Finite Group Databasebeta

← #37547832

Subgroups — #37547832

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#26042014C213yesmaximal, complemented
2#26042005C211yesmaximal, characteristic, complemented
3#26041992C213yesmaximal, complemented
4#26041973C213yesmaximal, complemented
5#26041970C213yesmaximal, complemented
6#26041951C211yesmaximal, characteristic, complemented
7#26041940C211yesmaximal, characteristic, complemented
8#26041940C211yesmaximal, characteristic, complemented
9#26041901C213yesmaximal, complemented
10#26041884C213yesmaximal, complemented
11#26041816C213yesmaximal, complemented
12#26041807C213yesmaximal, complemented
13#26041761C211yesmaximal, characteristic, complemented
14C4×#15668450C211yesmaximal, characteristic, complemented
15#519×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668450C2211yescharacteristic
18#19153815C2213yes—
19#19153813C2213yes—
20#19153811C2213yes—
21#19153810C2216yes—
22#19153809C2211yescharacteristic
23#19153808C2213yes—
24#19153807C2213yes—
25#19153806C2213yes—
26#19153805C2213yes—
27#19153805C2213yes—
28#19153804C2216yes—
29#19153802C2213yes—
30#19153801C2216yes—
31#19153801C2216yes—
32#19153801C2213yes—
33#19153798C2213yes—
34#19153794C2213yes—
35#19153792C2213yes—
36#19153792C2213yes—
37#19153791C2213yes—
38#19153791C2213yes—
39#19153790C2213yes—
40#19153790C2213yes—
41#19153789C2213yes—
42#19153789C2216yes—
43Q8×Dic181C2211yescharacteristic
44Q8×Dic181C2211yescharacteristic
45C4×Dic362C2213yes—
46C4×D724C2213yes—
47C4×D724C2213yes—
48C4×D724C2213yes—
49C4×D724C2216yes—
50#19153778C2213yes—
51#19153778C2213yes—
52C4×C22⋊D181C2213yes—
53C4×C22⋊D181C2213yes—
54#19153774C2213yes—
55#19153773C2213yes—
56#19153773C2216yes—
57C4⋊C4×D181C2213yes—
58C42×D181C2213yes—
59C22×Q8×C181C2211yescharacteristic
60C4×Q8×C181C2213yes—
61C4×Q8×C181C2211yescharacteristic
62C4×Q8×C181C2213yes—
63C4×Q8×C181C2211yescharacteristic
64#117×C181C2213yes—
65#117×C181C2213yes—
66#15668450—24no—
67#15668450—24no—
68C2×Dic362C2313yes—
69C2×D724C2313yes—
70C2×D724C2313yes—
71C2×D724C2316yes—
72C2×D724C2313yes—
73C2×C22⋊D181C2313yes—
74C2×C22⋊D181C2313yes—
75C2×C4×D181C2313yes—
76C2×C4×D181C2313yes—
77C2×C4×D181C2313yes—
78C2×C4×D181C2313yes—
79C2×C4×D181C2316yes—
80C4⋊Dic181C2313yes—
81C4⋊Dic181C2313yes—
82C4⋊Dic181C2311yescharacteristic
83C4⋊Dic181C2313yes—
84#15668428C2313yes—
85#15668428C2311yescharacteristic
86#15668428C2313yes—
87#15668428C2316yes—
88#15668428C2313yes—
89#15668428C2311yescharacteristic
90#15668428C2313yes—
91C22⋊Dic181C2311yescharacteristic
92C22⋊Dic181C2311yescharacteristic
93#15668426C2313yes—
94#15668426C2313yes—
95#15668426C2313yes—
96#15668426C2316yes—
97#15668426C2316yes—
98#15668426C2313yes—
99#15668426C2313yes—
100#15668426C2316yes—
101#15668426C2313yes—
102C4×Dic181C2311yescharacteristic
