Finite Group Databasebeta

← #38752899

Subgroups — #38752899

Order 25216.

406 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
1#27161702C211yesmaximal, characteristic, complemented
2#27161553C211yesmaximal, characteristic, complemented
3#27161403C211yesmaximal, characteristic, complemented
4#27161368C211yesmaximal, characteristic, complemented
5#27161358C211yesmaximal, characteristic, complemented
6#27161297C211yesmaximal, characteristic, complemented
7#27161297C211yesmaximal, characteristic, complemented
8#27161229C211yesmaximal, characteristic, complemented
9#27161215C211yesmaximal, characteristic
10#27161190C211yesmaximal, characteristic, complemented
11#27161182C211yesmaximal, characteristic, complemented
12#27161164C211yesmaximal, characteristic, complemented
13#27161161C211yesmaximal, characteristic, complemented
14#27161137C211yesmaximal, characteristic, complemented
15#485×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C2×#15691345C2211yescharacteristic
18C2×#15691345C2211yescharacteristic
19C2×D1576C2211yescharacteristic
20C2×D1576C2211yescharacteristic
21#20263325C2211yescharacteristic, complemented
22#20263321—21no—
23#20263315C2211yescharacteristic, complemented
24#20263312—21no—
25#20263308—21no—
26#20263306—21no—
27#20263300C2211yescharacteristic
28Q8×Dic197C2211yescharacteristic, complemented
29C4×D788—21no—
30C4×D788—21no—
31#20263288C2211yescharacteristic
32#20263288C2211yescharacteristic
33#20263288C2211yescharacteristic
34#20263288C2211yescharacteristic
35#20263288C2211yescharacteristic, complemented
36#20263285C2211yescharacteristic
37C22⋊C4×D197—21no—
38C22⋊C4×D197—21no—
39#20263259C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic
45#20263246C2211yescharacteristic
46#20263246C2211yescharacteristic
47#20263246C2211yescharacteristic, complemented
48#20263246C2211yescharacteristic, complemented
49C22⋊#13096802C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22⋊Q8×C197C2211yescharacteristic
53#117×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C23×D197—21no—
60C2×Dic394—21no—
61C2×D788—22no—
62C2×D788—22no—
63C2×D788—21no—
64C2×D788C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73#15691345—22no—
74#15691345—22no—
75D1576—22no—
76D1576—22no—
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C4⋊Dic197—21no—
80C4⋊Dic197D411yescharacteristic
81C4⋊Dic197D411yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84C22⋊Dic197D411yescharacteristic
85C22⋊Dic197D411yescharacteristic
86#15691335—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94C4×Dic197—21no—
95C4×Dic197C2311yescharacteristic
96C4×Dic197C2311yescharacteristic
97C2×Q8×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C2×C8×C197C2311yescharacteristic
101C4⋊C4×C197—21no—
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197—21no—
108C22×D197—21no—
109C22×D197—21no—
110C22×D197—24no—
111C22×D197—21no—
112C22×D197—81no—
113C22×D197—81no—
114Dic394—81no—
115D788—24no—
116D788—21no—
117D788—21no—
118D788—21no—
119D788—21no—
120D788#4112yes—
121D788#4112yes—
122C22⋊D197—81no—
123C2×Dic197—21no—
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C2×Dic197C2×D411yescharacteristic
127C2×Dic197C2×D411yescharacteristic
128C4×D197—81no—
129C4×D197—81no—
130C4×D197—81no—
131C4×D197—81no—
132C4×D197—81no—
133#13096802—21no—
134#13096802—21no—
135C23×C197C2×D411yescharacteristic
136Q8×C197—21no—
137Q8×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×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C8×C197—21no—
147C8×C197—21no—
148C22×C197—21no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197C22×D411yescharacteristic
152C2×D197—22no—
153C2×D197—22no—
154C2×D197—41no—
155C2×D197—41no—
156C2×D197—41no—
157C2×D197—41no—
158C2×D197—41no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—84no—
162C4×C197—21no—
163C4×C197—41no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197—41no—
167C4×C197C2×#4111yescharacteristic
168C4×C197C2×#4111yescharacteristic
169Dic197—21no—
170Dic197—21no—
171Dic197—21no—
172Dic197—21no—
173Dic197—41no—
174C2×C197—41no—
175C2×C197#52511yescharacteristic
176C2×C197#57611yescharacteristic
177C2×C197#57711yescharacteristic
178D197—42no—
179D197—42no—
180D197—81no—
181C197#311811yescharacteristic, complemented, Sylow
182#3118—1971nomaximal, Sylow
183#542—1971no—
184#518—1971no—
185#485D19711yescharacteristic, complemented
186#482—1971no—
