Finite Group Databasebeta

← #38754322

Subgroups — #38754322

Order 25216.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×#20263350C212yesmaximal, complemented
2#27161724C212yesmaximal, complemented
3#27161463C211yesmaximal, characteristic, complemented
4#27161326C212yesmaximal, complemented
5#27161294C212yesmaximal, complemented
6#27161242C212yesmaximal, complemented
7#27161214C212yesmaximal, complemented
8#27161158C211yesmaximal, characteristic, complemented
9C4×D8×C197C211yesmaximal, characteristic, complemented
10C2×#15691357C2212yes—
11C2×D4×D197C2212yescomplemented
12#20263350—28no—
13C2×Dic788C2211yescharacteristic, complemented
14C2×#15691347C2212yescomplemented
15C2×#15691346C2212yescomplemented
16C2×D1576C2211yescharacteristic
17C2×#15691343C2212yes—
18C2×#15691341C2212yescomplemented
19#20263314C2212yescomplemented
20#20263310—22no—
21#20263306C2212yescomplemented
22#20263301C2212yes—
23#20263300—22no—
24#20263296—22no—
25C4×D788C2211yescharacteristic
26C2×#15691335—22no—
27C4⋊#13096802C2211yescharacteristic, complemented
28#20263249C2212yescomplemented
29#20263246C2212yescomplemented
30#20263238C2212yescomplemented
31C2×D8×C197C2211yescharacteristic
32C4×D4×C197C2212yescomplemented
33#107×C197C2211yescharacteristic, complemented
34#102×C197C2212yescomplemented
35C4×C8×C197C2211yescharacteristic, complemented
36C23×D197—22no—
37#15691357—24no—
38#15691357—42no—
39C2×Dic394C2312yes—
40C2×D788—22no—
41C2×D788D411yescharacteristic
42C2×D788C2312yes—
43C2×C22⋊D197—22no—
44C2×C22⋊D197—22no—
45D4×D197—24no—
46D4×D197—42no—
47C22×Dic197—22no—
48C2×C4×D197C2312yes—
49Dic788—41no—
50#15691347—24no—
51#15691346—24no—
52D1576—41no—
53#15691343—24no—
54#15691341—24no—
55C2×#13096802C2312yes—
56C4⋊Dic197—22no—
57C4⋊Dic197D411yescharacteristic
58#15691335—22no—
59#15691335—22no—
60#15691335—22no—
61#15691335—22no—
62#15691335D412yes—
63C2×D4×C197C2312yes—
64C22×C4×C197—22no—
65D8×C197D414yescomplemented
66C2×C8×C197C2311yescharacteristic
67C2×C8×C197C2311yescharacteristic
68C4⋊C4×C197C2312yes—
69C22⋊C4×C197—22no—
70C42×C197C2311yescharacteristic
71C22×D197—22no—
72C22×D197—44no—
73C22×D197—44no—
74C22×D197—82no—
75C22×D197C2×D412yes—
76Dic394—22no—
77Dic394—22no—
78D788—22no—
79D788—22no—
80D788—41no—
81D788—82no—
82C22⋊D197—44no—
83C22⋊D197—44no—
84C2×Dic197—22no—
85C2×Dic197—44no—
86C2×Dic197C2×D412yes—
87C4×D197—22no—
88C4×D197—22no—
89#13096802—22no—
90C23×C197—22no—
91D4×C197—22no—
92D4×C197C2×D414yes—
93C2×C4×C197—22no—
94C2×C4×C197—82no—
95C2×C4×C197C2×D411yescharacteristic
96C2×C4×C197C2×D411yescharacteristic
97C2×C4×C197C2411yescharacteristic
98C8×C197—21no—
99C8×C197C2×D412yes—
100C22×C197—24no—
101C22×C197—42no—
102C22×C197—42no—
103C22×C197C22×D411yescharacteristic
104C2×D197—22no—
105C2×D197—22no—
106C2×D197—22no—
107C2×D197—42no—
108C2×D197—42no—
109C2×D197—42no—
110C2×D197—88no—
111C4×C197—21no—
112C4×C197—21no—
113C4×C197—42no—
114C4×C197C22×D411yescharacteristic
115C4×C197H(5,𝔽2)11yescharacteristic
116Dic197—22no—
117Dic197—42no—
118C2×C197—44no—
119C2×C197D4211yescharacteristic
120C2×C197#57511yescharacteristic
121C2×C197#57611yescharacteristic
122D197—42no—
123D197—82no—
124C197#327311yescharacteristic, complemented, Sylow
125#3273—1971nomaximal, Sylow
126C2×AGL(1,ℤ/8)—1972no—
127#546—1972no—
128#495—1971no—
129#468—1972no—
130#463—1972no—
131#452—1972no—
132#447—1972no—
