Finite Group Databasebeta

← #37547319

Subgroups — #37547319

Order 23168.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042012C213yesmaximal, complemented
2#26042005C211yesmaximal, characteristic, complemented
3#26041997C213yesmaximal, complemented
4#26041979C213yesmaximal, complemented
5#26041973C213yesmaximal, complemented
6#26041948C211yesmaximal, characteristic, complemented
7#26041941C211yesmaximal, characteristic, complemented
8#26041936C211yesmaximal, characteristic, complemented
9#26041826C213yesmaximal, complemented
10#26041813C213yesmaximal, complemented
11#26041811C213yesmaximal, complemented
12C4×#15668450C211yesmaximal, characteristic, complemented
13C4×#15668449C213yesmaximal, complemented
14C4×Q8×D181C211yesmaximal, characteristic, complemented
15C4×#41×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668449C2213yes—
18C2×Q8×D181C2211yescharacteristic
19#19153815C2213yes—
20#19153812C2213yes—
21#19153810C2216yes—
22#19153809C2216yes—
23#19153808C2213yes—
24#19153807C2213yes—
25#19153804C2213yes—
26#19153803C2213yes—
27#19153802C2216yes—
28#19153799C2216yes—
29#19153798C2216yes—
30#19153796C2213yes—
31#19153795C2213yes—
32#19153791C2213yes—
33#19153790C2213yes—
34#19153789C2216yes—
35#19153787C2213yes—
36#19153787C2213yes—
37Q8×Dic181C2211yescharacteristic
38Q8×Dic181C2211yescharacteristic
39C4×Dic362C2213yes—
40C4×Dic362C2213yes—
41C4×Dic362C2213yes—
42#19153781C2213yes—
43C4×D724C2213yes—
44C4×D724C2213yes—
45C4×D724C2213yes—
46#19153778C2213yes—
47C4×C22⋊D181C2213yes—
48C4×C22⋊D181C2213yes—
49C4×C22⋊D181C2213yes—
50C4×C22⋊D181C2213yes—
51C4×C22⋊D181C2216yes—
52#19153775C2213yes—
53#19153774C2213yes—
54#19153773C2211yescharacteristic
55#19153773C2213yes—
56C4⋊C4×D181C2213yes—
57C42×D181C2213yes—
58C42×D181C2213yes—
59C2×#41×C181C2211yescharacteristic
60C4×Q8×C181C2211yescharacteristic
61C4×Q8×C181C2211yescharacteristic
62C4×D4×C181C2213yes—
63C4×D4×C181C2213yes—
64#117×C181C2213yes—
65C2×C42×C181C2213yes—
66#15668450—24no—
67#15668449—212no—
68C2×Dic362C2313yes—
69C2×Dic362C2313yes—
70Q8×D181—24no—
71C2×D724C2313yes—
72C2×D724C2313yes—
73C2×C22⋊D181C2313yes—
74C2×C22⋊D181C2313yes—
75C2×C22⋊D181C2316yes—
76C2×C4×D181C2313yes—
77C2×C4×D181C2313yes—
78C2×C4×D181C2313yes—
79C2×C4×D181C2313yes—
80C2×C4×D181C2313yes—
81C2×C4×D181C2311yescharacteristic
82C4⋊Dic181C2313yes—
83C4⋊Dic181C2313yes—
84C4⋊Dic181C2313yes—
85C4⋊Dic181C2313yes—
86#15668428C2316yes—
87#15668428C2311yescharacteristic
88#15668428C2311yescharacteristic
89#15668428C2313yes—
90#15668428C2313yes—
91#15668428C2316yes—
92C22⋊Dic181C2313yes—
93C22⋊Dic181C2313yes—
94C22⋊Dic181C2316yes—
95#15668426C2311yescharacteristic
96#15668426C2311yescharacteristic
97#15668426C2313yes—
98#15668426C2313yes—
99#15668426C2316yes—
100#15668426C2316yes—
101C4×Dic181C2311yescharacteristic
102C4×Dic181C2313yes—
103C4×Dic181C2313yes—
104C4×Dic181C2313yes—
105C4×Dic181C2313yes—
106C4×Dic181C2313yes—
107#41×C181—24no—
108C2×Q8×C181C2311yescharacteristic
