Finite Group Databasebeta

← #38752645

Subgroups — #38752645

Order 25216.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#27161590C211yesmaximal, characteristic
2C2×#20263285C211yesmaximal, characteristic, complemented
3#27161421C214yesmaximal, complemented
4#27161404C214yesmaximal, complemented
5#27161121C212yesmaximal, complemented
6#27161081C212yesmaximal, complemented
7#407×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2211yescharacteristic
9#20263309—21no—
10#20263305C2212yes—
11#20263300C2212yes—
12#20263295—21no—
13C4×C22⋊D197—22no—
14#20263285—22no—
15#20263285C2211yescharacteristic
16#20263285C2214yes—
17C2×C4⋊Dic197C2212yes—
18C2×C22⋊Dic197—21no—
19C2×C4×Dic197—21no—
20#20263260C2214yescomplemented
21#20263258C2214yescomplemented
22#20263251C2214yescomplemented
23#20263248C2214yescomplemented
24C23×C4×C197C2211yescharacteristic
25C2×M16×C197C2212yes—
26C22⋊C8×C197C2214yes—
27#15691358—24no—
28C2×Dic394C2311yescharacteristic
29C2×D788C2311yescharacteristic
30C2×C22⋊D197—22no—
31C2×C22⋊D197—21no—
32C22×Dic197—22no—
33C2×C4×D197—21no—
34C4⋊Dic197—22no—
35C4⋊Dic197C2314yes—
36C4⋊Dic197C2312yes—
37#15691337—22no—
38C22⋊Dic197—24no—
39C22⋊Dic197—21no—
40C22⋊Dic197—22no—
41#15691335—22no—
42C4×Dic197—24no—
43C4×Dic197—21no—
44C24×C197D411yescharacteristic
45C22×C4×C197—24no—
46C22×C4×C197D411yescharacteristic
47C22×C4×C197D412yes—
48C22×C4×C197C2312yes—
49C22×C4×C197C2311yescharacteristic
50M16×C197—24no—
51C2×C8×C197C2314yes—
52C22×D197—21no—
53Dic394—22no—
54D788—22no—
55C22⋊D197—44no—
56C22⋊D197—42no—
57C2×Dic197—24no—
58C2×Dic197—22no—
59C2×Dic197—21no—
60C2×Dic197—44no—
61C4×D197—42no—
62C23×C197—22no—
63C23×C197—42no—
64C23×C197C2×D412yes—
65C23×C197C2×D411yescharacteristic
66C2×C4×C197—22no—
67C2×C4×C197—24no—
68C2×C4×C197—24no—
69C2×C4×C197C2×D412yes—
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197#4114yes—
72C2×C4×C197C2411yescharacteristic
73C8×C197—24no—
74C22×C197—22no—
75C22×C197—22no—
76C22×C197—22no—
77C22×C197—22no—
78C22×C197—44no—
79C22×C197#12312yes—
80C22×C197C22×D411yescharacteristic
81C2×D197—41no—
82C2×D197—42no—
83C4×C197—22no—
84C4×C197#12312yes—
85C4×C197C2×#4112yes—
86Dic197—41no—
87Dic197—44no—
88Dic197—42no—
89C2×C197—22no—
90C2×C197—42no—
91C2×C197C2×#12311yescharacteristic
92C2×C197#57512yes—
93D197—81no—
94C197#309311yescharacteristic, complemented, Sylow
95#3093—1971nomaximal, Sylow
96#525—1971no—
97C2×#117—1971no—
98#485—1974no—
99#482—1974no—
100#428—1972no—
101#417—1972no—
102#407D19711yescharacteristic, complemented
103C2×#41—1971no—
104C23×C4C2×D19711yescharacteristic
105C2×M16C2×D19712yes—
106#123—3941no—
107C22⋊Q8—1972no—
108#121—1972no—
109#120—3941no—
110C4×D4—3942no—
111#117—3942no—
112#117—1974no—
113#117—1971no—
114C2×C4⋊C4—1972no—
115C2×C22⋊C4—3941no—
116C2×C42—3941no—
117#107—1974no—
118#106—1974no—
119#103—1974no—
120#102—1974no—
121C22⋊C8C2×D19714yes—
122C24D78811yescharacteristic
123#41—3944no—
124C2×Q8—1971no—
125C2×D4—3942no—
126C2×D4—3941no—
127C2×D4—1971no—
128C22×C4—24no—
129C22×C4—3942no—
130C22×C4—3941no—
131C22×C4D78811yescharacteristic
132C22×C4D78812yes—
133C22×C4C22×D19712yes—
134C22×C4C22×D19711yescharacteristic
135M16—24no—
136C2×C8C22×D19714yes—
137C4⋊C4—3942no—
138C4⋊C4—3942no—
139C4⋊C4—1974no—
140C4⋊C4—1972no—
141C22⋊C4—3944no—
142C22⋊C4—3941no—
143C22⋊C4—3942no—
144C22⋊C4—3942no—
145C42—3944no—
146C42—3941no—
147C23—22no—
148C23—42no—
149C23—3941no—
150C23C2×D78812yes—
151C23C2×D78811yescharacteristic
152Q8—3942no—
153D4—7884no—
154D4—7882no—
155D4—3942no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—7884no—
160C2×C4—7882no—
161C2×C4—3944no—
162C2×C4—3942no—
163C2×C4—3941no—
164C2×C4C2×D78812yes—
165C2×C4C2×D78811yescharacteristic
166C2×C4#1569135714yes—
167C2×C4C23×D19711yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—7882no—
175C22—7881no—
176C22#2026331012yes—
177C22C22×D78811yescharacteristic
178C4—22no—
179C4—7884no—
180C4—7882no—
181C4—7881no—
182C4#2026331012yes—
183C4C2×#1569135712yes—
184C2—22no—
185C2—42no—
186C2—15761no—
187C2C2×#2026331011yescharacteristic
188C2#2716187912yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#407×C197C2
socleC22×C197C22×D4
prime radical#27161590C2
hypercenter#407D197
hypocenterC197#3093
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#20263285C2semidirect
3. #27161421C2semidirect
4. #27161404C2semidirect
5. #27161121C2semidirect
6. #27161081C2semidirect
7. #407×C197C2semidirect
20. #20263260C22semidirect
21. #20263258C22semidirect
22. #20263251C22semidirect
23. #20263248C22semidirect
94. C197#3093semidirect
102. #407D197semidirect

Maximal subgroups

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

subgroupindex
#271615902
C2×#202632852
#271614212
#271614042
#271611212
#271610812
#407×C1972
#3093197