Finite Group Databasebeta

← #38753176

Subgroups — #38753176

Order 25216.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20263357C212yesmaximal, complemented
2C2×#20263352C212yesmaximal, complemented
3#27161629C211yesmaximal, characteristic, complemented
4#27161474C212yesmaximal, complemented
5#27161469C212yesmaximal, complemented
6#27161461C214yesmaximal, complemented
7C4×#15691342C211yesmaximal, characteristic, complemented
8C2×#124×C197C211yesmaximal, characteristic, complemented
9C2×#15691358C2212yes—
10#20263357—28no—
11#20263352—28no—
12C2×#15691348C2214yescomplemented
13C2×#15691346C2214yes—
14C2×#15691345C2214yescomplemented
15C2×#15691343C2214yes—
16C2×#15691342C2212yes—
17#20263326C2211yescharacteristic, complemented
18#20263323C2211yescharacteristic, complemented
19#20263314C2212yescomplemented
20#20263300—22no—
21C4×D788—22no—
22#20263237C2212yescomplemented
23C4×#13096802C2212yescomplemented
24C22×Q8×C197C2211yescharacteristic
25C22×D4×C197C2211yescharacteristic
26#124×C197—22no—
27#124×C197C2212yes—
28#124×C197C2212yes—
29C2×C22⋊C4×C197—22no—
30C2×C42×C197C2211yescharacteristic
31#15691358—24no—
32#15691358—24no—
33C2×Dic394C2312yes—
34C2×D788—22no—
35C2×D788C2312yes—
36C2×C22⋊D197—22no—
37C2×C4×D197—22no—
38#15691348—28no—
39#15691346—28no—
40#15691345—28no—
41#15691343—28no—
42#15691342D418yescomplemented
43C2×#13096802C2314yes—
44C4⋊Dic197—22no—
45#15691335—24no—
46C24×C197—21no—
47C2×Q8×C197—21no—
48C2×Q8×C197—24no—
49C2×Q8×C197C2312yes—
50C2×D4×C197—24no—
51C2×D4×C197—21no—
52C2×D4×C197C2312yes—
53C22×C4×C197—21no—
54C22×C4×C197D412yes—
55C22×C4×C197C2311yescharacteristic
56C22⋊C4×C197—24no—
57C22⋊C4×C197—24no—
58C42×C197D412yes—
59C42×C197C2311yescharacteristic
60C42×C197C2311yescharacteristic
61C22×D197—22no—
62Dic394—22no—
63Dic394—22no—
64D788—22no—
65D788—22no—
66D788—44no—
67C22⋊D197—44no—
68C2×Dic197—22no—
69C4×D197—44no—
70#13096802C2×D418yes—
71C23×C197—22no—
72C23×C197—44no—
73C23×C197—42no—
74C23×C197C2×D411yescharacteristic
75Q8×C197—24no—
76Q8×C197—42no—
77D4×C197—24no—
78D4×C197—42no—
79C2×C4×C197—22no—
80C2×C4×C197—42no—
81C2×C4×C197—42no—
82C2×C4×C197C2×D414yes—
83C2×C4×C197C2×D412yes—
84C2×C4×C197C2×D412yes—
85C2×C4×C197C2×D411yescharacteristic
86C2×C4×C197C2411yescharacteristic
87C22×C197—22no—
88C22×C197—42no—
89C22×C197—44no—
90C22×C197—84no—
91C22×C197#12712yes—
92C22×C197C22×D411yescharacteristic
93C2×D197—42no—
94C2×D197—44no—
95C4×C197—22no—
96C4×C197—22no—
97C4×C197—42no—
98C4×C197#12712yes—
99C4×C197C22×D412yes—
100Dic197—42no—
101C2×C197—22no—
102C2×C197—82no—
103C2×C197C2×#12711yescharacteristic
104C2×C197#57512yes—
105D197—82no—
106C197#315311yescharacteristic, complemented, Sylow
107#3153—1971nomaximal, Sylow
108C2×#137—1972no—
109C2×AGL(1,ℤ/8)—1972no—
110#532—1971no—
111C2×#124D19711yescharacteristic, complemented
112#497—1972no—
113#496—1972no—
114#495—1974no—
115C4×M16—1971no—
116C2×#41—1972no—
117C22×Q8C2×D19711yescharacteristic
118C22×D4C2×D19711yescharacteristic
119#137—3948no—
120AGL(1,ℤ/8)—3948no—
121C2×Q16—1974no—
122C2×SD16—1974no—
123C2×SD16—1974no—
124C2×D8—1974no—
125C2×M16—1972no—
126C4⋊Q8—1971no—
127#127—1971no—
128#124—22no—
129#124—1972no—
130#124C2×D19712yes—
131#124C2×D19712yes—
132#121—3942no—
133C4×D4—3942no—
134C2×C22⋊C4—22no—
135C2×C42C2×D19711yescharacteristic
136#97—1972no—
137C4×C8—1972no—
138C24—21no—
139#41—3944no—
140#41—3944no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—1972no—
144C2×Q8C22×D19712yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3942no—
148C2×D4—3942no—
149C2×D4—1972no—
150C2×D4C22×D19712yes—
151C22×C4—21no—
152C22×C4—3942no—
153C22×C4C22⋊D19712yes—
154C22×C4C22×D19711yescharacteristic
155Q16—3948no—
156SD16—3948no—
157SD16—3948no—
158D8—3948no—
159M16—1978no—
160C2×C8—1974no—
161C4⋊C4—3942no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—3944no—
165C42C22⋊D19712yes—
166C42C22×D19711yescharacteristic
167C42C22×D19711yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—3942no—
172C23C2×C22⋊D19711yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—3942no—
176Q8—3942no—
177D4—24no—
178D4—42no—
179D4—7884no—
180D4—7884no—
181D4—3942no—
182D4—3942no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—7884no—
187C2×C4—3942no—
188C2×C4D4×D19714yes—
189C2×C4C2×C22⋊D19712yes—
190C2×C4C2×C22⋊D19712yes—
191C2×C4C2×C22⋊D19711yescharacteristic
192C2×C4C23×D19711yescharacteristic
193C8—1978no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—7882no—
199C22—7884no—
200C22#2026332212yes—
201C22C22×C22⋊D19711yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—7882no—
206C4#2026332212yes—
207C4C2×D4×D19712yes—
208C2—22no—
209C2—82no—
210C2—15762no—
211C2C2×#2026332211yescharacteristic
212C2#2716187712yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#124×C197C2
socleC22×C197C22×D4
hypercenterC2×#124D197
hypocenterC197#3153
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#20263357C2semidirect
2. C2×#20263352C2semidirect
3. #27161629C2semidirect
4. #27161474C2semidirect
5. #27161469C2semidirect
6. #27161461C2semidirect
7. C4×#15691342C2semidirect
8. C2×#124×C197C2semidirect
12. C2×#15691348C22semidirect
14. C2×#15691345C22semidirect
17. #20263326C22semidirect
18. #20263323C22semidirect
19. #20263314C22semidirect
22. #20263237C22semidirect
23. C4×#13096802C22semidirect
42. #15691342D4semidirect
106. C197#3153semidirect
111. C2×#124D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
C2×#202633522
#271616292
#271614742
#271614692
#271614612
C4×#156913422
C2×#124×C1972
#3153197