Finite Group Databasebeta

← #38745571

Subgroups — #38745571

Order 25216.

532 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
1C2×#20263284C211yesmaximal, characteristic
2#27160965C211yesmaximal, characteristic, complemented
3#27160964C211yesmaximal, characteristic, complemented
4#27160962C211yesmaximal, characteristic, complemented
5#27160953C211yesmaximal, characteristic
6#27160950C211yesmaximal, characteristic
7#27160946C211yesmaximal, characteristic, complemented
8#27160944C211yesmaximal, characteristic, complemented
9#27160941C211yesmaximal, characteristic
10C4×#15691337C211yesmaximal, characteristic, complemented
11C4×#15691335C211yesmaximal, characteristic
12#27160926C211yesmaximal, characteristic
13#27160924C211yesmaximal, characteristic, complemented
14#95×D197C211yesmaximal, characteristic
15#387×C197C211yesmaximal, characteristic, complemented
16C22×C4×D197C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19#20263284C418yescomplemented
20C2×C4⋊Dic197C2211yescharacteristic
21C2×C4⋊Dic197C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691335C2211yescharacteristic
25C2×#15691335C2211yescharacteristic
26C2×#15691335C2211yescharacteristic
27C2×#15691335C2211yescharacteristic
28C2×C4×Dic197C2211yescharacteristic
29C2×C4×Dic197C2211yescharacteristic
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
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
53C22×Dic197C2311yescharacteristic
54C22×Dic197C2311yescharacteristic
55C22×Dic197C2311yescharacteristic
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C2×C4×D197—22no—
61C2×C4×D197—22no—
62C2×C4×D197—22no—
63C2×C4×D197—22no—
64C2×C4×D197—22no—
65C2×C4×D197—22no—
66C2×C4×D197—41no—
67C2×C4×D197—41no—
68C2×C4×D197C2×C414yes—
69C4⋊Dic197—41no—
70C4⋊Dic197C2×C414yescomplemented
71#15691337—22no—
72#15691337—22no—
73#15691335—22no—
74#15691335—22no—
75#15691335C2×C414yes—
76#15691335C2×C414yes—
77C4×Dic197—22no—
78C4×Dic197—22no—
79C4×Dic197—41no—
80C4×Dic197C2×C414yescomplemented
81C4×Dic197C2×C414yescomplemented
82C22×C4×C197C2311yescharacteristic
83C22×C4×C197C2311yescharacteristic
84C22×C4×C197C2311yescharacteristic
85C22×C4×C197C2311yescharacteristic
86C22×C4×C197C2311yescharacteristic
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C4⋊C4×C197—41no—
90C4⋊C4×C197C2×C414yescomplemented
91C42×C197—41no—
92C22×D197—21no—
93C22×D197—21no—
94C22×D197—21no—
95C22×D197—21no—
96C22×D197C22×C412yes—
97C22×D197#4112yes—
98C22×D197#4112yes—
99C2×Dic197—21no—
100C2×Dic197—21no—
101C2×Dic197—21no—
102C2×Dic197—21no—
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×Dic197C22×C412yes—
117C2×Dic197C22×C412yes—
118C2×Dic197C22×C412yes—
119C2×Dic197#4112yes—
120C4×D197—28no—
121C4×D197—44no—
122C4×D197—44no—
123C23×C197C2411yescharacteristic
124C2×C4×C197—21no—
125C2×C4×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
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×C197C22×C412yes—
142C2×C4×C197C22×C412yes—
143C2×C4×C197C22×C412yes—
144C2×C4×C197#4112yes—
145C22×C197C23×C411yescharacteristic
146C22×C197C2×#4111yescharacteristic
147C22×C197C2×#4111yescharacteristic
148C22×C197C2×#4111yescharacteristic
149C22×C197C2×#4111yescharacteristic
150C22×C197H(5,𝔽2)11yescharacteristic
151C22×C1972−1+411yescharacteristic
152C2×D197—22no—
153C2×D197—22no—
154C2×D197—22no—
155C2×D197—22no—
156C2×D197—22no—
157C2×D197—22no—
158C2×D197#11714yes—
159C4×C197—22no—
160C4×C197—22no—
161C4×C197—22no—
162C4×C197—22no—
163C4×C197—41no—
164C4×C197—41no—
165C4×C197—41no—
166Dic197—22no—
167Dic197—22no—
168Dic197—22no—
169Dic197—22no—
170Dic197—41no—
171Dic197—41no—
172Dic197—41no—
173C2×C197C2×#11711yescharacteristic
174C2×C197C4×#4111yescharacteristic
175C2×C197#52011yescharacteristic
176C2×C197#55111yescharacteristic
177C2×C197#55111yescharacteristic
178C2×C197#55211yescharacteristic
179C2×C197#55211yescharacteristic
180D197—24no—
181C197#234311yescharacteristic, complemented, Sylow
182#2343—1971nomaximal, Sylow
