Finite Group Databasebeta

← #38752989

Subgroups — #38752989

Order 25216.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161700C211yesmaximal, characteristic, complemented
2#27161409C211yesmaximal, characteristic, complemented
3#27161397C211yesmaximal, characteristic, complemented
4#27161369C211yesmaximal, characteristic, complemented
5#27161364C211yesmaximal, characteristic, complemented
6#27161301C211yesmaximal, characteristic, complemented
7#27161294C211yesmaximal, characteristic, complemented
8#27161241C211yesmaximal, characteristic, complemented
9#27161224C211yesmaximal, characteristic
10#27161189C211yesmaximal, characteristic, complemented
11#27161184C211yesmaximal, characteristic, complemented
12C4×#15691345C211yesmaximal, characteristic, complemented
13C4×#15691346C211yesmaximal, characteristic, complemented
14#27161140C211yesmaximal, characteristic, complemented
15#520×C197C211yesmaximal, characteristic, complemented
16C2×#15691348C2211yescharacteristic
17C2×#15691346C2211yescharacteristic
18C2×#15691345C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20#20263326C2211yescharacteristic
21#20263321—21no—
22#20263314C2211yescharacteristic
23#20263312—21no—
24#20263308—21no—
25#20263305C2211yescharacteristic
26#20263304—21no—
27#20263300C2211yescharacteristic
28C4×Dic394C2211yescharacteristic
29C4×D788C2211yescharacteristic
30#20263257C2211yescharacteristic
31#20263257C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35C4⋊#13096802C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263249C2211yescharacteristic
39#20263249C2211yescharacteristic
40#20263247C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44C22⋊#13096802C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46#20263237C2211yescharacteristic
47C4×#13096802C2211yescharacteristic
48C2×#41×C197C2211yescharacteristic
49C4×Q8×C197C2211yescharacteristic
50C4×Q8×C197C2211yescharacteristic
51C4×D4×C197—21no—
52C4×D4×C197—21no—
53C4×D4×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic
55#117×C197—21no—
56#117×C197C2211yescharacteristic
57C2×C4⋊C4×C197—21no—
58C2×C4⋊C4×C197C2211yescharacteristic
59C2×Dic394—21no—
60C2×Dic394C2311yescharacteristic
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C4×D197—21no—
64#15691348—22no—
65#15691346—22no—
66#15691345—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—
83#41×C197—22no—
84#41×C197—22no—
85C2×Q8×C197C2311yescharacteristic
86C2×D4×C197—21no—
87C2×D4×C197C2311yescharacteristic
88C22×C4×C197—21no—
89C22×C4×C197—21no—
90C22×C4×C197—21no—
91C22×C4×C197—21no—
92C22×C4×C197C2311yescharacteristic
93C4⋊C4×C197—21no—
94C4⋊C4×C197—21no—
95C4⋊C4×C197—21no—
96C4⋊C4×C197D411yescharacteristic
97C4⋊C4×C197D411yescharacteristic
98C4⋊C4×C197C2311yescharacteristic
99C4⋊C4×C197C2311yescharacteristic
100C22⋊C4×C197—21no—
101C22⋊C4×C197—21no—
102C22⋊C4×C197D411yescharacteristic
103C22⋊C4×C197D411yescharacteristic
104C42×C197—21no—
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#4112yes—
127D4×C197—24no—
128D4×C197—21no—
129D4×C197#4112yes—
130C2×C4×C197—24no—
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×C197—81no—
138C2×C4×C197—81no—
139C2×C4×C197C2×D411yescharacteristic
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2411yescharacteristic
145C22×C197—21no—
146C22×C197—22no—
147C22×C197—21no—
148C22×C197—21no—
149C22×C197—41no—
150C22×C197—41no—
151C22×C197C22×D411yescharacteristic
152C2×D197—41no—
153C2×D197—41no—
154C2×D197—41no—
155C4×C197—21no—
156C4×C197—22no—
157C4×C197—21no—
158C4×C197—21no—
159C4×C197—21no—
160C4×C197—21no—
161C4×C197—41no—
162C4×C197—41no—
163C4×C197C2×#4111yescharacteristic
164C4×C197C2×#4111yescharacteristic
165Dic197—41no—
166Dic197—41no—
167Dic197—41no—
168C2×C197—41no—
169C2×C197—42no—
170C2×C197#52511yescharacteristic
171C2×C197#57711yescharacteristic
172C2×C197#57711yescharacteristic
173D197—81no—
174C197#312511yescharacteristic, complemented, Sylow
175#3125—1971nomaximal, Sylow
176#542—1971no—