133C4×D8D19711yescharacteristic, complemented
134#436—1971no—
135C2×#41—1972no—
136C22×D4—1972no—
137AGL(1,ℤ/8)—3948no—
138C2×Q16—1971no—
139C2×SD16—1972no—
140C2×SD16—1972no—
141C2×D8—1972no—
142C2×D8—1971no—
143C2×D8C2×D19711yescharacteristic
144C2×M16—1972no—
145#124—1972no—
146#123—3942no—
147C22⋊Q8—1972no—
148#121—3942no—
149#121—1972no—
150#120—3942no—
151C4×D4—1971no—
152C4×D4C2×D19712yescomplemented
153C2×C22⋊C4—3942no—
154#107C2×D19711yescharacteristic, complemented
155C4⋊C8—1971no—
156#103—1972no—
157#102—1972no—
158#102C2×D19712yescomplemented
159C22⋊C8—1972no—
160C4×C8C2×D19711yescharacteristic, complemented
161C24—3942no—
162#41—7882no—
163#41—3944no—
164C2×Q8—1972no—
165C2×D4—7882no—
166C2×D4—3942no—
167C2×D4—3942no—
168C2×D4—3944no—
169C2×D4—3942no—
170C2×D4—1971no—
171C2×D4—1972no—
172C2×D4C22×D19712yes—
173C22×C4—22no—
174C22×C4—3942no—
175C22×C4—1972no—
176Q16—7881no—
177SD16—3944no—
178SD16—3944no—
179D8—7881no—
180D8—3944no—
181D8D78814yescomplemented
182M16—3944no—
183C2×C8—1972no—
184C2×C8C22×D19711yescharacteristic
185C2×C8C22×D19711yescharacteristic
186C4⋊C4—3942no—
187C4⋊C4—1971no—
188C4⋊C4C22×D19712yes—
189C22⋊C4—22no—
190C22⋊C4—3942no—
191C22⋊C4—3942no—
192C22⋊C4—3942no—
193C22⋊C4—3942no—
194C22⋊C4—1972no—
195C42C22×D19711yescharacteristic
196C23—22no—
197C23—7884no—
198C23—7884no—
199C23—15762no—
200C23—3942no—
201C23—1972no—
202Q8—3942no—
203Q8—3942no—
204D4—22no—
205D4—7881no—
206D4—7884no—
207D4—7884no—
208D4—15762no—
209D4—3942no—
210D4—3942no—
211D4C2×D78814yes—
212C2×C4—22no—
213C2×C4—82no—
214C2×C4—7884no—
215C2×C4—3942no—
216C2×C4—3942no—
217C2×C4—3942no—
218C2×C4—1972no—
219C2×C4D4×D19711yescharacteristic
220C2×C4D4×D19711yescharacteristic
221C2×C4C23×D19711yescharacteristic
222C8—21no—
223C8—3942no—
224C8C2×D78812yes—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—7882no—
229C22—7882no—
230C22—7882no—
231C22—15768no—
232C22—3942no—
233C22—3942no—
234C22—3942no—
235C22C2×D4×D19711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—7882no—
240C4—3942no—
241C4C22×D78811yescharacteristic
242C4#2026337211yescharacteristic
243C2—44no—
244C2—7882no—
245C2—15762no—
246C2D4×D78811yescharacteristic
247C2#2716187611yescharacteristic
248C2#2716188011yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×D8×C197C2
socleC22×C197C22×D4
hypercenterC4×D8D197
hypocenterC197#3273
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. C2×#20263350C2semidirect
2. #27161724C2semidirect
3. #27161463C2semidirect
4. #27161326C2semidirect
5. #27161294C2semidirect
6. #27161242C2semidirect
7. #27161214C2semidirect
8. #27161158C2semidirect
9. C4×D8×C197C2semidirect
11. C2×D4×D197C22semidirect
13. C2×Dic788C22semidirect
14. C2×#15691347C22semidirect
15. C2×#15691346C22semidirect
18. C2×#15691341C22semidirect
19. #20263314C22semidirect
21. #20263306C22semidirect
27. C4⋊#13096802C22semidirect
28. #20263249C22semidirect
29. #20263246C22semidirect
30. #20263238C22semidirect
32. C4×D4×C197C22semidirect
33. #107×C197C22semidirect
34. #102×C197C22semidirect
35. C4×C8×C197C22semidirect
65. D8×C197D4semidirect
124. C197#3273semidirect
133. C4×D8D197semidirect
152. C4×D4C2×D197semidirect
154. #107C2×D197semidirect
158. #102C2×D197semidirect
160. C4×C8C2×D197semidirect
181. D8D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
#271617242
#271614632
#271613262
#271612942
#271612422
#271612142
#271611582
C4×D8×C1972
#3273197