Finite Group Databasebeta

← #38752952

Subgroups — #38752952

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#27161680C211yesmaximal, characteristic, complemented
2#27161551C211yesmaximal, characteristic, complemented
3#27161406C211yesmaximal, characteristic, complemented
4#27161401C211yesmaximal, characteristic, complemented
5#27161382C211yesmaximal, characteristic, complemented
6#27161377C211yesmaximal, characteristic, complemented
7#27161273C211yesmaximal, characteristic, complemented
8#27161272C211yesmaximal, characteristic, complemented
9#27161214C211yesmaximal, characteristic, complemented
10#27161211C211yesmaximal, characteristic, complemented
11#27161189C211yesmaximal, characteristic, complemented
12#27161189C211yesmaximal, characteristic, complemented
13C4×D1576C211yesmaximal, characteristic, complemented
14#27161138C211yesmaximal, characteristic, complemented
15C4×D8×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C2×D1576C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20#20263322C2211yescharacteristic, complemented
21#20263317C2211yescharacteristic, complemented
22#20263309—21no—
23#20263308—21no—
24#20263301C2211yescharacteristic
25#20263300—21no—
26#20263299—21no—
27#20263297C2211yescharacteristic
28C4×D788C2211yescharacteristic
29C4×D788C2211yescharacteristic, complemented
30D4×Dic197C2211yescharacteristic
31#20263288C2211yescharacteristic
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263284C2211yescharacteristic
35#20263283C2211yescharacteristic
36C2×#15691335—21no—
37C22⋊C4×D197—21no—
38#20263260C2211yescharacteristic, complemented
39#20263257C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic, complemented
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic, complemented
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263237C2211yescharacteristic, complemented
52C2×D8×C197C2211yescharacteristic
53C4×D4×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic, complemented
55#107×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic, complemented
58C4×C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×D788—21no—
61C2×D788C2311yescharacteristic
62C2×D788C2311yescharacteristic
63C2×C22⋊D197—21no—
64C2×C22⋊D197—21no—
65C2×C22⋊D197—21no—
66C2×C22⋊D197—21no—
67D4×D197—22no—
68D4×D197—22no—
69C22×Dic197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197C2311yescharacteristic
74D1576—22no—
75#15691343—22no—
76#15691343—22no—
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C4⋊Dic197—21no—
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82#15691337—21no—
83#15691337D411yescharacteristic
84#15691337D411yescharacteristic
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335D411yescharacteristic
93#15691335D411yescharacteristic
94C4×Dic197—21no—
95C4×Dic197C2311yescharacteristic
96C2×D4×C197C2311yescharacteristic
97C2×D4×C197C2311yescharacteristic
98C22×C4×C197—21no—
99C22×C4×C197—21no—
100D8×C197—22no—
101C2×C8×C197C2311yescharacteristic
102C2×C8×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—24no—
110C22×D197—21no—
111C22×D197—81no—
112C22×D197—81no—
113C22×D197C2×D411yescharacteristic
114D788—21no—
115D788—21no—
116D788—21no—
117D788—81no—
118D788#4112yes—
119C22⋊D197—24no—
120C22⋊D197—81no—
121C22⋊D197—81no—
122C22⋊D197—81no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—81no—
126C2×Dic197C2×D411yescharacteristic
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C4×D197—21no—
130C4×D197—21no—
131C4×D197—81no—
132C4×D197—81no—
133C4×D197—81no—
134#13096802—21no—
135#13096802—21no—
136C23×C197—21no—
137C23×C197—21no—
138D4×C197—21no—
139D4×C197—21no—
140D4×C197—21no—
141D4×C197#4112yes—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—81no—
145C2×C4×C197—81no—
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2411yescharacteristic
149C8×C197—21no—
150C8×C197—21no—
151C22×C197—22no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197—41no—
156C22×C197—41no—
157C22×C197C22×D411yescharacteristic
158C2×D197—21no—
159C2×D197—22no—
160C2×D197—21no—
161C2×D197—21no—
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—84no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197—41no—
171C4×C197—41no—
172C4×C197C2×#4111yescharacteristic
