Finite Group Databasebeta

← #38754671

Subgroups — #38754671

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#27161730C211yesmaximal, characteristic, complemented
2#27161720C211yesmaximal, characteristic, complemented
3#27161440C211yesmaximal, characteristic, complemented
4#27161399C211yesmaximal, characteristic, complemented
5#27161368C211yesmaximal, characteristic, complemented
6#27161333C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161297C211yesmaximal, characteristic, complemented
9#27161227C211yesmaximal, characteristic, complemented
10#27161215C211yesmaximal, characteristic, complemented
11#27161190C211yesmaximal, characteristic, complemented
12#27161147C211yesmaximal, characteristic, complemented
13#27161079C211yesmaximal, characteristic, complemented
14C2×#20263246C211yesmaximal, characteristic, complemented
15#483×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691358—21no—
18C22×D788C2211yescharacteristic
19C2×#15691345C2211yescharacteristic, complemented
20C2×D1576C2211yescharacteristic
21C2×D1576C2211yescharacteristic
22C2×#15691341C2211yescharacteristic, complemented
23C22×#13096802C2211yescharacteristic
24#20263325C2211yescharacteristic, complemented
25#20263315C2211yescharacteristic
26#20263311—21no—
27#20263311—21no—
28#20263309—21no—
29#20263304—21no—
30#20263301—21no—
31#20263301C2211yescharacteristic
32#20263301C2211yescharacteristic, complemented
33#20263300C2211yescharacteristic
34#20263296—21no—
35#20263288C2211yescharacteristic
36#20263288C2211yescharacteristic, complemented
37C4×C22⋊D197C2211yescharacteristic
38C4⋊C4×D197C2211yescharacteristic, complemented
39C22⋊C4×D197—21no—
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic
45#20263246C2211yescharacteristic
46#20263246C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic
49#20263246C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51#20263236C2211yescharacteristic, complemented
52C22⋊Q8×C197C2211yescharacteristic
53C2×C4⋊C4×C197C2211yescharacteristic
54#106×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C23×D197—21no—
60#15691359—22no—
61#15691359—41no—
62#15691358—44no—
63C2×Dic394—21no—
64C2×D788—22no—
65C2×D788—21no—
66C2×D788—21no—
67C2×D788—21no—
68C2×D788—21no—
69C2×D788—41no—
70C2×D788C2311yescharacteristic
71C2×D788C2311yescharacteristic
72C2×D788C2311yescharacteristic
73C2×C22⋊D197—21no—
74C2×C22⋊D197D411yescharacteristic
75C2×C4×D197—21no—
76C2×C4×D197—21no—
77C2×C4×D197—21no—
78C2×C4×D197—21no—
79C2×C4×D197C2311yescharacteristic
80#15691345—41no—
81D1576—22no—
82D1576—41no—
83#15691341—41no—
84C2×#13096802—41no—
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C4⋊Dic197C2311yescharacteristic
88#15691337—21no—
89#15691337—21no—
90#15691337D411yescharacteristic
91C22⋊Dic197D411yescharacteristic
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335—21no—
98#15691335—21no—
99#15691335D411yescharacteristic
100C4×Dic197C2311yescharacteristic
101C2×Q8×C197C2311yescharacteristic
102C22×C4×C197—21no—
103C22×C4×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C4⋊C4×C197—21no—
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C4⋊C4×C197C2311yescharacteristic
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—21no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197—81no—
118C22×D197C2×D411yescharacteristic
119Dic394—42no—
120D788—21no—
121D788—21no—
122D788—21no—
123D788—21no—
124D788—21no—
125D788—42no—
126D788—42no—
127D788—42no—
128D788—81no—
129D788#4112yescomplemented
130C22⋊D197—22no—
131C22⋊D197—42no—
132C2×Dic197—21no—
133C2×Dic197C2×D411yescharacteristic
134C2×Dic197C2×D411yescharacteristic
135C2×Dic197C2×D411yescharacteristic
136C4×D197—21no—
137C4×D197—21no—
138C4×D197—42no—
139C4×D197—42no—
140C4×D197—42no—
141C4×D197—81no—
142#13096802—21no—
143#13096802—21no—
144C23×C197C2×D411yescharacteristic
145Q8×C197—21no—
146Q8×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—42no—
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197AGL(1,ℤ/8)12yes—
161C22×C197C22×D411yescharacteristic
162C2×D197—21no—
163C2×D197—22no—
164C2×D197—21no—
165C2×D197—21no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—84no—
175C4×C197—21no—
176C4×C197—41no—
177C4×C197—41no—
178C4×C197—41no—
179C4×C197—41no—
180C4×C197C2×#4111yescharacteristic
181C4×C197H(5,𝔽2)11yescharacteristic
182Dic197—21no—
183Dic197—21no—
184Dic197—21no—
185Dic197—41no—
186C2×C197—22no—
187C2×C197#54611yescharacteristic
