Finite Group Databasebeta

← #38745873

Subgroups — #38745873

Order 25216.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 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#27160971C211yesmaximal, characteristic
6#27160970C211yesmaximal, characteristic
7#27160969C211yesmaximal, characteristic
8#27160968C211yesmaximal, characteristic
9#27160964C211yesmaximal, characteristic, complemented
10#27160963C211yesmaximal, characteristic, complemented
11#27160962C211yesmaximal, characteristic, complemented
12#27160961C211yesmaximal, characteristic, complemented
13#27160941C211yesmaximal, characteristic
14#27160938C211yesmaximal, characteristic
15#391×C197C211yesmaximal, characteristic, complemented
16C22×C4×D197C2211yescharacteristic
17#20263302C418yescomplemented
18C2×C4⋊Dic197C2211yescharacteristic
19C2×C4⋊Dic197C2211yescharacteristic
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691335C2211yescharacteristic
27C2×#15691335C2211yescharacteristic
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—41no—
61C2×C4×D197—41no—
62C2×C4×D197C2×C414yes—
63C4⋊Dic197—41no—
64C4⋊Dic197C2×C414yes—
65#15691337—41no—
66#15691337—41no—
67#15691337—41no—
68#15691337—41no—
69#15691337C2×C414yes—
70#15691337C2×C414yes—
71#15691335—41no—
72#15691335—41no—
73#15691335—41no—
74#15691335—41no—
75#15691335C2×C414yes—
76#15691335C2×C414yes—
77C4×Dic197—41no—
78C4×Dic197—41no—
79C4×Dic197—41no—
80C2×Q8×C197—41no—
81C2×Q8×C197—41no—
82C2×Q8×C197C2×C414yes—
83C22×C4×C197C2311yescharacteristic
84C22×C4×C197C2311yescharacteristic
85C22×C4×C197C2311yescharacteristic
86C22×C4×C197C2311yescharacteristic
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C4⋊C4×C197—41no—
91C42×C197—41no—
92C22×D197—21no—
93C22×D197—21no—
94C22×D197—21no—
95C22×D197—21no—
96C22×D197—21no—
97C22×D197—21no—
98C22×D197C22×C412yes—
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×Dic197—21no—
117C2×Dic197C22×C412yes—
118C2×Dic197C22×C412yes—
119C2×Dic197C22×C412yes—
120C4×D197—44no—
121C23×C197C2411yescharacteristic
122Q8×C197—44no—
123C2×C4×C197—21no—
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—
144C22×C197C23×C411yescharacteristic
145C22×C197H(5,𝔽2)11yescharacteristic
146C22×C197H(5,𝔽2)11yescharacteristic
147C22×C197H(5,𝔽2)11yescharacteristic
148C22×C1972−1+411yescharacteristic
149C22×C1972−1+411yescharacteristic
150C22×C1972−1+411yescharacteristic
151C2×D197—22no—
152C2×D197—22no—
153C2×D197—22no—
154C2×D197—41no—
155C2×D197—41no—
156C2×D197—41no—
157C2×D197—41no—
158C4×C197—22no—
159C4×C197—22no—
160C4×C197—22no—
161C4×C197—41no—
162C4×C197—41no—
163C4×C197—41no—
164C4×C197—41no—
165Dic197—22no—
166Dic197—22no—
167Dic197—22no—
168Dic197—41no—
169Dic197—41no—
170Dic197—41no—
171Dic197—41no—
172C2×C197#51811yescharacteristic
173C2×C197#51911yescharacteristic
174C2×C197#52011yescharacteristic
175C2×C197#56211yescharacteristic
176C2×C197#56211yescharacteristic
177C2×C197#56311yescharacteristic
178C2×C197#56311yescharacteristic
179D197—42no—
180C197#237811yescharacteristic, complemented, Sylow
181#2378—1971nomaximal, Sylow
182C2×C22⋊Q8—1971no—
183#391—1971no—
184#391D19711yescharacteristic, complemented
185#389—1971no—
186#389—1971no—
187#388—1971no—
188#388—1971no—
189#388—1971no—
190#388—1971no—
191#387—1971no—
192#387—1971no—
