Finite Group Databasebeta

← #38754177

Subgroups — #38754177

Order 25216.

346 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#27161700C211yesmaximal, characteristic, complemented
2#27161369C211yesmaximal, characteristic, complemented
3#27161366C211yesmaximal, characteristic, complemented
4#27161364C211yesmaximal, characteristic, complemented
5#27161361C211yesmaximal, characteristic, complemented
6#27161335C211yesmaximal, characteristic, complemented
7#27161330C211yesmaximal, characteristic
8#27161310C211yesmaximal, characteristic, complemented
9#27161306C211yesmaximal, characteristic, complemented
10#27161301C211yesmaximal, characteristic, complemented
11#27161298C211yesmaximal, characteristic, complemented
12#27161294C211yesmaximal, characteristic, complemented
13#27161291C211yesmaximal, characteristic, complemented
14#27161106C211yesmaximal, characteristic, complemented
15#542×C197C211yesmaximal, characteristic, complemented
16C2×#15691348C2211yescharacteristic
17C2×#15691346C2211yescharacteristic
18C2×#15691345C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×#15691342C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22#20263326C2211yescharacteristic
23#20263321—21no—
24#20263314C2211yescharacteristic
25#20263312—21no—
26#20263308—21no—
27#20263305C2211yescharacteristic
28#20263304—21no—
29#20263300C2211yescharacteristic
30C4×Dic394C2211yescharacteristic
31C4×D788C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34#20263257C2211yescharacteristic
35#20263257C2211yescharacteristic
36C4⋊#13096802C2211yescharacteristic
37C4⋊#13096802C2211yescharacteristic
38C4⋊#13096802C2211yescharacteristic
39C4⋊#13096802C2211yescharacteristic
40#20263250C2211yescharacteristic
41#20263250C2211yescharacteristic
42#20263249C2211yescharacteristic
43#20263249C2211yescharacteristic
44#20263247C2211yescharacteristic
45#20263247C2211yescharacteristic
46#20263246C2211yescharacteristic
47#20263246C2211yescharacteristic
48C4⋊Q8×C197C2211yescharacteristic
49#126×C197—21no—
50#124×C197—21no—
51#124×C197C2211yescharacteristic
52#123×C197—21no—
53C22⋊Q8×C197—21no—
54C22⋊Q8×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic
56C4×Q8×C197C2211yescharacteristic
57C4×D4×C197C2211yescharacteristic
58#117×C197C2211yescharacteristic
59C2×Dic394—21no—
60C2×Dic394C2311yescharacteristic
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C4×D197—21no—
64#15691348—41no—
65#15691346—41no—
66#15691345—41no—
67#15691343—41no—
68#15691342—41no—
69C2×#13096802—41no—
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C4⋊Dic197—21no—
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
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×C197—21no—
86C2×Q8×C197C2311yescharacteristic
87C2×D4×C197—21no—
88C2×D4×C197C2311yescharacteristic
89C22×C4×C197—21no—
90C22×C4×C197C2311yescharacteristic
91C4⋊C4×C197—21no—
92C4⋊C4×C197—21no—
93C4⋊C4×C197—21no—
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—21no—
124C23×C197C2×D411yescharacteristic
125Q8×C197—21no—
126Q8×C197—21no—
127Q8×C197—81no—
128D4×C197—21no—
129D4×C197—21no—
130D4×C197—81no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—81no—
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—21no—
146C22×C197—41no—
147C22×C197—41no—
148C22×C197—41no—
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—21no—
158C4×C197—41no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197H(5,𝔽2)11yescharacteristic
162C4×C1972−1+411yescharacteristic
163Dic197—41no—
164Dic197—41no—
165Dic197—41no—
166C2×C197—41no—
167C2×C197—81no—
168C2×C197#53711yescharacteristic
169C2×C197#57711yescharacteristic
170C2×C197#57711yescharacteristic