187#476—1971no—
188#474—1971no—
189#463—1971no—
190#463—1971no—
191#450—1971no—
192#447—1971no—
193#442—1971no—
194#440—1971no—
195C4×SD16—1971no—
196C4×D8—1971no—
197#433—1971no—
198C22×D4—1971no—
199C2×SD16—1971no—
200C2×SD16—1971no—
201C2×D8—1971no—
202C2×D8—1971no—
203C4⋊Q8—1971no—
204#126—3941no—
205#124—3941no—
206#124—1971no—
207#123—3941no—
208C22⋊Q8—3941no—
209C22⋊Q8C2×D19711yescharacteristic
210#121—1971no—
211C4×Q8—1971no—
212C4×D4—3941no—
213C4×D4—3941no—
214C4×D4—1971no—
215C4×D4—1971no—
216C4×D4—1971no—
217C4×D4—1971no—
218C4×D4—1971no—
219#117—1971no—
220#117C2×D19711yescharacteristic
221C2×C22⋊C4—3941no—
222C2×C22⋊C4—3941no—
223#107—1971no—
224#107C2×D19711yescharacteristic, complemented
225#106—1971no—
226#106C2×D19711yescharacteristic, complemented
227C4⋊C8—1971no—
228C4⋊C8—1971no—
229#103C2×D19711yescharacteristic, complemented
230#103C2×D19711yescharacteristic, complemented
231#102—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102—1971no—
236#102—1971no—
237C22⋊C8—1971no—
238C22⋊C8C2×D19711yescharacteristic
239#97—1971no—
240C4×C8—1971no—
241C24—3941no—
242C2×Q8—3941no—
243C2×Q8C22×D19711yescharacteristic
244C2×D4—3942no—
245C2×D4—3942no—
246C2×D4—3941no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4—1971no—
250C2×D4—1971no—
251C22×C4—3941no—
252C22×C4—3941no—
253C22×C4—3941no—
254C22×C4—3941no—
255C22×C4—3941no—
256C22×C4C22×D19711yescharacteristic
257SD16—3942no—
258SD16—3942no—
259D8—3942no—
260D8—3942no—
261C2×C8—1971no—
262C2×C8—1971no—
263C2×C8C22×D19711yescharacteristic
264C2×C8C22×D19711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—1971no—
269C4⋊C4—1971no—
270C4⋊C4—1971no—
271C4⋊C4C22×D19711yescharacteristic
272C4⋊C4C22×D19711yescharacteristic
273C4⋊C4C22×D19711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—3941no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—1971no—
285C22⋊C4—1971no—
286C42—21no—
287C42—3941no—
288C42—1971no—
289C42—1971no—
290C23—15761no—
291C23—15761no—
292C23—3944no—
293C23—3941no—
294C23—3941no—
295C23—3941no—
296C23D4×D19711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—15761no—
300D4—15761no—
301D4—3944no—
302D4—3941no—
303D4—3941no—
304D4—3941no—
305D4—3941no—
306D4—1972no—
307D4—1972no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—15761no—
315C2×C4—15761no—
316C2×C4—15761no—
317C2×C4—15761no—
318C2×C4—15761no—
319C2×C4—3941no—
320C2×C4—1971no—
321C2×C4—1971no—
322C2×C4—1971no—
323C2×C4—1971no—
324C2×C4D4×D19711yescharacteristic
325C2×C4C23×D19711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3941no—
329C8—3941no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—7881no—
334C22—7881no—
335C22—7881no—
336C22—7881no—
337C22—7881no—
338C22—7881no—
339C22—7881no—
340C22—15764no—
341C22—3942no—
342C22—3942no—
343C22C2×D4×D19711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—7881no—
350C4—3941no—
351C4—3941no—
352C4—3941no—
353C4—3941no—
354C4#41×D19711yescharacteristic
355C4#41×D19711yescharacteristic
356C2—41no—
357C2—7882no—
358C2—7882no—
359C2—15761no—
360C2#2716158711yescharacteristic
361C2#2716188111yescharacteristic
362C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#485×C197C2
socleC22×C197C22×D4
prime radical#27161215C2
hypercenter#485D197
hypocenterC197#3118
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #27161702C2semidirect
2. #27161553C2semidirect
3. #27161403C2semidirect
4. #27161368C2semidirect
5. #27161358C2semidirect
6. #27161297C2semidirect
7. #27161297C2semidirect
8. #27161229C2semidirect
10. #27161190C2semidirect
11. #27161182C2semidirect
12. #27161164C2semidirect
13. #27161161C2semidirect
14. #27161137C2semidirect
15. #485×C197C2semidirect
21. #20263325C22semidirect
23. #20263315C22semidirect
28. Q8×Dic197C22semidirect
35. #20263288C22semidirect
39. #20263259C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
47. #20263246C22semidirect
48. #20263246C22semidirect
50. #20263236C22semidirect
51. C8×Dic197C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. #103×C197C22semidirect
57. #103×C197C22semidirect
181. C197#3118semidirect
185. #485D197semidirect
224. #107C2×D197semidirect
226. #106C2×D197semidirect
229. #103C2×D197semidirect
230. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617022
#271615532
#271614032
#271613682
#271613582
#271612972
#271612292
#271612152
#271611902
#271611822
#271611642
#271611612
#271611372
#485×C1972
#3118197