Finite Group Databasebeta

← #37546108

Subgroups — #37546108

Order 23168.

356 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041987C211yesmaximal, characteristic
2#26041984C211yesmaximal, characteristic
3#26041683C211yesmaximal, characteristic, complemented
4#26041616C211yesmaximal, characteristic, complemented
5#26041615C211yesmaximal, characteristic, complemented
6#26041613C211yesmaximal, characteristic, complemented
7#26041573C211yesmaximal, characteristic, complemented
8#26041572C211yesmaximal, characteristic, complemented
9#26041519C211yesmaximal, characteristic, complemented
10#26041518C211yesmaximal, characteristic, complemented
11#26041371C211yesmaximal, characteristic, complemented
12#26041342C211yesmaximal, characteristic
13C2×#19153749C211yesmaximal, characteristic, complemented
14#107×D181C211yesmaximal, characteristic, complemented
15#484×C181C211yesmaximal, characteristic, complemented
16C2×#15668439C2211yescharacteristic
17C22×#13089678C2211yescharacteristic
18C2×C8×D181C2211yescharacteristic
19#19153815C2211yescharacteristic
20#19153815C2211yescharacteristic
21#19153806—21no—
22#19153806—21no—
23#19153796—21no—
24#19153796C2211yescharacteristic
25#19153795—21no—
26#19153795C2211yescharacteristic
27#19153794—21no—
28#19153793—21no—
29#19153792C2211yescharacteristic
30Q8×Dic181C2211yescharacteristic
31#19153781C2211yescharacteristic
32C4×C22⋊D181—21no—
33C4×C22⋊D181C2211yescharacteristic
34#19153776—21no—
35C2×C4⋊Dic181C2211yescharacteristic
36C4⋊C4×D181C2211yescharacteristic
37C4⋊C4×D181C2211yescharacteristic
38#19153750C2211yescharacteristic
39#19153750C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153749C2211yescharacteristic
42#19153749C2211yescharacteristic
43#19153749C2211yescharacteristic
44#19153749C2211yescharacteristic
45#19153745C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153741C2211yescharacteristic
48#19153740C2211yescharacteristic
49#19153739C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C22⋊Q8×C181C2211yescharacteristic
53C2×C4⋊C4×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362C2311yescharacteristic
62C2×C22⋊D181D411yescharacteristic
63C22×Dic181—21no—
64C2×C4×D181—21no—
65C2×C4×D181—21no—
66C2×C4×D181C2311yescharacteristic
67#15668439—22no—
68C2×#13089678—22no—
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C8×D181—22no—
72C4⋊Dic181—21no—
73C4⋊Dic181—21no—
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77C4⋊Dic181C2311yescharacteristic
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428D411yescharacteristic
86C22⋊Dic181—21no—
87C22⋊Dic181D411yescharacteristic
88#15668426—21no—
89#15668426—21no—
90#15668426D411yescharacteristic
91C4×Dic181—21no—
92C4×Dic181—21no—
93C4×Dic181C2311yescharacteristic
94C2×Q8×C181C2311yescharacteristic
95C22×C4×C181—21no—
96C22×C4×C181C2311yescharacteristic
97C2×C8×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181—21no—
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C22×D181C2×D411yescharacteristic
106Dic362—21no—
107Dic362—21no—
108Dic362—81no—
109Dic362—81no—
110C22⋊D181Q1614yescomplemented
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—81no—
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C4×D181—21no—
120C4×D181—21no—
121C4×D181—81no—
122C4×D181—81no—
123#13089678—21no—
124#13089678#4112yes—
125C23×C181C2×D411yescharacteristic
126Q8×C181—21no—
127Q8×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—81no—
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2411yescharacteristic
137C8×C181—21no—
138C8×C181—21no—
139C22×C181—21no—
140C22×C181—21no—
141C22×C181C2×Q1612yes—
142C22×C181C22×D411yescharacteristic
143C2×D181—21no—
144C2×D181—21no—
145C2×D181C2×Q1612yes—
146C4×C181—21no—
147C4×C181—41no—
148C4×C181—41no—
149C4×C181—41no—
150C4×C181—41no—
151C4×C181C2×#4111yescharacteristic
152C4×C1812−1+411yescharacteristic
153Dic181—21no—
154Dic181—21no—
155Dic181—41no—
