Finite Group Databasebeta

← #38752949

Subgroups — #38752949

Order 25216.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161682C211yesmaximal, characteristic, complemented
2#27161554C211yesmaximal, characteristic, complemented
3#27161404C211yesmaximal, characteristic, complemented
4#27161400C211yesmaximal, characteristic, complemented
5#27161381C211yesmaximal, characteristic, complemented
6#27161377C211yesmaximal, characteristic, complemented
7#27161272C211yesmaximal, characteristic, complemented
8#27161272C211yesmaximal, characteristic, complemented
9#27161213C211yesmaximal, characteristic, complemented
10#27161191C211yesmaximal, characteristic, complemented
11#27161188C211yesmaximal, characteristic, complemented
12D8×Dic197C211yesmaximal, characteristic, complemented
13#27161159C211yesmaximal, characteristic, complemented
14#27161137C211yesmaximal, characteristic, complemented
15#447×C197C211yesmaximal, characteristic, complemented
16C2×#15691343C2211yescharacteristic
17C2×#15691343C2211yescharacteristic
18#20263322C2211yescharacteristic, complemented
19#20263317C2211yescharacteristic, complemented
20#20263310—21no—
21#20263308—21no—
22#20263300C2211yescharacteristic
23#20263298—21no—
24#20263297—21no—
25D4×Dic197—21no—
26D4×Dic197—21no—
27D4×Dic197C2211yescharacteristic
28D4×Dic197C2211yescharacteristic
29D4×Dic197C2211yescharacteristic
30D4×Dic197C2211yescharacteristic
31D4×Dic197C2211yescharacteristic, complemented
32#20263288C2211yescharacteristic, complemented
33#20263285C2211yescharacteristic
34#20263285C2211yescharacteristic
35C2×C22⋊Dic197—21no—
36C2×C22⋊Dic197—21no—
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic, complemented
48#20263247C2211yescharacteristic, complemented
49C22⋊#13096802C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C2×D8×C197C2211yescharacteristic
54C2×D8×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic
59C2×D788C2311yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197—21no—
62C2×C22⋊D197—21no—
63C22×Dic197—21no—
64C22×Dic197—21no—
65C22×Dic197—21no—
66C22×Dic197—21no—
67C22×Dic197—21no—
68C2×C4×D197—21no—
69#15691343—22no—
70#15691343—22no—
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C4⋊Dic197—21no—
74C4⋊Dic197D411yescharacteristic
75C4⋊Dic197D411yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79#15691337—21no—
80C22⋊Dic197—21no—
81C22⋊Dic197—21no—
82C22⋊Dic197—21no—
83C22⋊Dic197—21no—
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197D411yescharacteristic
87C22⋊Dic197D411yescharacteristic
88#15691335—21no—
89#15691335—21no—
90C4×Dic197—21no—
91C4×Dic197C2311yescharacteristic
92C4×Dic197C2311yescharacteristic
93C24×C197—21no—
94C2×D4×C197—22no—
95C2×D4×C197—22no—
96C2×D4×C197—21no—
97C2×D4×C197—21no—
98C2×D4×C197C2311yescharacteristic
99C2×D4×C197C2311yescharacteristic
100C2×D4×C197C2311yescharacteristic
101C22×C4×C197C2311yescharacteristic
102D8×C197—22no—
103D8×C197—22no—
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C4⋊C4×C197C2311yescharacteristic
107C22⋊C4×C197—21no—
108C22×D197—21no—
109D788—21no—
110D788—21no—
111C22⋊D197—81no—
112C22⋊D197—81no—
113C22⋊D197—81no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—81no—
118C2×Dic197—81no—
119C2×Dic197—81no—
120C2×Dic197—81no—
121C2×Dic197—81no—
122C2×Dic197C2×D411yescharacteristic
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C4×D197—81no—
127#13096802—21no—
128#13096802—21no—
129C23×C197—24no—
130C23×C197—21no—
131C23×C197—21no—
132C23×C197—21no—
133C23×C197—81no—
134C23×C197—81no—
135C23×C197C2×D411yescharacteristic
136D4×C197—24no—
137D4×C197—21no—
138D4×C197—21no—
139D4×C197—21no—
140D4×C197—21no—
141D4×C197—81no—
142D4×C197#4112yes—
143D4×C197#4112yes—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2411yescharacteristic
149C8×C197—21no—
150C8×C197—21no—
151C22×C197—21no—
152C22×C197—22no—
153C22×C197—22no—
154C22×C197—21no—
155C22×C197—21no—
156C22×C197—41no—
157C22×C197—41no—
158C22×C197—41no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—84no—
164C22×C197C22×D411yescharacteristic
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C4×C197—21no—
169C4×C197—41no—
170C4×C197C2×#4111yescharacteristic
171C4×C197C2×#4111yescharacteristic
172Dic197—21no—
173Dic197—21no—
174Dic197—21no—
175Dic197—21no—
176Dic197—41no—
177Dic197—41no—
178Dic197—41no—
179C2×C197—41no—
180C2×C197—42no—
181C2×C197—42no—
182C2×C197—81no—
183C2×C197#52511yescharacteristic
184C2×C197#57611yescharacteristic
185C2×C197#57611yescharacteristic
186D197—81no—
