Finite Group Databasebeta

← #37547823

Subgroups — #37547823

Order 23168.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042011C213yesmaximal, complemented
2#26042005C211yesmaximal, characteristic, complemented
3#26041993C213yesmaximal, complemented
4#26041975C213yesmaximal, complemented
5#26041971C213yesmaximal, complemented
6#26041948C211yesmaximal, characteristic, complemented
7#26041940C211yesmaximal, characteristic, complemented
8#26041902C213yesmaximal, complemented
9#26041882C213yesmaximal, complemented
10#26041811C213yesmaximal, complemented
11#26041808C213yesmaximal, complemented
12#26041762C211yesmaximal, characteristic, complemented
13#26041757C211yesmaximal, characteristic, complemented
14C4×#15668450C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17#19153815C2213yes—
18#19153812C2213yes—
19#19153811C2216yes—
20#19153809C2213yes—
21#19153809C2211yescharacteristic
22#19153808C2213yes—
23#19153807C2213yes—
24#19153807C2213yes—
25#19153805C2213yes—
26#19153803C2213yes—
27#19153802C2216yes—
28#19153800C2216yes—
29#19153798C2216yes—
30#19153798C2213yes—
31#19153798C2213yes—
32#19153796C2213yes—
33#19153792C2213yes—
34#19153790C2213yes—
35#19153790C2213yes—
36#19153788C2216yes—
37#19153787C2213yes—
38#19153787C2213yes—
39Q8×Dic181C2213yes—
40Q8×Dic181C2211yescharacteristic
41Q8×Dic181C2211yescharacteristic
42Q8×Dic181C2213yes—
43Q8×Dic181C2211yescharacteristic
44C4×Dic362C2213yes—
45C4×D724C2213yes—
46D4×Dic181C2213yes—
47D4×Dic181C2213yes—
48#19153778C2213yes—
49C4×C22⋊D181C2213yes—
50C4×C22⋊D181C2213yes—
51#19153776C2216yes—
52#19153775C2213yes—
53#19153775C2213yes—
54#19153775C2213yes—
55#19153774C2213yes—
56#19153772C2216yes—
57C2×C4⋊Dic181C2213yes—
58C42×D181C2213yes—
59C2×#41×C181C2211yescharacteristic
60C22×Q8×C181C2211yescharacteristic
61#124×C181C2213yes—
62C22⋊Q8×C181C2213yes—
63#121×C181C2213yes—
64C4×Q8×C181C2211yescharacteristic
65C4×D4×C181C2213yes—
66#15668450—24no—
67C2×Dic362C2313yes—
68C2×D724C2313yes—
69C2×C22⋊D181C2316yes—
70C2×C22⋊D181C2313yes—
71C2×C22⋊D181C2313yes—
72C22×Dic181C2316yes—
73C2×C4×D181C2313yes—
74C2×C4×D181C2313yes—
75C2×C4×D181C2313yes—
76C4⋊Dic181C2313yes—
77C4⋊Dic181C2313yes—
78C4⋊Dic181C2313yes—
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2313yes—
81C4⋊Dic181C2313yes—
82C4⋊Dic181C2313yes—
83C4⋊Dic181C2313yes—
84#15668428C2313yes—
85#15668428C2316yes—
86#15668428C2313yes—
87#15668428C2311yescharacteristic
88#15668428C2311yescharacteristic
89C22⋊Dic181C2313yes—
90C22⋊Dic181C2311yescharacteristic
91C22⋊Dic181C2313yes—
92C22⋊Dic181C2313yes—
93C22⋊Dic181C2313yes—
94C22⋊Dic181C2311yescharacteristic
95#15668426C2316yes—
96#15668426C2316yes—
97#15668426C2313yes—
98#15668426C2313yes—
99C4×Dic181C2311yescharacteristic
100C4×Dic181C2313yes—
101C4×Dic181C2313yes—
102C4×Dic181C2316yes—
103C4×Dic181C2313yes—
104C4×Dic181C2313yes—
105#41×C181—24no—
