Finite Group Databasebeta

← #38754733

Subgroups — #38754733

Order 25216.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161769C211yesmaximal, characteristic, complemented
2#27161443C211yesmaximal, characteristic, complemented
3#27161400C211yesmaximal, characteristic, complemented
4#27161390C211yesmaximal, characteristic, complemented
5#27161379C211yesmaximal, characteristic, complemented
6#27161323C211yesmaximal, characteristic, complemented
7#27161306C211yesmaximal, characteristic, complemented
8#27161276C211yesmaximal, characteristic, complemented
9#27161230C211yesmaximal, characteristic, complemented
10#27161230C211yesmaximal, characteristic, complemented
11#27161179C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161084C211yesmaximal, characteristic, complemented
14C2×#20263247C211yesmaximal, characteristic, complemented
15D42×C197C211yesmaximal, characteristic
16C2×#15691346C2211yescharacteristic
17C2×#15691346C2211yescharacteristic, complemented
18C2×#15691346C2211yescharacteristic, complemented
19C2×#15691342C2211yescharacteristic, complemented
20C22×#13096802C2211yescharacteristic, complemented
21#20263319—21no—
22#20263318C2211yescharacteristic, complemented
23#20263316—21no—
24#20263307—21no—
25#20263305C2211yescharacteristic, complemented
26#20263305C2211yescharacteristic, complemented
27C4×Dic394C2211yescharacteristic, complemented
28#20263285C2211yescharacteristic, complemented
29#20263282—21no—
30C2×C4⋊Dic197C2211yescharacteristic, complemented
31#20263259C2211yescharacteristic, complemented
32#20263259C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263249C2211yescharacteristic, complemented
36#20263247C2211yescharacteristic
37#20263247C2211yescharacteristic
38#20263247C2211yescharacteristic
39#20263247C2211yescharacteristic, complemented
40#20263247C2211yescharacteristic, complemented
41#20263247C2211yescharacteristic, complemented
42#20263247C2211yescharacteristic, complemented
43#20263247C2211yescharacteristic, complemented
44#20263247C2211yescharacteristic, complemented
45C22⋊#13096802C2211yescharacteristic, complemented
46C22⋊#13096802C2211yescharacteristic, complemented
47#20263237C2211yescharacteristic, complemented
48C22×D4×C197—21no—
49C22×D4×C197C2211yescharacteristic
50C22×D4×C197C2211yescharacteristic
51#127×C197C2211yescharacteristic
52#121×C197—21no—
53#121×C197C2211yescharacteristic
54#121×C197C2211yescharacteristic
55#120×C197—21no—
56#120×C197—21no—
57C4×D4×C197C2211yescharacteristic
58C4×D4×C197C2211yescharacteristic
59C2×Dic394C2311yescharacteristic
60C22×Dic197—21no—
61#15691346—22no—
62#15691346—41no—
63#15691346—41no—
64#15691342—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—
85C24×C197—21no—
86C2×D4×C197—22no—
87C2×D4×C197—22no—
88C2×D4×C197—21no—
89C2×D4×C197—21no—
90C2×D4×C197—21no—
91C2×D4×C197—21no—
92C2×D4×C197—21no—
93C2×D4×C197—21no—
94C2×D4×C197—41no—
95C2×D4×C197—41no—
96C2×D4×C197—44no—
97C2×D4×C197D411yescharacteristic
98C2×D4×C197C2311yescharacteristic
99C2×D4×C197C2311yescharacteristic
100C2×D4×C197C2311yescharacteristic
101C22×C4×C197—21no—
102C22×C4×C197C2311yescharacteristic
103C22×C4×C197C2311yescharacteristic
104C4⋊C4×C197D411yescharacteristic
105C4⋊C4×C197C2311yescharacteristic
106C22⋊C4×C197—21no—
107C22⋊C4×C197—21no—
108C22⋊C4×C197—21no—
109C22⋊C4×C197D411yescharacteristic
110C22⋊C4×C197D411yescharacteristic
111C42×C197C2311yescharacteristic
112Dic394—21no—
113Dic394—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—42no—
119#13096802—21no—
120#13096802—21no—
121#13096802—21no—
122#13096802—21no—
123C23×C197—21no—
124C23×C197—21no—
125C23×C197—21no—
126C23×C197—42no—
127C23×C197—42no—
128C23×C197—42no—
129C23×C197—42no—
130C23×C197—81no—
131C23×C197—81no—
132C23×C197C2×D411yescharacteristic
133C23×C197C2×D411yescharacteristic
134D4×C197—22no—
135D4×C197—21no—
136D4×C197—21no—
137D4×C197—21no—
138D4×C197—21no—
139D4×C197—21no—
140D4×C197—42no—
141D4×C197—42no—
142D4×C197—42no—
143D4×C197—42no—
144D4×C197—42no—
145D4×C197—81no—
146D4×C197#4112yes—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—42no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2×D411yescharacteristic
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2411yescharacteristic
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—22no—
162C22×C197—21no—
163C22×C197—21no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197—41no—
168C22×C197—41no—
169C22×C197—41no—
170C22×C197—41no—
171C22×C197—41no—
172C22×C197—84no—
173C22×C197—84no—
174C22×C197AGL(1,ℤ/8)12yes—
175C22×C197C22×D411yescharacteristic
176C4×C197—21no—
177C4×C197—21no—
178C4×C197—21no—
179C4×C197—21no—
180C4×C197—41no—
181C4×C197C2×#4111yescharacteristic
