Finite Group Databasebeta

← #38754556

Subgroups — #38754556

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161725C211yesmaximal, characteristic, complemented
2#27161436C211yesmaximal, characteristic, complemented
3#27161414C211yesmaximal, characteristic, complemented
4#27161403C211yesmaximal, characteristic, complemented
5#27161376C211yesmaximal, characteristic, complemented
6#27161313C211yesmaximal, characteristic, complemented
7#27161306C211yesmaximal, characteristic, complemented
8#27161294C211yesmaximal, characteristic, complemented
9#27161222C211yesmaximal, characteristic, complemented
10#27161212C211yesmaximal, characteristic, complemented
11C4×#15691343C211yesmaximal, characteristic, complemented
12D4×#13096802C211yesmaximal, characteristic, complemented
13#27161073C211yesmaximal, characteristic, complemented
14C2×#20263246C211yesmaximal, characteristic, complemented
15#553×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×#15691346C2211yescharacteristic, complemented
19C2×#15691343C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22C22×#13096802C2211yescharacteristic, complemented
23#20263314C2211yescharacteristic
24#20263311—21no—
25#20263305C2211yescharacteristic
26#20263300C2211yescharacteristic
27#20263299—21no—
28#20263294—21no—
29C4×D788C2211yescharacteristic
30C22⋊C4×D197—21no—
31#20263259C2211yescharacteristic, complemented
32#20263259C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263249C2211yescharacteristic, complemented
36#20263249C2211yescharacteristic, complemented
37#20263249C2211yescharacteristic, complemented
38#20263247C2211yescharacteristic, complemented
39#20263246C2211yescharacteristic
40#20263246C2211yescharacteristic
41#20263246C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic, complemented
47C4×#13096802C2211yescharacteristic, complemented
48#126×C197—21no—
49#125×C197C2211yescharacteristic, complemented
50#123×C197—21no—
51#121×C197—21no—
52#121×C197C2211yescharacteristic
53#121×C197C2211yescharacteristic, complemented
54C4×D4×C197—21no—
55C4×D4×C197C2211yescharacteristic
56C4×D4×C197C2211yescharacteristic, complemented
57#117×C197C2211yescharacteristic, complemented
58C2×C4⋊C4×C197C2211yescharacteristic
59C23×D197—21no—
60#15691358—22no—
61#15691358—41no—
62C2×Dic394C2311yescharacteristic
63C2×D788—22no—
64C2×D788—21no—
65C2×D788—41no—
66C2×D788C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72#15691346—41no—
73#15691343—22no—
74#15691343—41no—
75C2×#13096802—41no—
76C2×#13096802—41no—
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C2×#13096802C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82#15691337—21no—
83C22⋊Dic197—21no—
84#15691335—21no—
85#15691335—21no—
86#15691335—21no—
87#15691335—21no—
88C2×D4×C197—21no—
89C2×D4×C197—21no—
90C2×D4×C197D411yescharacteristic
91C2×D4×C197C2311yescharacteristic
92C22×C4×C197—21no—
93C22×C4×C197—21no—
94C22×C4×C197C2311yescharacteristic
95C22×C4×C197C2311yescharacteristic
96C4⋊C4×C197—21no—
97C4⋊C4×C197—21no—
98C4⋊C4×C197—21no—
99C4⋊C4×C197D411yescharacteristic
100C4⋊C4×C197C2311yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C22⋊C4×C197—21no—
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116Dic394—21no—
117Dic394—21no—
118D788—21no—
119D788—21no—
120D788—21no—
121D788—42no—
122D788#4112yes—
123C22⋊D197—42no—
124C22⋊D197—81no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C4×D197—42no—
128C4×D197—81no—
129#13096802—21no—
130#13096802—21no—
131#13096802—21no—
132#13096802—21no—
133C23×C197—21no—
134C23×C197C2×D411yescharacteristic
135C23×C197C2×D411yescharacteristic
136D4×C197—22no—
137D4×C197—21no—
138D4×C197—21no—
139D4×C197—42no—
140D4×C197—81no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—81no—
149C2×C4×C197—81no—
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2×D411yescharacteristic
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2411yescharacteristic
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197#13512yes—
164C22×C197C22×D411yescharacteristic
165C2×D197—22no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—84no—
172C4×C197—21no—
173C4×C197—21no—
174C4×C197—21no—
175C4×C197—21no—
176C4×C197—41no—
177C4×C197—41no—
178C4×C197—41no—
179C4×C197C2×#4111yescharacteristic
180C4×C197H(5,𝔽2)11yescharacteristic
181Dic197—41no—
182Dic197—41no—
183C2×C197—22no—
184C2×C197—41no—
185C2×C197—81no—
186C2×C197#54611yescharacteristic
187C2×C197C2×#13511yescharacteristic
188C2×C197#57611yescharacteristic
189D197—42no—
190D197—81no—
191C197#329711yescharacteristic, complemented, Sylow
192#3297—1971nomaximal, Sylow
