Finite Group Databasebeta

← #38745859

Subgroups — #38745859

Order 25216.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263311C211yesmaximal, characteristic
2#27160981C211yesmaximal, characteristic
3#27160978C211yesmaximal, characteristic, complemented
4#27160976C211yesmaximal, characteristic, complemented
5#27160971C211yesmaximal, characteristic, complemented
6#27160969C211yesmaximal, characteristic, complemented
7#27160968C211yesmaximal, characteristic, complemented
8#27160968C211yesmaximal, characteristic, complemented
9#27160963C211yesmaximal, characteristic, complemented
10#27160957C211yesmaximal, characteristic
11#27160953C211yesmaximal, characteristic, complemented
12#27160953C211yesmaximal, characteristic, complemented
13#27160938C211yesmaximal, characteristic, complemented
14#27160936C211yesmaximal, characteristic, complemented
15#387×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19#20263311C418yescomplemented
20C2×C4⋊Dic197C2211yescharacteristic
21C2×C4⋊Dic197C2211yescharacteristic
22C2×C4⋊Dic197C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×#15691335C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic
37C2×C4×Dic197C2211yescharacteristic
38C2×C4×Dic197C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C2×C4⋊C4×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C42×C197C2211yescharacteristic
48#95×C197C2211yescharacteristic
49#95×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C23×D197C2311yescharacteristic
53C23×D197C2311yescharacteristic
54C2×D788—41no—
55C2×D788—41no—
56C2×D788C2×C414yes—
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C2×C4×D197—41no—
64C2×C4×D197—41no—
65C2×C4×D197—41no—
66C2×C4×D197—41no—
67C2×C4×D197—41no—
68C2×C4×D197C2×C414yescomplemented
69C4⋊Dic197—41no—
70C4⋊Dic197—41no—
71C4⋊Dic197—41no—
72#15691337—41no—
73#15691337—41no—
74#15691337—41no—
75#15691337C2×C414yescomplemented
76#15691335—41no—
77#15691335—41no—
78#15691335—41no—
79#15691335—41no—
80#15691335—41no—
81#15691335—41no—
82#15691335—41no—
83#15691335C2×C414yes—
84#15691335C2×C414yes—
85#15691335C2×C414yescomplemented
86C4×Dic197—41no—
87C4×Dic197—41no—
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C22×C4×C197C2311yescharacteristic
95C4⋊C4×C197—41no—
96C4⋊C4×C197C2×C414yescomplemented
97C42×C197—41no—
98C22×D197—21no—
99C22×D197—21no—
100C22×D197—21no—
101C22×D197—21no—
102C22×D197—21no—
103C22×D197—21no—
104C22×D197—21no—
105C22×D197—21no—
106C22×D197—21no—
107C22×D197—21no—
108C22×D197—21no—
109C22×D197—21no—
110C22×D197C22×C412yes—
111C22×D197C22×C412yes—
112D788—44no—
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—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197C22×C412yes—
130C2×Dic197C22×C412yes—
131C4×D197—44no—
132C4×D197—44no—
133C23×C197C2411yescharacteristic
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
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×C197C22×C412yes—
153C2×C4×C197C22×C412yes—
154C2×C4×C197C22×C412yes—
155C22×C197C23×C411yescharacteristic
156C22×C197H(5,𝔽2)11yescharacteristic
157C22×C197H(5,𝔽2)11yescharacteristic
158C22×C197H(5,𝔽2)11yescharacteristic
159C22×C197H(5,𝔽2)11yescharacteristic
160C22×C1972−1+411yescharacteristic
161C22×C1972−1+411yescharacteristic
162C2×D197—22no—
163C2×D197—22no—
164C2×D197—22no—
165C2×D197—22no—
166C2×D197—22no—
167C2×D197—22no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—41no—
175C2×D197—41no—
176C4×C197—22no—
177C4×C197—22no—
178C4×C197—22no—
179C4×C197—41no—
180C4×C197—41no—
181C4×C197—41no—
182C4×C197—41no—
183Dic197—22no—
184Dic197—22no—
185Dic197—41no—
186Dic197—41no—
187Dic197—41no—
188Dic197—41no—
189C2×C197#51811yescharacteristic
190C2×C197#52011yescharacteristic
191C2×C197#52011yescharacteristic
192C2×C197#56011yescharacteristic
193C2×C197#56211yescharacteristic
194C2×C197#56211yescharacteristic
195C2×C197#56311yescharacteristic
196D197—42no—
197D197—42no—
198C197#237611yescharacteristic, complemented, Sylow
199#2376—1971nomaximal, Sylow
200C2×#123—1971no—
201#390—1971no—
202#389—1971no—
203#389—1971no—