106C2×Q8×C181—24no—
107C2×Q8×C181C2313yes—
108C2×Q8×C181C2311yescharacteristic
109C2×D4×C181C2313yes—
110C2×D4×C181C2313yes—
111C22×C4×C181C2313yes—
112C22×C4×C181C2313yes—
113C4⋊C4×C181C2313yes—
114C4⋊C4×C181C2313yes—
115C22⋊C4×C181C2311yescharacteristic
116C22⋊C4×C181C2313yes—
117C22⋊C4×C181C2316yes—
118C42×C181C2313yes—
119C22×D181C2413yes—
120Dic362—43no—
121D724—26no—
122C22⋊D181—43no—
123C22⋊D181—43no—
124C22⋊D181—46no—
125C2×Dic181—46no—
126C2×Dic181C2413yes—
127C2×Dic181C2411yescharacteristic
128C2×Dic181C2411yescharacteristic
129C2×Dic181C2411yescharacteristic
130C2×Dic181C2413yes—
131C2×Dic181C2411yescharacteristic
132C2×Dic181C2413yes—
133C4×D181—26no—
134C4×D181—43no—
135C4×D181—43no—
136C23×C181C2413yes—
137C23×C181C2411yescharacteristic
138Q8×C181—26no—
139Q8×C181#4114yes—
140D4×C181—26no—
141D4×C181—43no—
142C2×C4×C181—26no—
143C2×C4×C181—26no—
144C2×C4×C181C2411yescharacteristic
145C2×C4×C181C2413yes—
146C2×C4×C181C2413yes—
147C2×C4×C181C2411yescharacteristic
148C2×C4×C181C2413yes—
149C22×C181—21no—
150C22×C181—23no—
151C22×C181—21no—
152C22×C181—23no—
153C22×C181—23no—
154C22×C181—21no—
155C22×C181C2511yescharacteristic
156C2×D181—23no—
157C2×D181—23no—
158C2×D181—23no—
159C4×C181—21no—
160C4×C181—23no—
161C4×C181—23no—
162C4×C181—21no—
163C4×C181C2×#4116yes—
164Dic181—21no—
165Dic181—21no—
166Dic181—23no—
167Dic181—23no—
168Dic181—21no—
169Dic181—23no—
170Dic181—21no—
171C2×C181—41no—
172C2×C181—43no—
173C2×C181C22×#4111yescharacteristic
174C2×C181#58511yescharacteristic
175C2×C181#58511yescharacteristic
176D181—43no—
177C181#352411yescharacteristic, complemented, Sylow
178#3524—1811nomaximal, Sylow
179#559—1813no—
180#557—1811no—
181#555—1813no—
182#553—1813no—
183#552—1813no—
184#550—1811no—
185#548—1811no—
186#548D18111yescharacteristic, complemented
187#542—1813no—
188#540—1813no—
189#529—1813no—
190#529—1813no—
191#520—1811no—
192#519—1811no—
193C4×#41—1811no—
194C2×#41—1811no—
195C2×#41C2×D18111yescharacteristic
196C22×Q8C2×D18111yescharacteristic
197C4⋊Q8—1813no—
198#127—1813no—
199#126—1816no—
200#126—1811no—
201#126—1813no—
202#125—1813no—
203#125—1813no—
204#125—1813no—
205#124—1813no—
206#124—1813no—
207#124—1816no—
208#124C2×D18113yes—
209#123—1816no—
210#123—1813no—
211#123—1813no—
212#123—1816no—
213C22⋊Q8—1813no—
214C22⋊Q8—1813no—
215C22⋊Q8C2×D18113yes—
216#121—1813no—
217#121—1813no—
218#121—1816no—
219#121—1813no—
220#121—1813no—
221#121C2×D18113yes—
222C4×Q8—1811no—
223C4×Q8—1813no—
224C4×Q8—1811no—
225C4×Q8—1813no—
226C4×Q8—1811no—
227C4×Q8—1813no—
228C4×Q8C2×D18111yescharacteristic
229C4×D4—1813no—
230C4×D4—1813no—
231C4×D4—1813no—
232C4×D4—1813no—
233C4×D4—1813no—
234C4×D4—1816no—
235C4×D4—1813no—
236C4×D4C2×D18113yes—
237#117—1813no—
238#117—1813no—
239#117—1813no—
240#117—1813no—
241#117—1816no—
242C2×C4⋊C4—1813no—
243C2×C42—1813no—
244#41—24no—
245#41—3624no—