103C4×Dic181C2313yes—
104C4×Dic181C2313yes—
105C4×Dic181C2313yes—
106C2×Q8×C181—24no—
107C2×Q8×C181C2313yes—
108C2×Q8×C181C2311yescharacteristic
109C22×C4×C181C2313yes—
110C4⋊C4×C181C2313yes—
111C4⋊C4×C181C2313yes—
112C4⋊C4×C181C2313yes—
113C4⋊C4×C181C2313yes—
114C22⋊C4×C181C2311yescharacteristic
115C22⋊C4×C181C2313yes—
116C42×C181C2313yes—
117C42×C181C2316yes—
118C42×C181C2313yes—
119C22×D181C2413yes—
120C22×D181C2413yes—
121Dic362—43no—
122D724—26no—
123D724—26no—
124D724—43no—
125D724—46no—
126C22⋊D181—43no—
127C22⋊D181—43no—
128C2×Dic181C2411yescharacteristic
129C2×Dic181C2411yescharacteristic
130C2×Dic181C2411yescharacteristic
131C2×Dic181C2413yes—
132C2×Dic181C2413yes—
133C2×Dic181C2411yescharacteristic
134C4×D181—26no—
135C4×D181—26no—
136C4×D181—43no—
137C4×D181—46no—
138C4×D181—43no—
139C23×C181C2411yescharacteristic
140Q8×C181—26no—
141Q8×C181#4114yes—
142C2×C4×C181—26no—
143C2×C4×C181C2411yescharacteristic
144C2×C4×C181C2413yes—
145C2×C4×C181C2413yes—
146C2×C4×C181C2411yescharacteristic
147C2×C4×C181C2413yes—
148C2×C4×C181C2413yes—
149C22×C181—21no—
150C22×C181—21no—
151C22×C181—21no—
152C22×C181C2511yescharacteristic
153C2×D181—23no—
154C2×D181—23no—
155C2×D181—23no—
156C2×D181—23no—
157C2×D181—23no—
158C2×D181—23no—
159C4×C181—23no—
160C4×C181—23no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—23no—
164C4×C181C2×#4116yes—
165Dic181—21no—
166Dic181—21no—
167Dic181—21no—
168Dic181—23no—
169Dic181—23no—
170Dic181—21no—
171C2×C181—41no—
172C2×C181C22×#4111yescharacteristic
173C2×C181#58511yescharacteristic
174C2×C181#58511yescharacteristic
175D181—43no—
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#548—1811no—
187#542—1813no—
188#540—1813no—
189#529—1813no—
190#529—1813no—
191#520—1811no—
192#519D18111yescharacteristic, complemented
193C4×#41—1811no—
194C2×#41—1811no—
195C2×#41—1811no—
196C22×Q8C2×D18111yescharacteristic
197C4⋊Q8—1813no—
198#127—1813no—
199#126—1813no—
200#126—1811no—
201#126—1816no—
202#125—1813no—
203#125—1813no—
204#125—1813no—
205#124—1813no—
206#124—1813no—
207#124—1813no—
208#124—1816no—
209#123—1816no—
210#123—1813no—
211#123—1816no—
212#123—1813no—
213C22⋊Q8—1813no—
214C22⋊Q8—1813no—
215C22⋊Q8—1813no—
216#121—1813no—
217#121—1813no—
218#121—1813no—
219#121—1813no—
220#121—1813no—
221#121—1816no—
222C4×Q8—1811no—
223C4×Q8—1811no—
224C4×Q8—1813no—
225C4×Q8C2×D18111yescharacteristic
226C4×Q8C2×D18113yes—
227C4×Q8C2×D18111yescharacteristic
228C4×Q8C2×D18113yes—
229C4×D4—1813no—
230C4×D4—1813no—
231C4×D4—1813no—
232C4×D4—1813no—
233C4×D4—1813no—
234C4×D4—1813no—
235C4×D4—1813no—
236C4×D4—1816no—
237#117—1813no—
238#117—1813no—
239#117—1816no—
240#117C2×D18113yes—
241#117C2×D18113yes—
242C2×C4⋊C4—1813no—
243C2×C42—1813no—
244#41—3624no—
245#41—3624no—
246C2×Q8—24no—
247C2×Q8—1813no—
