Finite Group Databasebeta

← #38754555

Subgroups — #38754555

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#27161776C211yesmaximal, characteristic, complemented
2#27161717C211yesmaximal, characteristic, complemented
3#27161435C211yesmaximal, characteristic, complemented
4#27161413C211yesmaximal, characteristic, complemented
5#27161404C211yesmaximal, characteristic, complemented
6#27161381C211yesmaximal, characteristic, complemented
7#27161313C211yesmaximal, characteristic, complemented
8#27161305C211yesmaximal, characteristic, complemented
9#27161296C211yesmaximal, characteristic, complemented
10#27161211C211yesmaximal, characteristic, complemented
11D8×Dic197C211yesmaximal, characteristic, complemented
12#27161142C211yesmaximal, characteristic, complemented
13#27161075C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#449×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C2×#15691343C2211yescharacteristic
18C22×#13096802C2211yescharacteristic, complemented
19C2×C8×D197C2211yescharacteristic
20#20263319—21no—
21#20263317C2211yescharacteristic, complemented
22#20263315C2211yescharacteristic
23#20263309—21no—
24#20263307—21no—
25#20263302C2211yescharacteristic
26#20263300C2211yescharacteristic
27#20263298—21no—
28#20263297—21no—
29#20263297C2211yescharacteristic
30#20263294—21no—
31D4×Dic197C2211yescharacteristic
32D4×Dic197C2211yescharacteristic, complemented
33C4×C22⋊D197C2211yescharacteristic
34#20263286—21no—
35#20263285C2211yescharacteristic, complemented
36C4⋊C4×D197C2211yescharacteristic
37C2×#15691335—21no—
38#20263260C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263250C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic
50#20263238C2211yescharacteristic
51C8×Dic197C2211yescharacteristic, complemented
52C2×#41×C197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic, complemented
54C2×D8×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×D788C2311yescharacteristic
61C2×C22⋊D197—21no—
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64C2×C22⋊D197D411yescharacteristic
65D4×D197—22no—
66D4×D197—41no—
67C22×Dic197—21no—
68C22×Dic197—21no—
69C2×C4×D197—21no—
70C2×C4×D197C2311yescharacteristic
71#15691343—41no—
72C2×#13096802—41no—
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C8×D197—41no—
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79C4⋊Dic197C2311yescharacteristic
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691337D411yescharacteristic
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87C22⋊Dic197—21no—
88C22⋊Dic197D411yescharacteristic
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335D411yescharacteristic
94C4×Dic197—21no—
95C4×Dic197C2311yescharacteristic
96#41×C197—22no—
97#41×C197—41no—
98C2×Q8×C197C2311yescharacteristic
99C2×D4×C197—21no—
100C2×D4×C197—21no—
101C2×D4×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197—21no—
104C22×C4×C197C2311yescharacteristic
105SD16×C197—41no—
106D8×C197—22no—
107C2×C8×C197C2311yescharacteristic
108C2×C8×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C22×D197—21no—
112C22×D197—42no—
113C22×D197—42no—
114C22×D197—81no—
115C22×D197C2×D411yescharacteristic
116D788—21no—
117D788—21no—
118C22⋊D197—22no—
119C22⋊D197—42no—
120C22⋊D197—42no—
121C22⋊D197—81no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—81no—
126C2×Dic197—81no—
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C2×Dic197C2×D411yescharacteristic
130C4×D197—21no—
131C4×D197—21no—
132C4×D197—81no—
133#13096802—21no—
134#13096802—21no—
135C23×C197—21no—
136C23×C197—21no—
137C23×C197C2×D411yescharacteristic
138Q8×C197—21no—
139Q8×C197—21no—
140D4×C197—21no—
141D4×C197—21no—
142D4×C197—21no—
143D4×C197—42no—
144D4×C197—81no—
145D4×C197#4112yes—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—42no—
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2411yescharacteristic
153C8×C197—21no—
154C8×C197—21no—
155C22×C197—21no—
156C22×C197—22no—
157C22×C197—21no—
158C22×C197—21no—
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—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—84no—
171C2×D197#13512yes—
172C4×C197—21no—
173C4×C197—41no—
174C4×C197—41no—
175C4×C197C2×#4111yescharacteristic
176C4×C197H(5,𝔽2)11yescharacteristic
177Dic197—21no—
178Dic197—21no—
179Dic197—21no—
180Dic197—41no—
181Dic197—41no—
182Dic197—41no—
183C2×C197—42no—
184C2×C197—41no—
185C2×C197—81no—
186C2×C197#54611yescharacteristic