246C2×Q8—24no—
247C2×Q8—1813no—
248C2×Q8C22×D18113yes—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—1813no—
251C2×D4—1816no—
252C2×D4—1813no—
253C2×D4—1813no—
254C2×D4C22×D18113yes—
255C2×D4C22×D18113yes—
256C22×C4—1813no—
257C22×C4—1816no—
258C22×C4—1813no—
259C22×C4—1813no—
260C22×C4C22×D18113yes—
261C22×C4C22×D18113yes—
262C4⋊C4—1813no—
263C4⋊C4—1813no—
264C4⋊C4—1813no—
265C4⋊C4—1811no—
266C4⋊C4—1813no—
267C4⋊C4—1813no—
268C4⋊C4—1813no—
269C4⋊C4—1813no—
270C4⋊C4—1816no—
271C4⋊C4—1811no—
272C4⋊C4—1813no—
273C4⋊C4—1811no—
274C4⋊C4—1813no—
275C4⋊C4C22×D18113yes—
276C4⋊C4C22×D18113yes—
277C22⋊C4—1811no—
278C22⋊C4—1813no—
279C22⋊C4—1813no—
280C22⋊C4—1813no—
281C22⋊C4—1813no—
282C22⋊C4—1811no—
283C22⋊C4—1816no—
284C22⋊C4—1813no—
285C22⋊C4—1816no—
286C22⋊C4—1813no—
287C22⋊C4C22×D18116yes—
288C22⋊C4C22×D18113yes—
289C22⋊C4C22×D18111yescharacteristic
290C42—1811no—
291C42—1813no—
292C42—1813no—
293C42—1816no—
294C42—1813no—
295C42—1813no—
296C42C22×D18113yes—
297C23—1813no—
298C23C23×D18111yescharacteristic
299C23C23×D18113yes—
300Q8—26no—
301Q8—7243no—
302Q8#1566844814yes—
303D4—26no—
304D4—43no—
305D4—3626no—
306D4—7246no—
307D4—7243no—
308D4—7243no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—3626no—
312C2×C4—1811no—
313C2×C4—1813no—
314C2×C4—1811no—
315C2×C4—1811no—
316C2×C4—1813no—
317C2×C4—1811no—
318C2×C4—1813no—
319C2×C4—7243no—
320C2×C4—7243no—
321C2×C4—7246no—
322C2×C4C23×D18113yes—
323C2×C4C23×D18111yescharacteristic
324C2×C4C23×D18113yes—
325C2×C4C23×D18111yescharacteristic
326C2×C4C23×D18113yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—21no—
332C22—23no—
333C22—3623no—
334C22—3623no—
335C22—3623no—
336C22C24×D18111yescharacteristic
337C4—23no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—3621no—
342C4—3623no—
343C4—3621no—
344C4—3623no—
345C4—3623no—
346C4—3621no—
347C4—3621no—
348C4C2×#1566844816yes—
349C2—41no—
350C2—43no—
351C2—7243no—
352C2C22×#1566844811yescharacteristic
353C2#2604211511yescharacteristic
354C2#2604211611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#548×C181C2
hypercenter#548D181
hypocenterC181#3524
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #26042011C2semidirect
2. #26042005C2semidirect
3. #26041993C2semidirect
4. #26041975C2semidirect
5. #26041971C2semidirect
6. #26041948C2semidirect
7. #26041940C2semidirect
8. #26041902C2semidirect
9. #26041882C2semidirect
10. #26041811C2semidirect
11. #26041808C2semidirect
12. #26041762C2semidirect
13. #26041757C2semidirect
14. C4×#15668450C2semidirect
15. #548×C181C2semidirect
177. C181#3524semidirect
186. #548D181semidirect

Maximal subgroups

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

subgroupindex
#260420112
#260420052
#260419932
#260419752
#260419712
#260419482
#260419402
#260419022
#260418822
#260418112
#260418082
#260417622
#260417572
C4×#156684502
#548×C1812
#3524181