Finite Group Databasebeta

← #38752898

Subgroups — #38752898

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#27161700C211yesmaximal, characteristic, complemented
2#27161553C211yesmaximal, characteristic, complemented
3#27161421C211yesmaximal, characteristic, complemented
4#27161404C211yesmaximal, characteristic, complemented
5#27161370C211yesmaximal, characteristic, complemented
6#27161360C211yesmaximal, characteristic, complemented
7#27161295C211yesmaximal, characteristic, complemented
8#27161295C211yesmaximal, characteristic, complemented
9#27161232C211yesmaximal, characteristic, complemented
10#27161215C211yesmaximal, characteristic
11#27161192C211yesmaximal, characteristic, complemented
12#27161181C211yesmaximal, characteristic, complemented
13C4×#15691347C211yesmaximal, characteristic, complemented
14C4×D1576C211yesmaximal, characteristic, complemented
15#433×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C2×#15691347C2211yescharacteristic
18C2×#15691347C2211yescharacteristic
19C2×D1576C2211yescharacteristic
20C2×D1576C2211yescharacteristic
21#20263326C2211yescharacteristic, complemented
22#20263321—21no—
23#20263314C2211yescharacteristic, complemented
24#20263312—21no—
25#20263308—21no—
26#20263305C2211yescharacteristic
27#20263304—21no—
28#20263300C2211yescharacteristic
29C4×Dic394C2211yescharacteristic, complemented
30C4×D788C2211yescharacteristic
31C4×D788C2211yescharacteristic
32C4×D788C2211yescharacteristic
33C4×D788C2211yescharacteristic
34C4×D788C2211yescharacteristic, complemented
35#20263288—21no—
36#20263288—21no—
37#20263285C2211yescharacteristic
38C22⋊C4×D197—21no—
39C22⋊C4×D197—21no—
40#20263260C2211yescharacteristic, complemented
41#20263260C2211yescharacteristic, complemented
42#20263258C2211yescharacteristic, complemented
43#20263258C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263251C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic
47#20263248C2211yescharacteristic
48#20263248C2211yescharacteristic
49#20263248C2211yescharacteristic
50#20263248C2211yescharacteristic, complemented
51#20263248C2211yescharacteristic, complemented
52#117×C197C2211yescharacteristic
53C4⋊C8×C197C2211yescharacteristic, complemented
54C4⋊C8×C197C2211yescharacteristic, complemented
55C22⋊C8×C197C2211yescharacteristic
56C22⋊C8×C197C2211yescharacteristic
57#97×C197C2211yescharacteristic, complemented
58C4×C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×Dic394—21no—
61C2×Dic394C2311yescharacteristic
62C2×D788—22no—
63C2×D788—22no—
64C2×D788—21no—
65C2×D788C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74#15691347—22no—
75#15691347—22no—
76D1576—22no—
77D1576—22no—
78C4⋊Dic197—21no—
79C4⋊Dic197C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84#15691337—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
97C22×C4×C197C2311yescharacteristic
98C2×C8×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C2×C8×C197C2311yescharacteristic
101C2×C8×C197C2311yescharacteristic
102C4⋊C4×C197D411yescharacteristic
103C4⋊C4×C197D411yescharacteristic
104C22⋊C4×C197D411yescharacteristic
105C22⋊C4×C197D411yescharacteristic
106C42×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—21no—
110C22×D197—24no—
111C22×D197—21no—
112C22×D197—81no—
113C22×D197—81no—
114Dic394—21no—
115Dic394—21no—
116Dic394—81no—
117D788—24no—
118D788—21no—
119D788—21no—
120D788—21no—
121D788—21no—
122D788#4112yes—
123D788#4112yes—
124C22⋊D197—81no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C4×D197—81no—
131C4×D197—81no—
132C4×D197—81no—
133C4×D197—81no—
134C4×D197—81no—
135C23×C197C2×D411yescharacteristic
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197C2×D411yescharacteristic
139C2×C4×C197C2×D411yescharacteristic
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2411yescharacteristic
144C8×C197—21no—
145C8×C197—21no—
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—21no—
164C4×C197—21no—
165C4×C197—21no—
166C4×C197—21no—
