Finite Group Databasebeta

← #38745444

Subgroups — #38745444

Order 25216.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263302C211yesmaximal, characteristic
2#27160986C211yesmaximal, characteristic, complemented
3#27160976C211yesmaximal, characteristic, complemented
4#27160974C211yesmaximal, characteristic, complemented
5#27160964C211yesmaximal, characteristic, complemented
6#27160963C211yesmaximal, characteristic, complemented
7#27160962C211yesmaximal, characteristic, complemented
8#27160961C211yesmaximal, characteristic, complemented
9#27160953C211yesmaximal, characteristic
10#27160950C211yesmaximal, characteristic
11#27160938C211yesmaximal, characteristic
12C4×#15691335C211yesmaximal, characteristic
13#27160926C211yesmaximal, characteristic
14#95×D197C211yesmaximal, characteristic
15#391×C197C211yesmaximal, characteristic, complemented
16C22×C4×D197C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19#20263302C418yescomplemented
20C2×C4⋊Dic197C2211yescharacteristic
21C2×C4⋊Dic197C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×#15691337C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33C2×C4×Dic197C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C22×Q8×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C42×C197C2211yescharacteristic
48#95×C197C2211yescharacteristic
49#95×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C23×D197C2311yescharacteristic
53C22×Dic197C2311yescharacteristic
54C22×Dic197C2311yescharacteristic
55C22×Dic197C2311yescharacteristic
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C2×C4×D197—22no—
61C2×C4×D197—41no—
62C2×C4×D197—41no—
63C2×C4×D197—41no—
64C2×C4×D197—41no—
65C2×C4×D197—41no—
66C2×C4×D197—41no—
67C2×C4×D197C2×C414yes—
68C2×C4×D197Q814yescomplemented
69C4⋊Dic197—41no—
70C4⋊Dic197C2×C414yescomplemented
71#15691337—41no—
72#15691337—41no—
73#15691337—41no—
74#15691337—41no—
75#15691337C2×C414yescomplemented
76#15691337C2×C414yescomplemented
77#15691335—22no—
78#15691335—22no—
79#15691335C2×C414yes—
80#15691335C2×C414yes—
81C4×Dic197—41no—
82C4×Dic197—41no—
83C4×Dic197—41no—
84C2×Q8×C197—41no—
85C2×Q8×C197—41no—
86C2×Q8×C197C2×C414yescomplemented
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C4⋊C4×C197—41no—
95C42×C197—41no—
96C22×D197—21no—
97C22×D197—21no—
98C22×D197—21no—
99C22×D197—21no—
100C22×D197C22×C412yes—
101C22×D197C2×Q812yes—
102C22×D197#4112yes—
103C2×Dic197—21no—
104C2×Dic197—21no—
105C2×Dic197—21no—
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×Dic197C22×C412yes—
121C2×Dic197C22×C412yes—
122C2×Dic197C22×C412yes—
123C2×Dic197C2×Q812yes—
124C4×D197—44no—
125C4×D197—44no—
126C4×D197—44no—
127C23×C197C2411yescharacteristic
128Q8×C197—44no—
129C2×C4×C197—21no—
130C2×C4×C197—21no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
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×C197C22×C412yes—
147C2×C4×C197C22×C412yes—
148C2×C4×C197C22×C412yes—
149C2×C4×C197C2×Q812yes—
150C22×C197C23×C411yescharacteristic
151C22×C197C22×Q811yescharacteristic
152C22×C197C2×#4111yescharacteristic
153C22×C197H(5,𝔽2)11yescharacteristic
154C22×C197H(5,𝔽2)11yescharacteristic
155C22×C197H(5,𝔽2)11yescharacteristic
156C22×C1972−1+411yescharacteristic
157C2×D197—22no—
158C2×D197—22no—
159C2×D197—22no—
160C2×D197—22no—
161C2×D197—22no—
162C2×D197—22no—
163C2×D197C4×Q814yes—
164C4×C197—22no—
165C4×C197—22no—
166C4×C197—22no—
167C4×C197—22no—
168C4×C197—41no—
169C4×C197—41no—
170C4×C197—41no—
171Dic197—22no—
172Dic197—22no—
173Dic197—22no—
174Dic197—22no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197C2×C4×Q811yescharacteristic
179C2×C197#51811yescharacteristic
180C2×C197#52011yescharacteristic
181C2×C197#53811yescharacteristic
182C2×C197#53911yescharacteristic
183C2×C197#54311yescharacteristic
184C2×C197#54411yescharacteristic
185D197—24no—
186C197#232911yescharacteristic, complemented, Sylow
187#2329—1971nomaximal, Sylow
188C2×C22⋊Q8—1971no—
189#391—1971no—
190#391D19711yescharacteristic, complemented
191#389—1971no—
192#389—1971no—
193#387—1971no—
