Finite Group Databasebeta

← #38754519

Subgroups — #38754519

Order 25216.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D788C211yesmaximal, characteristic, complemented
2#27161427C211yesmaximal, characteristic, complemented
3#27161392C212yesmaximal, complemented
4#27161356C211yesmaximal, characteristic, complemented
5#27161313C212yesmaximal, complemented
6#27161274C211yesmaximal, characteristic, complemented
7#27161212C212yesmaximal, complemented
8C4×#15691343C211yesmaximal, characteristic, complemented
9D4×#13096802C211yesmaximal, characteristic, complemented
10C2×#20263246C212yesmaximal, complemented
11#547×C197C211yesmaximal, characteristic, complemented
12C22×D788C2212yes—
13C2×D4×D197—21no—
14C2×#15691343C2211yescharacteristic
15C2×#15691343C2212yescomplemented
16C22×#13096802C2212yescomplemented
17#20263323C2211yescharacteristic
18#20263301—21no—
19#20263300C2212yes—
20#20263296—22no—
21C4×D788C2211yescharacteristic
22#20263259C2212yescomplemented
23#20263259C2211yescharacteristic, complemented
24C4⋊#13096802C2211yescharacteristic, complemented
25#20263247C2211yescharacteristic, complemented
26#20263246C2212yes—
27#20263246C2211yescharacteristic
28#20263246C2212yescomplemented
29#20263246C2212yescomplemented
30#20263246C2212yescomplemented
31C22⋊#13096802C2212yescomplemented
32C4×#13096802C2211yescharacteristic, complemented
33C2×#41×C197—21no—
34C4⋊Q8×C197C2211yescharacteristic, complemented
35#123×C197—22no—
36C22⋊Q8×C197—21no—
37#121×C197C2212yescomplemented
38C4×D4×C197C2211yescharacteristic
39C4×D4×C197C2211yescharacteristic, complemented
40C2×C4⋊C4×C197C2212yescomplemented
41C23×D197—22no—
42C2×D788—24no—
43C2×D788—22no—
44C2×D788—21no—
45C2×D788—21no—
46C2×D788—42no—
47C2×D788C2312yes—
48C2×D788C2311yescharacteristic
49C2×C22⋊D197—22no—
50D4×D197—44no—
51C2×C4×D197—21no—
52#15691343—22no—
53#15691343—42no—
54C2×#13096802—42no—
55C2×#13096802C2311yescharacteristic
56C2×#13096802C2311yescharacteristic
57C2×#13096802C2312yes—
58C4⋊Dic197C2311yescharacteristic
59#15691335—21no—
60#15691335—22no—
61#41×C197—44no—
62C2×Q8×C197—21no—
63C2×D4×C197—22no—
64C2×D4×C197D411yescharacteristic
65C2×D4×C197C2311yescharacteristic
66C22×C4×C197—22no—
67C22×C4×C197—21no—
68C22×C4×C197C2312yes—
69C4⋊C4×C197—22no—
70C4⋊C4×C197—21no—
71C4⋊C4×C197D411yescharacteristic
72C4⋊C4×C197C2311yescharacteristic
73C4⋊C4×C197C2312yes—
74C22⋊C4×C197—21no—
75C22⋊C4×C197—22no—
76C22⋊C4×C197D412yes—
77C42×C197C2311yescharacteristic
78C22×D197—21no—
79C22×D197—22no—
80C22×D197—44no—
81C22×D197—44no—
82C22×D197—82no—
83D788—22no—
84D788—21no—
85D788—22no—
86D788—44no—
87D788—42no—
88D788—81no—
89D788#4112yescomplemented
90C22⋊D197—44no—
91C2×Dic197—21no—
92C4×D197—42no—
93#13096802—21no—
94#13096802—21no—
95#13096802—22no—
96C23×C197—21no—
97C23×C197C2×D412yes—
98Q8×C197—42no—
99D4×C197—21no—
100D4×C197—21no—
101D4×C197—44no—
102D4×C197D814yescomplemented
103C2×C4×C197—22no—
104C2×C4×C197—22no—
105C2×C4×C197—21no—
106C2×C4×C197—22no—
107C2×C4×C197—44no—
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—41no—
116C22×C197—41no—
117C22×C197—41no—
118C22×C197C2×D814yes—
119C22×C197C22×D411yescharacteristic
120C2×D197—22no—
121C2×D197—41no—
122C2×D197—41no—
123C2×D197—42no—
124C2×D197—42no—
125C2×D197—42no—
126C2×D197—88no—
127C4×C197—22no—
128C4×C197—21no—
129C4×C197—41no—
130C4×C197—42no—
131C4×C197C2×D812yes—
132C4×C197C2×#4111yescharacteristic
133C4×C197H(5,𝔽2)11yescharacteristic
134Dic197—41no—
135C2×C197—24no—
136C2×C197—81no—
137C2×C197#54611yescharacteristic
138C2×C197C22×D811yescharacteristic
139C2×C197#57711yescharacteristic
140D197—42no—
141D197—82no—
142C197#329411yescharacteristic, complemented, Sylow
143#3294—1971nomaximal, Sylow
