Finite Group Databasebeta

← #38754560

Subgroups — #38754560

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#27161777C211yesmaximal, characteristic, complemented
2#27161720C211yesmaximal, characteristic, complemented
3#27161435C211yesmaximal, characteristic, complemented
4#27161406C211yesmaximal, characteristic, complemented
5#27161382C211yesmaximal, characteristic, complemented
6#27161316C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161297C211yesmaximal, characteristic, complemented
9#27161227C211yesmaximal, characteristic, complemented
10#27161215C211yesmaximal, characteristic, complemented
11#27161161C211yesmaximal, characteristic, complemented
12#27161142C211yesmaximal, characteristic, complemented
13#27161074C211yesmaximal, characteristic, complemented
14C2×#20263246C211yesmaximal, characteristic, complemented
15#484×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×#15691345C2211yescharacteristic, complemented
19C2×D1576C2211yescharacteristic
20C2×D1576C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22C2×C8×D197C2211yescharacteristic, complemented
23#20263321—21no—
24#20263317C2211yescharacteristic, complemented
25#20263315C2211yescharacteristic
26#20263311—21no—
27#20263311—21no—
28#20263301—21no—
29#20263301C2211yescharacteristic
30#20263301C2211yescharacteristic, complemented
31#20263300—21no—
32#20263300C2211yescharacteristic
33#20263296—21no—
34#20263288C2211yescharacteristic
35#20263288C2211yescharacteristic, complemented
36C4×C22⋊D197—21no—
37C4×C22⋊D197C2211yescharacteristic
38#20263284C2211yescharacteristic, 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
51C8×Dic197C2211yescharacteristic, complemented
52C22⋊Q8×C197C2211yescharacteristic
53C2×C4⋊C4×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#107×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C23×D197—21no—
60#15691359—22no—
61#15691359—41no—
62C2×D788—22no—
63C2×D788—21no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788—21no—
67C2×D788—41no—
68C2×D788C2311yescharacteristic
69C2×D788C2311yescharacteristic
70C2×D788C2311yescharacteristic
71C2×C22⋊D197—21no—
72C2×C22⋊D197D411yescharacteristic
73C2×C4×D197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197C2311yescharacteristic
77#15691345—41no—
78D1576—22no—
79D1576—41no—
80C2×#13096802—41no—
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2311yescharacteristic
83C8×D197—41no—
84C4⋊Dic197—21no—
85C4⋊Dic197C2311yescharacteristic
86#15691337—21no—
87#15691337D411yescharacteristic
88C22⋊Dic197D411yescharacteristic
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335D411yescharacteristic
98C4×Dic197—21no—
99C4×Dic197C2311yescharacteristic
100C2×Q8×C197C2311yescharacteristic
101C22×C4×C197—21no—
102C22×C4×C197C2311yescharacteristic
103C2×C8×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C4⋊C4×C197—21no—
106C4⋊C4×C197—21no—
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—81no—
117C22×D197C2×D411yescharacteristic
118D788—21no—
119D788—21no—
120D788—21no—
121D788—21no—
122D788—21no—
123D788—42no—
124D788—42no—
125D788—81no—
126D788—81no—
127D788#4112yes—
128C22⋊D197—22no—
129C22⋊D197—42no—
130C2×Dic197—21no—
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C4×D197—21no—
135C4×D197—21no—
136C4×D197—42no—
137C4×D197—81no—
138C4×D197—81no—
139#13096802—21no—
140#13096802—21no—
141C23×C197C2×D411yescharacteristic
142Q8×C197—21no—
143Q8×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—81no—
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2411yescharacteristic
153C8×C197—21no—
154C8×C197—21no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197#13512yes—
158C22×C197C22×D411yescharacteristic
159C2×D197—21no—
160C2×D197—22no—
161C2×D197—21no—
162C2×D197—21no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—84no—
172C4×C197—21no—
173C4×C197—41no—
174C4×C197—41no—
175C4×C197—41no—
176C4×C197—41no—
177C4×C197C2×#4111yescharacteristic
178C4×C197H(5,𝔽2)11yescharacteristic
179Dic197—21no—
180Dic197—21no—
181Dic197—21no—
182Dic197—41no—
183C2×C197—22no—
184C2×C197#54611yescharacteristic
185C2×C197C2×#13511yescharacteristic
186C2×C197#57611yescharacteristic
187D197—41no—
188D197—42no—
189D197—81no—
190D197—81no—
191C197#329711yescharacteristic, complemented, Sylow
192#3297—1971nomaximal, Sylow
193#553—1971no—
