Finite Group Databasebeta

← #37547717

Subgroups — #37547717

Order 23168.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042002C213yesmaximal, complemented
2#26041990C213yesmaximal, complemented
3#26041982C213yesmaximal, complemented
4#26041967C213yesmaximal, complemented
5Q8×D724C211yesmaximal, characteristic, complemented
6#26041941C211yesmaximal, characteristic, complemented
7#26041941C211yesmaximal, characteristic, complemented
8#26041903C213yesmaximal, complemented
9#26041892C213yesmaximal, complemented
10#26041816C213yesmaximal, complemented
11#26041814C213yesmaximal, complemented
12#26041761C211yesmaximal, characteristic, complemented
13#26041755C211yesmaximal, characteristic, complemented
14C4×Q8×D181C211yesmaximal, characteristic, complemented
15#557×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19#19153816C2213yes—
20#19153811C2216yes—
21#19153811C2216yes—
22#19153811C2213yes—
23#19153811C2213yes—
24#19153807C2213yes—
25#19153807C2216yes—
26#19153805C2213yes—
27#19153804C2213yes—
28#19153804C2213yes—
29#19153801C2216yes—
30#19153801C2213yes—
31#19153801C2211yescharacteristic
32#19153796C2213yes—
33#19153796C2213yes—
34#19153796C2213yes—
35#19153796C2213yes—
36#19153792C2216yes—
37#19153792C2213yes—
38#19153791C2213yes—
39#19153791C2213yes—
40#19153791C2213yes—
41Q8×Dic181C2211yescharacteristic
42C4×Dic362C2213yes—
43C4×Dic362C2213yes—
44#19153781C2213yes—
45#19153781C2213yes—
46C4×D724C2213yes—
47C4×D724C2213yes—
48C4×D724C2213yes—
49C4×D724C2213yes—
50#19153778C2213yes—
51#19153778C2216yes—
52#19153774C2213yes—
53#19153774C2216yes—
54#19153773C2213yes—
55#19153773C2213yes—
56C4⋊C4×D181C2213yes—
57C4⋊C4×D181C2213yes—
58C42×D181C2213yes—
59C4⋊Q8×C181C2213yes—
60#125×C181C2213yes—
61#125×C181C2213yes—
62C4×Q8×C181C2211yescharacteristic
63C4×Q8×C181C2211yescharacteristic
64C4×Q8×C181C2211yescharacteristic
65C4×Q8×C181C2213yes—
66#15668450—24no—
67C2×Dic362C2313yes—
68C2×Dic362C2313yes—
69Q8×D181—24no—
70Q8×D181—24no—
71C2×D724C2313yes—
72C2×D724C2313yes—
73C2×D724C2311yescharacteristic
74C2×D724C2313yes—
75C2×C4×D181C2311yescharacteristic
76C2×C4×D181C2313yes—
77C2×C4×D181C2313yes—
78C2×C4×D181C2313yes—
79C2×C4×D181C2313yes—
80C2×C4×D181C2316yes—
81C4⋊Dic181C2313yes—
82C4⋊Dic181C2316yes—
83C4⋊Dic181C2313yes—
84C4⋊Dic181C2313yes—
85C4⋊Dic181C2313yes—
86#15668428C2313yes—
87#15668428C2316yes—
88#15668428C2311yescharacteristic
89#15668428C2313yes—
90#15668428C2313yes—
91#15668426C2311yescharacteristic
92#15668426C2313yes—
93#15668426C2313yes—
94#15668426C2313yes—
95#15668426C2313yes—
96#15668426C2316yes—
97#15668426C2313yes—
98#15668426C2316yes—
99#15668426C2311yescharacteristic
100#15668426C2311yescharacteristic
101#15668426C2313yes—
102#15668426C2313yes—
103C4×Dic181C2313yes—
104C4×Dic181C2313yes—
105C4×Dic181C2313yes—