171D197—81no—
172C197#326311yescharacteristic, complemented, Sylow
173#3263—1971nomaximal, Sylow
174#542—1971no—
175#542D19711yescharacteristic, complemented
176#476—1971no—
177#476—1971no—
178#475—1971no—
179#475—1971no—
180#470—1971no—
181#469—1971no—
182#465—1971no—
183#465—1971no—
184#464—1971no—
185#464—1971no—
186#463—1971no—
187#463—1971no—
188#424—1971no—
189C2×Q16—1971no—
190C2×SD16—1971no—
191C2×SD16—1971no—
192C2×D8—1971no—
193C2×M16—1971no—
194C22×C8—1971no—
195C4⋊Q8—1971no—
196C4⋊Q8C2×D19711yescharacteristic
197#126—21no—
198#126—3941no—
199#124—21no—
200#124—3941no—
201#124—1971no—
202#124C2×D19711yescharacteristic
203#123—21no—
204#123—3941no—
205C22⋊Q8—21no—
206C22⋊Q8—3941no—
207C22⋊Q8—1971no—
208C22⋊Q8C2×D19711yescharacteristic
209#121—1971no—
210#121C2×D19711yescharacteristic
211C4×Q8—1971no—
212C4×Q8C2×D19711yescharacteristic
213C4×D4—1971no—
214C4×D4C2×D19711yescharacteristic
215#117C2×D19711yescharacteristic
216#106—1971no—
217#106—1971no—
218#106—1971no—
219#106—1971no—
220C4⋊C8—1971no—
221C4⋊C8—1971no—
222C4⋊C8—1971no—
223C4⋊C8—1971no—
224#103—1971no—
225#103—1971no—
226#103—1971no—
227#103—1971no—
228#102—1971no—
229#102—1971no—
230#102—1971no—
231#102—1971no—
232C2×Q8—21no—
233C2×Q8—3941no—
234C2×Q8—1971no—
235C2×Q8C22×D19711yescharacteristic
236C2×D4—21no—
237C2×D4—3941no—
238C2×D4—1971no—
239C2×D4C22×D19711yescharacteristic
240C22×C4—21no—
241C22×C4—3941no—
242C22×C4C22×D19711yescharacteristic
243Q16—7881no—
244SD16—7881no—
245SD16—7881no—
246D8—7881no—
247M16—7881no—
248C2×C8—7881no—
249C2×C8—1971no—
250C2×C8—1971no—
251C2×C8—1971no—
252C2×C8—1971no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—3941no—
257C4⋊C4—3941no—
258C4⋊C4—3941no—
259C4⋊C4—1971no—
260C4⋊C4—1971no—
261C4⋊C4C22⋊D19711yescharacteristic
262C4⋊C4C22⋊D19711yescharacteristic
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—
275C22⋊C4C22⋊D19711yescharacteristic
276C22⋊C4C22⋊D19711yescharacteristic
277C42—21no—
278C42—3941no—
279C42C22×D19711yescharacteristic
280C42C22×D19711yescharacteristic
281C23—21no—
282C23—3941no—
283C23C2×C22⋊D19711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—15761no—
288Q8—3941no—
289Q8—3941no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—15761no—
294D4—3941no—
295D4—3941no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—15761no—
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—3941no—
315C8—3941no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—7881no—
323C22—7881no—
324C22—7881no—
325C22C22×C22⋊D19711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—7881no—
335C4—7881no—
336C4—7881no—
337C4#2026337211yescharacteristic
338C4#2026337311yescharacteristic
339C2—41no—
340C2—81no—
341C2—15761no—
342C2#2716165711yescharacteristic
343C2#2716188711yescharacteristic
344C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#542×C197C2
socleC22×C197C22×D4
prime radical#27161330C2
hypercenter#542D197
hypocenterC197#3263
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #27161700C2semidirect
2. #27161369C2semidirect
3. #27161366C2semidirect
4. #27161364C2semidirect
5. #27161361C2semidirect
6. #27161335C2semidirect
8. #27161310C2semidirect
9. #27161306C2semidirect
10. #27161301C2semidirect
11. #27161298C2semidirect
12. #27161294C2semidirect
13. #27161291C2semidirect
14. #27161106C2semidirect
15. #542×C197C2semidirect
172. C197#3263semidirect
175. #542D197semidirect

Maximal subgroups

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

subgroupindex
#271617002
#271613692
#271613662
#271613642
#271613612
#271613352
#271613302
#271613102
#271613062
#271613012
#271612982
#271612942
#271612912
#271611062
#542×C1972
#3263197