156Dic181—41no—
157Dic181—41no—
158Dic181—41no—
159Dic181C2×Q1612yes—
160C2×C181—22no—
161C2×C181#54711yescharacteristic
162C2×C181C22×Q1611yescharacteristic
163C2×C181#57611yescharacteristic
164D181—22no—
165C181#333811yescharacteristic, complemented, Sylow
166#3338—1811nomaximal, Sylow
167#554—1811no—
168#554—1811no—
169#500—1811no—
170#484—1811no—
171#484—1811no—
172#484—1811no—
173#484D18111yescharacteristic, complemented
174#477—1811no—
175#477—1811no—
176#467—1811no—
177#467—1811no—
178C4×Q16—1811no—
179C8×D4—1811no—
180C2×#107—1811no—
181C2×#107—1811no—
182C2×Q16—1811no—
183C22×C8—1811no—
184C22×C8—1811no—
185C4⋊Q8—1811no—
186C4⋊Q8—1811no—
187#125—3621no—
188#125—3621no—
189C22⋊Q8—3621no—
190C22⋊Q8—3621no—
191C22⋊Q8—3621no—
192C22⋊Q8—3621no—
193C22⋊Q8—1811no—
194C22⋊Q8—1811no—
195C22⋊Q8—1811no—
196C22⋊Q8C2×D18111yescharacteristic
197C4×Q8—1811no—
198C4×Q8—1811no—
199C4×D4—3621no—
200C4×D4—3621no—
201C4×D4—1811no—
202C2×C4⋊C4—1811no—
203C2×C4⋊C4—1811no—
204C2×C4⋊C4—1811no—
205C2×C4⋊C4C2×D18111yescharacteristic
206#107—1811no—
207#107—1811no—
208#107—1811no—
209#107—1811no—
210#107—1811no—
211#107—1811no—
212#107—1811no—
213#107C2×D18111yescharacteristic
214#107C2×D18111yescharacteristic
215C4⋊C8—1811no—
216#103—1811no—
217#103—1811no—
218#103—1811no—
219#103—1811no—
220#103C2×D18111yescharacteristic
221#103C2×D18111yescharacteristic
222C22⋊C8—1811no—
223C22⋊C8C2×D18111yescharacteristic
224C4×C8—1811no—
225C2×Q8—3621no—
226C2×Q8—3621no—
227C2×Q8—1811no—
228C2×Q8C22×D18111yescharacteristic
229C2×D4—1811no—
230C22×C4—21no—
231C22×C4—3621no—
232C22×C4—3621no—
233C22×C4—3621no—
234C22×C4—1811no—
235C22×C4C22×D18111yescharacteristic
236Q16—3622no—
237C2×C8—3622no—
238C2×C8—3622no—
239C2×C8—1811no—
240C2×C8—1811no—
241C2×C8C22×D18111yescharacteristic
242C2×C8C22×D18111yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4C22×D18111yescharacteristic
260C4⋊C4C22×D18111yescharacteristic
261C4⋊C4C22×D18111yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C22⋊C4—3621no—
266C22⋊C4—1811no—
267C22⋊C4—1811no—
268C42—3621no—
269C42—3621no—
270C42—1811no—
271C23—1811no—
272C23D4×D18111yescharacteristic
273Q8—21no—
274Q8—21no—
275Q8—3621no—
276Q8—3621no—
277Q8—14481no—
278Q8—14481no—
279D4—1814no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—3621no—
288C2×C4—3621no—
289C2×C4—3621no—
290C2×C4—3621no—
291C2×C4—3621no—
292C2×C4—3621no—
293C2×C4—14481no—
294C2×C4—14481no—
295C2×C4—14481no—
296C2×C4—1811no—
297C2×C4—1811no—
298C2×C4—1811no—
299C2×C4D4×D18111yescharacteristic
300C2×C4C23×D18111yescharacteristic
301C8—21no—
302C8—21no—
303C8—3621no—
304C8—1812no—
305C22—21no—
306C22—21no—
307C22—3621no—
308C22—3621no—
309C22—1812no—
310C22Q16×D18112yes—
311C22C2×D4×D18111yescharacteristic
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—3621no—
318C4—3621no—
319C4—1812no—
320C4—7241no—
321C4—7241no—
322C4—7241no—
323C4—7241no—
324C4#41×D18111yescharacteristic
325C4#1915386311yescharacteristic
326C2—22no—
327C2—3622no—
328C2#2604192711yescharacteristic
329C2C2×Q16×D18111yescharacteristic
330C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#484×C181C2
socleC22×C181C22×D4
prime radicalC2×C22⋊D181D4
hypercenter#484D181
hypocenterC181#3338
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #26041683C2semidirect
4. #26041616C2semidirect
5. #26041615C2semidirect
6. #26041613C2semidirect
7. #26041573C2semidirect
8. #26041572C2semidirect
9. #26041519C2semidirect
10. #26041518C2semidirect
11. #26041371C2semidirect
13. C2×#19153749C2semidirect
14. #107×D181C2semidirect
15. #484×C181C2semidirect
110. C22⋊D181Q16semidirect
165. C181#3338semidirect
173. #484D181semidirect

Maximal subgroups

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

subgroupindex
#260419872
#260419842
#260416832
#260416162
#260416152
#260416132
#260415732
#260415722
#260415192
#260415182
#260413712
#260413422
C2×#191537492
#107×D1812
#484×C1812
#3338181