Finite Group Databasebeta

← #38745342

Subgroups — #38745342

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#27160972C211yesmaximal, characteristic, complemented
3#27160963C211yesmaximal, characteristic, complemented
4#27160961C211yesmaximal, characteristic, complemented
5#27160955C211yesmaximal, characteristic
6#27160955C211yesmaximal, characteristic
7#27160953C211yesmaximal, characteristic, complemented
8#27160950C211yesmaximal, characteristic, complemented
9#27160946C211yesmaximal, characteristic, complemented
10#27160944C211yesmaximal, characteristic, complemented
11#27160940C211yesmaximal, characteristic, complemented
12#27160937C211yesmaximal, characteristic, complemented
13C4×#15691335C211yesmaximal, characteristic, complemented
14#95×D197C211yesmaximal, characteristic, complemented
15#388×C197C211yesmaximal, characteristic, complemented
16C22×C22⋊D197C2211yescharacteristic
17C23×Dic197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19C22×C4×D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21#20263286C418yescomplemented
22C2×C4⋊Dic197C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×C22⋊Dic197C2211yescharacteristic
27C2×C22⋊Dic197C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
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×C22⋊C4×C197C2211yescharacteristic
47C2×C22⋊C4×C197C2211yescharacteristic
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C42×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C23×D197C2311yescharacteristic
53C2×C22⋊D197—22no—
54C2×C22⋊D197—22no—
55C2×C22⋊D197C2×C414yes—
56C22×Dic197—41no—
57C22×Dic197—41no—
58C22×Dic197C2×C414yescomplemented
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66C2×C4×D197—22no—
67C2×C4×D197—22no—
68C2×C4×D197—41no—
69C2×C4×D197—41no—
70C2×C4×D197—41no—
71C2×C4×D197—41no—
72C2×C4×D197—41no—
73C2×C4×D197—41no—
74C2×C4×D197C2×C414yes—
75C4⋊Dic197—41no—
76#15691337—41no—
77#15691337—41no—
78#15691337C2×C414yescomplemented
79C22⋊Dic197—41no—
80C22⋊Dic197—41no—
81#15691335—22no—
82#15691335—22no—
83#15691335C2×C414yes—
84C4×Dic197—41no—
85C4×Dic197C2×C414yescomplemented
86C24×C197C2311yescharacteristic
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C4⋊C4×C197—41no—
94C22⋊C4×C197—41no—
95C22⋊C4×C197—41no—
96C22⋊C4×C197C2×C414yescomplemented
97C42×C197—41no—
98C22×D197—21no—
99C22×D197—21no—
100C22×D197—21no—
101C22×D197—21no—
102C22×D197C22×C412yes—
103C22×D197#4112yes—
104C22×D197#4112yes—
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×Dic197C22×C412yes—
126C2×Dic197C22×C412yes—
127C2×Dic197C22×C412yes—
128C4×D197—44no—
129C4×D197—44no—
130C4×D197—44no—
131C23×C197—21no—
132C23×C197—21no—
133C23×C197—21no—
134C23×C197—21no—
135C23×C197—21no—
136C23×C197—21no—
137C23×C197C22×C412yes—
138C23×C197C2411yescharacteristic
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—21no—
155C2×C4×C197C22×C412yes—
156C2×C4×C197C22×C412yes—
157C22×C197—22no—
158C22×C197—22no—
159C22×C197—22no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—41no—
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—22no—
175C2×D197—22no—
176C2×D197—22no—
177C2×D197#11714yes—
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—42no—
192C2×C197C2×#11711yescharacteristic
193C2×C197#51811yescharacteristic
194C2×C197#52011yescharacteristic
195C2×C197#53811yescharacteristic
196C2×C197#53811yescharacteristic
197C2×C197#53911yescharacteristic
198C2×C197#53911yescharacteristic
199D197—24no—
200C197#231811yescharacteristic, complemented, Sylow
201#2318—1971nomaximal, Sylow
202C2×C4×D4—1971no—
203#388—1971no—
204#388D19711yescharacteristic, complemented
205#387—1971no—
206#387—1971no—
207#386—1971no—
208#386—1971no—
209#385—1971no—
210#385—1971no—
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×C4—1971no—
222C4×D4—1978no—
223C2×C4⋊C4—1971no—
224C2×C4⋊C4—1971no—
225C2×C4⋊C4—1971no—
226C2×C4⋊C4—1971no—
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⋊C4C2×D19711yescharacteristic
234C2×C22⋊C4C2×D19711yescharacteristic
235C2×C22⋊C4C2×D19711yescharacteristic
236C2×C42—1971no—
237C2×C42—1971no—
238C2×C42C2×D19711yescharacteristic
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—7881no—
259C22×C4—7881no—
260C22×C4—7881no—
261C22×C4—7881no—
262C22×C4—7881no—
263C22×C4—7881no—
264C22×C4—7881no—
265C22×C4—7881no—
266C22×C4—3942no—
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—7881no—
285C4⋊C4—7881no—
286C4⋊C4—7881no—
287C4⋊C4—1974no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—7881no—
291C22⋊C4—7881no—
292C22⋊C4—3942no—
293C22⋊C4—3942no—
294C22⋊C4—1974no—
295C22⋊C4C4×D19714yescomplemented
296C42—41no—
297C42—7881no—
298C42—1974no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—3941no—
306C23—3941no—
307C23—3941no—
308C23—3941no—
309C23—1972no—
310C23—1972no—
311C23—1972no—
312C23C2×C4×D19712yes—
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—7884no—
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—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3942no—
366C22—3942no—
367C22—3942no—
368C22—3942no—
369C22—3942no—
370C22—3942no—
371C22—1974no—
372C22C22×C4×D19711yescharacteristic
373C22#41×D19711yescharacteristic
374C22#41×D19711yescharacteristic
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—42no—
393C2—3944no—
394C2#117×D19711yescharacteristic
395C2#2716155111yescharacteristic
396C2#2716155811yescharacteristic
397C2#2716166111yescharacteristic
398C2#2716166311yescharacteristic
399C2#2716167011yescharacteristic
400C2#2716167211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#388×C197C2
socle, cosocleC23×C197C24
prime radicalC22×C22⋊D197C22
hypercenter#388D197
hypocenterC197#2318
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #27160972C2semidirect
3. #27160963C2semidirect
4. #27160961C2semidirect
7. #27160953C2semidirect
8. #27160950C2semidirect
9. #27160946C2semidirect
10. #27160944C2semidirect
11. #27160940C2semidirect
12. #27160937C2semidirect
13. C4×#15691335C2semidirect
14. #95×D197C2semidirect
15. #388×C197C2semidirect
21. #20263286C4semidirect
58. C22×Dic197C2×C4semidirect
78. #15691337C2×C4semidirect
85. C4×Dic197C2×C4semidirect
96. C22⋊C4×C197C2×C4semidirect
200. C197#2318semidirect
204. #388D197semidirect
295. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632862
#271609722
#271609632
#271609612
#271609552
#271609532
#271609502
#271609462
#271609442
#271609402
#271609372
C4×#156913352
#95×D1972
#388×C1972
#2318197