177#520D19711yescharacteristic, complemented
178#483—1971no—
179#481—1971no—
180#476—1971no—
181#475—1971no—
182#464—1971no—
183#463—1971no—
184#452—1971no—
185#449—1971no—
186#442—1971no—
187#441—1971no—
188C4×SD16—1971no—
189C4×SD16—1971no—
190#433—1971no—
191C2×#41C2×D19711yescharacteristic
192C2×Q16—1971no—
193C2×SD16—1971no—
194C2×SD16—1971no—
195C2×D8—1971no—
196C4⋊Q8—1971no—
197#126—3941no—
198#124—3941no—
199#124—1971no—
200#123—3941no—
201C22⋊Q8—3941no—
202C22⋊Q8—1971no—
203#121—1971no—
204C4×Q8—1971no—
205C4×Q8C2×D19711yescharacteristic
206C4×Q8C2×D19711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1971no—
210C4×D4C2×D19711yescharacteristic
211C4×D4C2×D19711yescharacteristic
212#117—21no—
213#117C2×D19711yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D19711yescharacteristic
216#106—1971no—
217#106—1971no—
218#106—1971no—
219#106—1971no—
220C4⋊C8—1971no—
221C4⋊C8—1971no—
222#103—1971no—
223#103—1971no—
224#103—1971no—
225#103—1971no—
226#102—1971no—
227#102—1971no—
228#102—1971no—
229#102—1971no—
230C22⋊C8—1971no—
231C22⋊C8—1971no—
232#97—1971no—
233C4×C8—1971no—
234#41—22no—
235#41—22no—
236C2×Q8—3941no—
237C2×Q8—1971no—
238C2×Q8C22×D19711yescharacteristic
239C2×D4—21no—
240C2×D4—3941no—
241C2×D4—1971no—
242C2×D4C22×D19711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3941no—
248C22×C4C22×D19711yescharacteristic
249Q16—3942no—
250SD16—3942no—
251SD16—3942no—
252D8—3942no—
253C2×C8—1971no—
254C2×C8—1971no—
255C2×C8—1971no—
256C2×C8—1971no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3941no—
261C4⋊C4—3941no—
262C4⋊C4—3941no—
263C4⋊C4—1971no—
264C4⋊C4—1971no—
265C4⋊C4C22⋊D19711yescharacteristic
266C4⋊C4C22⋊D19711yescharacteristic
267C4⋊C4C22×D19711yescharacteristic
268C4⋊C4C22×D19711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3941no—
272C22⋊C4—3941no—
273C22⋊C4—3941no—
274C22⋊C4—3941no—
275C22⋊C4—3941no—
276C22⋊C4C22⋊D19711yescharacteristic
277C22⋊C4C22⋊D19711yescharacteristic
278C42—21no—
279C42—21no—
280C42—3941no—
281C42C22×D19711yescharacteristic
282C42C22×D19711yescharacteristic
283C23—21no—
284C23—3941no—
285C23C2×C22⋊D19711yescharacteristic
286Q8—21no—
287Q8—15761no—
288Q8—3941no—
289Q8—3941no—
290Q8#1569135812yes—
291D4—24no—
292D4—21no—
293D4—15761no—
294D4—3941no—
295D4—3941no—
296D4#1569135812yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—15761no—
307C2×C4—3941no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4C2×C22⋊D19711yescharacteristic
311C2×C4C2×C22⋊D19711yescharacteristic
312C2×C4C2×C22⋊D19711yescharacteristic
313C2×C4C2×C22⋊D19711yescharacteristic
314C2×C4C2×C22⋊D19711yescharacteristic
315C2×C4C23×D19711yescharacteristic
316C8—3941no—
317C8—3941no—
318C8—3941no—
319C8—3941no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—7881no—
327C22—7881no—
328C22—7881no—
329C22C22×C22⋊D19711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—7881no—
339C4—7881no—
340C4—7881no—
341C4C2×#1569135811yescharacteristic
342C4C2×#1569135811yescharacteristic
343C2—41no—
344C2—42no—
345C2—15761no—
346C2#2716159011yescharacteristic
347C2#2716188711yescharacteristic
348C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#520×C197C2
socleC22×C197C22×D4
prime radical#27161224C2
hypercenter#520D197
hypocenterC197#3125
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #27161700C2semidirect
2. #27161409C2semidirect
3. #27161397C2semidirect
4. #27161369C2semidirect
5. #27161364C2semidirect
6. #27161301C2semidirect
7. #27161294C2semidirect
8. #27161241C2semidirect
10. #27161189C2semidirect
11. #27161184C2semidirect
12. C4×#15691345C2semidirect
13. C4×#15691346C2semidirect
14. #27161140C2semidirect
15. #520×C197C2semidirect
174. C197#3125semidirect
177. #520D197semidirect

Maximal subgroups

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

subgroupindex
#271617002
#271614092
#271613972
#271613692
#271613642
#271613012
#271612942
#271612412
#271612242
#271611892
#271611842
C4×#156913452
C4×#156913462
#271611402
#520×C1972
#3125197