Finite Group Databasebeta

← #38754528

Subgroups — #38754528

Order 25216.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#27161734C211yesmaximal, characteristic, complemented
2#27161714C211yesmaximal, characteristic, complemented
3#27161425C211yesmaximal, characteristic, complemented
4#27161391C212yesmaximal, complemented
5#27161360C211yesmaximal, characteristic, complemented
6#27161312C212yesmaximal, complemented
7#27161211C212yesmaximal, complemented
8C4×#15691343C211yesmaximal, characteristic, complemented
9#27161144C211yesmaximal, characteristic, complemented
10#102×D197C212yesmaximal, complemented
11#459×C197C211yesmaximal, characteristic, complemented
12C2×#15691358—21no—
13C2×D4×D197—21no—
14C2×D4×D197C2212yes—
15C2×#15691343C2211yescharacteristic
16C2×C8×D197C2212yescomplemented
17#20263326C2211yescharacteristic, complemented
18#20263308—22no—
19#20263305—21no—
20#20263297—21no—
21#20263297C2212yes—
22#20263297C2212yescomplemented
23#20263294—22no—
24C4×D788C2211yescharacteristic
25C4×D788C2211yescharacteristic
26C4⋊C4×D197C2212yescomplemented
27#20263260C2212yescomplemented
28#20263259C2211yescharacteristic, complemented
29#20263248C2212yes—
30#20263247C2212yescomplemented
31#20263247C2211yescharacteristic, complemented
32#20263247C2212yescomplemented
33#20263246C2211yescharacteristic
34#20263238C2212yescomplemented
35C4×#13096802C2211yescharacteristic, complemented
36C2×D8×C197C2212yescomplemented
37#127×C197C2211yescharacteristic
38C4×D4×C197C2211yescharacteristic, complemented
39C4⋊C8×C197C2211yescharacteristic, complemented
40#102×C197C2212yescomplemented
41C23×D197—22no—
42#15691358—44no—
43C2×Dic394—21no—
44C2×D788D411yescharacteristic
45C2×D788C2311yescharacteristic
46C2×C22⋊D197—22no—
47C2×C22⋊D197—22no—
48C2×C22⋊D197—22no—
49D4×D197—24no—
50D4×D197—42no—
51D4×D197—44no—
52C2×C4×D197—22no—
53C2×C4×D197—21no—
54C2×C4×D197C2312yes—
55#15691343—22no—
56C2×#13096802C2311yescharacteristic
57C2×#13096802C2311yescharacteristic
58C8×D197—42no—
59C4⋊Dic197—21no—
60C4⋊Dic197D411yescharacteristic
61C4⋊Dic197C2312yes—
62C4⋊Dic197C2311yescharacteristic
63#15691337—22no—
64C22⋊Dic197—22no—
65C22⋊Dic197—22no—
66#15691335—21no—
67#15691335D412yes—
68C2×D4×C197—21no—
69C2×D4×C197—21no—
70C2×D4×C197C2312yes—
71C2×D4×C197C2311yescharacteristic
72C22×C4×C197—21no—
73D8×C197—42no—
74C2×C8×C197C2312yes—
75C4⋊C4×C197C2311yescharacteristic
76C22⋊C4×C197—21no—
77C42×C197C2311yescharacteristic
78C22×D197—21no—
79C22×D197—44no—
80C22×D197—44no—
81C22×D197—82no—
82C22×D197C2×D412yes—
83Dic394—42no—
84D788—21no—
85D788D814yescomplemented
86D788#4112yescomplemented
87C22⋊D197—44no—
88C22⋊D197—44no—
89C22⋊D197—44no—
90C2×Dic197—21no—
91C2×Dic197—22no—
92C2×Dic197C2×D412yes—
93C4×D197—22no—
94C4×D197—22no—
95C4×D197—44no—
96C4×D197—42no—
97#13096802—21no—
98#13096802—21no—
99C23×C197—21no—
100C23×C197—22no—
101D4×C197—21no—
102D4×C197—22no—
103D4×C197—21no—
104D4×C197—22no—
105D4×C197—42no—
106D4×C197—81no—
107C2×C4×C197—21no—
108C2×C4×C197—42no—
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2411yescharacteristic
112C8×C197—22no—
113C22×C197—41no—
114C22×C197—42no—
115C22×C197—41no—
116C22×C197—42no—
117C22×C197—42no—
118C22×C197—41no—
119C22×C197C22×D411yescharacteristic
120C2×D197—22no—
121C2×D197—22no—
122C2×D197—22no—
123C2×D197—41no—
124C2×D197—41no—
125C2×D197—88no—
126C2×D197C2×D814yes—
127C4×C197—21no—
128C4×C197—41no—
129C4×C197C2×D812yes—
130C4×C197C2×#4111yescharacteristic
131C4×C197H(5,𝔽2)11yescharacteristic
132Dic197—22no—
133Dic197—41no—
134Dic197—42no—
135C2×C197—82no—
136C2×C197—81no—
137C2×C197#54611yescharacteristic
138C2×C197C22×D811yescharacteristic
139C2×C197#57711yescharacteristic
140D197—24no—
141D197—42no—
142C197#329411yescharacteristic, complemented, Sylow
143#3294—1971nomaximal, Sylow
144#547—1971no—