204#388—1971no—
205#388—1971no—
206#388—1971no—
207#388—1971no—
208#387—1971no—
209#387D19711yescharacteristic, complemented
210#386—1971no—
211#385—1971no—
212#385—1971no—
213#380—1971no—
214#380—1971no—
215C22×D4—1971no—
216C23×C4—1971no—
217C23×C4—1971no—
218#123—1978no—
219C2×C4⋊C4—1971no—
220C2×C4⋊C4—1971no—
221C2×C4⋊C4—1971no—
222C2×C4⋊C4—1971no—
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⋊C4—1971no—
235C2×C22⋊C4—1971no—
236C2×C22⋊C4—1971no—
237C2×C22⋊C4—1971no—
238C2×C42—1971no—
239C2×C42—1971no—
240C2×C42C2×D19711yescharacteristic
241#95—1971no—
242#95—1971no—
243#95—1971no—
244#95—1971no—
245#95—1971no—
246#95—1971no—
247#95C2×D19711yescharacteristic
248#95C2×D19711yescharacteristic
249#95C2×D19711yescharacteristic
250#95C2×D19711yescharacteristic
251C24—1971no—
252C24—1971no—
253C2×D4—7881no—
254C2×D4—7881no—
255C2×D4—1974no—
256C22×C4—7881no—
257C22×C4—7881no—
258C22×C4—7881no—
259C22×C4—7881no—
260C22×C4—7881no—
261C22×C4—1974no—
262C22×C4—1971no—
263C22×C4—1971no—
264C22×C4—1971no—
265C22×C4—1971no—
266C22×C4—1971no—
267C22×C4—1971no—
268C22×C4C22×D19711yescharacteristic
269C22×C4C22×D19711yescharacteristic
270C22×C4C22×D19711yescharacteristic
271C22×C4C22×D19711yescharacteristic
272C22×C4C22×D19711yescharacteristic
273C22×C4C22×D19711yescharacteristic
274C22×C4C22×D19711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—7881no—
277C4⋊C4—7881no—
278C4⋊C4—7881no—
279C4⋊C4—7881no—
280C4⋊C4—7881no—
281C4⋊C4—7881no—
282C4⋊C4—1974no—
283C4⋊C4C4×D19714yescomplemented
284C22⋊C4—7881no—
285C22⋊C4—7881no—
286C22⋊C4—7881no—
287C22⋊C4—7881no—
288C22⋊C4—7881no—
289C22⋊C4—7881no—
290C22⋊C4—7881no—
291C22⋊C4—1974no—
292C22⋊C4—1974no—
293C22⋊C4—1974no—
294C42—41no—
295C42—7881no—
296C42—7881no—
297C23—3941no—
298C23—3941no—
299C23—3941no—
300C23—3941no—
301C23—3941no—
302C23—3941no—
303C23—3941no—
304C23—3941no—
305C23—3941no—
306C23—3941no—
307C23—3941no—
308C23—3941no—
309C23—1972no—
310C23—1972no—
311C23C23×D19711yescharacteristic
312D4—7884no—
313C2×C4—21no—
314C2×C4—21no—
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—7884no—
332C2×C4—7884no—
333C2×C4—3941no—
334C2×C4—3941no—
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—1972no—
350C2×C4—1972no—
351C2×C4C2×C4×D19712yes—
352C2×C4C2×C4×D19712yes—
353C2×C4C2×C4×D19712yes—
354C22—7881no—
355C22—7881no—
356C22—7881no—
357C22—7881no—
358C22—7881no—
359C22—7881no—
360C22—7881no—
361C22—7881no—
362C22—3942no—
363C22—3942no—
364C22—3942no—
365C22—3942no—
366C22—3942no—
367C22—3942no—
368C22C22×C4×D19711yescharacteristic
369C22#2026337111yescharacteristic
370C22#2026337211yescharacteristic
371C22#2026337211yescharacteristic
372C22#2026337211yescharacteristic
373C22#2026337311yescharacteristic
374C22#2026337411yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—7881no—
383C4—7881no—
384C4—7881no—
385C4—7881no—
386C4—3942no—
387C4—3942no—
388C2—7882no—
389C2—7882no—
390C2#2716155111yescharacteristic
391C2#2716155911yescharacteristic
392C2#2716156111yescharacteristic
393C2#2716182011yescharacteristic
394C2#2716182511yescharacteristic
395C2#2716182611yescharacteristic
396C2#2716183311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#387×C197C2
socle, cosocleC23×C197C24
prime radicalC22×D788C22
hypercenter#387D197
hypocenterC197#2376
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #27160978C2semidirect
4. #27160976C2semidirect
5. #27160971C2semidirect
6. #27160969C2semidirect
7. #27160968C2semidirect
8. #27160968C2semidirect
9. #27160963C2semidirect
11. #27160953C2semidirect
12. #27160953C2semidirect
13. #27160938C2semidirect
14. #27160936C2semidirect
15. #387×C197C2semidirect
19. #20263311C4semidirect
68. C2×C4×D197C2×C4semidirect
75. #15691337C2×C4semidirect
85. #15691335C2×C4semidirect
96. C4⋊C4×C197C2×C4semidirect
198. C197#2376semidirect
209. #387D197semidirect
283. C4⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633112
#271609812
#271609782
#271609762
#271609712
#271609692
#271609682
#271609632
#271609572
#271609532
#271609382
#271609362
#387×C1972
#2376197