Finite Group Databasebeta

← #38753178

Subgroups — #38753178

Order 25216.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#20263359C212yesmaximal, complemented
2C2×#20263354C212yesmaximal, complemented
3#27161629C211yesmaximal, characteristic, complemented
4C2×#20263314C211yesmaximal, characteristic, complemented
5#27161473C212yesmaximal, complemented
6#27161468C212yesmaximal, complemented
7#27161463C214yesmaximal, complemented
8C4×M16×C197C211yesmaximal, characteristic, complemented
9C2×#15691358C2212yes—
10C22×Dic394C2211yescharacteristic
11C22×D788C2211yescharacteristic
12#20263359—28no—
13#20263354—28no—
14C2×Dic788C2214yescomplemented
15C2×#15691347C2214yes—
16C2×#15691347C2214yescomplemented
17C2×D1576C2214yes—
18#20263326C2211yescharacteristic, complemented
19#20263323C2211yescharacteristic, complemented
20#20263314—22no—
21#20263314C2212yes—
22#20263314C2212yes—
23#20263314C2212yescomplemented
24#20263300—22no—
25C4×D788—22no—
26C2×#15691335—22no—
27C2×M16×C197C2212yes—
28C2×C42×C197C2211yescharacteristic
29#97×C197C2212yescomplemented
30C4×C8×C197C2212yescomplemented
31C23×D197—21no—
32#15691358—24no—
33#15691358—24no—
34C2×Dic394—21no—
35C2×Dic394—24no—
36C2×Dic394C2312yes—
37C2×Dic394C2312yes—
38C2×D788—24no—
39C2×D788—22no—
40C2×D788—21no—
41C2×D788C2312yes—
42C2×D788C2312yes—
43C2×C22⋊D197—22no—
44C22×Dic197—21no—
45C2×C4×D197—22no—
46Dic788—28no—
47#15691347—28no—
48#15691347—28no—
49D1576—28no—
50C4⋊Dic197—22no—
51#15691335—24no—
52#15691335—24no—
53#15691335—24no—
54C22×C4×C197D412yes—
55C22×C4×C197C2311yescharacteristic
56M16×C197D418yescomplemented
57C2×C8×C197C2314yes—
58C42×C197D412yes—
59C42×C197C2311yescharacteristic
60C42×C197C2311yescharacteristic
61C22×D197—22no—
62C22×D197—22no—
63C22×D197—44no—
64C22×D197—42no—
65Dic394—22no—
66Dic394—24no—
67Dic394—22no—
68Dic394—42no—
69D788—22no—
70D788—24no—
71D788—22no—
72D788—44no—
73D788—42no—
74C22⋊D197—44no—
75C2×Dic197—22no—
76C2×Dic197—22no—
77C2×Dic197—42no—
78C4×D197—44no—
79C23×C197C2×D411yescharacteristic
80C2×C4×C197—42no—
81C2×C4×C197C2×D412yes—
82C2×C4×C197C2×D412yes—
83C2×C4×C197C2×D411yescharacteristic
84C2×C4×C197C2×D414yes—
85C2×C4×C197C2411yescharacteristic
86C8×C197C2×D418yes—
87C22×C197—22no—
88C22×C197#12712yes—
89C22×C197C22×D411yescharacteristic
90C2×D197—42no—
91C2×D197—42no—
92C2×D197—44no—
93C2×D197—44no—
94C2×D197—84no—
95C4×C197—22no—
96C4×C197—22no—
97C4×C197#12712yes—
98C4×C197C22×D412yes—
99Dic197—42no—
100Dic197—42no—
101C2×C197—22no—
102C2×C197C2×#12711yescharacteristic
103C2×C197#57512yes—
104D197—82no—
105D197—82no—
106C197#315311yescharacteristic, complemented, Sylow
107#3153—1971nomaximal, Sylow
108C2×#137—1972no—
109C2×AGL(1,ℤ/8)—1972no—
110#532—1971no—
111C2×#124—1971no—
112#497—1972no—
113#496—1972no—
