Finite Group Databasebeta

← #38754749

Subgroups — #38754749

Order 25216.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161797C211yesmaximal, characteristic, complemented
2#27161723C211yesmaximal, characteristic
3#27161437C211yesmaximal, characteristic, complemented
4#27161420C211yesmaximal, characteristic, complemented
5#27161402C211yesmaximal, characteristic, complemented
6#27161367C211yesmaximal, characteristic, complemented
7#27161337C211yesmaximal, characteristic, complemented
8#27161307C211yesmaximal, characteristic, complemented
9#27161300C211yesmaximal, characteristic, complemented
10#27161242C211yesmaximal, characteristic, complemented
11#27161230C211yesmaximal, characteristic, complemented
12#27161176C211yesmaximal, characteristic, complemented
13#27161148C211yesmaximal, characteristic, complemented
14#27161072C211yesmaximal, characteristic, complemented
15#418×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×Dic788C2211yescharacteristic, complemented
18C2×#15691346C2211yescharacteristic
19C2×#15691346C2211yescharacteristic
20C2×C8×D197C2211yescharacteristic, complemented
21#20263325C2211yescharacteristic, complemented
22#20263320—21no—
23#20263319—21no—
24#20263313C2211yescharacteristic
25#20263312—21no—
26#20263307—21no—
27#20263306C2211yescharacteristic, complemented
28#20263305C2211yescharacteristic
29#20263305C2211yescharacteristic
30#20263298—21no—
31#20263297C2211yescharacteristic
32#20263295—21no—
33C4×Dic394—21no—
34#20263291C2211yescharacteristic, complemented
35D4×Dic197C2211yescharacteristic
36C4×C22⋊D197C2211yescharacteristic
37#20263284C2211yescharacteristic, complemented
38C2×C22⋊Dic197—21no—
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263257C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263249C2211yescharacteristic, complemented
46#20263249C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263238C2211yescharacteristic, complemented
51#20263236C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C2×M16×C197C2211yescharacteristic
54#117×C197C2211yescharacteristic
55#102×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic
59#15691357—22no—
60#15691357—41no—
61C2×Dic394—21no—
62C2×Dic394C2311yescharacteristic
63C2×Dic394C2311yescharacteristic
64C2×C22⋊D197—21no—
65C2×C22⋊D197—21no—
66C2×C22⋊D197D411yescharacteristic
67C22×Dic197—21no—
68C22×Dic197—21no—
69C2×C4×D197C2311yescharacteristic
70Dic788—41no—
71#15691346—22no—
72#15691346—41no—
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C8×D197—41no—
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79#15691337—21no—
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691337D411yescharacteristic
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87C22⋊Dic197—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90C22⋊Dic197D411yescharacteristic
91#15691335—21no—
92#15691335—21no—
93#15691335D411yescharacteristic
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197C2311yescharacteristic
97C24×C197—21no—
98C2×D4×C197—22no—
99C2×D4×C197—21no—
100C2×D4×C197—41no—
101C2×D4×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197C2311yescharacteristic
104M16×C197—41no—
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C22⋊C4×C197—21no—
110C42×C197—21no—
111C22×D197C2×D411yescharacteristic
112Dic394—21no—
113Dic394—21no—
114Dic394—21no—
115Dic394—21no—
116Dic394—42no—
117C22⋊D197—41no—
118C22⋊D197—42no—
119C22⋊D197—81no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—42no—
125C2×Dic197—81no—
126C2×Dic197C2×D411yescharacteristic
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C4×D197—21no—
130C4×D197—21no—
131#13096802—21no—
132#13096802—21no—
133C23×C197—21no—
134C23×C197—21no—
135C23×C197—42no—
136C23×C197—42no—
137C23×C197—81no—
138C23×C197C2×D411yescharacteristic
139D4×C197—21no—
140D4×C197—21no—
141D4×C197—21no—
142D4×C197—42no—
143D4×C197#4112yes—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2411yescharacteristic
150C8×C197—21no—
151C8×C197—21no—
152C22×C197—21no—
153C22×C197—22no—
154C22×C197—21no—
155C22×C197—21no—
156C22×C197—41no—
157C22×C197—41no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197—84no—
162C22×C197C22×D411yescharacteristic
163C2×D197—21no—
164C2×D197—21no—
165C2×D197—21no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197C2×#4111yescharacteristic
170C4×C197H(5,𝔽2)11yescharacteristic
171Dic197—21no—
172Dic197—21no—
173Dic197—21no—
174Dic197—41no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—41no—
179C2×C197—42no—
180C2×C197—81no—