188C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C197#57711yescharacteristic
190D197—41no—
191D197—42no—
192D197—81no—
193D197—81no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547—1971no—
197#546—1971no—
198#489—1971no—
199#483D19711yescharacteristic, complemented
200#481—1971no—
201#476—1971no—
202#469—1971no—
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×D8—1971no—
217C2×M16—1971no—
218C22×C8—1971no—
219C4⋊Q8—1971no—
220#124—1971no—
221#123—3941no—
222#123—3941no—
223#123—3941no—
224C22⋊Q8—3941no—
225C22⋊Q8C2×D19711yescharacteristic
226#121—3941no—
227#121—1971no—
228#121—1971no—
229#121—1971no—
230#120—3941no—
231C4×D4—1971no—
232C4×D4—1971no—
233C4×D4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4C2×D19711yescharacteristic
236C2×C22⋊C4—3941no—
237#106—1971no—
238#106C2×D19711yescharacteristic, complemented
239#106C2×D19711yescharacteristic, complemented
240C4⋊C8—1971no—
241#103—1971no—
242#103C2×D19711yescharacteristic, complemented
243#103C2×D19711yescharacteristic, complemented
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251C22⋊C8—1971no—
252C22⋊C8C2×D19711yescharacteristic
253#97—1971no—
254C24—3941no—
255#41—7884no—
256#41—7881no—
257#41—3942no—
258C2×Q8—3941no—
259C2×Q8C22×D19711yescharacteristic
260C2×D4—7881no—
261C2×D4—3941no—
262C2×D4—3941no—
263C2×D4—3942no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4—1971no—
269C2×D4—1971no—
270C2×D4—1971no—
271C22×C4—21no—
272C22×C4—3941no—
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4—1971no—
277C22×C4C22×D19711yescharacteristic
278SD16—7881no—
279D8—7881no—
280D8—3942no—
281M16—7881no—
282C2×C8—7881no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8C22×D19711yescharacteristic
286C2×C8C22×D19711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4—1971no—
293C4⋊C4C22×D19711yescharacteristic
294C4⋊C4C22×D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3941no—
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—7882no—
308C23—7882no—
309C23—15761no—
310C23—3941no—
311C23—3941no—
312C23—3941no—
313C23—1971no—
314C23D4×D19711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—7882no—
318D4—7882no—
319D4—7882no—
320D4—7882no—
321D4—7882no—
322D4—15761no—
323D4—3942no—
324D4—3941no—
325D4—3941no—
326D4—3941no—
327D4—3941no—
328D4—3941no—
329D4—1972no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—7882no—
338C2×C4—7882no—
339C2×C4—7882no—
340C2×C4—15761no—
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—7881no—
356C22—7881no—
357C22—7881no—
358C22—7881no—
359C22—7881no—
360C22—7881no—
361C22—7881no—
362C22—7881no—
363C22—15764no—
364C22—3941no—
365C22—3941no—
366C22—3942no—
367C22—3941no—
368C22#2026335312yes—
369C22C2×D4×D19711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—7881no—
376C4—3941no—
377C4—3941no—
378C4—3941no—
379C4#41×D19711yescharacteristic
380C4#2026337211yescharacteristic
381C2—22no—
382C2—7882no—
383C2—7881no—
384C2—15761no—
385C2—15761no—
386C2#2716172111yescharacteristic
387C2C2×#2026335311yescharacteristic
388C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#483×C197C2
socleC22×C197C22×D4
hypercenter#483D197
hypocenterC197#3306
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #27161730C2semidirect
2. #27161720C2semidirect
3. #27161440C2semidirect
4. #27161399C2semidirect
5. #27161368C2semidirect
6. #27161333C2semidirect
7. #27161310C2semidirect
8. #27161297C2semidirect
9. #27161227C2semidirect
10. #27161215C2semidirect
11. #27161190C2semidirect
12. #27161147C2semidirect
13. #27161079C2semidirect
14. C2×#20263246C2semidirect
15. #483×C197C2semidirect
19. C2×#15691345C22semidirect
22. C2×#15691341C22semidirect
24. #20263325C22semidirect
32. #20263301C22semidirect
36. #20263288C22semidirect
38. C4⋊C4×D197C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263250C22semidirect
49. #20263246C22semidirect
50. #20263238C22semidirect
51. #20263236C22semidirect
54. #106×C197C22semidirect
55. #106×C197C22semidirect
56. #103×C197C22semidirect
57. #103×C197C22semidirect
129. D788#41semidirect
194. C197#3306semidirect
199. #483D197semidirect
238. #106C2×D197semidirect
239. #106C2×D197semidirect
242. #103C2×D197semidirect
243. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617302
#271617202
#271614402
#271613992
#271613682
#271613332
#271613102
#271612972
#271612272
#271612152
#271611902
#271611472
#271610792
C2×#202632462
#483×C1972
#3306197