109C2×D4×C181C2313yes—
110C22×C4×C181C2313yes—
111C22×C4×C181C2313yes—
112C22×C4×C181C2313yes—
113C4⋊C4×C181C2313yes—
114C4⋊C4×C181C2313yes—
115C22⋊C4×C181C2316yes—
116C42×C181C2311yescharacteristic
117C42×C181C2313yes—
118C42×C181C2313yes—
119C42×C181C2313yes—
120C22×D181C2413yes—
121C22×D181C2411yescharacteristic
122Dic362—26no—
123Dic362—26no—
124D724—26no—
125D724—26no—
126C22⋊D181—26no—
127C22⋊D181—26no—
128C22⋊D181—46no—
129C2×Dic181C2411yescharacteristic
130C2×Dic181C2411yescharacteristic
131C2×Dic181C2413yes—
132C2×Dic181C2413yes—
133C2×Dic181C2413yes—
134C2×Dic181C2411yescharacteristic
135C4×D181—26no—
136C4×D181—26no—
137C4×D181—26no—
138C4×D181—43no—
139C4×D181—43no—
140C4×D181#4114yescomplemented
141C23×C181C2413yes—
142Q8×C181#4114yescomplemented
143D4×C181—26no—
144C2×C4×C181—26no—
145C2×C4×C181—26no—
146C2×C4×C181—43no—
147C2×C4×C181C2411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C2×C4×C181C2413yes—
150C2×C4×C181C2413yes—
151C2×C4×C181C2413yes—
152C2×C4×C181C2411yescharacteristic
153C22×C181—23no—
154C22×C181—23no—
155C22×C181—23no—
156C22×C181C2511yescharacteristic
157C2×D181—21no—
158C2×D181—21no—
159C2×D181—23no—
160C2×D181—23no—
161C2×D181—23no—
162C2×D181C2×#4112yes—
163C4×C181—21no—
164C4×C181—23no—
165C4×C181—21no—
166C4×C181—23no—
167C4×C181C2×#4112yes—
168C4×C181C2×#4116yes—
169Dic181—21no—
170Dic181—23no—
171Dic181—23no—
172Dic181—21no—
173Dic181—23no—
174Dic181C2×#4112yes—
175C2×C181—43no—
176C2×C181C22×#4111yescharacteristic
177C2×C181C22×#4111yescharacteristic
178C2×C181#58511yescharacteristic
179D181—22no—
180D181—43no—
181C181#348011yescharacteristic, complemented, Sylow
182#3480—1811nomaximal, Sylow
183#559—1813no—
184#557—1811no—
185#556—1813no—
186#553—1813no—
187#553—1813no—
188#550—1811no—
189#548—1811no—
190#548—1811no—
191#532—1813no—
192#529—1813no—
193#529—1813no—
194C4×#41—1811no—
195C4×#41—1813no—
196C4×#41D18111yescharacteristic, complemented
197C2×C4×Q8—1811no—
198C2×#41—1811no—
199C2×#41—1813no—
200C2×#41C2×D18111yescharacteristic
201C22×Q8—1811no—
202C4⋊Q8—1813no—
203#127—1813no—
204#126—1816no—
205#126—1816no—
206#125—1813no—
207#125—1813no—
208#124—1813no—
209#124—1816no—
210#124—1813no—
211#123—1816no—
212#123—1816no—
213C22⋊Q8—1813no—
214C22⋊Q8—1813no—
215#121—1813no—
216#121—1813no—
217#121—1813no—
218#121—1813no—
219#121—1816no—
220C4×Q8—1811no—
221C4×Q8—1811no—
222C4×Q8—1813no—
223C4×Q8—1813no—
224C4×Q8—1813no—
225C4×Q8—1813no—
226C4×Q8C2×D18111yescharacteristic
227C4×Q8C2×D18111yescharacteristic
228C4×D4—1813no—
229C4×D4—1813no—
230C4×D4—1813no—
231C4×D4—1813no—
232C4×D4—1816no—
233C4×D4—1813no—
234C4×D4—1813no—
235C4×D4—1813no—
236C4×D4—1813no—
237C4×D4C2×D18113yes—
238C4×D4C2×D18113yes—
239#117—1813no—
240#117—1813no—
241#117—1811no—
242#117—1813no—
243#117C2×D18113yes—
244C2×C4⋊C4—1813no—
245C2×C42—1813no—
246C2×C42—1813no—
247C2×C42C2×D18113yes—
248#41—24no—
249#41—36212no—
250#41—3624no—
251C2×Q8—3624no—
252C2×Q8—1813no—
253C2×Q8—1813no—
