Finite Group Databasebeta

← #37547546

Subgroups — #37547546

Order 23168.

988 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#26042008C211yesmaximal, characteristic, complemented
2#26042005C211yesmaximal, characteristic, complemented
3#26042005C211yesmaximal, characteristic, complemented
4#26041990C213yesmaximal, complemented
5#26041989C213yesmaximal, complemented
6#26041985C213yesmaximal, complemented
7#26041985C213yesmaximal, complemented
8#26041909C213yesmaximal, complemented
9#26041906C213yesmaximal, complemented
10#26041833C213yesmaximal, complemented
11#125×D181C213yesmaximal, complemented
12#26041761C211yesmaximal, characteristic
13#26041761C211yesmaximal, characteristic
14C4×#15668450C211yesmaximal, characteristic
15#557×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17#19153816C2213yes—
18#19153815C2213yes—
19#19153815C2213yes—
20#19153815C2213yes—
21#19153814C2213yes—
22#19153808C2213yes—
23#19153808C2213yes—
24#19153807C2213yes—
25#19153807C2213yes—
26#19153807C2213yes—
27#19153807C2213yes—
28#19153806C2216yes—
29#19153806C2213yes—
30#19153806C2211yescharacteristic
31#19153796C2216yes—
32#19153796C2213yes—
33#19153794C2216yes—
34#19153794C2216yes—
35#19153794C2213yes—
36#19153794C2213yes—
37#19153792C2216yes—
38#19153792C2213yes—
39Q8×Dic181C2211yescharacteristic
40Q8×Dic181C2211yescharacteristic
41Q8×Dic181C2211yescharacteristic
42C4×Dic362C2213yes—
43C4×Dic362C2213yes—
44C4×Dic362C2213yes—
45C4×D724C2213yes—
46C4×D724C2213yes—
47C4×D724C2213yes—
48#19153778C2213yes—
49#19153778C2213yes—
50#19153778C2213yes—
51#19153774C2213yes—
52#19153773C2213yes—
53#19153773C2213yes—
54C4⋊C4×D181C2213yes—
55C4⋊C4×D181C2213yes—
56C4⋊C4×D181C2216yes—
57C4⋊C4×D181C2216yes—
58C42×D181C2213yes—
59C4⋊Q8×C181C2213yes—
60#125×C181C2213yes—
61#125×C181C2213yes—
62C4×Q8×C181C2213yes—
63C4×Q8×C181C2211yescharacteristic
64C4×Q8×C181C2211yescharacteristic
65C4×Q8×C181C2211yescharacteristic
66#15668450Q818yescomplemented
67C2×Dic362C2313yes—
68C2×Dic362C2313yes—
69C2×Dic362C2313yes—
70C2×D724C2313yes—
71C2×C4×D181C2313yes—
72C2×C4×D181C2313yes—
73C2×C4×D181C2316yes—
74C2×C4×D181C2316yes—
75C2×C4×D181C2313yes—
76C4⋊Dic181C2313yes—
77C4⋊Dic181C2313yes—
78C4⋊Dic181C2313yes—
79C4⋊Dic181C2316yes—
80C4⋊Dic181C2313yes—
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2313yes—
83#15668428C2313yes—
84#15668428C2313yes—
85#15668428C2313yes—
86#15668428C2313yes—
87#15668428C2316yes—
88#15668428C2316yes—
89#15668428C2311yescharacteristic
90#15668428C2311yescharacteristic
91#15668428C2311yescharacteristic
92#15668428C2311yescharacteristic
93#15668428C2313yes—
94#15668428C2313yes—
95#15668426C2313yes—
96#15668426C2313yes—
97#15668426C2313yes—
98#15668426C2313yes—
99#15668426C2316yes—
