Finite Group Databasebeta

← #38754547

Subgroups — #38754547

Order 25216.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161769C211yesmaximal, characteristic, complemented
2#27161433C211yesmaximal, characteristic, complemented
3#27161407C211yesmaximal, characteristic, complemented
4#27161400C211yesmaximal, characteristic, complemented
5#27161379C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161306C211yesmaximal, characteristic, complemented
8#27161298C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161230C211yesmaximal, characteristic, complemented
11C4×#15691346C211yesmaximal, characteristic, complemented
12D4×#13096802C211yesmaximal, characteristic, complemented
13#27161075C211yesmaximal, characteristic, complemented
14C2×#20263247C211yesmaximal, characteristic, complemented
15#546×C197C211yesmaximal, characteristic
16C2×#15691348C2211yescharacteristic, complemented
17C2×#15691346C2211yescharacteristic
18C2×#15691346C2211yescharacteristic
19C22×#13096802C2211yescharacteristic
20C22×#13096802C2211yescharacteristic, complemented
21#20263319—21no—
22#20263318C2211yescharacteristic, complemented
23#20263316—21no—
24#20263307—21no—
25#20263305C2211yescharacteristic
26#20263305C2211yescharacteristic, complemented
27C4×Dic394C2211yescharacteristic, complemented
28#20263285C2211yescharacteristic, complemented
29#20263282—21no—
30C2×C4⋊Dic197C2211yescharacteristic
31#20263259C2211yescharacteristic, complemented
32#20263257C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263250C2211yescharacteristic, complemented
36#20263250C2211yescharacteristic, complemented
37#20263250C2211yescharacteristic, complemented
38#20263249C2211yescharacteristic, complemented
39#20263247C2211yescharacteristic
40#20263247C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263247C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263247C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic, complemented
47C4×#13096802C2211yescharacteristic, complemented
48C2×#41×C197C2211yescharacteristic
49C22×D4×C197C2211yescharacteristic
50#124×C197C2211yescharacteristic
51#123×C197—21no—
52C22⋊Q8×C197C2211yescharacteristic
53#121×C197—21no—
54#121×C197C2211yescharacteristic
55#120×C197—21no—
56C4×D4×C197C2211yescharacteristic
57C4×D4×C197C2211yescharacteristic
58C2×C22⋊C4×C197—21no—
59C2×Dic394C2311yescharacteristic
60C22×Dic197—21no—
61#15691348—41no—
62#15691346—22no—
63#15691346—41no—
64C2×#13096802—41no—
65C2×#13096802—41no—
66C2×#13096802C2311yescharacteristic
67C2×#13096802C2311yescharacteristic
68C2×#13096802C2311yescharacteristic
69C2×#13096802C2311yescharacteristic
70C4⋊Dic197—21no—
71C4⋊Dic197C2311yescharacteristic
72C4⋊Dic197C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74#15691337—21no—
75#15691337—21no—
76#15691337—21no—
77#15691337—21no—
78#15691337—21no—
79C22⋊Dic197—21no—
80C22⋊Dic197—21no—
81C22⋊Dic197—21no—
82C4×Dic197—21no—
83C4×Dic197—21no—
84C24×C197—21no—
85#41×C197—22no—
86#41×C197—41no—
87C2×Q8×C197C2311yescharacteristic
88C2×D4×C197—22no—
89C2×D4×C197—21no—
90C2×D4×C197—21no—
91C2×D4×C197—21no—
92C2×D4×C197—41no—
93C2×D4×C197D411yescharacteristic
94C2×D4×C197C2311yescharacteristic
95C2×D4×C197C2311yescharacteristic
96C22×C4×C197—21no—
97C22×C4×C197—21no—
98C22×C4×C197C2311yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C4⋊C4×C197—21no—
101C4⋊C4×C197D411yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C22⋊C4×C197—21no—
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197—21no—
108C22⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C42×C197C2311yescharacteristic
111Dic394—21no—
112Dic394—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—42no—
118#13096802—21no—
119#13096802—21no—
120#13096802—21no—
121#13096802—21no—
122C23×C197—21no—
123C23×C197—21no—
124C23×C197—42no—
125C23×C197—42no—
126C23×C197—81no—
127C23×C197C2×D411yescharacteristic
128C23×C197C2×D411yescharacteristic
129Q8×C197—21no—
130Q8×C197—21no—
131D4×C197—22no—
132D4×C197—21no—
133D4×C197—21no—
134D4×C197—21no—
135D4×C197—42no—
136D4×C197—42no—
137D4×C197—81no—
138D4×C197#4112yes—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—42no—
146C2×C4×C197—81no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2411yescharacteristic
152C22×C197—21no—
153C22×C197—21no—
154C22×C197—21no—
155C22×C197—22no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—84no—
164C22×C197#13512yes—
165C22×C197C22×D411yescharacteristic
166C4×C197—21no—
167C4×C197—21no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197—41no—
171C4×C197—41no—
172C4×C197C2×#4111yescharacteristic
