Finite Group Databasebeta

← #38754525

Subgroups — #38754525

Order 25216.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161733C211yesmaximal, characteristic, complemented
2#27161425C211yesmaximal, characteristic, complemented
3#27161390C212yesmaximal, complemented
4#27161359C211yesmaximal, characteristic, complemented
5#27161313C212yesmaximal, complemented
6#27161271C211yesmaximal, characteristic, complemented
7#27161213C212yesmaximal, complemented
8C4×#15691343C211yesmaximal, characteristic, complemented
9D4×#13096802C211yesmaximal, characteristic, complemented
10C2×#20263247C212yesmaximal, complemented
11D42×C197C211yesmaximal, characteristic, complemented
12C2×#15691357—21no—
13C2×#15691343C2211yescharacteristic
14C2×#15691343C2212yescomplemented
15C22×#13096802C2212yescomplemented
16#20263326C2211yescharacteristic, complemented
17#20263310—22no—
18#20263306—21no—
19#20263300C2212yescomplemented
20C4×D788C2211yescharacteristic, complemented
21C2×C4⋊Dic197C2212yescomplemented
22#20263259C2212yescomplemented
23#20263259C2211yescharacteristic, complemented
24C4⋊#13096802C2211yescharacteristic, complemented
25#20263247C2212yes—
26#20263247C2211yescharacteristic
27#20263247C2212yescomplemented
28#20263247C2212yescomplemented
29#20263247C2212yescomplemented
30#20263246C2211yescharacteristic, complemented
31C22⋊#13096802C2212yescomplemented
32C4×#13096802C2211yescharacteristic, complemented
33C22×D4×C197—21no—
34C22×D4×C197C2212yes—
35#127×C197C2211yescharacteristic
36#121×C197—21no—
37#121×C197C2212yes—
38#120×C197—22no—
39C4×D4×C197C2211yescharacteristic
40C4×D4×C197C2211yescharacteristic
41#15691357—44no—
42C2×Dic394—21no—
43C2×D788C2311yescharacteristic
44C2×C22⋊D197—22no—
45C22×Dic197—22no—
46C2×C4×D197—21no—
47#15691343—22no—
48#15691343—42no—
49C2×#13096802—42no—
50C2×#13096802C2311yescharacteristic
51C2×#13096802C2311yescharacteristic
52C2×#13096802C2312yes—
53C4⋊Dic197—22no—
54C4⋊Dic197—21no—
55C4⋊Dic197C2311yescharacteristic
56C4⋊Dic197C2312yes—
57#15691335—21no—
58#15691335—22no—
59C24×C197—22no—
60C2×D4×C197—24no—
61C2×D4×C197—22no—
62C2×D4×C197—22no—
63C2×D4×C197—21no—
64C2×D4×C197—21no—
65C2×D4×C197—42no—
66C2×D4×C197—44no—
67C2×D4×C197D411yescharacteristic
68C2×D4×C197C2312yes—
69C2×D4×C197C2311yescharacteristic
70C22×C4×C197—21no—
71C22×C4×C197C2312yes—
72C4⋊C4×C197D411yescharacteristic
73C4⋊C4×C197C2311yescharacteristic
74C22⋊C4×C197—21no—
75C22⋊C4×C197—22no—
76C22⋊C4×C197D412yes—
77C42×C197C2311yescharacteristic
78C22×D197—21no—
79Dic394—42no—
80D788—21no—
81D788—21no—
82C22⋊D197—44no—
83C2×Dic197—21no—
84C2×Dic197—22no—
85C2×Dic197—44no—
86C4×D197—42no—
87#13096802—21no—
88#13096802—21no—
89#13096802—22no—
90C23×C197—21no—
91C23×C197—22no—
92C23×C197—44no—
93C23×C197—44no—
94C23×C197—82no—
95C23×C197C2×D412yes—
96D4×C197—22no—
97D4×C197—21no—
98D4×C197—22no—
99D4×C197—44no—
100D4×C197—44no—
101D4×C197—42no—
102D4×C197—81no—
103D4×C197D814yescomplemented
104D4×C197#4112yescomplemented
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197—21no—
108C2×C4×C197—42no—
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2×D412yes—
111C2×C4×C197C2×D411yescharacteristic
112C2×C4×C197C2411yescharacteristic
113C22×C197—22no—
114C22×C197—22no—
115C22×C197—22no—
116C22×C197—41no—
117C22×C197—41no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197—42no—
121C22×C197—88no—
122C22×C197C2×D814yes—
123C22×C197C22×D411yescharacteristic
124C2×D197—41no—
125C2×D197—41no—
126C2×D197—41no—
127C4×C197—22no—
128C4×C197—21no—
129C4×C197—41no—
130C4×C197C2×D812yes—
131C4×C197C2×#4111yescharacteristic
132C4×C197H(5,𝔽2)11yescharacteristic
133Dic197—41no—
134Dic197—42no—
135C2×C197—24no—
136C2×C197—42no—
137C2×C197—82no—
138C2×C197#54611yescharacteristic
139C2×C197C22×D811yescharacteristic
