescharacteristic
205C4⋊C8—1811no—
206C4⋊C8—1811no—
207#103—1811no—
208#103—1811no—
209#103—1811no—
210#103—1811no—
211#102—1811no—
212#102—1811no—
213#102—1811no—
214#102—1811no—
215C22⋊C8—1811no—
216C22⋊C8—1811no—
217C4×C8—1811no—
218C4×C8C2×D18111yescharacteristic
219C2×Q8—1811no—
220C2×D4—3621no—
221C2×D4—1811no—
222C22×C4—3621no—
223C22×C4—3621no—
224C22×C4—3621no—
225C22×C4—1811no—
226C22×C4—1811no—
227C22×C4—1811no—
228C2×C8—1811no—
229C2×C8—1811no—
230C2×C8C22×D18111yescharacteristic
231C2×C8C22×D18111yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—3621no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4—3621no—
246C4⋊C4—1811no—
247C4⋊C4—1811no—
248C4⋊C4C22×D18111yescharacteristic
249C4⋊C4C22×D18111yescharacteristic
250C4⋊C4C22×D18111yescharacteristic
251C4⋊C4C22×D18111yescharacteristic
252C22⋊C4—3621no—
253C22⋊C4—3621no—
254C22⋊C4—3621no—
255C22⋊C4—3621no—
256C22⋊C4—3621no—
257C42—3621no—
258C42—1811no—
259C42—1811no—
260C42—1811no—
261C42C22×D18111yescharacteristic
262C23—3621no—
263C23—1811no—
264Q8—3621no—
265Q8—3621no—
266D4—3621no—
267D4—3621no—
268D4—7242no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—3622no—
274C2×C4—3621no—
275C2×C4—3621no—
276C2×C4—3621no—
277C2×C4—3621no—
278C2×C4—3621no—
279C2×C4—1811no—
280C2×C4—1811no—
281C2×C4—1811no—
282C2×C4—7241no—
283C2×C4—7242no—
284C2×C4—7242no—
285C2×C4—7242no—
286C2×C4D4×D18111yescharacteristic
287C2×C4D4×D18111yescharacteristic
288C2×C4C23×D18111yescharacteristic
289C8—21no—
290C8—21no—
291C8—3621no—
292C8—3621no—
293C22—3621no—
294C22—3621no—
295C22—1812no—
296C22—7241no—
297C22—7241no—
298C22—7241no—
299C22C2×D4×D18111yescharacteristic
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—3621no—
307C4—3621no—
308C4—3621no—
309C4—7241no—
310C4—7241no—
311C4—7241no—
312C4#1915386411yescharacteristic
313C4#1915386411yescharacteristic
314C2—3622no—
315C2—14481no—
316C2#2604185411yescharacteristic
317C2#135×D18111yescharacteristic
318C2#2604207611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#499×C181C2
socleC22×C181C22×D4
prime radical#19153791C22
hypercenter#499D181
hypocenterC181#3232
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
2. #125×D181C2semidirect
3. #26041678C2semidirect
4. #26041636C2semidirect
5. #26041634C2semidirect
6. #26041616C2semidirect
7. #26041611C2semidirect
10. #26041589C2semidirect
11. #26041585C2semidirect
12. #26041582C2semidirect
13. #26041578C2semidirect
14. #26041330C2semidirect
15. #499×C181C2semidirect
159. C181#3232semidirect
164. #499D181semidirect

Maximal subgroups

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

subgroupindex
#260418142
#125×D1812
#260416782
#260416362
#260416342
#260416162
#260416112
#260415992
#260415942
#260415892
#260415852
#260415822
#260415782
#260413302
#499×C1812
#3232181