182C4×C197H(5,𝔽2)11yescharacteristic
183Dic197—41no—
184Dic197—41no—
185Dic197—41no—
186Dic197—41no—
187C2×C197—22no—
188C2×C197—41no—
189C2×C197—42no—
190C2×C197—81no—
191C2×C197—81no—
192C2×C197#54611yescharacteristic
193C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
194C2×C197#57711yescharacteristic
195C197#331011yescharacteristic, complemented, Sylow
196#3310—1971nomaximal, Sylow
197#552—1971no—
198D42D19711yescharacteristic
199#490—1971no—
200#482—1971no—
201#480—1971no—
202#478—1971no—
203#468—1971no—
204#465—1971no—
205#460—1971no—
206#450—1971no—
207#450—1971no—
208#440—1971no—
209#435—1971no—
210#418—1971no—
211C2×#102—1971no—
212C22×D4—21no—
213C22×D4C2×D19711yescharacteristic
214C22×D4C2×D19711yescharacteristic
215C2×SD16—1971no—
216C2×SD16—1971no—
217C2×SD16—1971no—
218C2×M16—1971no—
219C22×C8—1971no—
220#127C2×D19711yescharacteristic
221#126—3941no—
222#125—3941no—
223#125—1971no—
224#123—3941no—
225C22⋊Q8—1971no—
226C22⋊Q8—1971no—
227#121—21no—
228#121C2×D19711yescharacteristic
229#121C2×D19711yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8—1971no—
233C4×D4C2×D19711yescharacteristic
234C4×D4C2×D19711yescharacteristic
235#117—3941no—
236#117—1971no—
237C2×C4⋊C4—1971no—
238#107—1971no—
239#107—1971no—
240#106—1971no—
241C4⋊C8—1971no—
242#103—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251#102—1971no—
252C22⋊C8—1971no—
253C22⋊C8—1971no—
254#97—1971no—
255C24—21no—
256C24—21no—
257C2×Q8—1971no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D19711yescharacteristic
270C2×D4C22×D19711yescharacteristic
271C2×D4C22×D19711yescharacteristic
272C2×D4C22×D19711yescharacteristic
273C22×C4—21no—
274C22×C4—3941no—
275C22×C4C22×D19711yescharacteristic
276C22×C4C22×D19711yescharacteristic
277SD16—7881no—
278SD16—7881no—
279SD16—3942no—
280M16—7881no—
281C2×C8—7881no—
282C2×C8—1971no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8—1971no—
286C4⋊C4—3941no—
287C4⋊C4—3941no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—3941no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4—1971no—
295C4⋊C4C22⋊D19711yescharacteristic
296C4⋊C4C22×D19711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4C22⋊D19711yescharacteristic
304C22⋊C4C22⋊D19711yescharacteristic
305C42—3941no—
306C42—3941no—
307C42C22×D19711yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D19711yescharacteristic
318C23C2×C22⋊D19711yescharacteristic
319Q8—3941no—
320Q8—3941no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#1569135712yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—7882no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4C2×C22⋊D19711yescharacteristic
346C2×C4C2×C22⋊D19711yescharacteristic
347C2×C4C2×C22⋊D19711yescharacteristic
348C2×C4C2×C22⋊D19711yescharacteristic
349C2×C4C23×D19711yescharacteristic
350C8—3941no—
351C8—3941no—
352C8—3941no—
353C8—3941no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#2026335212yes—
371C22C22×C22⋊D19711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—7881no—
378C4—7881no—
379C4—7881no—
380C4—7881no—
381C4C2×#1569135711yescharacteristic
382C4#2026337111yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#2716172311yescharacteristic
389C2C2×#2026335211yescharacteristic
390C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radicalD42×C197C2
socleC22×C197C22×D4
hypercenterD42D197
hypocenterC197#3310
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #27161769C2semidirect
2. #27161443C2semidirect
3. #27161400C2semidirect
4. #27161390C2semidirect
5. #27161379C2semidirect
6. #27161323C2semidirect
7. #27161306C2semidirect
8. #27161276C2semidirect
9. #27161230C2semidirect
10. #27161230C2semidirect
11. #27161179C2semidirect
12. #27161151C2semidirect
13. #27161084C2semidirect
14. C2×#20263247C2semidirect
17. C2×#15691346C22semidirect
18. C2×#15691346C22semidirect
19. C2×#15691342C22semidirect
20. C22×#13096802C22semidirect
22. #20263318C22semidirect
25. #20263305C22semidirect
26. #20263305C22semidirect
27. C4×Dic394C22semidirect
28. #20263285C22semidirect
30. C2×C4⋊Dic197C22semidirect
31. #20263259C22semidirect
32. #20263259C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263249C22semidirect
39. #20263247C22semidirect
40. #20263247C22semidirect
41. #20263247C22semidirect
42. #20263247C22semidirect
43. #20263247C22semidirect
44. #20263247C22semidirect
45. C22⋊#13096802C22semidirect
46. C22⋊#13096802C22semidirect
47. #20263237C22semidirect
195. C197#3310semidirect

Maximal subgroups

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

subgroupindex
#271617692
#271614432
#271614002
#271613902
#271613792
#271613232
#271613062
#271612762
#271612302
#271611792
#271611512
#271610842
C2×#202632472
D42×C1972
#3310197