Finite Group Databasebeta

← #38752896

Subgroups — #38752896

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#27161420C211yesmaximal, characteristic, complemented
3#27161403C211yesmaximal, characteristic, complemented
4#27161369C211yesmaximal, characteristic, complemented
5#27161359C211yesmaximal, characteristic, complemented
6#27161294C211yesmaximal, characteristic, complemented
7#27161294C211yesmaximal, characteristic, complemented
8#27161230C211yesmaximal, characteristic, complemented
9#27161213C211yesmaximal, characteristic
10#27161189C211yesmaximal, characteristic, complemented
11#27161179C211yesmaximal, characteristic, complemented
12C4×#15691346C211yesmaximal, characteristic, complemented
13C4×#15691343C211yesmaximal, characteristic, complemented
14#27161140C211yesmaximal, characteristic, complemented
15#518×C197C211yesmaximal, characteristic, complemented
16C2×#15691346C2211yescharacteristic
17C2×#15691346C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20#20263326C2211yescharacteristic, complemented
21#20263321—21no—
22#20263314C2211yescharacteristic, complemented
23#20263312—21no—
24#20263308—21no—
25#20263305C2211yescharacteristic
26#20263304—21no—
27#20263300C2211yescharacteristic
28C4×Dic394C2211yescharacteristic, complemented
29C4×D788C2211yescharacteristic, complemented
30#20263259C2211yescharacteristic, complemented
31#20263259C2211yescharacteristic, complemented
32#20263257C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35C4⋊#13096802C2211yescharacteristic, complemented
36#20263249C2211yescharacteristic, complemented
37#20263249C2211yescharacteristic, complemented
38#20263247C2211yescharacteristic
39#20263247C2211yescharacteristic
40#20263247C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263246C2211yescharacteristic, complemented
43#20263246C2211yescharacteristic, complemented
44C22⋊#13096802C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46#20263237C2211yescharacteristic, complemented
47C4×#13096802C2211yescharacteristic, complemented
48C22×D4×C197C2211yescharacteristic
49C4×D4×C197—21no—
50C4×D4×C197—21no—
51C4×D4×C197C2211yescharacteristic
52C4×D4×C197C2211yescharacteristic
53C4×D4×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic
55#117×C197C2211yescharacteristic
56#117×C197C2211yescharacteristic
57C2×C22⋊C4×C197—21no—
58C2×C22⋊C4×C197—21no—
59C2×Dic394—21no—
60C2×Dic394C2311yescharacteristic
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C4×D197—21no—
64#15691346—22no—
65#15691346—22no—
66#15691343—22no—
67#15691343—22no—
68C2×#13096802C2311yescharacteristic
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C4⋊Dic197—21no—
73C4⋊Dic197C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75#15691337—21no—
76#15691337—21no—
77C22⋊Dic197—21no—
78#15691335—21no—
79#15691335—21no—
80#15691335—21no—
81#15691335—21no—
82C4×Dic197—21no—
83C24×C197—21no—
84C2×D4×C197—22no—
85C2×D4×C197—22no—
86C2×D4×C197—21no—
87C2×D4×C197C2311yescharacteristic
88C2×D4×C197C2311yescharacteristic
89C22×C4×C197—21no—
90C22×C4×C197—21no—
91C22×C4×C197—21no—
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
94C4⋊C4×C197D411yescharacteristic
95C4⋊C4×C197D411yescharacteristic
96C4⋊C4×C197C2311yescharacteristic
97C4⋊C4×C197C2311yescharacteristic
98C22⋊C4×C197—21no—
99C22⋊C4×C197—21no—
100C22⋊C4×C197—21no—
101C22⋊C4×C197—21no—
102C22⋊C4×C197—21no—
103C22⋊C4×C197D411yescharacteristic
104C22⋊C4×C197D411yescharacteristic
105C42×C197—21no—
106C42×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109Dic394—21no—
110Dic394—21no—
111Dic394—81no—
112D788—21no—
113D788—21no—
114C22⋊D197—81no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C4×D197—81no—
119#13096802—21no—
120#13096802—21no—
121#13096802—21no—
122#13096802—21no—
123C23×C197—24no—
124C23×C197—21no—
125C23×C197—21no—
126C23×C197—81no—
127C23×C197—81no—
128C23×C197C2×D411yescharacteristic
129D4×C197—24no—
130D4×C197—21no—
131D4×C197—21no—
132D4×C197#4112yes—
133D4×C197#4112yes—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—81no—
139C2×C4×C197—81no—
140C2×C4×C197—81no—
141C2×C4×C197—81no—
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C22×C197—21no—
149C22×C197—22no—
150C22×C197—22no—
151C22×C197—21no—
152C22×C197—21no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197—41no—
156C22×C197—41no—
157C22×C197—84no—
158C22×C197C22×D411yescharacteristic