187C2×C197C2×#13511yescharacteristic
188C2×C197#57611yescharacteristic
189D197—22no—
190D197—81no—
191C197#329711yescharacteristic, complemented, Sylow
192#3297—1971nomaximal, Sylow
193#553—1971no—
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#449D19711yescharacteristic, complemented
203#447—1971no—
204C4×D8—1971no—
205C8×D4—1971no—
206#416—1971no—
207C2×#102—1971no—
208C2×#41C2×D19711yescharacteristic
209C22×D4—1971no—
210C2×SD16C2×D19711yescharacteristic, complemented
211C2×D8—1971no—
212C2×D8C2×D19711yescharacteristic
213C22×C8—1971no—
214C22×C8—1971no—
215#126—3941no—
216#125—1971no—
217#124—1971no—
218#123—3941no—
219#123—3941no—
220C22⋊Q8—1971no—
221#121—3941no—
222#121—3941no—
223#121—1971no—
224#121—1971no—
225#121C2×D19711yescharacteristic, complemented
226#120—3941no—
227C4×D4—3941no—
228C4×D4—1971no—
229C4×D4—1971no—
230C4×D4—1971no—
231#117—1971no—
232C2×C4⋊C4—1971no—
233C2×C22⋊C4—3941no—
234#107—1971no—
235#107—1971no—
236#106—1971no—
237C4⋊C8—1971no—
238#103—1971no—
239#103—1971no—
240#103C2×D19711yescharacteristic, complemented
241#102—1971no—
242#102—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102C2×D19711yescharacteristic
248C22⋊C8—1971no—
249C22⋊C8C2×D19711yescharacteristic, complemented
250C4×C8—1971no—
251C24—3941no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D19711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—7881no—
258C2×D4—3941no—
259C2×D4—3942no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—1971no—
263C2×D4—1971no—
264C2×D4C22×D19711yescharacteristic
265C2×D4C22×D19711yescharacteristic
266C22×C4—21no—
267C22×C4—3941no—
268C22×C4—3941no—
269C22×C4—3941no—
270C22×C4—1971no—
271C22×C4C22×D19711yescharacteristic
272SD16—41no—
273D8—22no—
274D8—7881no—
275C2×C8—7881no—
276C2×C8—7881no—
277C2×C8—1971no—
278C2×C8—1971no—
279C2×C8C22×D19711yescharacteristic
280C2×C8C22×D19711yescharacteristic
281C4⋊C4—3941no—
282C4⋊C4—3941no—
283C4⋊C4—3941no—
284C4⋊C4—3941no—
285C4⋊C4—1971no—
286C4⋊C4—1971no—
287C4⋊C4—1971no—
288C4⋊C4—1971no—
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—21no—
304C23—21no—
305C23—7882no—
306C23—7882no—
307C23—15761no—
308C23—3941no—
309C23—1971no—
310C23D4×D19711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—7882no—
319D4—7882no—
320D4—15761no—
321D4—3942no—
322D4—3941no—
323D4—3941no—
324D4#1569135712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—15761no—
331C2×C4—15761no—
332C2×C4—15761no—
333C2×C4—3941no—
334C2×C4—3941no—
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—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—7881no—
357C22—7881no—
358C22—7881no—
359C22—15764no—
360C22—3941no—
361C22—3941no—
362C22—1972no—
363C22C2×D4×D19711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—7881no—
368C4—7881no—
369C4—7881no—
370C4—3941no—
371C4—3941no—
372C4—3941no—
373C4C2×#1569135711yescharacteristic
374C4#2026337111yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—15761no—
379C2—3942no—
380C2#2716171611yescharacteristic
381C2#135×D19711yescharacteristic
382C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#449×C197C2
socleC22×C197C22×D4
hypercenter#449D197
hypocenterC197#3297
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #27161776C2semidirect
2. #27161717C2semidirect
3. #27161435C2semidirect
4. #27161413C2semidirect
5. #27161404C2semidirect
6. #27161381C2semidirect
7. #27161313C2semidirect
8. #27161305C2semidirect
9. #27161296C2semidirect
10. #27161211C2semidirect
11. D8×Dic197C2semidirect
12. #27161142C2semidirect
13. #27161075C2semidirect
14. #102×D197C2semidirect
15. #449×C197C2semidirect
18. C22×#13096802C22semidirect
21. #20263317C22semidirect
32. D4×Dic197C22semidirect
35. #20263285C22semidirect
38. #20263260C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263255C22semidirect
42. #20263250C22semidirect
43. #20263250C22semidirect
48. #20263247C22semidirect
51. C8×Dic197C22semidirect
53. C2×SD16×C197C22semidirect
55. #121×C197C22semidirect
56. #103×C197C22semidirect
58. C22⋊C8×C197C22semidirect
191. C197#3297semidirect
202. #449D197semidirect
210. C2×SD16C2×D197semidirect
225. #121C2×D197semidirect
240. #103C2×D197semidirect
249. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617762
#271617172
#271614352
#271614132
#271614042
#271613812
#271613132
#271613052
#271612962
#271612112
D8×Dic1972
#271611422
#271610752
#102×D1972
#449×C1972
#3297197