Finite Group Databasebeta

← #38754668

Subgroups — #38754668

Order 25216.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161729C211yesmaximal, characteristic, complemented
2#27161726C211yesmaximal, characteristic, complemented
3#27161437C211yesmaximal, characteristic, complemented
4#27161411C211yesmaximal, characteristic, complemented
5#27161395C211yesmaximal, characteristic, complemented
6#27161368C211yesmaximal, characteristic, complemented
7#27161307C211yesmaximal, characteristic, complemented
8#27161292C211yesmaximal, characteristic, complemented
9#27161222C211yesmaximal, characteristic, complemented
10#27161211C211yesmaximal, characteristic, complemented
11#27161188C211yesmaximal, characteristic, complemented
12#27161148C211yesmaximal, characteristic, complemented
13#27161081C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#469×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×#15691357—21no—
18C2×D4×D197C2211yescharacteristic
19C2×#15691346C2211yescharacteristic, complemented
20C2×#15691343C2211yescharacteristic
21C2×C8×D197C2211yescharacteristic
22#20263325C2211yescharacteristic, complemented
23#20263313C2211yescharacteristic
24#20263308—21no—
25#20263308—21no—
26#20263307—21no—
27#20263306C2211yescharacteristic
28#20263303—21no—
29#20263297—21no—
30#20263297C2211yescharacteristic
31#20263294—21no—
32D4×Dic197C2211yescharacteristic, complemented
33#20263288C2211yescharacteristic
34C4×C22⋊D197C2211yescharacteristic
35C2×C4⋊Dic197C2211yescharacteristic, complemented
36C4⋊C4×D197C2211yescharacteristic
37C22⋊C4×D197—21no—
38#20263257C2211yescharacteristic, complemented
39#20263257C2211yescharacteristic, complemented
40#20263255C2211yescharacteristic, complemented
41#20263251C2211yescharacteristic, complemented
42#20263251C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic
48#20263246C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic
51#20263236C2211yescharacteristic, complemented
52C2×D8×C197C2211yescharacteristic
53C2×M16×C197C2211yescharacteristic, complemented
54#121×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic, complemented
56#106×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60#15691358—22no—
61#15691358—41no—
62#15691357—44no—
63C2×Dic394—21no—
64C2×Dic394C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C2×C22⋊D197D411yescharacteristic
70D4×D197—22no—
71D4×D197—41no—
72C22×Dic197—21no—
73C22×Dic197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197—21no—
77C2×C4×D197C2311yescharacteristic
78#15691346—41no—
79#15691343—22no—
80C2×#13096802C2311yescharacteristic
81C2×#13096802C2311yescharacteristic
82C8×D197—41no—
83C4⋊Dic197—21no—
84C4⋊Dic197—21no—
85C4⋊Dic197C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87#15691337—21no—
88#15691337—21no—
89#15691337D411yescharacteristic
90C22⋊Dic197—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197—21no—
93C22⋊Dic197—21no—
94C22⋊Dic197D411yescharacteristic
95#15691335—21no—
96#15691335—21no—
97#15691335D411yescharacteristic
98C4×Dic197C2311yescharacteristic
99C2×D4×C197—21no—
100C2×D4×C197—21no—
101C2×D4×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197C2311yescharacteristic
104D8×C197—41no—
105M16×C197—41no—
106C2×C8×C197C2311yescharacteristic
107C2×C8×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116C22×D197C2×D411yescharacteristic
117Dic394—21no—
118Dic394—21no—
119Dic394—42no—
120D788—21no—
121D788#4112yescomplemented
122C22⋊D197—22no—
123C22⋊D197—42no—
124C22⋊D197—42no—
125C22⋊D197—42no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—42no—
130C2×Dic197—42no—
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C4×D197—21no—
135C4×D197—21no—
136C4×D197—42no—
137C4×D197—42no—
138C4×D197—81no—
139#13096802—21no—
140#13096802—21no—
141C23×C197—21no—
142C23×C197—21no—
143C23×C197C2×D411yescharacteristic
144D4×C197—21no—
145D4×C197—21no—
146D4×C197—21no—
147D4×C197—21no—
148D4×C197—42no—
149D4×C197—81no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197C22×D411yescharacteristic
168C2×D197—21no—
169C2×D197—21no—
170C2×D197—22no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—84no—
174C2×D197AGL(1,ℤ/8)12yes—
175C4×C197—21no—
176C4×C197—41no—
177C4×C197—41no—
178C4×C197C2×#4111yescharacteristic
179C4×C197H(5,𝔽2)11yescharacteristic
180Dic197—21no—
181Dic197—21no—
182Dic197—21no—
183Dic197—41no—
184Dic197—41no—
185Dic197—41no—
186C2×C197—41no—
187C2×C197—81no—
188C2×C197—81no—
189C2×C197#54611yescharacteristic
190C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C197#57711yescharacteristic
192D197—22no—
193D197—42no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547—1971no—