114#495—1974no—
115C4×M16D19711yescharacteristic, complemented
116C2×#41—1972no—
117C22×Q8—1971no—
118C22×D4—1971no—
119#137—3948no—
120AGL(1,ℤ/8)—3948no—
121C2×Q16—1974no—
122C2×SD16—1974no—
123C2×SD16—1974no—
124C2×D8—1974no—
125C2×M16C2×D19712yes—
126C4⋊Q8—1971no—
127#127—1971no—
128#124—3942no—
129#124—1972no—
130#124—1972no—
131#124—1972no—
132#121—3942no—
133C4×D4—3942no—
134C2×C22⋊C4—3942no—
135C2×C42C2×D19711yescharacteristic
136#97C2×D19712yescomplemented
137C4×C8C2×D19712yescomplemented
138C24—3941no—
139#41—3944no—
140#41—3944no—
141C2×Q8—3944no—
142C2×Q8—3941no—
143C2×Q8—1972no—
144C2×Q8—1972no—
145C2×D4—3942no—
146C2×D4—3942no—
147C2×D4—3941no—
148C2×D4—3944no—
149C2×D4—1972no—
150C2×D4—1972no—
151C22×C4—3941no—
152C22×C4—3942no—
153C22×C4D78812yes—
154C22×C4C22×D19711yescharacteristic
155Q16—3948no—
156SD16—3948no—
157SD16—3948no—
158D8—3948no—
159M16D78818yescomplemented
160C2×C8C22×D19714yes—
161C4⋊C4—3942no—
162C22⋊C4—3944no—
163C22⋊C4—3944no—
164C22⋊C4—3944no—
165C42D78812yes—
166C42C22×D19711yescharacteristic
167C42C22×D19711yescharacteristic
168C23—7884no—
169C23—7882no—
170C23—3942no—
171C23—3942no—
172C23C2×D78811yescharacteristic
173Q8—7882no—
174Q8—3942no—
175Q8—3944no—
176Q8—3942no—
177D4—7884no—
178D4—7884no—
179D4—7882no—
180D4—3942no—
181D4—3944no—
182D4—3942no—
183C2×C4—42no—
184C2×C4—7882no—
185C2×C4—7884no—
186C2×C4—3942no—
187C2×C4—3942no—
188C2×C4C2×D78812yes—
189C2×C4C2×D78812yes—
190C2×C4C2×D78811yescharacteristic
191C2×C4C2×D78814yes—
192C2×C4C23×D19711yescharacteristic
193C8C2×D78818yes—
194C22—22no—
195C22—7882no—
196C22—7882no—
197C22—7884no—
198C22—7884no—
199C22—15764no—
200C22#2026332312yes—
201C22C22×D78811yescharacteristic
202C4—22no—
203C4—22no—
204C4—7882no—
205C4—7882no—
206C4#2026332312yes—
207C4C22×D78812yes—
208C2—22no—
209C2—15762no—
210C2—15762no—
211C2C2×#2026332311yescharacteristic
212C2#2716187912yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×M16×C197C2
socleC22×C197C22×D4
hypercenterC4×M16D197
hypocenterC197#3153
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#20263359C2semidirect
2. C2×#20263354C2semidirect
3. #27161629C2semidirect
4. C2×#20263314C2semidirect
5. #27161473C2semidirect
6. #27161468C2semidirect
7. #27161463C2semidirect
8. C4×M16×C197C2semidirect
14. C2×Dic788C22semidirect
16. C2×#15691347C22semidirect
18. #20263326C22semidirect
19. #20263323C22semidirect
23. #20263314C22semidirect
29. #97×C197C22semidirect
30. C4×C8×C197C22semidirect
56. M16×C197D4semidirect
106. C197#3153semidirect
115. C4×M16D197semidirect
136. #97C2×D197semidirect
137. C4×C8C2×D197semidirect
159. M16D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633592
C2×#202633542
#271616292
C2×#202633142
#271614732
#271614682
#271614632
C4×M16×C1972
#3153197