173C4×C197H(5,𝔽2)11yescharacteristic
174Dic197—41no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—22no—
179C2×C197—41no—
180C2×C197—42no—
181C2×C197—81no—
182C2×C197#54611yescharacteristic
183C2×C197C2×#13511yescharacteristic
184C2×C197#57711yescharacteristic
185C197#329611yescharacteristic, complemented, Sylow
186#3296—1971nomaximal, Sylow
187#552—1971no—
188#546D19711yescharacteristic
189#488—1971no—
190#483—1971no—
191#482—1971no—
192#478—1971no—
193#467—1971no—
194#465—1971no—
195#464—1971no—
196#452—1971no—
197#450—1971no—
198C4×SD16—1971no—
199C8×D4—1971no—
200#416—1971no—
201C2×#102—1971no—
202C2×#41C2×D19711yescharacteristic
203C22×D4C2×D19711yescharacteristic
204C2×Q16—1971no—
205C2×SD16—1971no—
206C2×SD16—1971no—
207C22×C8—1971no—
208C22×C8—1971no—
209#126—3941no—
210#125—3941no—
211#125—1971no—
212#124C2×D19711yescharacteristic
213#123—21no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8—1971no—
217C22⋊Q8C2×D19711yescharacteristic
218#121—21no—
219#121C2×D19711yescharacteristic
220#120—21no—
221C4×Q8—1971no—
222C4×D4C2×D19711yescharacteristic
223C4×D4C2×D19711yescharacteristic
224#117—3941no—
225#117—1971no—
226C2×C4⋊C4—1971no—
227C2×C22⋊C4—21no—
228#107—1971no—
229#106—1971no—
230#106—1971no—
231C4⋊C8—1971no—
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103—1971no—
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241#102—1971no—
242C22⋊C8—1971no—
243C22⋊C8—1971no—
244C4×C8—1971no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—1971no—
249C2×Q8C22×D19711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D19711yescharacteristic
256C2×D4C22×D19711yescharacteristic
257C2×D4C22×D19711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3941no—
261C22×C4C22×D19711yescharacteristic
262C22×C4C22×D19711yescharacteristic
263Q16—7881no—
264SD16—7881no—
265SD16—3942no—
266C2×C8—7881no—
267C2×C8—7881no—
268C2×C8—1971no—
269C2×C8—1971no—
270C2×C8—1971no—
271C2×C8—1971no—
272C4⋊C4—21no—
273C4⋊C4—3941no—
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—3941no—
277C4⋊C4—3941no—
278C4⋊C4—3941no—
279C4⋊C4—1971no—
280C4⋊C4—1971no—
281C4⋊C4—1971no—
282C4⋊C4C22⋊D19711yescharacteristic
283C4⋊C4C22×D19711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4C22⋊D19711yescharacteristic
293C22⋊C4C22⋊D19711yescharacteristic
294C42—3941no—
295C42—3941no—
296C42C22×D19711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D19711yescharacteristic
303C23C2×C22⋊D19711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3941no—
307Q8—3941no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1569135712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—7882no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4C2×C22⋊D19711yescharacteristic
330C2×C4C2×C22⋊D19711yescharacteristic
331C2×C4C2×C22⋊D19711yescharacteristic
332C2×C4C2×C22⋊D19711yescharacteristic
333C2×C4C23×D19711yescharacteristic
334C8—3941no—
335C8—3941no—
336C8—3941no—
337C8—3941no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#2026334612yes—
351C22C22×C22⋊D19711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—7881no—
359C4—7881no—
360C4—7881no—
361C4—7881no—
362C4C2×#1569135711yescharacteristic
363C4#2026337111yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2716172311yescharacteristic
369C2C2×#2026334611yescharacteristic
370C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radical#546×C197C2
socleC22×C197C22×D4
hypercenter#546D197
hypocenterC197#3296
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #27161769C2semidirect
2. #27161433C2semidirect
3. #27161407C2semidirect
4. #27161400C2semidirect
5. #27161379C2semidirect
6. #27161318C2semidirect
7. #27161306C2semidirect
8. #27161298C2semidirect
9. #27161241C2semidirect
10. #27161230C2semidirect
11. C4×#15691346C2semidirect
12. D4×#13096802C2semidirect
13. #27161075C2semidirect
14. C2×#20263247C2semidirect
16. C2×#15691348C22semidirect
20. C22×#13096802C22semidirect
22. #20263318C22semidirect
26. #20263305C22semidirect
27. C4×Dic394C22semidirect
28. #20263285C22semidirect
31. #20263259C22semidirect
32. #20263257C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263250C22semidirect
36. #20263250C22semidirect
37. #20263250C22semidirect
38. #20263249C22semidirect
46. C22⋊#13096802C22semidirect
47. C4×#13096802C22semidirect
185. C197#3296semidirect

Maximal subgroups

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

subgroupindex
#271617692
#271614332
#271614072
#271614002
#271613792
#271613182
#271613062
#271612982
#271612412
#271612302
C4×#156913462
D4×#130968022
#271610752
C2×#202632472
#546×C1972
#3296197