167C4×C197C2×#4111yescharacteristic
168C4×C197C2×#4111yescharacteristic
169Dic197—41no—
170Dic197—41no—
171Dic197—41no—
172Dic197—41no—
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#485—1971no—
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#433D19711yescharacteristic, complemented
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⋊Q8—1971no—
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#107—1971no—
225#106—1971no—
226#106—1971no—
227C4⋊C8C2×D19711yescharacteristic, complemented
228C4⋊C8C2×D19711yescharacteristic, complemented
229#103—1971no—
230#103—1971no—
231#102—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102—1971no—
236#102—1971no—
237C22⋊C8C2×D19711yescharacteristic
238C22⋊C8C2×D19711yescharacteristic
239#97C2×D19711yescharacteristic, complemented
240C4×C8C2×D19711yescharacteristic, complemented
241C24—3941no—
242C2×Q8—3941no—
243C2×Q8—1971no—
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×C8C22×D19711yescharacteristic
262C2×C8C22×D19711yescharacteristic
263C2×C8C22×D19711yescharacteristic
264C2×C8C22×D19711yescharacteristic
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—1971no—
269C4⋊C4—1971no—
270C4⋊C4—1971no—
271C4⋊C4—1971no—
272C4⋊C4D78811yescharacteristic
273C4⋊C4D78811yescharacteristic
274C22⋊C4—3941no—
275C22⋊C4—3941no—
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⋊C4D78811yescharacteristic
285C22⋊C4D78811yescharacteristic
286C42—3941no—
287C42—3941no—
288C42C22×D19711yescharacteristic
289C42C22×D19711yescharacteristic
290C23—15761no—
291C23—15761no—
292C23—3944no—
293C23—3941no—
294C23—3941no—
295C23—3941no—
296C23C2×D78811yescharacteristic
297Q8—15761no—
298Q8—3941no—
299Q8—3941no—
300D4—15761no—
301D4—3944no—
302D4—3941no—
303D4—3941no—
304D4—3941no—
305D4—3941no—
306D4—1972no—
307D4—1972no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—15761no—
311C2×C4—15761no—
312C2×C4—15761no—
313C2×C4—15761no—
314C2×C4—15761no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4C2×D78811yescharacteristic
321C2×C4C2×D78811yescharacteristic
322C2×C4C2×D78811yescharacteristic
323C2×C4C2×D78811yescharacteristic
324C2×C4C2×D78811yescharacteristic
325C2×C4C23×D19711yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
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—
343C22C22×D78811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—7881no—
350C4—7881no—
351C4—7881no—
352C4—7881no—
353C4—7881no—
354C4#41×D19711yescharacteristic
355C4#41×D19711yescharacteristic
356C2—41no—
357C2—7882no—
358C2—7882no—
359C2—15761no—
360C2#2716159111yescharacteristic
361C2#2716188411yescharacteristic
362C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#433×C197C2
socleC22×C197C22×D4
prime radical#27161215C2
hypercenter#433D197
hypocenterC197#3118
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #27161700C2semidirect
2. #27161553C2semidirect
3. #27161421C2semidirect
4. #27161404C2semidirect
5. #27161370C2semidirect
6. #27161360C2semidirect
7. #27161295C2semidirect
8. #27161295C2semidirect
9. #27161232C2semidirect
11. #27161192C2semidirect
12. #27161181C2semidirect
13. C4×#15691347C2semidirect
14. C4×D1576C2semidirect
15. #433×C197C2semidirect
21. #20263326C22semidirect
23. #20263314C22semidirect
29. C4×Dic394C22semidirect
34. C4×D788C22semidirect
40. #20263260C22semidirect
41. #20263260C22semidirect
42. #20263258C22semidirect
43. #20263258C22semidirect
44. #20263251C22semidirect
45. #20263251C22semidirect
50. #20263248C22semidirect
51. #20263248C22semidirect
53. C4⋊C8×C197C22semidirect
54. C4⋊C8×C197C22semidirect
57. #97×C197C22semidirect
58. C4×C8×C197C22semidirect
181. C197#3118semidirect
197. #433D197semidirect
227. C4⋊C8C2×D197semidirect
228. C4⋊C8C2×D197semidirect
239. #97C2×D197semidirect
240. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617002
#271615532
#271614212
#271614042
#271613702
#271613602
#271612952
#271612322
#271612152
#271611922
#271611812
C4×#156913472
C4×D15762
#433×C1972
#3118197