Finite Group Databasebeta

← #38752983

Subgroups — #38752983

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#27161559C211yesmaximal, characteristic, complemented
2#27161407C211yesmaximal, characteristic, complemented
3#27161395C211yesmaximal, characteristic, complemented
4#27161366C211yesmaximal, characteristic, complemented
5#27161361C211yesmaximal, characteristic, complemented
6#27161298C211yesmaximal, characteristic, complemented
7#27161291C211yesmaximal, characteristic, complemented
8#27161239C211yesmaximal, characteristic, complemented
9#27161222C211yesmaximal, characteristic
10#27161189C211yesmaximal, characteristic, complemented
11#27161184C211yesmaximal, characteristic, complemented
12C4×#15691345C211yesmaximal, characteristic, complemented
13C4×#15691346C211yesmaximal, characteristic, complemented
14#27161140C211yesmaximal, characteristic, complemented
15#542×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×#15691348C2211yescharacteristic
18C2×#15691346C2211yescharacteristic
19C2×#15691345C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C4×Dic394C2211yescharacteristic
22C4×Dic394C2211yescharacteristic
23C4×D788C2211yescharacteristic
24C4×D788C2211yescharacteristic
25#20263286—21no—
26#20263286—21no—
27#20263284—21no—
28C2×C4⋊Dic197C2211yescharacteristic
29C4⋊C4×D197—21no—
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
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
59#15691358—22no—
60#15691358—22no—
61C2×Dic394C2311yescharacteristic
62C2×D788C2311yescharacteristic
63C2×C22⋊D197—21no—
64C22×Dic197—21no—
65C2×C4×D197—21no—
66C2×C4×D197—21no—
67C2×C4×D197—21no—
68#15691348—22no—
69#15691346—22no—
70#15691345—22no—
71#15691343—22no—
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C2×#13096802C2311yescharacteristic
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79#15691337—21no—
80#15691337—21no—
81#15691335—21no—
82#15691335—21no—
83C4×Dic197—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#4112yes—
111D788—21no—
112D788#4112yes—
113C22⋊D197—24no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—81no—
118C4×D197—24no—
119C4×D197—81no—
120C4×D197—81no—
121#13096802—21no—
122#13096802—21no—
123#13096802—21no—
124#13096802—21no—
125C23×C197—21no—
126C23×C197C2×D411yescharacteristic
127Q8×C197—21no—
128Q8×C197—21no—
129Q8×C197—81no—
130D4×C197—21no—
131D4×C197—21no—
132D4×C197—81no—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
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—21no—
147C22×C197—21no—
148C22×C197—41no—
149C22×C197—41no—
150C22×C197—41no—
151C22×C197C22×D411yescharacteristic
152C2×D197—22no—
153C2×D197—41no—
154C2×D197—41no—
155C4×C197—21no—
156C4×C197—21no—
157C4×C197—21no—
158C4×C197—21no—
159C4×C197—21no—
160C4×C197—41no—
161C4×C197—41no—
162C4×C197—41no—
163C4×C197C2×#4111yescharacteristic
164C4×C197C2×#4111yescharacteristic
165Dic197—22no—
166Dic197—41no—
167Dic197—41no—
168C2×C197—41no—
169C2×C197—81no—
170C2×C197#52511yescharacteristic
171C2×C197#57711yescharacteristic
172C2×C197#57711yescharacteristic
173D197—42no—
174C197#312511yescharacteristic, complemented, Sylow
175#3125—1971nomaximal, Sylow
176#542D19711yescharacteristic, complemented
177#520—1971no—
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×#41—1971no—
192C2×Q16—1971no—
193C2×SD16—1971no—
194C2×SD16—1971no—
195C2×D8—1971no—
196C4⋊Q8C2×D19711yescharacteristic
197#126—21no—
198#124—21no—
199#124C2×D19711yescharacteristic
200#123—21no—
201C22⋊Q8—21no—
202C22⋊Q8C2×D19711yescharacteristic
203#121C2×D19711yescharacteristic
204C4×Q8—1971no—
205C4×Q8—1971no—
206C4×Q8C2×D19711yescharacteristic
207C4×D4—3941no—
208C4×D4—3941no—
209C4×D4—1971no—
210C4×D4—1971no—
211C4×D4C2×D19711yescharacteristic
212#117—3941no—
213#117C2×D19711yescharacteristic
214C2×C4⋊C4—3941no—
215C2×C4⋊C4—1971no—
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—3942no—
235#41—3942no—
236C2×Q8—21no—
237C2×Q8—1971no—
238C2×Q8C22×D19711yescharacteristic
239C2×D4—21no—
240C2×D4—3941no—
241C2×D4—1971no—
242C2×D4C22×D19711yescharacteristic
243C22×C4—21no—
244C22×C4—3941no—
245C22×C4—3941no—
246C22×C4—3941no—
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—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—3941no—
275C22⋊C4—3941no—
276C22⋊C4C22⋊D19711yescharacteristic
277C22⋊C4C22⋊D19711yescharacteristic
278C42—21no—
279C42—3941no—
280C42—3941no—
281C42C22×D19711yescharacteristic
282C42C22×D19711yescharacteristic
283C23—21no—
284C23—3941no—
285C23C2×C22⋊D19711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—81no—
289Q8—3941no—
290Q8—1972no—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—3944no—
295D4—3941no—
296D4—1972no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—15761no—
304C2×C4—15761no—
305C2×C4—15761no—
306C2×C4—3944no—
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—41no—
324C22—41no—
325C22—41no—
326C22—7881no—
327C22—7881no—
328C22—3942no—
329C22C22×C22⋊D19711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—7881no—
339C4—7881no—
340C4—3942no—
341C4#41×D19711yescharacteristic
342C4#41×D19711yescharacteristic
343C2—41no—
344C2—81no—
345C2—7882no—
346C2#2716158611yescharacteristic
347C2#2716188711yescharacteristic
348C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#542×C197C2
socleC22×C197C22×D4
prime radical#27161222C2
hypercenter#542D197
hypocenterC197#3125
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. #27161559C2semidirect
2. #27161407C2semidirect
3. #27161395C2semidirect
4. #27161366C2semidirect
5. #27161361C2semidirect
6. #27161298C2semidirect
7. #27161291C2semidirect
8. #27161239C2semidirect
10. #27161189C2semidirect
11. #27161184C2semidirect
12. C4×#15691345C2semidirect
13. C4×#15691346C2semidirect
14. #27161140C2semidirect
15. #542×C197C2semidirect
174. C197#3125semidirect
176. #542D197semidirect

Maximal subgroups

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

subgroupindex
#271615592
#271614072
#271613952
#271613662
#271613612
#271612982
#271612912
#271612392
#271612222
#271611892
#271611842
C4×#156913452
C4×#156913462
#271611402
#542×C1972
#3125197