194#546—1971no—
195#488—1971no—
196#484D19711yescharacteristic, complemented
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—3941no—
216#125—1971no—
217#124—1971no—
218#123—3941no—
219#123—3941no—
220C22⋊Q8C2×D19711yescharacteristic
221#121—3941no—
222#121—3941no—
223#121—1971no—
224#121—1971no—
225#121—1971no—
226#120—3941no—
227C4×D4—3941no—
228C4×D4—1971no—
229C4×D4—1971no—
230C4×D4—1971no—
231#117—1971no—
232C2×C4⋊C4C2×D19711yescharacteristic
233C2×C22⋊C4—3941no—
234#107C2×D19711yescharacteristic, complemented
235#107C2×D19711yescharacteristic, complemented
236#106—1971no—
237C4⋊C8—1971no—
238#103—1971no—
239#103C2×D19711yescharacteristic, complemented
240#103C2×D19711yescharacteristic, complemented
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⋊C8C2×D19711yescharacteristic
250C4×C8—1971no—
251C24—3941no—
252#41—7881no—
253#41—3942no—
254C2×Q8C22×D19711yescharacteristic
255C2×D4—7881no—
256C2×D4—3941no—
257C2×D4—3942no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—1971no—
263C2×D4—1971no—
264C2×D4—1971no—
265C2×D4—1971no—
266C22×C4—21no—
267C22×C4—3941no—
268C22×C4—3941no—
269C22×C4—3941no—
270C22×C4—1971no—
271C22×C4C22×D19711yescharacteristic
272SD16—7881no—
273D8—7881no—
274D8—3942no—
275C2×C8—7881no—
276C2×C8—7881no—
277C2×C8—1971no—
278C2×C8—1971no—
279C2×C8C22×D19711yescharacteristic
280C2×C8C22×D19711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3941no—
284C4⋊C4—3941no—
285C4⋊C4—1971no—
286C4⋊C4—1971no—
287C4⋊C4C22×D19711yescharacteristic
288C4⋊C4C22×D19711yescharacteristic
289C4⋊C4C22×D19711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3941no—
292C22⋊C4—3941no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4—1971no—
300C22⋊C4—1971no—
301C42—3941no—
302C42—1971no—
303C23—7882no—
304C23—7882no—
305C23—15761no—
306C23—3941no—
307C23—3941no—
308C23—3941no—
309C23—1971no—
310C23D4×D19711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—7882no—
314D4—7882no—
315D4—7882no—
316D4—15761no—
317D4—15761no—
318D4—3942no—
319D4—3941no—
320D4—3941no—
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—81no—
332C2×C4—7882no—
333C2×C4—15761no—
334C2×C4—15761no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4—1971no—
339C2×C4—1971no—
340C2×C4—1971no—
341C2×C4D4×D19711yescharacteristic
342C2×C4C23×D19711yescharacteristic
343C8—21no—
344C8—21no—
345C8—3941no—
346C8—3941no—
347C22—21no—
348C22—21no—
349C22—7881no—
350C22—7881no—
351C22—7881no—
352C22—7881no—
353C22—7881no—
354C22—7881no—
355C22—7881no—
356C22—7881no—
357C22—15764no—
358C22—3941no—
359C22—3941no—
360C22—3942no—
361C22—3941no—
362C22#2026334912yes—
363C22C2×D4×D19711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—7881no—
370C4—3941no—
371C4—3941no—
372C4—3941no—
373C4#41×D19711yescharacteristic
374C4#2026337211yescharacteristic
375C2—22no—
376C2—7882no—
377C2—7881no—
378C2—15761no—
379C2—15761no—
380C2#2716172111yescharacteristic
381C2C2×#2026334911yescharacteristic
382C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#484×C197C2
socleC22×C197C22×D4
hypercenter#484D197
hypocenterC197#3297
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #27161777C2semidirect
2. #27161720C2semidirect
3. #27161435C2semidirect
4. #27161406C2semidirect
5. #27161382C2semidirect
6. #27161316C2semidirect
7. #27161310C2semidirect
8. #27161297C2semidirect
9. #27161227C2semidirect
10. #27161215C2semidirect
11. #27161161C2semidirect
12. #27161142C2semidirect
13. #27161074C2semidirect
14. C2×#20263246C2semidirect
15. #484×C197C2semidirect
18. C2×#15691345C22semidirect
22. C2×C8×D197C22semidirect
24. #20263317C22semidirect
30. #20263301C22semidirect
35. #20263288C22semidirect
38. #20263284C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263250C22semidirect
49. #20263246C22semidirect
50. #20263238C22semidirect
51. C8×Dic197C22semidirect
54. #107×C197C22semidirect
55. #107×C197C22semidirect
56. #103×C197C22semidirect
57. #103×C197C22semidirect
191. C197#3297semidirect
196. #484D197semidirect
234. #107C2×D197semidirect
235. #107C2×D197semidirect
239. #103C2×D197semidirect
240. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617772
#271617202
#271614352
#271614062
#271613822
#271613162
#271613102
#271612972
#271612272
#271612152
#271611612
#271611422
#271610742
C2×#202632462
#484×C1972
#3297197