Finite Group Databasebeta

← #38754505

Subgroups — #38754505

Order 25216.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#27161885C214yesmaximal, complemented
2#27161339C214yesmaximal, complemented
3#27161261C212yesmaximal, complemented
4#27161246C212yesmaximal, complemented
5#27161197C212yesmaximal, complemented
6#492×C197C211yesmaximal, characteristic, complemented
7#20263374C2212yes—
8#20263371C2212yescomplemented
9#20263357C2214yescomplemented
10#20263355—24no—
11#20263353—24no—
12#20263352C2214yes—
13#20263348C2212yes—
14#20263347—24no—
15SD16×D197—24no—
16#20263332C2214yescomplemented
17#20263262C2212yescomplemented
18#20263254C2214yescomplemented
19#20263245C2212yescomplemented
20#20263243C2212yescomplemented
21C2×SD16×C197C2214yescomplemented
22C4⋊Q8×C197C2211yescharacteristic, complemented
23#127×C197C2211yescharacteristic
24C4×C8×C197C2211yescharacteristic, complemented
25#15691359—24no—
26#15691358—42no—
27#15691358C2314yes—
28#15691357—24no—
29Q8×D197—24no—
30C2×C22⋊D197—42no—
31D4×D197—24no—
32Dic788D412yescomplemented
33#15691348—24no—
34#15691347D414yescomplemented
35#15691346—24no—
36#15691345—24no—
37D1576D412yescomplemented
38#15691343—24no—
39#15691342C2314yes—
40#15691341—24no—
41C8×D197—24no—
42C2×Q8×C197C2312yes—
43C2×D4×C197—41no—
44C2×D4×C197C2312yes—
45SD16×C197—24no—
46SD16×C197—24no—
47C2×C8×C197C2312yes—
48C4⋊C4×C197—41no—
49C42×C197C2311yescharacteristic
50C22×D197—42no—
51Dic394—42no—
52Dic394C2×D414yes—
53D788—42no—
54D788C2×D414yes—
55C22⋊D197—24no—
56C22⋊D197—44no—
57C2×Dic197—42no—
58C4×D197—24no—
59C4×D197—44no—
60#13096802—24no—
61C23×C197—22no—
62Q8×C197—24no—
63D4×C197—24no—
64D4×C197—82no—
65C2×C4×C197—21no—
66C2×C4×C197—22no—
67C2×C4×C197C2411yescharacteristic
68C8×C197C2×D412yes—
69C8×C197C2×D412yes—
70C22×C197—42no—
71C22×C197—82no—
72C22×C197H(5,𝔽2)11yescharacteristic
73C2×D197—24no—
74C2×D197—162no—
75C4×C197—22no—
76C4×C197—42no—
77C4×C197C22×D412yes—
78Dic197—24no—
79C2×C197—21no—
80C2×C197—82no—
81C2×C197D4211yescharacteristic
82D197—44no—
83C197#329111yescharacteristic, complemented, Sylow
84#3291—1971nomaximal, Sylow
85#577—1974no—
86#492D19711yescharacteristic, complemented
87#471—1974no—
88#456—1972no—
89#453—1972no—
90#443—1972no—
912−1+4—1972no—
92H(5,𝔽2)—1972no—
93#137—3944no—
94#137—1974no—
95AGL(1,ℤ/8)—3944no—
96AGL(1,ℤ/8)—1974no—
97#135—3944no—
98#135—1972no—
99C2×SD16—3944no—
100C2×SD16C2×D19714yescomplemented
101#131—1974no—
102C4⋊Q8C2×D19711yescharacteristic, complemented
103#127C2×D19711yescharacteristic
104#108—1972no—
105#104—1974no—
106#101—1972no—
107#100—1972no—
108C4×C8C2×D19711yescharacteristic, complemented
109#41—7882no—
110#41—3944no—
111#41—3944no—
112#41—1974no—
113C2×Q8—3944no—
114C2×Q8C22×D19712yes—
115C2×D4—41no—
116C2×D4—7882no—
117C2×D4—3944no—
118C2×D4C22×D19712yes—
119Q16—3944no—
120Q16—1972no—
121SD16—24no—
122SD16—24no—
123SD16—3944no—
124SD16—3944no—
125SD16—1974no—
126D8—3944no—
127D8—1972no—
128M16—3944no—
129M16—1974no—
130C2×C8—3944no—
131C2×C8C22×D19712yes—
132C4⋊C4—41no—
133C42C22×D19711yescharacteristic
134C23—22no—
135C23—7882no—
136Q8—24no—
137Q8—7882no—
138Q8—1974no—
139D4—24no—
140D4—82no—
141D4—7882no—
142D4—7884no—
143D4—3944no—
144D4—1974no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—7884no—
148C2×C4—7882no—
149C2×C4—3944no—
150C2×C4C23×D19711yescharacteristic
151C8—3944no—
152C8D4×D19712yes—
153C8D4×D19712yes—
154C22—42no—
155C22—82no—
156C22—31522no—
157C22—3944no—
158C22#2026337111yescharacteristic
159C4—22no—
160C4—42no—
161C4—3944no—
162C4C2×D4×D19712yes—
163C2—21no—
164C2—82no—
165C2—7884no—
166C2#2716171411yescharacteristic
subgroupG/H
centerC2#27161714
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#492×C197C2
socleC2×C197D42
hypercenter#492D197
hypocenterC197#3291
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #27161885C2semidirect
2. #27161339C2semidirect
3. #27161261C2semidirect
4. #27161246C2semidirect
5. #27161197C2semidirect
6. #492×C197C2semidirect
8. #20263371C22semidirect
9. #20263357C22semidirect
16. #20263332C22semidirect
17. #20263262C22semidirect
18. #20263254C22semidirect
19. #20263245C22semidirect
20. #20263243C22semidirect
21. C2×SD16×C197C22semidirect
22. C4⋊Q8×C197C22semidirect
24. C4×C8×C197C22semidirect
32. Dic788D4semidirect
34. #15691347D4semidirect
37. D1576D4semidirect
83. C197#3291semidirect
86. #492D197semidirect
100. C2×SD16C2×D197semidirect
102. C4⋊Q8C2×D197semidirect
108. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618852
#271613392
#271612612
#271612462
#271611972
#492×C1972
#3291197