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—41no—
162C4×C197—21no—
163C4×C197—21no—
164C4×C197—21no—
165C4×C197—21no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197C2×#4111yescharacteristic
170C4×C197C2×#4111yescharacteristic
171Dic197—41no—
172Dic197—41no—
173Dic197—41no—
174C2×C197—41no—
175C2×C197—42no—
176C2×C197—42no—
177C2×C197#52511yescharacteristic
178C2×C197#57611yescharacteristic
179C2×C197#57711yescharacteristic
180D197—81no—
181C197#311811yescharacteristic, complemented, Sylow
182#3118—1971nomaximal, Sylow
183#542—1971no—
184#518D19711yescharacteristic, complemented
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#433—1971no—
198C22×D4C2×D19711yescharacteristic
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—21no—
213C4×D4—21no—
214C4×D4—1971no—
215C4×D4C2×D19711yescharacteristic
216C4×D4C2×D19711yescharacteristic
217C4×D4C2×D19711yescharacteristic
218C4×D4C2×D19711yescharacteristic
219#117C2×D19711yescharacteristic
220#117C2×D19711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—1971no—
224#107—1971no—
225#106—1971no—
226#106—1971no—
227C4⋊C8—1971no—
228C4⋊C8—1971no—
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⋊C8—1971no—
238C22⋊C8—1971no—
239#97—1971no—
240C4×C8—1971no—
241C24—21no—
242C2×Q8—3941no—
243C2×Q8—1971no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4C22×D19711yescharacteristic
250C2×D4C22×D19711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3941no—
256C22×C4C22×D19711yescharacteristic
257SD16—3942no—
258SD16—3942no—
259D8—3942no—
260D8—3942no—
261C2×C8—1971no—
262C2×C8—1971no—
263C2×C8—1971no—
264C2×C8—1971no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—1971no—
269C4⋊C4—1971no—
270C4⋊C4C22⋊D19711yescharacteristic
271C4⋊C4C22⋊D19711yescharacteristic
272C4⋊C4C22×D19711yescharacteristic
273C4⋊C4C22×D19711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4C22⋊D19711yescharacteristic
285C22⋊C4C22⋊D19711yescharacteristic
286C42—21no—
287C42—3941no—
288C42C22×D19711yescharacteristic
289C42C22×D19711yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3941no—
296C23C2×C22⋊D19711yescharacteristic
297Q8—15761no—
298Q8—3941no—
299Q8—3941no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—15761no—
304D4—3941no—
305D4—3941no—
306D4#1569135812yes—
307D4#1569135812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—15761no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4C2×C22⋊D19711yescharacteristic
321C2×C4C2×C22⋊D19711yescharacteristic
322C2×C4C2×C22⋊D19711yescharacteristic
323C2×C4C2×C22⋊D19711yescharacteristic
324C2×C4C2×C22⋊D19711yescharacteristic
325C2×C4C23×D19711yescharacteristic
326C8—3941no—
327C8—3941no—
328C8—3941no—
329C8—3941no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—7881no—
341C22—7881no—
342C22—7881no—
343C22C22×C22⋊D19711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—7881no—
352C4—7881no—
353C4—7881no—
354C4C2×#1569135811yescharacteristic
355C4C2×#1569135811yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—15761no—
360C2#2716159011yescharacteristic
361C2#2716188211yescharacteristic
362C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#518×C197C2
socleC22×C197C22×D4
prime radical#27161213C2
hypercenter#518D197
hypocenterC197#3118
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #27161700C2semidirect
2. #27161420C2semidirect
3. #27161403C2semidirect
4. #27161369C2semidirect
5. #27161359C2semidirect
6. #27161294C2semidirect
7. #27161294C2semidirect
8. #27161230C2semidirect
10. #27161189C2semidirect
11. #27161179C2semidirect
12. C4×#15691346C2semidirect
13. C4×#15691343C2semidirect
14. #27161140C2semidirect
15. #518×C197C2semidirect
20. #20263326C22semidirect
22. #20263314C22semidirect
28. C4×Dic394C22semidirect
29. C4×D788C22semidirect
30. #20263259C22semidirect
31. #20263259C22semidirect
32. #20263257C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. C4⋊#13096802C22semidirect
36. #20263249C22semidirect
37. #20263249C22semidirect
42. #20263246C22semidirect
43. #20263246C22semidirect
46. #20263237C22semidirect
47. C4×#13096802C22semidirect
181. C197#3118semidirect
184. #518D197semidirect

Maximal subgroups

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

subgroupindex
#271617002
#271614202
#271614032
#271613692
#271613592
#271612942
#271612302
#271612132
#271611892
#271611792
C4×#156913462
C4×#156913432
#271611402
#518×C1972
#3118197