144#547D19711yescharacteristic, complemented
145D42—1971no—
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—21no—
156C22×D4—3941no—
157C22×D4—1972no—
158C2×D8—1972no—
159C2×D8—1971no—
160C22×C8—1972no—
161C4⋊Q8C2×D19711yescharacteristic, complemented
162#127—1971no—
163#123—22no—
164C22⋊Q8—21no—
165#121—3941no—
166#121—1972no—
167#121C2×D19712yescomplemented
168#120—3942no—
169C4×D4—1971no—
170C4×D4C2×D19711yescharacteristic
171C4×D4C2×D19711yescharacteristic, complemented
172C2×C4⋊C4C2×D19712yescomplemented
173#107—1971no—
174#107—1972no—
175C4⋊C8—1971no—
176#102—1971no—
177#102—1971no—
178#102—1972no—
179#102—1972no—
180#102—1972no—
181#102—1972no—
182C22⋊C8—1972no—
183C4×C8—1971no—
184C24—3942no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—7882no—
189C2×D4—7884no—
190C2×D4—3944no—
191C2×D4—3942no—
192C2×D4—3942no—
193C2×D4—3941no—
194C2×D4—3941no—
195C2×D4—1972no—
196C2×D4—1971no—
197C2×D4C22⋊D19711yescharacteristic
198C2×D4C22×D19711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—3941no—
202C22×C4C22×D19712yes—
203D8—7882no—
204D8—3942no—
205C2×C8—7882no—
206C2×C8—1971no—
207C2×C8—1971no—
208C2×C8—1972no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1971no—
212C4⋊C4C22⋊D19711yescharacteristic
213C4⋊C4C22×D19711yescharacteristic
214C4⋊C4C22×D19712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—3941no—
218C22⋊C4—3942no—
219C22⋊C4C22⋊D19712yes—
220C42C22×D19711yescharacteristic
221C23—21no—
222C23—7884no—
223C23—7884no—
224C23—15762no—
225C23—3942no—
226C23—3941no—
227C23C2×C22⋊D19712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—7884no—
233D4—7884no—
234D4—7882no—
235D4—15761no—
236D4—3942no—
237D4—3942no—
238D4—3941no—
239D4—1972no—
240D4#1569134314yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—7882no—
248C2×C4—3941no—
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—41no—
259C22—41no—
260C22—41no—
261C22—7881no—
262C22—7881no—
263C22—7882no—
264C22—7882no—
265C22—7882no—
266C22—15768no—
267C22—3942no—
268C22C2×#1569134314yes—
269C22C22×C22⋊D19711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—7881no—
275C4C2×#1569134312yes—
276C4#41×D19711yescharacteristic
277C4#2026337211yescharacteristic
278C2—24no—
279C2—81no—
280C2—7882no—
281C2—15762no—
282C2#2716171711yescharacteristic
283C2C22×#1569134311yescharacteristic
284C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#547×C197C2
socleC22×C197C22×D4
hypercenter#547D197
hypocenterC197#3294
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D788C2semidirect
2. #27161427C2semidirect
3. #27161392C2semidirect
4. #27161356C2semidirect
5. #27161313C2semidirect
6. #27161274C2semidirect
7. #27161212C2semidirect
8. C4×#15691343C2semidirect
9. D4×#13096802C2semidirect
10. C2×#20263246C2semidirect
11. #547×C197C2semidirect
15. C2×#15691343C22semidirect
16. C22×#13096802C22semidirect
22. #20263259C22semidirect
23. #20263259C22semidirect
24. C4⋊#13096802C22semidirect
25. #20263247C22semidirect
28. #20263246C22semidirect
29. #20263246C22semidirect
30. #20263246C22semidirect
31. C22⋊#13096802C22semidirect
32. C4×#13096802C22semidirect
34. C4⋊Q8×C197C22semidirect
37. #121×C197C22semidirect
39. C4×D4×C197C22semidirect
40. C2×C4⋊C4×C197C22semidirect
89. D788#41semidirect
102. D4×C197D8semidirect
142. C197#3294semidirect
144. #547D197semidirect
161. C4⋊Q8C2×D197semidirect
167. #121C2×D197semidirect
171. C4×D4C2×D197semidirect
172. C2×C4⋊C4C2×D197semidirect
240. D4#15691343semidirect

Maximal subgroups

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

subgroupindex
D4×D7882
#271614272
#271613922
#271613562
#271613132
#271612742
#271612122
C4×#156913432
D4×#130968022
C2×#202632462
#547×C1972
#3294197