145D42—1971no—
146#486—1971no—
147#480—1972no—
148#474—1971no—
149#466—1972no—
150#459D19711yescharacteristic, complemented
151#447—1972no—
152C4×D8—1971no—
153C8×D4—1971no—
154C2×#102—1972no—
155C2×#41—3941no—
156C22×D4—3941no—
157C22×D4—1972no—
158C2×D8—1971no—
159C2×D8C2×D19712yescomplemented
160C22×C8—1972no—
161C4⋊Q8—1971no—
162#127C2×D19711yescharacteristic
163#123—3942no—
164C22⋊Q8—3941no—
165#121—3941no—
166#121—1972no—
167#121—1972no—
168#120—3942no—
169C4×D4—1971no—
170C4×D4—1971no—
171C4×D4C2×D19711yescharacteristic, complemented
172C2×C4⋊C4—1972no—
173#107—1972no—
174#107—1971no—
175C4⋊C8C2×D19711yescharacteristic, complemented
176#102—1972no—
177#102—1971no—
178#102—1972no—
179#102—1972no—
180#102—1971no—
181#102C2×D19712yescomplemented
182C22⋊C8—1972no—
183C4×C8—1971no—
184C24—3942no—
185#41—7884no—
186C2×Q8—3941no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—7884no—
190C2×D4—7882no—
191C2×D4—3944no—
192C2×D4—3942no—
193C2×D4—3942no—
194C2×D4—3942no—
195C2×D4—1971no—
196C2×D4—1971no—
197C2×D4C22×D19712yes—
198C2×D4C22×D19711yescharacteristic
199C22×C4—21no—
200C22×C4—3942no—
201C22×C4—3941no—
202C22×C4—1972no—
203D8—42no—
204D8—3942no—
205C2×C8—7882no—
206C2×C8—1971no—
207C2×C8—1971no—
208C2×C8C22×D19712yes—
209C4⋊C4—3942no—
210C4⋊C4—3941no—
211C4⋊C4—1972no—
212C4⋊C4—1971no—
213C4⋊C4—1971no—
214C4⋊C4C22×D19711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—3942no—
217C22⋊C4—3942no—
218C22⋊C4—3941no—
219C22⋊C4—1972no—
220C42C22×D19711yescharacteristic
221C23—21no—
222C23—22no—
223C23—7884no—
224C23—7884no—
225C23—15762no—
226C23—3941no—
227C23—1972no—
228Q8—7882no—
229D4—21no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—7884no—
236D4—7884no—
237D4—7884no—
238D4—3941no—
239D4—1974no—
240D4—1972no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—7884no—
244C2×C4—7882no—
245C2×C4—3941no—
246C2×C4—3942no—
247C2×C4—3942no—
248C2×C4—3942no—
249C2×C4—1972no—
250C2×C4D4×D19711yescharacteristic
251C2×C4D4×D19711yescharacteristic
252C2×C4C23×D19711yescharacteristic
253C8—22no—
254C8—3941no—
255C8—3941no—
256C22—42no—
257C22—41no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—7881no—
263C22—7881no—
264C22—15768no—
265C22—3942no—
266C22—3942no—
267C22—3942no—
268C22—1974no—
269C22C2×D4×D19711yescharacteristic
270C4—21no—
271C4—41no—
272C4—7881no—
273C4—7882no—
274C4—3942no—
275C4D8×D19712yes—
276C4#41×D19711yescharacteristic
277C4#2026337111yescharacteristic
278C2—82no—
279C2—81no—
280C2—7882no—
281C2—3944no—
282C2#2716172611yescharacteristic
283C2C2×D8×D19711yescharacteristic
284C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#459×C197C2
socleC22×C197C22×D4
hypercenter#459D197
hypocenterC197#3294
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #27161734C2semidirect
2. #27161714C2semidirect
3. #27161425C2semidirect
4. #27161391C2semidirect
5. #27161360C2semidirect
6. #27161312C2semidirect
7. #27161211C2semidirect
8. C4×#15691343C2semidirect
9. #27161144C2semidirect
10. #102×D197C2semidirect
11. #459×C197C2semidirect
16. C2×C8×D197C22semidirect
17. #20263326C22semidirect
22. #20263297C22semidirect
26. C4⋊C4×D197C22semidirect
27. #20263260C22semidirect
28. #20263259C22semidirect
30. #20263247C22semidirect
31. #20263247C22semidirect
32. #20263247C22semidirect
34. #20263238C22semidirect
35. C4×#13096802C22semidirect
36. C2×D8×C197C22semidirect
38. C4×D4×C197C22semidirect
39. C4⋊C8×C197C22semidirect
40. #102×C197C22semidirect
85. D788D8semidirect
86. D788#41semidirect
142. C197#3294semidirect
150. #459D197semidirect
159. C2×D8C2×D197semidirect
171. C4×D4C2×D197semidirect
175. C4⋊C8C2×D197semidirect
181. #102C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617342
#271617142
#271614252
#271613912
#271613602
#271613122
#271612112
C4×#156913432
#271611442
#102×D1972
#459×C1972
#3294197