106C4×Dic181C2316yes—
107C2×Q8×C181C2313yes—
108C2×Q8×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C4⋊C4×C181C2313yes—
111C4⋊C4×C181C2313yes—
112C4⋊C4×C181C2313yes—
113C4⋊C4×C181C2313yes—
114C4⋊C4×C181C2313yes—
115C4⋊C4×C181C2316yes—
116C42×C181C2313yes—
117C42×C181C2313yes—
118C42×C181C2313yes—
119C22×D181C2411yescharacteristic
120C22×D181C2413yes—
121C22×D181C2411yescharacteristic
122Dic362—26no—
123Dic362—26no—
124D724—26no—
125D724—22no—
126D724—26no—
127D724—43no—
128C2×Dic181C2411yescharacteristic
129C2×Dic181C2413yes—
130C2×Dic181C2413yes—
131C2×Dic181C2413yes—
132C2×Dic181C2411yescharacteristic
133C4×D181—22no—
134C4×D181—26no—
135C4×D181—26no—
136C4×D181—26no—
137C4×D181—46no—
138C4×D181—43no—
139Q8×C181—43no—
140Q8×C181#4114yes—
141C2×C4×C181C2411yescharacteristic
142C2×C4×C181C2411yescharacteristic
143C2×C4×C181C2411yescharacteristic
144C2×C4×C181C2413yes—
145C2×C4×C181C2413yes—
146C2×C4×C181C2413yes—
147C2×C4×C181C2413yes—
148C22×C181C2511yescharacteristic
149C2×D181—21no—
150C2×D181—21no—
151C2×D181—21no—
152C2×D181—23no—
153C2×D181—21no—
154C2×D181—23no—
155C2×D181—23no—
156C2×D181—21no—
157C2×D1812−1+412yes—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—23no—
161C4×C181—21no—
162C4×C181—23no—
163C4×C181—23no—
164C4×C181C2×#4116yes—
165Dic181—21no—
166Dic181—23no—
167Dic181—23no—
168Dic181—21no—
169Dic181—23no—
170C2×C181C22×#4111yescharacteristic
171C2×C181C2×2−1+411yescharacteristic
172C2×C181#58511yescharacteristic
173D181—22no—
174D181—41no—
175D181—43no—
176C181#351611yescharacteristic, complemented, Sylow
177#3516—1811nomaximal, Sylow
178#557D18111yescharacteristic, complemented
179#556—1813no—
180#554—1813no—
181#553—1813no—
182#552—1813no—
183D4×Q8—1811no—
184#548—1811no—
185#548—1811no—
186#542—1813no—
187#541—1813no—
188#529—1813no—
189#529—1813no—
190#520—1811no—
191#519—1811no—
192C2×C4×Q8—1811no—
193C2×#41—1811no—
194C22×Q8—1811no—
195C22×Q8—1811no—
196C4⋊Q8—1813no—
197C4⋊Q8C2×D18113yes—
198#126—1816no—
199#126—1813no—
200#126—1813no—
201#126—1816no—
202#125—1813no—
203#125—1816no—
204#125C2×D18113yes—
205#125C2×D18113yes—
206#124—1813no—
207#124—1813no—
208#124—1813no—
209#123—1811no—
210#123—1813no—
211#123—1816no—
212C22⋊Q8—1813no—
213C22⋊Q8—1816no—
214C22⋊Q8—1813no—
215C22⋊Q8—1813no—
216C22⋊Q8—1813no—
217C22⋊Q8—1813no—
218#121—1813no—
219#121—1813no—
220#121—1813no—
221C4×Q8—1811no—
222C4×Q8—1813no—
223C4×Q8—1813no—
224C4×Q8—1813no—
225C4×Q8—1813no—
226C4×Q8C2×D18111yescharacteristic
227C4×Q8C2×D18111yescharacteristic
228C4×Q8C2×D18111yescharacteristic
229C4×Q8C2×D18113yes—
230C4×D4—1813no—
231C4×D4—1816no—
232C4×D4—1813no—
233C4×D4—1813no—
234C4×D4—1813no—
235C4×D4—1813no—
236#117—1813no—
237#117—1816no—
238#117—1813no—
239#117—1813no—
240C2×C4⋊C4—1813no—
241C2×C4⋊C4—1813no—
242C2×C42—1813no—
243#41—3624no—
