Finite Group Databasebeta

← #38753276

Subgroups — #38753276

Order 25216.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161613C211yesmaximal, characteristic, complemented
2#27161478C211yesmaximal, characteristic, complemented
3#27161461C211yesmaximal, characteristic, complemented
4#27161436C211yesmaximal, characteristic, complemented
5#27161433C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161313C211yesmaximal, characteristic
8#27161310C211yesmaximal, characteristic, complemented
9#27161306C211yesmaximal, characteristic, complemented
10C4×#15691348C211yesmaximal, characteristic, complemented
11C4×#15691345C211yesmaximal, characteristic, complemented
12C4×#15691346C211yesmaximal, characteristic, complemented
13C4×#15691343C211yesmaximal, characteristic, complemented
14C2×C4×#13096802C211yesmaximal, characteristic, complemented
15#529×C197C211yesmaximal, characteristic, complemented
16C2×#15691348C2211yescharacteristic
17C2×#15691346C2211yescharacteristic
18C2×#15691345C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C22×#13096802C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22#20263320—21no—
23#20263318C2211yescharacteristic
24#20263314C2211yescharacteristic
25#20263309—21no—
26#20263305C2211yescharacteristic
27#20263300C2211yescharacteristic
28C4×Dic394C2211yescharacteristic
29C4×D788C2211yescharacteristic
30C4×C22⋊D197—21no—
31#20263283—21no—
32#20263259C2211yescharacteristic
33#20263259C2211yescharacteristic
34#20263257C2211yescharacteristic
35#20263257C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263249C2211yescharacteristic
39#20263249C2211yescharacteristic
40#20263247C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44C4×#13096802C2211yescharacteristic
45C4×#13096802C2211yescharacteristic
46C4×#13096802C2211yescharacteristic
47C4×#13096802C2211yescharacteristic
48#126×C197—21no—
49#125×C197C2211yescharacteristic
50#124×C197C2211yescharacteristic
51#123×C197—21no—
52C22⋊Q8×C197C2211yescharacteristic
53#121×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic
55C4×D4×C197—21no—
56C4×D4×C197C2211yescharacteristic
57#117×C197—21no—
58C2×C42×C197C2211yescharacteristic
59C2×Dic394C2311yescharacteristic
60C2×D788C2311yescharacteristic
61C2×C22⋊D197—21no—
62C2×C4×D197—21no—
63#15691348—22no—
64#15691346—22no—
65#15691345—22no—
66#15691343—22no—
67C2×#13096802—22no—
68C2×#13096802—22no—
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75#15691337—21no—
76#15691337—21no—
77#15691337—21no—
78#15691337—21no—
79C22⋊Dic197—21no—
80#15691335—21no—
81#15691335—21no—
82#15691335—21no—
83#15691335—21no—
84C4×Dic197—21no—
85C2×Q8×C197C2311yescharacteristic
86C2×D4×C197—21no—
87C2×D4×C197C2311yescharacteristic
88C22×C4×C197—21no—
89C22×C4×C197D411yescharacteristic
90C22×C4×C197D411yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C4⋊C4×C197—21no—
93C4⋊C4×C197—21no—
94C4⋊C4×C197—21no—
95C4⋊C4×C197—21no—
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—
103C42×C197—21no—
104C42×C197D411yescharacteristic
105C42×C197D411yescharacteristic
106C42×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109Dic394—21no—
110Dic394—21no—
111D788—21no—
112D788—21no—
113C22⋊D197—42no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C4×D197—42no—
118#13096802—21no—
119#13096802—21no—
120#13096802#4112yes—
121#13096802#4112yes—
122C23×C197—21no—
123C23×C197C2×D411yescharacteristic
124Q8×C197—21no—
125Q8×C197—21no—
126D4×C197—21no—
127D4×C197—21no—
128D4×C197—42no—
129C2×C4×C197—22no—
130C2×C4×C197—22no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—42no—
137C2×C4×C197C2×D411yescharacteristic
138C2×C4×C197C2×D411yescharacteristic
139C2×C4×C197C2×D411yescharacteristic
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2411yescharacteristic
143C22×C197—21no—
144C22×C197—21no—
145C22×C197—41no—
146C22×C197—41no—
147C22×C197—41no—
148C22×C197#13512yes—
149C22×C197C22×D411yescharacteristic
150C2×D197—41no—
151C2×D197—41no—
152C2×D197—41no—
153C4×C197—21no—
154C4×C197—21no—
155C4×C197—21no—
156C4×C197—21no—
157C4×C197—41no—
158C4×C197—41no—
159C4×C197—41no—
160C4×C197#13512yes—
161C4×C197C2×#4111yescharacteristic
162C4×C197C2×#4111yescharacteristic
163Dic197—41no—
164Dic197—41no—
165Dic197—41no—
166C2×C197—22no—