193#387—1971no—
194#387—1971no—
195#381—1971no—
196#380—1971no—
197C22×Q8C2×D19711yescharacteristic
198C23×C4—1971no—
199C22⋊Q8—1978no—
200C2×C4⋊C4—1971no—
201C2×C4⋊C4—1971no—
202C2×C4⋊C4—1971no—
203C2×C4⋊C4—1971no—
204C2×C4⋊C4—1971no—
205C2×C4⋊C4—1971no—
206C2×C4⋊C4—1971no—
207C2×C4⋊C4—1971no—
208C2×C4⋊C4C2×D19711yescharacteristic
209C2×C22⋊C4—1971no—
210C2×C22⋊C4—1971no—
211C2×C22⋊C4—1971no—
212C2×C22⋊C4—1971no—
213C2×C22⋊C4—1971no—
214C2×C22⋊C4—1971no—
215C2×C42—1971no—
216C2×C42—1971no—
217C2×C42—1971no—
218C2×C42C2×D19711yescharacteristic
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#95C2×D19711yescharacteristic
230#95C2×D19711yescharacteristic
231#95C2×D19711yescharacteristic
232#95C2×D19711yescharacteristic
233C24—1971no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8C4×D19714yes—
237C22×C4—7881no—
238C22×C4—7881no—
239C22×C4—1974no—
240C22×C4—1971no—
241C22×C4—1971no—
242C22×C4—1971no—
243C22×C4—1971no—
244C22×C4—1971no—
245C22×C4—1971no—
246C22×C4—1971no—
247C22×C4C22×D19711yescharacteristic
248C22×C4C22×D19711yescharacteristic
249C22×C4C22×D19711yescharacteristic
250C22×C4C22×D19711yescharacteristic
251C22×C4C22×D19711yescharacteristic
252C22×C4C22×D19711yescharacteristic
253C22×C4C22×D19711yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—7881no—
256C4⋊C4—7881no—
257C4⋊C4—7881no—
258C4⋊C4—7881no—
259C4⋊C4—7881no—
260C4⋊C4—1974no—
261C4⋊C4—1974no—
262C4⋊C4—1974no—
263C22⋊C4—7881no—
264C22⋊C4—7881no—
265C22⋊C4—7881no—
266C22⋊C4—7881no—
267C22⋊C4—1974no—
268C22⋊C4—1974no—
269C42—41no—
270C42—7881no—
271C42—7881no—
272C42—7881no—
273C23—3941no—
274C23—3941no—
275C23—3941no—
276C23—3941no—
277C23—3941no—
278C23—3941no—
279C23—1972no—
280C23C23×D19711yescharacteristic
281Q8—44no—
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—21no—
300C2×C4—7884no—
301C2×C4—3941no—
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×C4C2×C4×D19712yes—
323C2×C4C2×C4×D19712yes—
324C2×C4C2×C4×D19712yes—
325C22—7881no—
326C22—7881no—
327C22—7881no—
328C22—7881no—
329C22—3942no—
330C22—3942no—
331C22—3942no—
332C22C22×C4×D19711yescharacteristic
333C22#2026337111yescharacteristic
334C22#2026337111yescharacteristic
335C22#2026337211yescharacteristic
336C22#2026337311yescharacteristic
337C22#2026337411yescharacteristic
338C22#2026337411yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—7881no—
347C4—7881no—
348C4—7881no—
349C4—7881no—
350C4—3942no—
351C4—3942no—
352C4—3942no—
353C2—7882no—
354C2#2716155211yescharacteristic
355C2#2716155511yescharacteristic
356C2#2716156111yescharacteristic
357C2#2716182311yescharacteristic
358C2#2716182511yescharacteristic
359C2#2716182811yescharacteristic
360C2#2716183011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#391×C197C2
socle, cosocleC23×C197C24
prime radicalC23×D197C23
hypercenter#391D197
hypocenterC197#2378
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
9. #27160964C2semidirect
10. #27160963C2semidirect
11. #27160962C2semidirect
12. #27160961C2semidirect
15. #391×C197C2semidirect
17. #20263302C4semidirect
180. C197#2378semidirect
184. #391D197semidirect

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
#271609712
#271609702
#271609692
#271609682
#271609642
#271609632
#271609622
#271609612
#271609412
#271609382
#391×C1972
#2378197