Finite Group Databasebeta

← #38752079

Subgroups — #38752079

Order 25216.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#20263374C211yesmaximal, characteristic, complemented
2#27161879C212yesmaximal, complemented
3#27161653C211yesmaximal, characteristic, complemented
4#27161264C214yesmaximal, complemented
5#27161254C214yesmaximal, complemented
6#27161095C212yesmaximal, complemented
7C2×#101×C197C211yesmaximal, characteristic, complemented
8#20263374—24no—
9#20263374—22no—
10#20263374C2214yes—
11C2×#15691359—22no—
12C2×#15691358—22no—
13C2×#15691358C2212yes—
14C2×#15691358C2211yescharacteristic
15C2×Q8×D197—22no—
16#20263359—22no—
17#20263359C2214yescomplemented
18#20263354—22no—
19#20263354C2214yescomplemented
20#20263348—24no—
21#20263324C2212yescomplemented
22#20263315C2212yescomplemented
23#20263309—22no—
24#20263302—22no—
25#20263285C2211yescharacteristic
26#20263252C2218yescomplemented
27C22×Q8×C197C2211yescharacteristic
28C2×M16×C197C2212yes—
29#101×C197C2214yescomplemented
30#15691359—28no—
31#15691359—28no—
32#15691358—28no—
33#15691358—28no—
34#15691358—22no—
35#15691358—22no—
36#15691358—22no—
37#15691358—22no—
38#15691358D414yes—
39#15691358C2314yes—
40C2×Dic394—22no—
41C2×Dic394D412yes—
42C2×Dic394C2311yescharacteristic
43Q8×D197—28no—
44Q8×D197—28no—
45C2×D788—22no—
46C2×D788D412yes—
47C2×D788C2311yescharacteristic
48C2×C22⋊D197—22no—
49C2×C22⋊D197—21no—
50C2×C4×D197—24no—
51C2×C4×D197—22no—
52C2×C4×D197—21no—
53Dic788—24no—
54#15691347—24no—
55#15691347—24no—
56D1576—24no—
57C4⋊Dic197—21no—
58#15691337—24no—
59C22⋊Dic197—21no—
60#15691335—24no—
61C4×Dic197C2312yes—
62C2×Q8×C197—22no—
63C2×Q8×C197—22no—
64C2×Q8×C197D412yes—
65C2×Q8×C197C2312yes—
66C22×C4×C197D412yes—
67C22×C4×C197C2311yescharacteristic
68M16×C197—22no—
69M16×C197C2314yes—
70C2×C8×C197—22no—
71C22×D197—22no—
72C22×D197—21no—
73Dic394—28no—
74Dic394—22no—
75Dic394—22no—
76Dic394C2×D414yes—
77D788—28no—
78D788—22no—
79D788—22no—
80D788C2×D414yes—
81C22⋊D197—24no—
82C22⋊D197—24no—
83C22⋊D197—41no—
84C22⋊D197—41no—
85C2×Dic197—22no—
86C2×Dic197—22no—
87C2×Dic197—21no—
88C4×D197—216no—
89C4×D197—24no—
90C4×D197—24no—
91C4×D197—42no—
92C23×C197C2×D411yescharacteristic
93Q8×C197—24no—
94Q8×C197—24no—
95C2×C4×C197—22no—
96C2×C4×C197—22no—
97C2×C4×C197C2×D412yes—
98C2×C4×C197C2×D412yes—
99C2×C4×C197C2×D414yes—
100C2×C4×C197C2×D411yescharacteristic
101C2×C4×C197C2411yescharacteristic
102C8×C197—24no—
103C22×C197—41no—
104C22×C197#12011yescharacteristic
105C22×C197#12011yescharacteristic
106C22×C197C22×D411yescharacteristic
107C2×D197—24no—
108C2×D197—44no—
109C2×D197—42no—
110C2×D197—41no—
111C4×C197—24no—
112C4×C197#12012yes—
113C4×C197C22×D412yes—
114Dic197—24no—
115Dic197—42no—
116Dic197—41no—
117C2×C197—21no—
118C2×C197—21no—
119C2×C197—21no—
120C2×C197C2×#12011yescharacteristic
121D197—44no—
122D197—81no—
123C197#302011yescharacteristic, complemented, Sylow
124#3020—1971nomaximal, Sylow
125C2×2−1+4—1971no—
126#575—1972no—
