Finite Group Databasebeta

← #38745340

Subgroups — #38745340

Order 25216.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263286C211yesmaximal, characteristic
2#27160971C211yesmaximal, characteristic, complemented
3#27160969C211yesmaximal, characteristic, complemented
4#27160966C211yesmaximal, characteristic, complemented
5#27160960C211yesmaximal, characteristic, complemented
6#27160958C211yesmaximal, characteristic
7#27160956C211yesmaximal, characteristic
8#27160954C211yesmaximal, characteristic, complemented
9#27160946C211yesmaximal, characteristic, complemented
10#27160944C211yesmaximal, characteristic, complemented
11#27160941C211yesmaximal, characteristic, complemented
12#27160936C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic197C211yesmaximal, characteristic, complemented
14C2×#20263233C211yesmaximal, characteristic, complemented
15#385×C197C211yesmaximal, characteristic, complemented
16C22×C22⋊D197C2211yescharacteristic
17C23×Dic197C2211yescharacteristic
18C23×Dic197C2211yescharacteristic
19C22×C4×D197C2211yescharacteristic
20#20263286C418yescomplemented
21C2×C4⋊Dic197C2211yescharacteristic
22C2×C4⋊Dic197C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×C22⋊Dic197C2211yescharacteristic
25C2×#15691335C2211yescharacteristic
26C2×#15691335C2211yescharacteristic
27C2×#15691335C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×C4×Dic197C2211yescharacteristic
31C2×C4×Dic197C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33#20263233C2211yescharacteristic
34#20263233C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C23×C4×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C4⋊C4×C197C2211yescharacteristic
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C22⋊C4×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C23×D197C2311yescharacteristic
53C2×C22⋊D197—22no—
54C2×C22⋊D197—22no—
55C2×C22⋊D197C2×C414yes—
56C22×Dic197—22no—
57C22×Dic197—41no—
58C22×Dic197—41no—
59C22×Dic197—41no—
60C22×Dic197—41no—
61C22×Dic197C2×C414yes—
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66C22×Dic197C2311yescharacteristic
67C22×Dic197C2311yescharacteristic
68C22×Dic197C2311yescharacteristic
69C2×C4×D197—41no—
70C2×C4×D197—41no—
71C2×C4×D197C2×C414yescomplemented
72C4⋊Dic197—41no—
73C4⋊Dic197—41no—
74#15691337C2×C414yescomplemented
75C22⋊Dic197—22no—
76#15691335—41no—
77#15691335—41no—
78#15691335—41no—
79#15691335—41no—
80#15691335C2×C414yescomplemented
81C4×Dic197—41no—
82C4×Dic197—41no—
83C4×Dic197C2×C414yescomplemented
84C24×C197C2311yescharacteristic
85C22×C4×C197—22no—
86C22×C4×C197—41no—
87C22×C4×C197—41no—
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C4⋊C4×C197—41no—
95C4⋊C4×C197—41no—
96C22⋊C4×C197—22no—
97C22⋊C4×C197C2×C414yes—
98C22×D197—21no—
99C22×D197—21no—
100C22×D197—21no—
101C22×D197—21no—
102C22×D197—21no—
103C22×D197—21no—
104C22×D197C22×C412yes—
105C22⋊D197—28no—
106C2×Dic197—21no—
107C2×Dic197—21no—
108C2×Dic197—21no—
109C2×Dic197—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—44no—
125C2×Dic197—44no—
126C2×Dic197C22×C412yes—
127C2×Dic197C22×C412yes—
128C2×Dic197C22×C412yes—
129C4×D197—44no—
130C23×C197—21no—
131C23×C197—21no—
132C23×C197—21no—
133C23×C197—21no—
134C23×C197C22×C412yes—
135C23×C197#4112yes—
136C23×C197#4112yes—
137C23×C197C2411yescharacteristic
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197—44no—
155C2×C4×C197C22×C412yes—
156C2×C4×C197C22×C412yes—
157C22×C197—22no—
158C22×C197—22no—
159C22×C197—22no—
160C22×C197—22no—
161C22×C197—22no—
162C22×C197—22no—
163C22×C197#11714yes—
164C22×C197C23×C411yescharacteristic
165C22×C197C2×#4111yescharacteristic
166C22×C197C2×#4111yescharacteristic
167C22×C197H(5,𝔽2)11yescharacteristic
168C22×C197H(5,𝔽2)11yescharacteristic
169C22×C197H(5,𝔽2)11yescharacteristic
170C22×C1972−1+411yescharacteristic
171C2×D197—22no—
172C2×D197—22no—
173C2×D197—22no—
174C2×D197—41no—
175C2×D197—41no—
176C2×D197—41no—
177C2×D197—41no—
178C4×C197—22no—
179C4×C197—22no—
180C4×C197—41no—
181C4×C197—41no—
182C4×C197—41no—
183C4×C197—41no—
184Dic197—22no—
185Dic197—22no—
186Dic197—22no—
187Dic197—41no—
188Dic197—41no—
189Dic197—41no—
190Dic197—41no—
191C2×C197—24no—
192C2×C197C2×#11711yescharacteristic
193C2×C197#51811yescharacteristic
194C2×C197#52011yescharacteristic
195C2×C197#53811yescharacteristic
196C2×C197#53811yescharacteristic
197C2×C197#53911yescharacteristic
198C2×C197#53911yescharacteristic
199D197—42no—