244C2×Q8—3624no—
245C2×Q8—3624no—
246C2×Q8—1813no—
247C2×Q8—1813no—
248C2×Q8C22×D18113yes—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—1813no—
251C2×D4—1811no—
252C2×D4—1813no—
253C2×D4—1813no—
254C22×C4—1811no—
255C22×C4—1813no—
256C22×C4—1813no—
257C22×C4—1813no—
258C22×C4—1813no—
259C22×C4—1816no—
260C4⋊C4—1813no—
261C4⋊C4—1813no—
262C4⋊C4—1816no—
263C4⋊C4—1811no—
264C4⋊C4—1813no—
265C4⋊C4—1813no—
266C4⋊C4—1816no—
267C4⋊C4—1813no—
268C4⋊C4—1813no—
269C4⋊C4—1813no—
270C4⋊C4C22×D18113yes—
271C4⋊C4C22×D18113yes—
272C4⋊C4C22×D18113yes—
273C4⋊C4C22×D18113yes—
274C4⋊C4C22×D18116yes—
275C4⋊C4C22×D18111yescharacteristic
276C4⋊C4C22×D18113yes—
277C22⋊C4—1813no—
278C22⋊C4—1811no—
279C22⋊C4—1813no—
280C22⋊C4—1813no—
281C22⋊C4—1813no—
282C22⋊C4—1816no—
283C22⋊C4—1813no—
284C22⋊C4—1811no—
285C22⋊C4—1811no—
286C22⋊C4—1816no—
287C22⋊C4—1813no—
288C22⋊C4—1813no—
289C42—1813no—
290C42—1813no—
291C42—1813no—
292C42—1816no—
293C42C22×D18113yes—
294C42C22×D18113yes—
295C42C22×D18113yes—
296C23—1813no—
297C23—1811no—
298C23—1811no—
299Q8—43no—
300Q8—3626no—
301Q8—3626no—
302Q8#1566845014yes—
303D4—3626no—
304D4—3622no—
305D4—3626no—
306D4—7243no—
307C2×C4—3622no—
308C2×C4—3626no—
309C2×C4—3626no—
310C2×C4—3626no—
311C2×C4—1811no—
312C2×C4—1813no—
313C2×C4—1813no—
314C2×C4—1813no—
315C2×C4—1811no—
316C2×C4—7246no—
317C2×C4—7243no—
318C2×C4C23×D18111yescharacteristic
319C2×C4C23×D18113yes—
320C2×C4C23×D18113yes—
321C2×C4C23×D18111yescharacteristic
322C2×C4C23×D18111yescharacteristic
323C2×C4C23×D18113yes—
324C2×C4C23×D18113yes—
325C22—3621no—
326C22—3621no—
327C22—3623no—
328C22—3621no—
329C22—3621no—
330C22—3623no—
331C22—3623no—
332C22—3621no—
333C22—1812no—
334C22C24×D18111yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—3623no—
342C4—3621no—
343C4—3623no—
344C4—3623no—
345C4—3621no—
346C4C2×#1566845016yes—
347C2—3622no—
348C2—7243no—
349C2—7241no—
350C2C22×#1566845011yescharacteristic
351C22−1+4×D18111yescharacteristic
352C2#2604211511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#557×C181C2
hypercenter#557D181
hypocenterC181#3516
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #26042002C2semidirect
2. #26041990C2semidirect
3. #26041982C2semidirect
4. #26041967C2semidirect
5. Q8×D724C2semidirect
6. #26041941C2semidirect
7. #26041941C2semidirect
8. #26041903C2semidirect
9. #26041892C2semidirect
10. #26041816C2semidirect
11. #26041814C2semidirect
12. #26041761C2semidirect
13. #26041755C2semidirect
14. C4×Q8×D181C2semidirect
15. #557×C181C2semidirect
176. C181#3516semidirect
178. #557D181semidirect

Maximal subgroups

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

subgroupindex
#260420022
#260419902
#260419822
#260419672
Q8×D7242
#260419412
#260419032
#260418922
#260418162
#260418142
#260417612
#260417552
C4×Q8×D1812
#557×C1812
#3516181