127#536—1971no—
128#456—1974no—
129#454—1974no—
130#421—1972no—
131C2×#101D19711yescharacteristic, complemented
1322−1+4—3944no—
1332−1+4—3942no—
1342−1+4—1974no—
135C2×#41—3942no—
136C2×#41—3942no—
137C2×#41—1971no—
138C2×#41—1972no—
139C22×Q8—3942no—
140C22×Q8C2×D19711yescharacteristic
141#137—3942no—
142#137—1974no—
143AGL(1,ℤ/8)—3942no—
144AGL(1,ℤ/8)—1974no—
145#135—3944no—
146C2×M16C2×D19712yes—
147C4⋊Q8—1972no—
148#124—1972no—
149#123—3942no—
150C22⋊Q8—3942no—
151#117—1971no—
152#104—1978no—
153#101C2×D19714yescomplemented
154#41—3948no—
155#41—3942no—
156#41—3942no—
157#41—3948no—
158#41—3948no—
159#41—3942no—
160#41—3948no—
161#41—3942no—
162#41—1974no—
163#41—1974no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—3942no—
167C2×Q8—3948no—
168C2×Q8—3948no—
169C2×Q8—1972no—
170C2×Q8—1971no—
171C2×Q8D78812yes—
172C2×Q8C22×D19712yes—
173C2×D4—3942no—
174C2×D4—3941no—
175C2×D4—3942no—
176C2×D4—1972no—
177C2×D4—1971no—
178C22×C4—3944no—
179C22×C4—3941no—
180C22×C4—3942no—
181C22×C4D78812yes—
182C22×C4C22×D19711yescharacteristic
183Q16—3944no—
184SD16—3944no—
185SD16—3944no—
186D8—3944no—
187M16—22no—
188M16C22×D19714yes—
189C2×C8—22no—
190C4⋊C4—3944no—
191C4⋊C4—3941no—
192C22⋊C4—3941no—
193C22⋊C4—3944no—
194C42—1972no—
195C23—3941no—
196C23—3942no—
197C23C2×D78811yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—3948no—
201Q8—3942no—
202Q8—3942no—
203Q8—1974no—
204D4—7881no—
205D4—7881no—
206D4—3948no—
207D4—3944no—
208D4—3944no—
209D4—3942no—
210D4—3942no—
211D4—1974no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—7882no—
215C2×C4—39416no—
216C2×C4—3942no—
217C2×C4—3941no—
218C2×C4—3942no—
219C2×C4—3944no—
220C2×C4—3944no—
221C2×C4D4×D19712yes—
222C2×C4D4×D19712yes—
223C2×C4C2×D78814yes—
224C2×C4C2×D78811yescharacteristic
225C2×C4C23×D19711yescharacteristic
226C8—24no—
227C22—41no—
228C22—7884no—
229C22—7882no—
230C22—7881no—
231C22—3944no—
232C22#2026329611yescharacteristic
233C22#2026329611yescharacteristic
234C22C22×D78811yescharacteristic
235C4—24no—
236C4—7882no—
237C4—7881no—
238C4—3944no—
239C4#2026329612yes—
240C4C2×D4×D19712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—7884no—
245C2—15761no—
246C2C2×#2026329611yescharacteristic
subgroupG/H
centerC2C2×#20263296
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#101×C197C2
socleC2×C197C2×#120
hypercenterC2×#101D197
hypocenterC197#3020
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
1. C2×#20263374C2semidirect
2. #27161879C2semidirect
3. #27161653C2semidirect
4. #27161264C2semidirect
5. #27161254C2semidirect
6. #27161095C2semidirect
7. C2×#101×C197C2semidirect
17. #20263359C22semidirect
19. #20263354C22semidirect
21. #20263324C22semidirect
22. #20263315C22semidirect
26. #20263252C22semidirect
29. #101×C197C22semidirect
123. C197#3020semidirect
131. C2×#101D197semidirect
153. #101C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633742
#271618792
#271616532
#271612642
#271612542
#271610952
C2×#101×C1972
#3020197