173C4×C197C2×#4111yescharacteristic
174Dic197—21no—
175Dic197—21no—
176Dic197—21no—
177Dic197—41no—
178Dic197—41no—
179C2×C197—42no—
180C2×C197—81no—
181C2×C197#52511yescharacteristic
182C2×C197#57611yescharacteristic
183C2×C197#57611yescharacteristic
184D197—41no—
185D197—42no—
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#447—1971no—
199#442—1971no—
200#442—1971no—
201C4×D8—1971no—
202C4×D8D19711yescharacteristic, complemented
203#433—1971no—
204C22×D4—1971no—
205C2×D8—1971no—
206C2×D8—1971no—
207C2×D8—1971no—
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#121—1971no—
217C4×D4—3941no—
218C4×D4—3941no—
219C4×D4—1971no—
220C4×D4—1971no—
221C4×D4—1971no—
222C4×D4—1971no—
223C4×D4C2×D19711yescharacteristic
224C4×D4C2×D19711yescharacteristic, complemented
225#117—1971no—
226#117—1971no—
227C2×C22⋊C4—3941no—
228C2×C22⋊C4—3941no—
229#107—1971no—
230#107C2×D19711yescharacteristic, complemented
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, complemented
243C22⋊C8—1971no—
244C22⋊C8—1971no—
245#97—1971no—
246C4×C8C2×D19711yescharacteristic, complemented
247C24—3941no—
248C2×D4—3941no—
249C2×D4—3942no—
250C2×D4—3941no—
251C2×D4—3942no—
252C2×D4—3941no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4—1971no—
257C2×D4C22×D19711yescharacteristic
258C2×D4C22×D19711yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—3941no—
262C22×C4—3941no—
263C22×C4—3941no—
264C22×C4—3941no—
265C22×C4—1971no—
266D8—22no—
267D8—3942no—
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⋊C4C22×D19711yescharacteristic
281C4⋊C4C22×D19711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
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—
295C42C22×D19711yescharacteristic
296C23—21no—
297C23—21no—
298C23—15761no—
299C23—15761no—
300C23—3944no—
301C23—3941no—
302C23—3941no—
303C23—1971no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—15761no—
308D4—15761no—
309D4—15761no—
310D4—15761no—
311D4—3944no—
312D4—3941no—
313D4—3941no—
314D4—3941no—
315D4—1972no—
316D4#1569135812yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—15761no—
322C2×C4—15761no—
323C2×C4—15761no—
324C2×C4—15761no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4—1971no—
330C2×C4—1971no—
331C2×C4—1971no—
332C2×C4D4×D19711yescharacteristic
333C2×C4D4×D19711yescharacteristic
334C2×C4C23×D19711yescharacteristic
335C8—21no—
336C8—21no—
337C8—3941no—
338C8—3941no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—7881no—
346C22—7881no—
347C22—7881no—
348C22—7881no—
349C22—7881no—
350C22—15764no—
351C22—3941no—
352C22—3941no—
353C22—3941no—
354C22—3942no—
355C22C2×D4×D19711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—7881no—
361C4—7881no—
362C4—3941no—
363C4—3941no—
364C4—3941no—
365C4#41×D19711yescharacteristic
366C4C2×#1569135811yescharacteristic
367C2—42no—
368C2—81no—
369C2—7882no—
370C2—7881no—
371C2—15761no—
372C2#2716158811yescharacteristic
373C2#2716188011yescharacteristic
374C2#2716188111yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×D8×C197C2
socleC22×C197C22×D4
hypercenterC4×D8D197
hypocenterC197#3122
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #27161680C2semidirect
2. #27161551C2semidirect
3. #27161406C2semidirect
4. #27161401C2semidirect
5. #27161382C2semidirect
6. #27161377C2semidirect
7. #27161273C2semidirect
8. #27161272C2semidirect
9. #27161214C2semidirect
10. #27161211C2semidirect
11. #27161189C2semidirect
12. #27161189C2semidirect
13. C4×D1576C2semidirect
14. #27161138C2semidirect
15. C4×D8×C197C2semidirect
20. #20263322C22semidirect
21. #20263317C22semidirect
29. C4×D788C22semidirect
38. #20263260C22semidirect
39. #20263257C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
44. #20263248C22semidirect
46. #20263247C22semidirect
48. #20263246C22semidirect
51. #20263237C22semidirect
54. C4×D4×C197C22semidirect
55. #107×C197C22semidirect
57. #102×C197C22semidirect
58. C4×C8×C197C22semidirect
187. C197#3122semidirect
202. C4×D8D197semidirect
224. C4×D4C2×D197semidirect
230. #107C2×D197semidirect
242. #102C2×D197semidirect
246. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616802
#271615512
#271614062
#271614012
#271613822
#271613772
#271612732
#271612722
#271612142
#271612112
#271611892
C4×D15762
#271611382
C4×D8×C1972
#3122197