193#553D19711yescharacteristic, complemented
194#546—1971no—
195#488—1971no—
196#484—1971no—
197#482—1971no—
198#478—1971no—
199#466—1971no—
200#465—1971no—
201#463—1971no—
202#449—1971no—
203#447—1971no—
204C4×D8—1971no—
205C8×D4—1971no—
206#416—1971no—
207C2×#102—1971no—
208C2×#41—1971no—
209C22×D4—1971no—
210C2×SD16—1971no—
211C2×D8—1971no—
212C2×D8—1971no—
213C22×C8—1971no—
214C22×C8—1971no—
215#126—21no—
216#125C2×D19711yescharacteristic, complemented
217#124—1971no—
218#123—21no—
219#123—3941no—
220C22⋊Q8—1971no—
221#121—21no—
222#121—3941no—
223#121—1971no—
224#121C2×D19711yescharacteristic
225#121C2×D19711yescharacteristic, complemented
226#120—3941no—
227C4×D4—21no—
228C4×D4—1971no—
229C4×D4C2×D19711yescharacteristic
230C4×D4C2×D19711yescharacteristic, complemented
231#117C2×D19711yescharacteristic, complemented
232C2×C4⋊C4C2×D19711yescharacteristic
233C2×C22⋊C4—3941no—
234#107—1971no—
235#107—1971no—
236#106—1971no—
237C4⋊C8—1971no—
238#103—1971no—
239#103—1971no—
240#103—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248C22⋊C8—1971no—
249C22⋊C8—1971no—
250C4×C8—1971no—
251C24—3941no—
252#41—7881no—
253#41—3942no—
254C2×Q8—1971no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—7881no—
258C2×D4—3942no—
259C2×D4—3941no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—1971no—
263C2×D4—1971no—
264C2×D4C22⋊D19711yescharacteristic
265C2×D4C22×D19711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—3941no—
269C22×C4—3941no—
270C22×C4C22×D19711yescharacteristic
271C22×C4C22×D19711yescharacteristic
272SD16—7881no—
273D8—7881no—
274D8—3942no—
275C2×C8—7881no—
276C2×C8—7881no—
277C2×C8—1971no—
278C2×C8—1971no—
279C2×C8—1971no—
280C2×C8—1971no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—3941no—
285C4⋊C4—1971no—
286C4⋊C4C22⋊D19711yescharacteristic
287C4⋊C4C22×D19711yescharacteristic
288C4⋊C4C22×D19711yescharacteristic
289C4⋊C4C22×D19711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4C22⋊D19711yescharacteristic
300C22⋊C4C22⋊D19711yescharacteristic
301C42—21no—
302C42C22×D19711yescharacteristic
303C23—21no—
304C23—7882no—
305C23—7882no—
306C23—15761no—
307C23—3941no—
308C23—3941no—
309C23C2×C22⋊D19711yescharacteristic
310C23C2×C22⋊D19711yescharacteristic
311Q8—3941no—
312Q8—3941no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—7882no—
319D4—7882no—
320D4—15761no—
321D4—3941no—
322D4—3941no—
323D4—3941no—
324D4—1972no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—7882no—
335C2×C4—15761no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4C2×C22⋊D19711yescharacteristic
339C2×C4C2×C22⋊D19711yescharacteristic
340C2×C4C2×C22⋊D19711yescharacteristic
341C2×C4C2×C22⋊D19711yescharacteristic
342C2×C4C23×D19711yescharacteristic
343C8—3941no—
344C8—3941no—
345C8—3941no—
346C8—3941no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—7881no—
356C22—7881no—
357C22—7881no—
358C22—7881no—
359C22—7881no—
360C22—15764no—
361C22—3942no—
362C22#2026334612yes—
363C22C22×C22⋊D19711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—7881no—
372C4—7881no—
373C4#41×D19711yescharacteristic
374C4#2026337211yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—7882no—
379C2—15761no—
380C2#2716171711yescharacteristic
381C2C2×#2026334611yescharacteristic
382C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#553×C197C2
socleC22×C197C22×D4
hypercenter#553D197
hypocenterC197#3297
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #27161725C2semidirect
2. #27161436C2semidirect
3. #27161414C2semidirect
4. #27161403C2semidirect
5. #27161376C2semidirect
6. #27161313C2semidirect
7. #27161306C2semidirect
8. #27161294C2semidirect
9. #27161222C2semidirect
10. #27161212C2semidirect
11. C4×#15691343C2semidirect
12. D4×#13096802C2semidirect
13. #27161073C2semidirect
14. C2×#20263246C2semidirect
15. #553×C197C2semidirect
18. C2×#15691346C22semidirect
22. C22×#13096802C22semidirect
31. #20263259C22semidirect
32. #20263259C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263249C22semidirect
36. #20263249C22semidirect
37. #20263249C22semidirect
38. #20263247C22semidirect
46. C22⋊#13096802C22semidirect
47. C4×#13096802C22semidirect
49. #125×C197C22semidirect
53. #121×C197C22semidirect
56. C4×D4×C197C22semidirect
57. #117×C197C22semidirect
191. C197#3297semidirect
193. #553D197semidirect
216. #125C2×D197semidirect
225. #121C2×D197semidirect
230. C4×D4C2×D197semidirect
231. #117C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617252
#271614362
#271614142
#271614032
#271613762
#271613132
#271613062
#271612942
#271612222
#271612122
C4×#156913432
D4×#130968022
#271610732
C2×#202632462
#553×C1972
#3297197