Finite Group Databasebeta

← #38754732

Subgroups — #38754732

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#27161767C211yesmaximal, characteristic, complemented
2#27161714C211yesmaximal, characteristic
3#27161401C211yesmaximal, characteristic, complemented
4#27161391C211yesmaximal, characteristic, complemented
5#27161376C211yesmaximal, characteristic, complemented
6#27161324C211yesmaximal, characteristic, complemented
7#27161307C211yesmaximal, characteristic, complemented
8#27161276C211yesmaximal, characteristic, complemented
9#27161231C211yesmaximal, characteristic, complemented
10#27161231C211yesmaximal, characteristic, complemented
11#27161181C211yesmaximal, characteristic, complemented
12#27161152C211yesmaximal, characteristic, complemented
13#27161082C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#490×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197—21no—
17C2×D4×D197C2211yescharacteristic
18C2×D4×D197C2211yescharacteristic
19C2×#15691347C2211yescharacteristic
20C2×#15691346C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C2×#15691341C2211yescharacteristic, complemented
23C2×C8×D197C2211yescharacteristic, complemented
24#20263321—21no—
25#20263317—21no—
26#20263311—21no—
27#20263306C2211yescharacteristic, complemented
28#20263306C2211yescharacteristic, complemented
29#20263297—21no—
30#20263297C2211yescharacteristic
31#20263297C2211yescharacteristic
32#20263294—21no—
33#20263294—21no—
34C4×Dic394C2211yescharacteristic, complemented
35C4×D788C2211yescharacteristic
36C4×D788C2211yescharacteristic
37#20263284—21no—
38#20263284C2211yescharacteristic, complemented
39C4⋊C4×D197C2211yescharacteristic, complemented
40#20263260C2211yescharacteristic, complemented
41#20263259C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43C4⋊#13096802C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic, complemented
48#20263246C2211yescharacteristic
49#20263246C2211yescharacteristic
50#20263238C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic, complemented
52#127×C197C2211yescharacteristic
53#125×C197C2211yescharacteristic, complemented
54#102×C197C2211yescharacteristic, complemented
55#102×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic, complemented
58#97×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C23×D197—21no—
61C2×Dic394C2311yescharacteristic
62C2×D788D411yescharacteristic
63C2×D788C2311yescharacteristic
64C2×C22⋊D197—21no—
65C2×C22⋊D197—21no—
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68D4×D197—22no—
69D4×D197—22no—
70D4×D197—41no—
71D4×D197—41no—
72D4×D197—44no—
73C2×C4×D197—21no—
74C2×C4×D197—21no—
75C2×C4×D197C2311yescharacteristic
76C2×C4×D197C2311yescharacteristic
77#15691347—22no—
78#15691346—41no—
79#15691346—41no—
80#15691341—41no—
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2311yescharacteristic
83C8×D197—41no—
84C4⋊Dic197—21no—
85C4⋊Dic197—21no—
86C4⋊Dic197D411yescharacteristic
87C4⋊Dic197C2311yescharacteristic
88C4⋊Dic197C2311yescharacteristic
89#15691337—21no—
90#15691337—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335D411yescharacteristic
98#15691335D411yescharacteristic
99C4×Dic197—21no—
100C4×Dic197—21no—
101C2×D4×C197—21no—
102C2×D4×C197—21no—
103C2×D4×C197C2311yescharacteristic
104C2×D4×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197—21no—
109C4⋊C4×C197C2311yescharacteristic
110C4⋊C4×C197C2311yescharacteristic
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197—81no—
118C22×D197—81no—
119C22×D197C2×D411yescharacteristic
120C22×D197C2×D411yescharacteristic
121Dic394—21no—
122Dic394—21no—
123D788—22no—
124D788—21no—
125D788#4112yes—
126C22⋊D197—42no—
127C22⋊D197—42no—
128C22⋊D197—42no—
129C22⋊D197—42no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197—21no—
133C2×Dic197C2×D411yescharacteristic
134C2×Dic197C2×D411yescharacteristic
135C4×D197—21no—
136C4×D197—21no—
137C4×D197—21no—
138C4×D197—21no—
139C4×D197—42no—
140C4×D197—42no—
141#13096802—21no—
142#13096802—21no—
143C23×C197—21no—
144C23×C197—21no—
145D4×C197—21no—
146D4×C197—21no—
147D4×C197—21no—
148D4×C197—21no—
149D4×C197—42no—
150D4×C197—81no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197C22×D411yescharacteristic
165C2×D197—21no—
166C2×D197—21no—
167C2×D197—21no—
168C2×D197—22no—
169C2×D197—21no—
170C2×D197—21no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—84no—
174C2×D197—84no—
175C2×D197AGL(1,ℤ/8)12yes—
176C4×C197—21no—
177C4×C197—21no—
178C4×C197—41no—
179C4×C197—41no—
180C4×C197C2×#4111yescharacteristic
181C4×C197H(5,𝔽2)11yescharacteristic
182Dic197—21no—
183Dic197—21no—
184Dic197—41no—
185Dic197—41no—
186Dic197—41no—
187C2×C197—81no—
188C2×C197—81no—
189C2×C197#54611yescharacteristic