100C4×Dic181C2311yescharacteristic
101C4×Dic181C2313yes—
102C4×Dic181C2313yes—
103C4×Dic181C2313yes—
104C4×Dic181C2313yes—
105C2×Q8×C181C2313yes—
106C2×Q8×C181C2311yescharacteristic
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2313yes—
109C4⋊C4×C181C2313yes—
110C4⋊C4×C181C2313yes—
111C4⋊C4×C181C2313yes—
112C4⋊C4×C181C2313yes—
113C4⋊C4×C181C2316yes—
114C42×C181C2313yes—
115C42×C181C2313yes—
116C42×C181C2313yes—
117C22×D181C2413yes—
118Dic362—43no—
119Dic362—43no—
120Dic362—43no—
121D724C2×Q8112yes—
122C2×Dic181C2411yescharacteristic
123C2×Dic181C2411yescharacteristic
124C2×Dic181C2411yescharacteristic
125C2×Dic181C2413yes—
126C2×Dic181C2413yes—
127C2×Dic181C2413yes—
128C2×Dic181C2411yescharacteristic
129C4×D181—43no—
130C4×D181—46no—
131C4×D181—46no—
132C4×D181—43no—
133C4×D181C2×Q8112yes—
134Q8×C181—43no—
135Q8×C181C2×Q814yes—
136C2×C4×C181C2411yescharacteristic
137C2×C4×C181C2411yescharacteristic
138C2×C4×C181C2411yescharacteristic
139C2×C4×C181C2413yes—
140C2×C4×C181C2413yes—
141C2×C4×C181C2413yes—
142C2×C4×C181C2413yes—
143C22×C181C2511yescharacteristic
144C2×D181—23no—
145C2×D181—23no—
146C2×D181C22×Q816yes—
147C4×C181—21no—
148C4×C181—23no—
149C4×C181—23no—
150C4×C181—21no—
151C4×C181—23no—
152C4×C181—21no—
153C4×C181C22×Q816yes—
154Dic181—21no—
155Dic181—23no—
156Dic181—23no—
157Dic181—23no—
158Dic181—21no—
159Dic181—21no—
160Dic181C22×Q812yes—
161C2×C181C23×Q811yescharacteristic
162C2×C181#58511yescharacteristic
163C2×C181#58511yescharacteristic
164D181—26no—
165C181#350411yescharacteristic, complemented, Sylow
166#3504—1811nomaximal, Sylow
167#557—1811no—
168#557—1811no—
169#557—1811no—
170#557D18111yescharacteristic, complemented
171#554—1813no—
172#554—1813no—
173#554—1813no—
174#554—1813no—
175#544—1813no—
176#544—1813no—
177#533—1813no—
178C2×#125—1813no—
179#520—1811no—
180#520—1811no—
181C4×#41—1811no—
182C2×#41—1811no—
183C4⋊Q8—1813no—
184C4⋊Q8—1813no—
185C4⋊Q8—1813no—
186C4⋊Q8—1813no—
187C4⋊Q8—1813no—
188C4⋊Q8C2×D18113yes—
189#125—1813no—
190#125—1816no—
191#125—1813no—
192#125—1813no—
193#125—1811no—
194#125—1813no—
195#125—1813no—
196#125—1813no—
197#125—1813no—
198#125C2×D18113yes—
199#125C2×D18113yes—
200C22⋊Q8—1813no—
201C22⋊Q8—1816no—
202C22⋊Q8—1813no—
203C22⋊Q8—1813no—
204C22⋊Q8—1813no—
205C22⋊Q8—1816no—
206C22⋊Q8—1816no—
207C22⋊Q8—1816no—
208C4×Q8—1811no—
209C4×Q8—1811no—
210C4×Q8—1811no—
211C4×Q8—1813no—
212C4×Q8—1813no—
213C4×Q8—1813no—
214C4×Q8C2×D18111yescharacteristic
215C4×Q8C2×D18111yescharacteristic
216C4×Q8C2×D18111yescharacteristic
217C4×Q8C2×D18113yes—
218C4×D4—1813no—
219C4×D4—1813no—
220C4×D4—1813no—
221C4×D4—1813no—
222C4×D4—1813no—
223C4×D4—1813no—
224#117—1813no—
225#117—1813no—
226#117—1813no—
227C2×C4⋊C4—1813no—
