Finite Group Databasebeta

← #38753164

Subgroups — #38753164

Order 25216.

656 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 173 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×#20263354C214yesmaximal, complemented
2#27161629C211yesmaximal, characteristic, complemented
3C2×#20263323C211yesmaximal, characteristic, complemented
4#27161468C214yesmaximal, complemented
5#27161456C212yesmaximal, complemented
6#27161452C212yesmaximal, complemented
7C4×M16×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2212yes—
9C22×D788—22no—
10C22×D788C2212yes—
11#20263354—216no—
12C2×#15691347C2218yescomplemented
13C2×D1576C2218yescomplemented
14#20263326C2211yescharacteristic, complemented
15#20263323—22no—
16#20263323C2212yes—
17#20263323C2212yes—
18#20263323C2211yescharacteristic, complemented
19#20263314C2212yescomplemented
20#20263300—22no—
21C4×D788—22no—
22C2×M16×C197C2212yes—
23C2×C42×C197C2211yescharacteristic
24#97×C197C2212yescomplemented
25C4×C8×C197C2212yescomplemented
26C23×D197—22no—
27#15691358—28no—
28C2×Dic394C2312yes—
29C2×D788—28no—
30C2×D788—22no—
31C2×D788—22no—
32C2×D788—24no—
33C2×D788—24no—
34C2×D788—48no—
35C2×D788C2312yes—
36C2×D788C2314yes—
37C2×C22⋊D197—22no—
38C2×C4×D197—22no—
39#15691347—216no—
40D1576—216no—
41C4⋊Dic197—22no—
42#15691335—24no—
43C22×C4×C197D412yes—
44C22×C4×C197C2311yescharacteristic
45M16×C197D418yescomplemented
46C2×C8×C197C2314yes—
47C42×C197D412yes—
48C42×C197C2311yescharacteristic
49C42×C197C2311yescharacteristic
50C22×D197—22no—
51C22×D197—24no—
52C22×D197—48no—
53C22×D197—44no—
54Dic394—24no—
55D788—24no—
56D788—28no—
57D788—48no—
58D788—44no—
59D788—44no—
60D788—48no—
61C22⋊D197—44no—
62C2×Dic197—22no—
63C4×D197—44no—
64C23×C197C2×D411yescharacteristic
65C2×C4×C197—24no—
66C2×C4×C197C2×D412yes—
67C2×C4×C197C2×D411yescharacteristic
68C2×C4×C197C2×D414yes—
69C2×C4×C197C2×D412yes—
70C2×C4×C197C2411yescharacteristic
71C8×C197C2×D418yescomplemented
72C22×C197—22no—
73C22×C197#12712yes—
74C22×C197C22×D411yescharacteristic
75C2×D197—42no—
76C2×D197—44no—
77C2×D197—48no—
78C2×D197—44no—
79C2×D197—88no—
80C4×C197—22no—
81C4×C197#12712yes—
82C4×C197AGL(1,ℤ/8)14yescomplemented
83C4×C197C22×D412yes—
84Dic197—42no—
85C2×C197—22no—
86C2×C197C2×#12711yescharacteristic
87C2×C197C2×AGL(1,ℤ/8)12yes—
88D197—82no—
89D197—84no—
90C197#315111yescharacteristic, complemented, Sylow
91#3151—1971nomaximal, Sylow
92C2×AGL(1,ℤ/8)—1974no—
93#532—1971no—
94C2×#127—1971no—
95#496—1974no—
96#493—1972no—
97#492—1972no—
98C4×M16D19711yescharacteristic, complemented
99C2×#41—1972no—
100C22×D4—3942no—
101C22×D4—1972no—
102AGL(1,ℤ/8)—39416no—
103C2×SD16—1978no—
104C2×D8—1978no—
105C2×M16C2×D19712yes—
106C4⋊Q8—1971no—
107#127—3942no—
108#127—1972no—
109#127—1972no—
110#127—1971no—
111#124—1972no—
112#121—3942no—
113C4×D4—3942no—
114C2×C42C2×D19711yescharacteristic
115#97C2×D19712yescomplemented
116C4×C8C2×D19712yescomplemented
117C24—3942no—
118#41—3948no—
119C2×Q8—1972no—
120C2×D4—7888no—
121C2×D4—3942no—
122C2×D4—3942no—
123C2×D4—3942no—
124C2×D4—3944no—
125C2×D4—3944no—
126C2×D4—3948no—
127C2×D4—1972no—
128C2×D4—1974no—
129C22×C4—3942no—
130C22×C4D78812yes—
131C22×C4C22×D19711yescharacteristic
132SD16—39416no—
133D8—39416no—
134M16D78818yescomplemented
135C2×C8C22×D19714yes—
136C4⋊C4—3942no—
137C22⋊C4—3944no—
138C42D78812yes—
139C42C22×D19711yescharacteristic
140C42C22×D19711yescharacteristic
141C23—7888no—
142C23—7884no—
143C23—3942no—
144C23—3944no—
145C23C2×D78811yescharacteristic
146Q8—3944no—
147D4—7888no—
148D4—7884no—
149D4—7884no—
150D4—7888no—
151D4—7884no—
152D4—3944no—
153D4—3948no—
154C2×C4—24no—
155C2×C4—7884no—
156C2×C4—3942no—
157C2×C4C2×D78812yes—
158C2×C4C2×D78811yescharacteristic
159C2×C4C2×D78814yes—
160C2×C4C2×D78812yes—
161C2×C4C23×D19711yescharacteristic
162C8C2×D78818yescomplemented
163C22—22no—
164C22—7888no—
165C22—7884no—
166C22—7882no—
167C22—7884no—
168C22—15768no—
169C22#2026332312yes—
170C22C22×D78811yescharacteristic
171C4—22no—
172C4—7882no—
173C4#2026332312yes—
174C4#2026335414yescomplemented
175C4C22×D78812yes—
176C2—22no—
177C2—15764no—
178C2—15762no—
179C2C2×#2026332311yescharacteristic
180C2C2×#2026335412yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC4×M16×C197C2
socleC22×C197C22×D4
hypercenterC4×M16D197
hypocenterC197#3151
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×#20263354C2semidirect
2. #27161629C2semidirect
3. C2×#20263323C2semidirect
4. #27161468C2semidirect
5. #27161456C2semidirect
6. #27161452C2semidirect
7. C4×M16×C197C2semidirect
12. C2×#15691347C22semidirect
13. C2×D1576C22semidirect
14. #20263326C22semidirect
18. #20263323C22semidirect
19. #20263314C22semidirect
24. #97×C197C22semidirect
25. C4×C8×C197C22semidirect
45. M16×C197D4semidirect
71. C8×C197C2×D4semidirect
82. C4×C197AGL(1,ℤ/8)semidirect
90. C197#3151semidirect
98. C4×M16D197semidirect
115. #97C2×D197semidirect
116. C4×C8C2×D197semidirect
134. M16D788semidirect
162. C8C2×D788semidirect
174. C4#20263354semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
#271616292
C2×#202633232
#271614682
#271614562
#271614522
C4×M16×C1972
#3151197