Finite Group Databasebeta

← #38751172

Subgroups — #38751172

Order 25216.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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×#20263371C211yesmaximal, characteristic
2#27161554C211yesmaximal, characteristic, complemented
3#27161052C214yesmaximal, complemented
4C2×#20263240C218yesmaximal, complemented
5#518×C197C211yesmaximal, characteristic, complemented
6#20263371—24no—
7#20263371—24no—
8C2×#15691358C2212yes—
9C2×#15691357C414yescomplemented
10C22×C22⋊D197C2214yes—
11C2×D4×D197C414yescomplemented
12D4×Dic197—24no—
13#20263285C2212yescomplemented
14C2×C22⋊Dic197C2214yescomplemented
15#20263240C22116yescomplemented
16C22×D4×C197C2211yescharacteristic
17C4×D4×C197—24no—
18#117×C197C2212yescomplemented
19C2×C22⋊C4×C197C2214yescomplemented
20C23×D197C2×C414yescomplemented
21#15691358—24no—
22#15691358—24no—
23#15691357—216no—
24C2×Dic394C2×C412yescomplemented
25C2×D788C2×C412yescomplemented
26C2×C22⋊D197—28no—
27C2×C22⋊D197—28no—
28C2×C22⋊D197C2×C418yescomplemented
29C2×C22⋊D197C2314yes—
30D4×D197—216no—
31C22×Dic197—24no—
32C22×Dic197C2×C414yescomplemented
33C2×C4×D197C2×C414yescomplemented
34C4⋊Dic197—22no—
35C22⋊Dic197—24no—
36C22⋊Dic197C2314yes—
37C4×Dic197—22no—
38C24×C197C2312yes—
39C2×D4×C197C2×C414yes—
40C2×D4×C197D414yes—
41C2×D4×C197D414yes—
42C22×C4×C197—24no—
43C22×C4×C197C2311yescharacteristic
44C4⋊C4×C197—22no—
45C22⋊C4×C197—24no—
46C22⋊C4×C197C2314yes—
47C42×C197—22no—
48C22×D197—28no—
49C22×D197—84no—
50C22×D197C22×C414yes—
51Dic394—24no—
52D788—24no—
53C22⋊D197—216no—
54C22⋊D197—24no—
55C22⋊D197—24no—
56C2×Dic197—28no—
57C2×Dic197—24no—
58C2×Dic197—84no—
59C2×Dic197C22×C414yes—
60C4×D197—28no—
61C23×C197—82no—
62C23×C197C22×C414yes—
63C23×C197C2×D414yes—
64C23×C197C2×D414yes—
65C23×C197C2411yescharacteristic
66D4×C197C22⋊C4116yes—
67C2×C4×C197—24no—
68C2×C4×C197—84no—
69C2×C4×C197C22×C412yes—
70C2×C4×C197C2×D412yes—
71C2×C4×C197C2×D412yes—
72C22×C197—44no—
73C22×C197—41no—
74C22×C197—44no—
75C22×C197—44no—
76C22×C197C2×C22⋊C418yes—
77C22×C197C23×C411yescharacteristic
78C22×C197C22×D411yescharacteristic
79C22×C197C22×D411yescharacteristic
80C2×D197—24no—
81C2×D197—44no—
82C2×D197—88no—
83C4×C197—44no—
84C4×C197C2×C22⋊C414yes—
85Dic197—24no—
86Dic197—44no—
87C2×C197—21no—
88C2×C197—21no—
89C2×C197—21no—
90C2×C197—28no—
91C2×C197C22×C22⋊C411yescharacteristic
92D197—44no—
93C197#288311yescharacteristic, complemented, Sylow
94#2883—1971nomaximal, Sylow
95C2×H(5,𝔽2)—1971no—
96#518—1971no—
97#518D19711yescharacteristic, complemented
98#410—1974no—
99C2×#99—1978no—
100H(5,𝔽2)—3944no—
101H(5,𝔽2)—3944no—
102C2×#41—1974no—
103C2×#41—1972no—
104C22×D4—1974no—
105C22×D4—1974no—
106C22×D4C2×D19711yescharacteristic
107C4×D4—24no—
108C4×D4—3944no—
109#117—1972no—
110#117C2×D19712yescomplemented
111C2×C22⋊C4—1974no—
112C2×C22⋊C4C2×D19714yescomplemented
113#99—19716no—
114C24—1974no—
115C24C22×D19712yes—
116#41—39416no—
117#41—3944no—
118#41—3944no—
119C2×Q8—1972no—
120C2×D4—3948no—
121C2×D4—3948no—
122C2×D4—39416no—
123C2×D4—1978no—
124C2×D4—1974no—
125C2×D4—1972no—
126C2×D4C4×D19714yes—
127C2×D4C22⋊D19714yes—
128C2×D4D78814yes—
129C22×C4—24no—
130C22×C4—3944no—
131C22×C4—1974no—
132C22×C4—1974no—
133C22×C4C22×D19711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—3942no—
136C22⋊C4—24no—
137C22⋊C4—3944no—
138C22⋊C4—1974no—
139C22⋊C4C22×D19714yes—
140C42—22no—
141C42—3942no—
142C23—82no—
143C23—15764no—
144C23—3948no—
145C23—1974no—
146C23C2×C4×D19714yes—
147C23C2×C22⋊D19714yes—
148C23C2×D78814yes—
149C23C23×D19711yescharacteristic
150Q8—3944no—
151D4—39416no—
152D4—3944no—
153D4—3944no—
154D4—3944no—
155D4#15691335116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—15764no—
159C2×C4—3944no—
160C2×C4—3948no—
161C2×C4—3948no—
162C2×C4—1974no—
163C2×C4C2×C4×D19712yes—
164C2×C4C2×C22⋊D19712yes—
165C2×C4C2×D78812yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—7884no—
171C22—15768no—
172C22—3944no—
173C22C2×#1569133518yes—
174C22C22×C4×D19711yescharacteristic
175C22C22×C22⋊D19711yescharacteristic
176C22C22×D78811yescharacteristic
177C4—44no—
178C4—7884no—
179C4—3944no—
180C4C2×#1569133514yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—7884no—
186C2C22×#1569133511yescharacteristic
subgroupG/H
centerC2C22×#15691335
commutatorC22×C197C23×C4
FrattiniC23C23×D197
Fitting#518×C197C2
socleC2×C197C22×C22⋊C4
prime radicalC2×#20263371C2
cosocleC23×C197C24
hypercenter#518D197
hypocenterC197#2883
LCSG ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #27161554C2semidirect
3. #27161052C2semidirect
4. C2×#20263240C2semidirect
5. #518×C197C2semidirect
9. C2×#15691357C4semidirect
11. C2×D4×D197C4semidirect
13. #20263285C22semidirect
14. C2×C22⋊Dic197C22semidirect
15. #20263240C22semidirect
18. #117×C197C22semidirect
19. C2×C22⋊C4×C197C22semidirect
20. C23×D197C2×C4semidirect
24. C2×Dic394C2×C4semidirect
25. C2×D788C2×C4semidirect
28. C2×C22⋊D197C2×C4semidirect
32. C22×Dic197C2×C4semidirect
33. C2×C4×D197C2×C4semidirect
93. C197#2883semidirect
97. #518D197semidirect
110. #117C2×D197semidirect
112. C2×C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633712
#271615542
#271610522
C2×#202632402
#518×C1972
#2883197