200C197#231811yescharacteristic, complemented, Sylow
201#2318—1971nomaximal, Sylow
202C2×C4×D4—1971no—
203#388—1971no—
204#388—1971no—
205#387—1971no—
206#387—1971no—
207#386—1971no—
208#386—1971no—
209#385—1971no—
210#385D19711yescharacteristic, complemented
211#383—1971no—
212#382—1971no—
213#381—1971no—
214#380—1971no—
215C4×C22⋊C4—1971no—
216C2×#95—1971no—
217C22×D4—1971no—
218C23×C4—1971no—
219C23×C4—1971no—
220C23×C4—1971no—
221C23×C4C2×D19711yescharacteristic
222C4×D4—1978no—
223C2×C4⋊C4—1971no—
224C2×C4⋊C4—1971no—
225C2×C4⋊C4—1971no—
226C2×C4⋊C4C2×D19711yescharacteristic
227C2×C4⋊C4C2×D19711yescharacteristic
228C2×C22⋊C4—1971no—
229C2×C22⋊C4—1971no—
230C2×C22⋊C4—1971no—
231C2×C22⋊C4—1971no—
232C2×C22⋊C4—1971no—
233C2×C22⋊C4—1971no—
234C2×C22⋊C4C2×D19711yescharacteristic
235C2×C22⋊C4C2×D19711yescharacteristic
236C2×C42—1971no—
237C2×C42—1971no—
238C2×C42—1971no—
239#95—1971no—
240#95—1971no—
241#95—1971no—
242#95—1971no—
243#95—1971no—
244#95—1971no—
245#95—1971no—
246#95—1971no—
247#95—1971no—
248#95—1971no—
249#95—1971no—
250#95—1971no—
251#95C2×D19711yescharacteristic
252#95C2×D19711yescharacteristic
253C24—1971no—
254C24C22×D19711yescharacteristic
255C2×D4—3942no—
256C2×D4—3942no—
257C2×D4—1974no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—7881no—
262C22×C4—7881no—
263C22×C4—7881no—
264C22×C4—7881no—
265C22×C4—7881no—
266C22×C4—7881no—
267C22×C4—3942no—
268C22×C4—1974no—
269C22×C4—1974no—
270C22×C4—1971no—
271C22×C4—1971no—
272C22×C4—1971no—
273C22×C4—1971no—
274C22×C4—1971no—
275C22×C4—1971no—
276C22×C4—1971no—
277C22×C4C22×D19711yescharacteristic
278C22×C4C22×D19711yescharacteristic
279C22×C4C22×D19711yescharacteristic
280C22×C4C22×D19711yescharacteristic
281C22×C4C22×D19711yescharacteristic
282C22×C4C22×D19711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—7881no—
286C4⋊C4—7881no—
287C4⋊C4—1974no—
288C22⋊C4—22no—
289C22⋊C4—7881no—
290C22⋊C4—7881no—
291C22⋊C4—7881no—
292C22⋊C4—7881no—
293C22⋊C4—3942no—
294C22⋊C4—1974no—
295C22⋊C4C4×D19714yes—
296C42—7881no—
297C42—7881no—
298C42—1974no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—3941no—
304C23—3941no—
305C23—3941no—
306C23—3941no—
307C23—3941no—
308C23—3941no—
309C23—1972no—
310C23C2×C4×D19712yes—
311C23#1569135712yes—
312C23#1569135912yes—
313C23C23×D19711yescharacteristic
314D4—3948no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—7884no—
333C2×C4—7884no—
334C2×C4—7884no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4—3941no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—3941no—
350C2×C4—3941no—
351C2×C4—3941no—
352C2×C4—3941no—
353C2×C4—1972no—
354C2×C4—1972no—
355C2×C4—1972no—
356C2×C4C2×C4×D19712yes—
357C2×C4C2×C4×D19712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—7881no—
365C22—7881no—
366C22—7881no—
367C22—7881no—
368C22—3942no—
369C22—3942no—
370C22—3942no—
371C22#2026328414yes—
372C22C22×C4×D19711yescharacteristic
373C22C2×#1569135711yescharacteristic
374C22C2×#1569135911yescharacteristic
375C22#2026337111yescharacteristic
376C22#2026337111yescharacteristic
377C22#2026337211yescharacteristic
378C22#2026337311yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—7881no—
386C4—7881no—
387C4—7881no—
388C4—7881no—
389C4—3942no—
390C4—3942no—
391C4—3942no—
392C2—24no—
393C2—7882no—
394C2C2×#2026328411yescharacteristic
395C2#2716155111yescharacteristic
396C2#2716155811yescharacteristic
397C2#2716166411yescharacteristic
398C2#2716166911yescharacteristic
399C2#2716167311yescharacteristic
400C2#2716167811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#385×C197C2
socle, cosocleC23×C197C24
prime radicalC22×C22⋊D197C22
hypercenter#385D197
hypocenterC197#2318
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #27160971C2semidirect
3. #27160969C2semidirect
4. #27160966C2semidirect
5. #27160960C2semidirect
8. #27160954C2semidirect
9. #27160946C2semidirect
10. #27160944C2semidirect
11. #27160941C2semidirect
12. #27160936C2semidirect
13. C22⋊C4×Dic197C2semidirect
14. C2×#20263233C2semidirect
15. #385×C197C2semidirect
20. #20263286C4semidirect
71. C2×C4×D197C2×C4semidirect
74. #15691337C2×C4semidirect
80. #15691335C2×C4semidirect
83. C4×Dic197C2×C4semidirect
200. C197#2318semidirect
210. #385D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632862
#271609712
#271609692
#271609662
#271609602
#271609582
#271609562
#271609542
#271609462
#271609442
#271609412
#271609362
C22⋊C4×Dic1972
C2×#202632332
#385×C1972
#2318197