254C2×Q8C22×D18111yescharacteristic
255C2×D4—1813no—
256C2×D4—1813no—
257C2×D4—1816no—
258C2×D4—1813no—
259C2×D4—1813no—
260C2×D4C22×D18113yes—
261C22×C4—1813no—
262C22×C4—1813no—
263C22×C4—1813no—
264C22×C4—1811no—
265C22×C4—1813no—
266C22×C4—1813no—
267C22×C4C22×D18113yes—
268C22×C4C22×D18113yes—
269C22×C4C22×D18113yes—
270C4⋊C4—1813no—
271C4⋊C4—1816no—
272C4⋊C4—1811no—
273C4⋊C4—1811no—
274C4⋊C4—1813no—
275C4⋊C4—1813no—
276C4⋊C4—1813no—
277C4⋊C4—1813no—
278C4⋊C4—1813no—
279C4⋊C4—1816no—
280C4⋊C4C22×D18113yes—
281C4⋊C4C22×D18113yes—
282C22⋊C4—1813no—
283C22⋊C4—1813no—
284C22⋊C4—1816no—
285C22⋊C4—1811no—
286C22⋊C4—1811no—
287C22⋊C4—1813no—
288C22⋊C4—1816no—
289C22⋊C4—1813no—
290C22⋊C4—1816no—
291C22⋊C4C22×D18116yes—
292C42—1813no—
293C42—1813no—
294C42—1813no—
295C42—1813no—
296C42—1811no—
297C42—1813no—
298C42C22×D18113yes—
299C42C22×D18113yes—
300C42C22×D18113yes—
301C42C22×D18111yescharacteristic
302C23—1813no—
303C23—1811no—
304C23C23×D18113yes—
305Q8—3626no—
306Q8—3626no—
307Q8#1566844914yescomplemented
308D4—26no—
309D4—3626no—
310D4—3626no—
311D4—3626no—
312D4—3626no—
313D4—7246no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—43no—
317C2×C4—3626no—
318C2×C4—3626no—
319C2×C4—3626no—
320C2×C4—1811no—
321C2×C4—1811no—
322C2×C4—1813no—
323C2×C4—1813no—
324C2×C4—1813no—
325C2×C4—1811no—
326C2×C4—1814no—
327C2×C4—7243no—
328C2×C4—7243no—
329C2×C4C23×D18113yes—
330C2×C4C23×D18111yescharacteristic
331C2×C4C23×D18111yescharacteristic
332C2×C4C23×D18113yes—
333C2×C4C23×D18113yes—
334C2×C4C23×D18111yescharacteristic
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3621no—
339C22—3621no—
340C22—3623no—
341C22—3623no—
342C22—3623no—
343C22—1812no—
344C22C24×D18111yescharacteristic
345C4—21no—
346C4—23no—
347C4—23no—
348C4—21no—
349C4—3621no—
350C4—3623no—
351C4—3623no—
352C4—3621no—
353C4—3623no—
354C4—1812no—
355C4#41×D18112yes—
356C4C2×#1566844916yes—
357C2—43no—
358C2—3622no—
359C2—7243no—
360C2C2×#41×D18111yescharacteristic
361C2C22×#1566844911yescharacteristic
362C2#2604211511yescharacteristic
subgroupG/H
centerC2×C4C23×D181
commutator, socle, cosocleC22×C181C25
FrattiniC22C24×D181
FittingC4×#41×C181C2
hypercenterC4×#41D181
hypocenterC181#3480
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #26042012C2semidirect
2. #26042005C2semidirect
3. #26041997C2semidirect
4. #26041979C2semidirect
5. #26041973C2semidirect
6. #26041948C2semidirect
7. #26041941C2semidirect
8. #26041936C2semidirect
9. #26041826C2semidirect
10. #26041813C2semidirect
11. #26041811C2semidirect
12. C4×#15668450C2semidirect
13. C4×#15668449C2semidirect
14. C4×Q8×D181C2semidirect
15. C4×#41×C181C2semidirect
140. C4×D181#41semidirect
142. Q8×C181#41semidirect
181. C181#3480semidirect
196. C4×#41D181semidirect
307. Q8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260420122
#260420052
#260419972
#260419792
#260419732
#260419482
#260419412
#260419362
#260418262
#260418132
#260418112
C4×#156684502
C4×#156684492
C4×Q8×D1812
C4×#41×C1812
#3480181