248C2×Q8C22×D18113yes—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—1813no—
251C2×D4—1813no—
252C2×D4—1813no—
253C2×D4—1813no—
254C2×D4—1816no—
255C2×D4—1813no—
256C22×C4—1813no—
257C22×C4—1813no—
258C22×C4—1813no—
259C22×C4—1813no—
260C22×C4—1816no—
261C22×C4C22×D18113yes—
262C4⋊C4—1813no—
263C4⋊C4—1813no—
264C4⋊C4—1811no—
265C4⋊C4—1813no—
266C4⋊C4—1813no—
267C4⋊C4—1816no—
268C4⋊C4—1811no—
269C4⋊C4—1813no—
270C4⋊C4—1813no—
271C4⋊C4—1811no—
272C4⋊C4—1813no—
273C4⋊C4C22×D18113yes—
274C4⋊C4C22×D18113yes—
275C4⋊C4C22×D18113yes—
276C4⋊C4C22×D18113yes—
277C22⋊C4—1811no—
278C22⋊C4—1811no—
279C22⋊C4—1813no—
280C22⋊C4—1813no—
281C22⋊C4—1816no—
282C22⋊C4—1813no—
283C22⋊C4—1816no—
284C22⋊C4—1813no—
285C22⋊C4—1813no—
286C22⋊C4—1816no—
287C22⋊C4—1813no—
288C22⋊C4C22×D18111yescharacteristic
289C22⋊C4C22×D18113yes—
290C42—1811no—
291C42—1813no—
292C42—1813no—
293C42—1813no—
294C42C22×D18113yes—
295C42C22×D18116yes—
296C42C22×D18113yes—
297C23—1813no—
298C23—1813no—
299C23C23×D18111yescharacteristic
300Q8—26no—
301Q8—7243no—
302Q8#1566844914yes—
303D4—3626no—
304D4—3626no—
305D4—7243no—
306D4—7246no—
307D4—7243no—
308D4—7243no—
309C2×C4—26no—
310C2×C4—3626no—
311C2×C4—3626no—
312C2×C4—1811no—
313C2×C4—1811no—
314C2×C4—1811no—
315C2×C4—1813no—
316C2×C4—1813no—
317C2×C4—1811no—
318C2×C4—7243no—
319C2×C4—7246no—
320C2×C4—7243no—
321C2×C4C23×D18111yescharacteristic
322C2×C4C23×D18113yes—
323C2×C4C23×D18113yes—
324C2×C4C23×D18111yescharacteristic
325C2×C4C23×D18113yes—
326C2×C4C23×D18113yes—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—3623no—
331C22—3623no—
332C22—3623no—
333C22—3623no—
334C22—3623no—
335C22—3623no—
336C22C24×D18111yescharacteristic
337C4—23no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—23no—
342C4—3621no—
343C4—3623no—
344C4—3623no—
345C4—3621no—
346C4—3621no—
347C4—3621no—
348C4C2×#1566844916yes—
349C2—41no—
350C2—7243no—
351C2—7243no—
352C2C22×#1566844911yescharacteristic
353C2#2604211611yescharacteristic
354C2#2604211611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#519×C181C2
hypercenter#519D181
hypocenterC181#3524
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #26042014C2semidirect
2. #26042005C2semidirect
3. #26041992C2semidirect
4. #26041973C2semidirect
5. #26041970C2semidirect
6. #26041951C2semidirect
7. #26041940C2semidirect
8. #26041940C2semidirect
9. #26041901C2semidirect
10. #26041884C2semidirect
11. #26041816C2semidirect
12. #26041807C2semidirect
13. #26041761C2semidirect
14. C4×#15668450C2semidirect
15. #519×C181C2semidirect
177. C181#3524semidirect
192. #519D181semidirect

Maximal subgroups

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

subgroupindex
#260420142
#260420052
#260419922
#260419732
#260419702
#260419512
#260419402
#260419012
#260418842
#260418162
#260418072
#260417612
C4×#156684502
#519×C1812
#3524181