197#546—1971no—
198#489—1971no—
199#483—1971no—
200#481—1971no—
201#476—1971no—
202#469D19711yescharacteristic, complemented
203#465—1971no—
204#463—1971no—
205#449—1971no—
206#447—1971no—
207#442—1971no—
208#435—1971no—
209#417—1971no—
210C2×#102—1971no—
211C2×#41—3941no—
212C2×#41—1971no—
213C22×D4—1971no—
214C2×SD16—1971no—
215C2×D8—1971no—
216C2×D8C2×D19711yescharacteristic
217C2×M16C2×D19711yescharacteristic, complemented
218C22×C8—1971no—
219C4⋊Q8—1971no—
220#124—1971no—
221#123—3941no—
222#123—3941no—
223#123—3941no—
224C22⋊Q8—3941no—
225C22⋊Q8—1971no—
226#121—3941no—
227#121—1971no—
228#121C2×D19711yescharacteristic
229#121C2×D19711yescharacteristic, complemented
230#120—3941no—
231C4×D4—1971no—
232C4×D4—1971no—
233C4×D4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C22⋊C4—3941no—
237#106—1971no—
238#106—1971no—
239#106C2×D19711yescharacteristic, complemented
240C4⋊C8—1971no—
241#103—1971no—
242#103—1971no—
243#103—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102C2×D19711yescharacteristic
250#102C2×D19711yescharacteristic, complemented
251C22⋊C8—1971no—
252C22⋊C8—1971no—
253#97—1971no—
254C24—3941no—
255#41—7881no—
256#41—7884no—
257#41—3942no—
258C2×Q8—3941no—
259C2×Q8—1971no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—7881no—
263C2×D4—3942no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4—1971no—
269C2×D4C22×D19711yescharacteristic
270C2×D4C22×D19711yescharacteristic
271C22×C4—3941no—
272C22×C4—3941no—
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4—1971no—
277C22×C4C22×D19711yescharacteristic
278SD16—7881no—
279D8—41no—
280D8—3942no—
281M16—41no—
282C2×C8—7881no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8C22×D19711yescharacteristic
286C2×C8C22×D19711yescharacteristic
287C4⋊C4—3941no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4C22×D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—3941no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4—1971no—
305C22⋊C4—1971no—
306C42—1971no—
307C23—21no—
308C23—21no—
309C23—7882no—
310C23—7882no—
311C23—15761no—
312C23—3941no—
313C23—1971no—
314C23D4×D19711yescharacteristic
315Q8—7882no—
316Q8—3941no—
317Q8—3941no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—81no—
324D4—7882no—
325D4—7882no—
326D4—7882no—
327D4—3942no—
328D4—3941no—
329D4—1972no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—7882no—
335C2×C4—7882no—
336C2×C4—7882no—
337C2×C4—7882no—
338C2×C4—15761no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—1971no—
345C2×C4—1971no—
346C2×C4—1971no—
347C2×C4D4×D19711yescharacteristic
348C2×C4C23×D19711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3941no—
352C8—3941no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—7881no—
363C22—7881no—
364C22—15764no—
365C22—3942no—
366C22—3941no—
367C22—3941no—
368C22—1972no—
369C22C2×D4×D19711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—7881no—
374C4—7881no—
375C4—7881no—
376C4—3941no—
377C4—3941no—
378C4—3941no—
379C4#41×D19711yescharacteristic
380C4#2026337111yescharacteristic
381C2—41no—
382C2—81no—
383C2—81no—
384C2—7882no—
385C2—3942no—
386C2#2716171811yescharacteristic
387C2AGL(1,ℤ/8)×D19711yescharacteristic
388C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#469×C197C2
socleC22×C197C22×D4
hypercenter#469D197
hypocenterC197#3306
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #27161729C2semidirect
2. #27161726C2semidirect
3. #27161437C2semidirect
4. #27161411C2semidirect
5. #27161395C2semidirect
6. #27161368C2semidirect
7. #27161307C2semidirect
8. #27161292C2semidirect
9. #27161222C2semidirect
10. #27161211C2semidirect
11. #27161188C2semidirect
12. #27161148C2semidirect
13. #27161081C2semidirect
14. #102×D197C2semidirect
15. #469×C197C2semidirect
19. C2×#15691346C22semidirect
22. #20263325C22semidirect
32. D4×Dic197C22semidirect
35. C2×C4⋊Dic197C22semidirect
38. #20263257C22semidirect
39. #20263257C22semidirect
40. #20263255C22semidirect
41. #20263251C22semidirect
42. #20263251C22semidirect
43. #20263249C22semidirect
49. C22⋊#13096802C22semidirect
51. #20263236C22semidirect
53. C2×M16×C197C22semidirect
55. #121×C197C22semidirect
56. #106×C197C22semidirect
58. #102×C197C22semidirect
121. D788#41semidirect
194. C197#3306semidirect
202. #469D197semidirect
217. C2×M16C2×D197semidirect
229. #121C2×D197semidirect
239. #106C2×D197semidirect
250. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617292
#271617262
#271614372
#271614112
#271613952
#271613682
#271613072
#271612922
#271612222
#271612112
#271611882
#271611482
#271610812
#102×D1972
#469×C1972
#3306197