187C197#312211yescharacteristic, complemented, Sylow
188#3122—1971nomaximal, Sylow
189#540—1971no—
190#518—1971no—
191#482—1971no—
192#482—1971no—
193#478—1971no—
194#478—1971no—
195#459—1971no—
196#459—1971no—
197#447—1971no—
198#447D19711yescharacteristic, complemented
199#442—1971no—
200#442—1971no—
201C4×D8—1971no—
202C4×D8—1971no—
203#433—1971no—
204C22×D4C2×D19711yescharacteristic
205C2×D8—1971no—
206C2×D8—1971no—
207C2×D8C2×D19711yescharacteristic
208C2×D8C2×D19711yescharacteristic
209#127—1971no—
210#125—1971no—
211#123—3941no—
212#123—3941no—
213#121—3941no—
214#121—3941no—
215#121—1971no—
216#121C2×D19711yescharacteristic
217C4×D4—3941no—
218C4×D4—3941no—
219C4×D4—1971no—
220C4×D4—1971no—
221C4×D4—1971no—
222C4×D4—1971no—
223C4×D4—1971no—
224C4×D4—1971no—
225#117—1971no—
226#117—1971no—
227C2×C22⋊C4—3941no—
228C2×C22⋊C4—3941no—
229#107—1971no—
230#107—1971no—
231#106—1971no—
232#106—1971no—
233C4⋊C8—1971no—
234C4⋊C8—1971no—
235#102—1971no—
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241#102C2×D19711yescharacteristic
242#102C2×D19711yescharacteristic
243C22⋊C8—1971no—
244C22⋊C8C2×D19711yescharacteristic
245#97—1971no—
246C4×C8—1971no—
247C24—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3941no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4C22×D19711yescharacteristic
257C2×D4C22×D19711yescharacteristic
258C2×D4C22×D19711yescharacteristic
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4—3941no—
262C22×C4—3941no—
263C22×C4—3941no—
264C22×C4—3941no—
265C22×C4C22×D19711yescharacteristic
266D8—22no—
267D8—22no—
268D8—3942no—
269D8—3942no—
270C2×C8—1971no—
271C2×C8—1971no—
272C2×C8C22×D19711yescharacteristic
273C2×C8C22×D19711yescharacteristic
274C4⋊C4—3941no—
275C4⋊C4—3941no—
276C4⋊C4—1971no—
277C4⋊C4—1971no—
278C4⋊C4—1971no—
279C4⋊C4—1971no—
280C4⋊C4—1971no—
281C4⋊C4C22×D19711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3941no—
284C22⋊C4—3941no—
285C22⋊C4—3941no—
286C22⋊C4—3941no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—1971no—
292C22⋊C4—1971no—
293C42—3941no—
294C42—1971no—
295C42—1971no—
296C23—24no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—81no—
301C23—81no—
302C23—3941no—
303C23D4×D19711yescharacteristic
304D4—24no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—15761no—
311D4—15761no—
312D4—15761no—
313D4—3941no—
314D4—3941no—
315D4#1569135712yes—
316D4#1569135712yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—15761no—
321C2×C4—15761no—
322C2×C4—15761no—
323C2×C4—15761no—
324C2×C4—15761no—
325C2×C4—15761no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4—1971no—
330C2×C4—1971no—
331C2×C4—1971no—
332C2×C4—1971no—
333C2×C4D4×D19711yescharacteristic
334C2×C4C23×D19711yescharacteristic
335C8—21no—
336C8—21no—
337C8—3941no—
338C8—3941no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—84no—
352C22—7881no—
353C22—7881no—
354C22—7881no—
355C22C2×D4×D19711yescharacteristic
356C4—21no—
357C4—41no—
358C4—7881no—
359C4—7881no—
360C4—7881no—
361C4—3941no—
362C4—3941no—
363C4—3941no—
364C4—3941no—
365C4C2×#1569135711yescharacteristic
366C4C2×#1569135711yescharacteristic
367C2—41no—
368C2—42no—
369C2—42no—
370C2—81no—
371C2—15761no—
372C2#2716158511yescharacteristic
373C2#2716188011yescharacteristic
374C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#447×C197C2
socleC22×C197C22×D4
hypercenter#447D197
hypocenterC197#3122
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #27161682C2semidirect
2. #27161554C2semidirect
3. #27161404C2semidirect
4. #27161400C2semidirect
5. #27161381C2semidirect
6. #27161377C2semidirect
7. #27161272C2semidirect
8. #27161272C2semidirect
9. #27161213C2semidirect
10. #27161191C2semidirect
11. #27161188C2semidirect
12. D8×Dic197C2semidirect
13. #27161159C2semidirect
14. #27161137C2semidirect
15. #447×C197C2semidirect
18. #20263322C22semidirect
19. #20263317C22semidirect
31. D4×Dic197C22semidirect
32. #20263288C22semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263258C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
43. #20263248C22semidirect
44. #20263248C22semidirect
47. #20263247C22semidirect
48. #20263247C22semidirect
50. #20263236C22semidirect
51. C8×Dic197C22semidirect
187. C197#3122semidirect
198. #447D197semidirect

Maximal subgroups

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

subgroupindex
#271616822
#271615542
#271614042
#271614002
#271613812
#271613772
#271612722
#271612132
#271611912
#271611882
D8×Dic1972
#271611592
#271611372
#447×C1972
#3122197