228C2×C4⋊C4—1813no—
229C2×C4⋊C4—1816no—
230C2×C4⋊C4—1816no—
231C2×C42—1813no—
232#41—1818no—
233C2×Q8—1813no—
234C2×Q8—1813no—
235C2×Q8—1813no—
236C2×Q8C22×D18113yes—
237C2×Q8C22×D18111yescharacteristic
238C2×D4—1813no—
239C22×C4—1813no—
240C22×C4—1813no—
241C22×C4—1813no—
242C22×C4—1816no—
243C22×C4—1816no—
244C4⋊C4—1813no—
245C4⋊C4—1813no—
246C4⋊C4—1813no—
247C4⋊C4—1813no—
248C4⋊C4—1813no—
249C4⋊C4—1813no—
250C4⋊C4—1813no—
251C4⋊C4—1816no—
252C4⋊C4—1816no—
253C4⋊C4—1816no—
254C4⋊C4—1811no—
255C4⋊C4—1811no—
256C4⋊C4—1813no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4—1811no—
260C4⋊C4—1813no—
261C4⋊C4—1813no—
262C4⋊C4—1813no—
263C4⋊C4C22×D18113yes—
264C4⋊C4C22×D18113yes—
265C4⋊C4C22×D18113yes—
266C4⋊C4C22×D18116yes—
267C4⋊C4C22×D18113yes—
268C4⋊C4C22×D18111yescharacteristic
269C4⋊C4C22×D18113yes—
270C22⋊C4—1813no—
271C22⋊C4—1813no—
272C22⋊C4—1816no—
273C22⋊C4—1813no—
274C22⋊C4—1813no—
275C42—1811no—
276C42—1813no—
277C42—1813no—
278C42—1813no—
279C42—1813no—
280C42C22×D18113yes—
281C42C22×D18113yes—
282C42C22×D18113yes—
283C23—1813no—
284Q8—43no—
285Q8—7243no—
286Q8—7243no—
287Q8—7243no—
288Q8Q8×D18114yes—
289D4—18112no—
290C2×C4—1811no—
291C2×C4—1811no—
292C2×C4—1811no—
293C2×C4—1813no—
294C2×C4—1813no—
295C2×C4—1813no—
296C2×C4—1811no—
297C2×C4—18112no—
298C2×C4—7246no—
299C2×C4—7243no—
300C2×C4—7243no—
301C2×C4—7246no—
302C2×C4C23×D18111yescharacteristic
303C2×C4C23×D18113yes—
304C2×C4C23×D18111yescharacteristic
305C2×C4C23×D18111yescharacteristic
306C2×C4C23×D18113yes—
307C2×C4C23×D18113yes—
308C2×C4C23×D18113yes—
309C22—3623no—
310C22—3623no—
311C22—1816no—
312C22C24×D18111yescharacteristic
313C4—23no—
314C4—23no—
315C4—21no—
316C4—21no—
317C4—23no—
318C4—21no—
319C4—3621no—
320C4—3623no—
321C4—3621no—
322C4—3623no—
323C4—3623no—
324C4—3621no—
325C4—1812no—
326C4C2×Q8×D18116yes—
327C2—3626no—
328C2C22×Q8×D18111yescharacteristic
329C2#2604211511yescharacteristic
330C2#2604211611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#557×C181C2
prime radicalC2×#15668450C22
hypercenter#557D181
hypocenterC181#3504
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #26042008C2semidirect
2. #26042005C2semidirect
3. #26042005C2semidirect
4. #26041990C2semidirect
5. #26041989C2semidirect
6. #26041985C2semidirect
7. #26041985C2semidirect
8. #26041909C2semidirect
9. #26041906C2semidirect
10. #26041833C2semidirect
11. #125×D181C2semidirect
15. #557×C181C2semidirect
66. #15668450Q8semidirect
165. C181#3504semidirect
170. #557D181semidirect

Maximal subgroups

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

subgroupindex
#260420082
#260420052
#260419902
#260419892
#260419852
#260419092
#260419062
#260418332
#125×D1812
#260417612
C4×#156684502
#557×C1812
#3504181