167C2×C197—81no—
168C2×C197#53211yescharacteristic
169C2×C197C2×#13511yescharacteristic
170C2×C197C2×#13511yescharacteristic
171D197—81no—
172C197#316811yescharacteristic, complemented, Sylow
173#3168—1971nomaximal, Sylow
174#529—1971no—
175#529D19711yescharacteristic, complemented
176#499—1971no—
177#495—1971no—
178#488—1971no—
179#488—1971no—
180#467—1971no—
181#466—1971no—
182#465—1971no—
183#465—1971no—
184C4×Q16—1971no—
185C4×SD16—1971no—
186C4×SD16—1971no—
187C4×D8—1971no—
188C2×C4×C8—1971no—
189C2×Q16—1971no—
190C2×SD16—1971no—
191C2×SD16—1971no—
192C2×D8—1971no—
193C22×C8—1971no—
194C22×C8—1971no—
195#126—21no—
196#126—3941no—
197#125—1971no—
198#125C2×D19711yescharacteristic
199#124—1971no—
200#124C2×D19711yescharacteristic
201#123—21no—
202#123—3941no—
203C22⋊Q8—1971no—
204C22⋊Q8C2×D19711yescharacteristic
205#121—1971no—
206#121C2×D19711yescharacteristic
207C4×Q8—1971no—
208C4×Q8C2×D19711yescharacteristic
209C4×D4—21no—
210C4×D4—3941no—
211C4×D4—1971no—
212C4×D4C2×D19711yescharacteristic
213#117—21no—
214#117—3941no—
215C2×C42C2×D19711yescharacteristic
216#107—1971no—
217#107—1971no—
218#106—1971no—
219#106—1971no—
220#103—1971no—
221#103—1971no—
222#103—1971no—
223#103—1971no—
224#102—1971no—
225#102—1971no—
226#102—1971no—
227#102—1971no—
228C4×C8—1971no—
229C4×C8—1971no—
230C4×C8—1971no—
231C4×C8—1971no—
232C2×Q8—1971no—
233C2×Q8C22×D19711yescharacteristic
234C2×D4—21no—
235C2×D4—3941no—
236C2×D4—1971no—
237C2×D4C22×D19711yescharacteristic
238C22×C4—21no—
239C22×C4—3941no—
240C22×C4C22⋊D19711yescharacteristic
241C22×C4C22⋊D19711yescharacteristic
242C22×C4C22×D19711yescharacteristic
243Q16—3942no—
244SD16—3942no—
245SD16—3942no—
246D8—3942no—
247C2×C8—3942no—
248C2×C8—3942no—
249C2×C8—1971no—
250C2×C8—1971no—
251C2×C8—1971no—
252C2×C8—1971no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—3941no—
258C4⋊C4—3941no—
259C4⋊C4—3941no—
260C4⋊C4—3941no—
261C4⋊C4—1971no—
262C4⋊C4—1971no—
263C4⋊C4C22×D19711yescharacteristic
264C4⋊C4C22×D19711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3941no—
271C22⋊C4—3941no—
272C22⋊C4—3941no—
273C22⋊C4—3941no—
274C22⋊C4—3941no—
275C42—21no—
276C42—3941no—
277C42C22⋊D19711yescharacteristic
278C42C22⋊D19711yescharacteristic
279C42C22×D19711yescharacteristic
280C42C22×D19711yescharacteristic
281C23—21no—
282C23—3941no—
283C23C2×C22⋊D19711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—3941no—
287Q8—3941no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—7882no—
292D4—3941no—
293D4—3941no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—7882no—
303C2×C4—3941no—
304C2×C4—3941no—
305C2×C4—3941no—
306C2×C4C2×C22⋊D19711yescharacteristic
307C2×C4C2×C22⋊D19711yescharacteristic
308C2×C4C2×C22⋊D19711yescharacteristic
309C2×C4C2×C22⋊D19711yescharacteristic
310C2×C4C2×C22⋊D19711yescharacteristic
311C2×C4C23×D19711yescharacteristic
312C8—3941no—
313C8—3941no—
314C8—1972no—
315C8—1972no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—7881no—
322C22—7881no—
323C22—7881no—
324C22#2026334612yes—
325C22C22×C22⋊D19711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—7881no—
334C4—7881no—
335C4—7881no—
336C4#2026334612yes—
337C4#41×D19711yescharacteristic
338C4#41×D19711yescharacteristic
339C2—22no—
340C2—81no—
341C2—15761no—
342C2#2716162511yescharacteristic
343C2C2×#2026334611yescharacteristic
344C2C2×#2026334611yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#529×C197C2
socleC22×C197C22×D4
prime radical#27161313C2
hypercenter#529D197
hypocenterC197#3168
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #27161613C2semidirect
2. #27161478C2semidirect
3. #27161461C2semidirect
4. #27161436C2semidirect
5. #27161433C2semidirect
6. #27161318C2semidirect
8. #27161310C2semidirect
9. #27161306C2semidirect
10. C4×#15691348C2semidirect
11. C4×#15691345C2semidirect
12. C4×#15691346C2semidirect
13. C4×#15691343C2semidirect
14. C2×C4×#13096802C2semidirect
15. #529×C197C2semidirect
172. C197#3168semidirect
175. #529D197semidirect

Maximal subgroups

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

subgroupindex
#271616132
#271614782
#271614612
#271614362
#271614332
#271613182
#271613132
#271613102
#271613062
C4×#156913482
C4×#156913452
C4×#156913462
C4×#156913432
C2×C4×#130968022
#529×C1972
#3168197