183C2×#117—1971no—
184#387—1971no—
185#387—1971no—
186#387—1971no—
187#387D19711yescharacteristic, complemented
188#385—1971no—
189#385—1971no—
190#383—1971no—
191#382—1971no—
192#381—1971no—
193C4×C4⋊C4—1971no—
194C4×C22⋊C4—1971no—
195C4×C22⋊C4—1971no—
196#376—1971no—
197C2×#95—1971no—
198C23×C4—1971no—
199C23×C4—1971no—
200C23×C4—1971no—
201#117—1978no—
202C2×C4⋊C4—1971no—
203C2×C4⋊C4—1971no—
204C2×C4⋊C4—1971no—
205C2×C4⋊C4—1971no—
206C2×C4⋊C4C2×D19711yescharacteristic
207C2×C4⋊C4C2×D19711yescharacteristic
208C2×C22⋊C4—1971no—
209C2×C22⋊C4—1971no—
210C2×C22⋊C4—1971no—
211C2×C22⋊C4—1971no—
212C2×C42—1971no—
213C2×C42—1971no—
214C2×C42—1971no—
215C2×C42—1971no—
216C2×C42—1971no—
217C2×C42C2×D19711yescharacteristic
218#95—1971no—
219#95—1971no—
220#95—1971no—
221#95—1971no—
222#95—1971no—
223#95—1971no—
224#95—1971no—
225#95—1971no—
226#95—1971no—
227#95—1971no—
228#95—1971no—
229#95—1971no—
230#95C2×D19711yescharacteristic
231#95C2×D19711yescharacteristic
232#95C2×D19711yescharacteristic
233#95C2×D19711yescharacteristic
234C24—1971no—
235C22×C4—7881no—
236C22×C4—7881no—
237C22×C4—3942no—
238C22×C4—3942no—
239C22×C4—3942no—
240C22×C4—3942no—
241C22×C4—3942no—
242C22×C4—3942no—
243C22×C4—1971no—
244C22×C4—1971no—
245C22×C4—1971no—
246C22×C4—1971no—
247C22×C4—1971no—
248C22×C4—1971no—
249C22×C4—1971no—
250C22×C4—1974no—
251C22×C4C22×D19711yescharacteristic
252C22×C4C22×D19711yescharacteristic
253C22×C4C22×D19711yescharacteristic
254C22×C4C22×D19711yescharacteristic
255C22×C4C22×D19711yescharacteristic
256C22×C4C22×D19711yescharacteristic
257C22×C4C22×D19711yescharacteristic
258C4⋊C4—41no—
259C4⋊C4—7881no—
260C4⋊C4—3942no—
261C4⋊C4—3942no—
262C4⋊C4—1974no—
263C4⋊C4C4×D19714yescomplemented
264C22⋊C4—3942no—
265C22⋊C4—3942no—
266C22⋊C4—1974no—
267C22⋊C4—1974no—
268C42—41no—
269C42—7881no—
270C42—3942no—
271C42—3942no—
272C42—1974no—
273C42—1974no—
274C23—3941no—
275C23—3941no—
276C23—3941no—
277C23—3941no—
278C23—1972no—
279C23—1972no—
280C23—1972no—
281C23C23×D19711yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—7884no—
300C2×C4—7884no—
301C2×C4—3948no—
302C2×C4—3941no—
303C2×C4—3941no—
304C2×C4—3941no—
305C2×C4—3941no—
306C2×C4—3941no—
307C2×C4—3941no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—1972no—
320C2×C4—1972no—
321C2×C4—1972no—
322C2×C4—1972no—
323C2×C4C2×C4×D19712yes—
324C2×C4C2×C4×D19712yes—
325C2×C4C2×C4×D19712yes—
326C2×C4#1569135912yes—
327C22—3942no—
328C22—3942no—
329C22—3942no—
330C22—3942no—
331C22—3942no—
332C22—3942no—
333C22—1974no—
334C22C22×C4×D19711yescharacteristic
335C22#41×D19711yescharacteristic
336C22#41×D19711yescharacteristic
337C22#41×D19711yescharacteristic
338C22C2×#1569135911yescharacteristic
339C22#2026337111yescharacteristic
340C22#2026337311yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—7881no—
349C4—7881no—
350C4—7881no—
351C4—3942no—
352C4—3942no—
353C4—3942no—
354C4—3942no—
355C2—3944no—
356C2#117×D19711yescharacteristic
357C2C4×#1569135911yescharacteristic
358C2#2716155811yescharacteristic
359C2#2716175611yescharacteristic
360C2#2716175711yescharacteristic
361C2#2716176511yescharacteristic
362C2#2716176711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#387×C197C2
socle, cosocleC23×C197C24
prime radicalC23×D197C23
hypercenter#387D197
hypocenterC197#2343
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #27160965C2semidirect
3. #27160964C2semidirect
4. #27160962C2semidirect
7. #27160946C2semidirect
8. #27160944C2semidirect
10. C4×#15691337C2semidirect
13. #27160924C2semidirect
15. #387×C197C2semidirect
19. #20263284C4semidirect
70. C4⋊Dic197C2×C4semidirect
80. C4×Dic197C2×C4semidirect
81. C4×Dic197C2×C4semidirect
90. C4⋊C4×C197C2×C4semidirect
181. C197#2343semidirect
187. #387D197semidirect
263. C4⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632842
#271609652
#271609642
#271609622
#271609532
#271609502
#271609462
#271609442
#271609412
C4×#156913372
C4×#156913352
#271609262
#271609242
#95×D1972
#387×C1972
#2343197