181C2×C197#54611yescharacteristic
182C2×C197#57511yescharacteristic
183C2×C197#57711yescharacteristic
184D197—41no—
185C197#331111yescharacteristic, complemented, Sylow
186#3311—1971nomaximal, Sylow
187#556—1971no—
188#546—1971no—
189#489—1971no—
190#485—1971no—
191#482—1971no—
192#476—1971no—
193#470—1971no—
194#465—1971no—
195#464—1971no—
196#452—1971no—
197#450—1971no—
198#440—1971no—
199#435—1971no—
200#418D19711yescharacteristic, complemented
201#416—1971no—
202C2×#41—1971no—
203C22×D4C2×D19711yescharacteristic
204C2×Q16—1971no—
205C2×SD16—1971no—
206C2×SD16—1971no—
207C2×M16C2×D19711yescharacteristic
208C22×C8—1971no—
209C4⋊Q8—1971no—
210#126—3941no—
211#126—3941no—
212#124—3941no—
213#124—1971no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8—1971no—
217C22⋊Q8—1971no—
218#121—3941no—
219#121—1971no—
220#120—3941no—
221C4×Q8—3941no—
222C4×Q8—1971no—
223C4×D4—1971no—
224C4×D4—1971no—
225#117—1971no—
226#117C2×D19711yescharacteristic
227C2×C22⋊C4—3941no—
228#107—1971no—
229#106—1971no—
230#106—1971no—
231C4⋊C8—1971no—
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103—1971no—
236#102—1971no—
237#102—1971no—
238#102C2×D19711yescharacteristic
239#102C2×D19711yescharacteristic
240#102C2×D19711yescharacteristic
241#102C2×D19711yescharacteristic
242C22⋊C8—1971no—
243C22⋊C8—1971no—
244#97—1971no—
245C24—21no—
246#41—7881no—
247#41—3942no—
248C2×Q8—3941no—
249C2×Q8—1971no—
250C2×Q8—1971no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—3941no—
255C2×D4—3941no—
256C2×D4—1971no—
257C2×D4C22×D19711yescharacteristic
258C2×D4C22×D19711yescharacteristic
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4—1971no—
262C22×C4C22×D19711yescharacteristic
263Q16—7881no—
264SD16—7881no—
265SD16—3942no—
266M16—41no—
267C2×C8—7881no—
268C2×C8—1971no—
269C2×C8—1971no—
270C2×C8C22×D19711yescharacteristic
271C2×C8C22×D19711yescharacteristic
272C4⋊C4—3941no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—3941no—
277C4⋊C4—1971no—
278C4⋊C4—1971no—
279C4⋊C4—1971no—
280C4⋊C4C22×D19711yescharacteristic
281C4⋊C4C22×D19711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—1971no—
292C22⋊C4—1971no—
293C42—21no—
294C42—3941no—
295C42—3941no—
296C42—1971no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1971no—
303C23D4×D19711yescharacteristic
304Q8—7882no—
305Q8—3941no—
306Q8—3941no—
307Q8—3941no—
308Q8—3941no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—7882no—
314D4—7881no—
315D4—15761no—
316D4#1569135812yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—7882no—
322C2×C4—15761no—
323C2×C4—3941no—
324C2×C4—3941no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4—1971no—
330C2×C4—1971no—
331C2×C4—1971no—
332C2×C4D4×D19711yescharacteristic
333C2×C4C23×D19711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3941no—
337C8—3941no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—3941no—
349C22—3941no—
350C22—3941no—
351C22C2×D4×D19711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—7881no—
356C4—7881no—
357C4—7881no—
358C4—7881no—
359C4—3941no—
360C4—3941no—
361C4—3941no—
362C4C2×#1569135811yescharacteristic
363C4#2026337111yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—7881no—
368C2#2716171911yescharacteristic
369C2#2716187511yescharacteristic
370C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#418×C197C2
socleC22×C197C22×D4
prime radical#27161723C2
hypercenter#418D197
hypocenterC197#3311
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #27161797C2semidirect
3. #27161437C2semidirect
4. #27161420C2semidirect
5. #27161402C2semidirect
6. #27161367C2semidirect
7. #27161337C2semidirect
8. #27161307C2semidirect
9. #27161300C2semidirect
10. #27161242C2semidirect
11. #27161230C2semidirect
12. #27161176C2semidirect
13. #27161148C2semidirect
14. #27161072C2semidirect
15. #418×C197C2semidirect
17. C2×Dic788C22semidirect
20. C2×C8×D197C22semidirect
21. #20263325C22semidirect
27. #20263306C22semidirect
34. #20263291C22semidirect
37. #20263284C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263257C22semidirect
42. #20263255C22semidirect
43. #20263251C22semidirect
44. #20263251C22semidirect
45. #20263249C22semidirect
46. #20263249C22semidirect
50. #20263238C22semidirect
51. #20263236C22semidirect
185. C197#3311semidirect
200. #418D197semidirect

Maximal subgroups

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

subgroupindex
#271617972
#271617232
#271614372
#271614202
#271614022
#271613672
#271613372
#271613072
#271613002
#271612422
#271612302
#271611762
#271611482
#271610722
#418×C1972
#3311197