194#387—1971no—
195#387—1971no—
196#387—1971no—
197#385—1971no—
198#385—1971no—
199#380—1971no—
200C4×C22⋊C4—1971no—
201C4×C22⋊C4—1971no—
202C2×#95—1971no—
203C22×Q8C2×D19711yescharacteristic
204C23×C4—1971no—
205C23×C4—1971no—
206C23×C4—1971no—
207C22⋊Q8—1978no—
208C2×C4⋊C4—1971no—
209C2×C4⋊C4—1971no—
210C2×C4⋊C4—1971no—
211C2×C4⋊C4—1971no—
212C2×C4⋊C4—1971no—
213C2×C4⋊C4—1971no—
214C2×C4⋊C4—1971no—
215C2×C4⋊C4—1971no—
216C2×C4⋊C4C2×D19711yescharacteristic
217C2×C22⋊C4—1971no—
218C2×C22⋊C4—1971no—
219C2×C22⋊C4—1971no—
220C2×C22⋊C4—1971no—
221C2×C42—1971no—
222C2×C42—1971no—
223C2×C42—1971no—
224C2×C42C2×D19711yescharacteristic
225#95—1971no—
226#95—1971no—
227#95—1971no—
228#95—1971no—
229#95—1971no—
230#95—1971no—
231#95—1971no—
232#95—1971no—
233#95—1971no—
234#95—1971no—
235#95C2×D19711yescharacteristic
236#95C2×D19711yescharacteristic
237#95C2×D19711yescharacteristic
238#95C2×D19711yescharacteristic
239C24—1971no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8C4×D19714yescomplemented
243C22×C4—7881no—
244C22×C4—7881no—
245C22×C4—7881no—
246C22×C4—7881no—
247C22×C4—7881no—
248C22×C4—7881no—
249C22×C4—3942no—
250C22×C4—1974no—
251C22×C4—1974no—
252C22×C4—1971no—
253C22×C4—1971no—
254C22×C4—1971no—
255C22×C4—1971no—
256C22×C4—1971no—
257C22×C4—1971no—
258C22×C4—1971no—
259C22×C4C22×D19711yescharacteristic
260C22×C4C22×D19711yescharacteristic
261C22×C4C22×D19711yescharacteristic
262C22×C4C22×D19711yescharacteristic
263C22×C4C22×D19711yescharacteristic
264C22×C4C22×D19711yescharacteristic
265C22×C4C22×D19711yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—7881no—
268C4⋊C4—7881no—
269C4⋊C4—7881no—
270C4⋊C4—7881no—
271C4⋊C4—7881no—
272C4⋊C4—1974no—
273C4⋊C4—1974no—
274C4⋊C4—1974no—
275C22⋊C4—3942no—
276C22⋊C4—3942no—
277C22⋊C4—1974no—
278C22⋊C4—1974no—
279C42—41no—
280C42—7881no—
281C42—7881no—
282C42—7881no—
283C23—3941no—
284C23—3941no—
285C23—3941no—
286C23—3941no—
287C23—1972no—
288C23—1972no—
289C23—1972no—
290C23C23×D19711yescharacteristic
291Q8—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—7884no—
310C2×C4—7884no—
311C2×C4—7884no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—3941no—
321C2×C4—3941no—
322C2×C4—3941no—
323C2×C4—3941no—
324C2×C4—3941no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4—1972no—
330C2×C4—1972no—
331C2×C4—1972no—
332C2×C4—1972no—
333C2×C4C2×C4×D19712yes—
334C2×C4C2×C4×D19712yes—
335C2×C4C2×C4×D19712yes—
336C2×C4Q8×D19712yes—
337C22—3942no—
338C22—3942no—
339C22—3942no—
340C22—3942no—
341C22—3942no—
342C22—3942no—
343C22—1974no—
344C22C22×C4×D19711yescharacteristic
345C22C2×Q8×D19711yescharacteristic
346C22#41×D19711yescharacteristic
347C22#2026337111yescharacteristic
348C22#2026337111yescharacteristic
349C22#2026337211yescharacteristic
350C22#2026337411yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—7881no—
359C4—7881no—
360C4—7881no—
361C4—3942no—
362C4—3942no—
363C4—3942no—
364C4—3942no—
365C2—3944no—
366C2C4×Q8×D19711yescharacteristic
367C2#2716155211yescharacteristic
368C2#2716156111yescharacteristic
369C2#2716166311yescharacteristic
370C2#2716167511yescharacteristic
371C2#2716170411yescharacteristic
372C2#2716170611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#391×C197C2
socle, cosocleC23×C197C24
prime radicalC23×D197C23
hypercenter#391D197
hypocenterC197#2329
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #27160986C2semidirect
3. #27160976C2semidirect
4. #27160974C2semidirect
5. #27160964C2semidirect
6. #27160963C2semidirect
7. #27160962C2semidirect
8. #27160961C2semidirect
15. #391×C197C2semidirect
19. #20263302C4semidirect
68. C2×C4×D197Q8semidirect
70. C4⋊Dic197C2×C4semidirect
75. #15691337C2×C4semidirect
76. #15691337C2×C4semidirect
86. C2×Q8×C197C2×C4semidirect
186. C197#2329semidirect
190. #391D197semidirect
242. C2×Q8C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633022
#271609862
#271609762
#271609742
#271609642
#271609632
#271609622
#271609612
#271609532
#271609502
#271609382
C4×#156913352
#271609262
#95×D1972
#391×C1972
#2329197