190C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C197#57711yescharacteristic
192D197—22no—
193D197—41no—
194D197—42no—
195C197#331011yescharacteristic, complemented, Sylow
196#3310—1971nomaximal, Sylow
197#552—1971no—
198D42—1971no—
199#490D19711yescharacteristic, complemented
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—3941no—
213C22×D4—1971no—
214C22×D4—1971no—
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#125C2×D19711yescharacteristic, complemented
224#123—3941no—
225C22⋊Q8—1971no—
226C22⋊Q8—1971no—
227#121—3941no—
228#121—1971no—
229#121—1971no—
230#120—3941no—
231#120—3941no—
232C4×Q8—1971no—
233C4×D4—1971no—
234C4×D4—1971no—
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#102C2×D19711yescharacteristic, complemented
249#102C2×D19711yescharacteristic, complemented
250#102C2×D19711yescharacteristic, complemented
251#102C2×D19711yescharacteristic, complemented
252C22⋊C8—1971no—
253C22⋊C8—1971no—
254#97C2×D19711yescharacteristic, complemented
255C24—3941no—
256C24—3941no—
257C2×Q8—1971no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—7884no—
261C2×D4—7881no—
262C2×D4—7881no—
263C2×D4—3942no—
264C2×D4—3942no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—3941no—
268C2×D4—3941no—
269C2×D4—1971no—
270C2×D4—1971no—
271C2×D4C22×D19711yescharacteristic
272C2×D4C22×D19711yescharacteristic
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—1971no—
276C22×C4—1971no—
277SD16—7881no—
278SD16—7881no—
279SD16—3942no—
280M16—7881no—
281C2×C8—7881no—
282C2×C8—1971no—
283C2×C8—1971no—
284C2×C8C22×D19711yescharacteristic
285C2×C8C22×D19711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
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—3941no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—1971no—
304C22⋊C4—1971no—
305C42—3941no—
306C42—3941no—
307C42C22×D19711yescharacteristic
308C23—21no—
309C23—21no—
310C23—7882no—
311C23—7882no—
312C23—7882no—
313C23—7882no—
314C23—15761no—
315C23—15761no—
316C23—3941no—
317C23—1971no—
318C23—1971no—
319Q8—3941no—
320Q8—3941no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—7882no—
328D4—7882no—
329D4—7882no—
330D4—7882no—
331D4—3942no—
332D4—3941no—
333D4—1972no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—7882no—
337C2×C4—7882no—
338C2×C4—3941no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—1971no—
346C2×C4—1971no—
347C2×C4D4×D19711yescharacteristic
348C2×C4D4×D19711yescharacteristic
349C2×C4C23×D19711yescharacteristic
350C8—21no—
351C8—21no—
352C8—3941no—
353C8—3941no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—7881no—
361C22—7881no—
362C22—15764no—
363C22—15764no—
364C22—3941no—
365C22—3941no—
366C22—3941no—
367C22—3942no—
368C22—3941no—
369C22—3941no—
370C22—1972no—
371C22C2×D4×D19711yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—7881no—
377C4—7881no—
378C4—7881no—
379C4—3941no—
380C4—3941no—
381C4#41×D19711yescharacteristic
382C4#2026337211yescharacteristic
383C2—81no—
384C2—81no—
385C2—7882no—
386C2—7881no—
387C2—3942no—
388C2#2716172211yescharacteristic
389C2AGL(1,ℤ/8)×D19711yescharacteristic
390C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#490×C197C2
socleC22×C197C22×D4
prime radical#27161714C2
hypercenter#490D197
hypocenterC197#3310
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #27161767C2semidirect
3. #27161401C2semidirect
4. #27161391C2semidirect
5. #27161376C2semidirect
6. #27161324C2semidirect
7. #27161307C2semidirect
8. #27161276C2semidirect
9. #27161231C2semidirect
10. #27161231C2semidirect
11. #27161181C2semidirect
12. #27161152C2semidirect
13. #27161082C2semidirect
14. #102×D197C2semidirect
15. #490×C197C2semidirect
20. C2×#15691346C22semidirect
21. C2×#15691346C22semidirect
22. C2×#15691341C22semidirect
23. C2×C8×D197C22semidirect
27. #20263306C22semidirect
28. #20263306C22semidirect
34. C4×Dic394C22semidirect
38. #20263284C22semidirect
39. C4⋊C4×D197C22semidirect
40. #20263260C22semidirect
41. #20263259C22semidirect
42. #20263257C22semidirect
43. C4⋊#13096802C22semidirect
44. #20263251C22semidirect
46. #20263247C22semidirect
47. #20263247C22semidirect
50. #20263238C22semidirect
51. #20263238C22semidirect
53. #125×C197C22semidirect
54. #102×C197C22semidirect
55. #102×C197C22semidirect
56. #102×C197C22semidirect
57. #102×C197C22semidirect
58. #97×C197C22semidirect
195. C197#3310semidirect
199. #490D197semidirect
223. #125C2×D197semidirect
248. #102C2×D197semidirect
249. #102C2×D197semidirect
250. #102C2×D197semidirect
251. #102C2×D197semidirect
254. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617672
#271617142
#271614012
#271613912
#271613762
#271613242
#271613072
#271612762
#271612312
#271611812
#271611522
#271610822
#102×D1972
#490×C1972
#3310197