140C2×C197#57711yescharacteristic
141D197—81no—
142C197#329411yescharacteristic, complemented, Sylow
143#3294—1971nomaximal, Sylow
144#547—1971no—
145D42D19711yescharacteristic, complemented
146#486—1971no—
147#480—1972no—
148#474—1971no—
149#466—1972no—
150#459—1971no—
151#447—1972no—
152C4×D8—1971no—
153C8×D4—1971no—
154C2×#102—1972no—
155C2×#41—3941no—
156C22×D4—21no—
157C22×D4C2×D19712yes—
158C2×D8—1972no—
159C2×D8—1971no—
160C22×C8—1972no—
161C4⋊Q8—1971no—
162#127C2×D19711yescharacteristic
163#123—3942no—
164C22⋊Q8—3941no—
165#121—21no—
166#121—1972no—
167#121C2×D19712yes—
168#120—22no—
169C4×D4—1971no—
170C4×D4C2×D19711yescharacteristic
171C4×D4C2×D19711yescharacteristic
172C2×C4⋊C4—1972no—
173#107—1971no—
174#107—1972no—
175C4⋊C8—1971no—
176#102—1971no—
177#102—1972no—
178#102—1972no—
179#102—1972no—
180#102—1972no—
181#102—1971no—
182C22⋊C8—1972no—
183C4×C8—1971no—
184C24—22no—
185#41—7884no—
186C2×Q8—3941no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—3942no—
195C2×D4—1971no—
196C2×D4C22⋊D19711yescharacteristic
197C2×D4C22×D19712yes—
198C2×D4C22×D19711yescharacteristic
199C22×C4—21no—
200C22×C4—3942no—
201C22×C4—3941no—
202C22×C4C22×D19712yes—
203D8—7882no—
204D8—3942no—
205C2×C8—7882no—
206C2×C8—1971no—
207C2×C8—1971no—
208C2×C8—1972no—
209C4⋊C4—3942no—
210C4⋊C4—3941no—
211C4⋊C4—1971no—
212C4⋊C4—1972no—
213C4⋊C4C22⋊D19711yescharacteristic
214C4⋊C4C22×D19711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—3941no—
218C22⋊C4—3942no—
219C22⋊C4C22⋊D19712yes—
220C42C22×D19711yescharacteristic
221C23—21no—
222C23—22no—
223C23—44no—
224C23—44no—
225C23—82no—
226C23—3941no—
227C23C2×C22⋊D19712yes—
228Q8—7882no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—7884no—
237D4—3941no—
238D4—3941no—
239D4#1569134314yescomplemented
240D4#1569135712yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—7884no—
246C2×C4—7882no—
247C2×C4—3941no—
248C2×C4—3942no—
249C2×C4C2×C22⋊D19711yescharacteristic
250C2×C4C2×C22⋊D19712yes—
251C2×C4C2×C22⋊D19711yescharacteristic
252C2×C4C23×D19711yescharacteristic
253C8—3941no—
254C8—3941no—
255C8—3942no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—88no—
265C22—7881no—
266C22—7881no—
267C22—7881no—
268C22C2×#1569134314yes—
269C22C22×C22⋊D19711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—7881no—
274C4—7882no—
275C4C2×#1569134312yes—
276C4C2×#1569135711yescharacteristic
277C4#2026337111yescharacteristic
278C2—24no—
279C2—42no—
280C2—82no—
281C2—15761no—
282C2#2716172311yescharacteristic
283C2C22×#1569134311yescharacteristic
284C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD42×C197C2
socleC22×C197C22×D4
hypercenterD42D197
hypocenterC197#3294
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #27161733C2semidirect
2. #27161425C2semidirect
3. #27161390C2semidirect
4. #27161359C2semidirect
5. #27161313C2semidirect
6. #27161271C2semidirect
7. #27161213C2semidirect
8. C4×#15691343C2semidirect
9. D4×#13096802C2semidirect
10. C2×#20263247C2semidirect
11. D42×C197C2semidirect
14. C2×#15691343C22semidirect
15. C22×#13096802C22semidirect
16. #20263326C22semidirect
19. #20263300C22semidirect
20. C4×D788C22semidirect
21. C2×C4⋊Dic197C22semidirect
22. #20263259C22semidirect
23. #20263259C22semidirect
24. C4⋊#13096802C22semidirect
27. #20263247C22semidirect
28. #20263247C22semidirect
29. #20263247C22semidirect
30. #20263246C22semidirect
31. C22⋊#13096802C22semidirect
32. C4×#13096802C22semidirect
103. D4×C197D8semidirect
104. D4×C197#41semidirect
142. C197#3294semidirect
145. D42D197semidirect
239. D4#15691343semidirect
240. D4#15691357semidirect

Maximal subgroups

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

subgroupindex
#271617332
#271614252
#271613902
#271613592
#271613132
#271612712
#271612132
C4×#156913432
D4×#130968022
C2×#202632472
D42×C1972
#3294197