Finite Group Databasebeta

← #38743153

Subgroups — #38743153

Order 25216.

442 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 97 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
2C22×C4×Dic197C211yesmaximal, characteristic, complemented
3C2×#20263252C212yesmaximal, complemented
4#27160840C212yesmaximal, complemented
5#407×C197C211yesmaximal, characteristic, complemented
6C2×#15691358C2211yescharacteristic
7C23×Dic197—21no—
8#20263309—21no—
9#20263305C412yescomplemented
10#20263300C412yescomplemented
11#20263295—21no—
12C4×C22⋊D197—22no—
13#20263285C2211yescharacteristic
14C2×C4×Dic197—24no—
15C2×C4×Dic197—21no—
16C2×C4×Dic197C2212yes—
17#20263252—24no—
18C23×C4×C197C2211yescharacteristic
19C2×M16×C197C2212yes—
20C22⋊C8×C197—22no—
21#15691358—22no—
22#15691358—22no—
23C2×Dic394C2×C411yescharacteristic, complemented
24C2×D788C2×C411yescharacteristic, complemented
25C2×C22⋊D197—22no—
26C2×C22⋊D197—21no—
27C22×Dic197—24no—
28C22×Dic197—28no—
29C22×Dic197—22no—
30C2×C4×D197—21no—
31C4⋊Dic197C2×C412yescomplemented
32#15691337—22no—
33C22⋊Dic197—22no—
34C22⋊Dic197—21no—
35#15691335—22no—
36C4×Dic197—24no—
37C4×Dic197—22no—
38C4×Dic197—21no—
39C4×Dic197D414yescomplemented
40C24×C197D411yescharacteristic
41C22×C4×C197—22no—
42C22×C4×C197—22no—
43C22×C4×C197C2×C412yes—
44C22×C4×C197D412yes—
45C22×C4×C197D411yescharacteristic
46C22×C4×C197C2311yescharacteristic
47M16×C197—22no—
48M16×C197—22no—
49C2×C8×C197—22no—
50C22×D197—21no—
51Dic394—22no—
52D788—22no—
53C22⋊D197—44no—
54C22⋊D197—41no—
55C22⋊D197—41no—
56C2×Dic197—216no—
57C2×Dic197—28no—
58C2×Dic197—24no—
59C2×Dic197—22no—
60C2×Dic197—21no—
61C4×D197—42no—
62C23×C197—24no—
63C23×C197—21no—
64C23×C197—21no—
65C23×C197C22⋊C412yes—
66C23×C197C2×D411yescharacteristic
67C2×C4×C197—21no—
68C2×C4×C197—21no—
69C2×C4×C197—24no—
70C2×C4×C197—24no—
71C2×C4×C197C22⋊C412yes—
72C2×C4×C197C22×C411yescharacteristic
73C2×C4×C197C2×D412yes—
74C2×C4×C197C2×D411yescharacteristic
75C2×C4×C197#4112yes—
76C8×C197—42no—
77C22×C197—24no—
78C22×C197—24no—
79C22×C197—22no—
80C22×C197—22no—
81C22×C197—22no—
82C22×C197#10414yescomplemented
83C22×C197C2×C22⋊C411yescharacteristic
84C22×C197#12011yescharacteristic
85C22×C197#12311yescharacteristic
86C2×D197—41no—
87C2×D197—42no—
88C4×C197—22no—
89C4×C197C4×D412yes—
90C4×C197#12112yes—
91Dic197—28no—
92Dic197—41no—
93Dic197—42no—
94C2×C197—21no—
95C2×C197—24no—
96C2×C197—21no—
97C2×C197#38611yescharacteristic
98C2×C197C2×#10412yes—
99D197—81no—
100C197#189611yescharacteristic, complemented, Sylow
101#1896—1971nomaximal, Sylow
102#525—1971no—
103C22×C42—1971no—
104C2×#104—1972no—
105#407D19711yescharacteristic, complemented
106#339—1972no—
107C2×#41—1971no—
108C23×C4—3941no—
109C23×C4C2×D19711yescharacteristic
110C2×M16C2×D19712yes—
111#123—3941no—
112C22⋊Q8—1972no—
113#121—1972no—
114#120—3941no—
115C4×D4—3942no—
116#117—1971no—
117C2×C42—3941no—
118C2×C42—3944no—
119C2×C42—1972no—
120#104—3944no—
121C22⋊C8—22no—
122C24D78811yescharacteristic
123#41—3942no—
124#41—3942no—
125C2×Q8—1971no—
126C2×D4—3942no—
127C2×D4—3941no—
128C2×D4—1971no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—3941no—
132C22×C4—3944no—
133C22×C4—3948no—
134C22×C4—3942no—
135C22×C4C4×D19712yes—
136C22×C4C22⋊D19712yes—
137C22×C4D78811yescharacteristic
138C22×C4C22×D19711yescharacteristic
139M16—22no—
140M16—22no—
141C2×C8—22no—
142C4⋊C4—3942no—
143C4⋊C4—1972no—
144C22⋊C4—3942no—
145C22⋊C4—3941no—
146C22⋊C4—3942no—
147C42—3944no—
148C42—3942no—
149C42—3941no—
150C42—1974no—
151C23—24no—
152C23—21no—
153C23—21no—
154C23—3941no—
155C23#1569133512yes—
156C23C2×D78811yescharacteristic
157Q8—3942no—
158D4—7884no—
159D4—7881no—
160D4—7881no—
161D4—3942no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—7882no—
167C2×C4—3942no—
168C2×C4—3941no—
169C2×C4—39416no—
170C2×C4—3948no—
171C2×C4—3944no—
172C2×C4#1569133512yes—
173C2×C4C2×C4×D19711yescharacteristic
174C2×C4D4×D19712yes—
175C2×C4C2×C22⋊D19711yescharacteristic
176C2×C4#1569135712yes—
177C8—42no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—7882no—
184C22—7881no—
185C22#2026325214yescomplemented
186C22C2×#1569133511yescharacteristic
187C22#2026329611yescharacteristic
188C22#2026331011yescharacteristic
189C4—22no—
190C4—7882no—
191C4—7881no—
192C4—3948no—
193C4#2026328612yes—
194C4#2026329812yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—15761no—
199C2#2716095611yescharacteristic
200C2C2×#2026325212yes—
subgroupG/H
centerC2×C4C2×C22⋊D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#407×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161590C2
cosocleC22×C4×C197C23
hypercenter#407D197
hypocenterC197#1896
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic197C2semidirect
3. C2×#20263252C2semidirect
4. #27160840C2semidirect
5. #407×C197C2semidirect
9. #20263305C4semidirect
10. #20263300C4semidirect
23. C2×Dic394C2×C4semidirect
24. C2×D788C2×C4semidirect
31. C4⋊Dic197C2×C4semidirect
39. C4×Dic197D4semidirect
82. C22×C197#104semidirect
100. C197#1896semidirect
105. #407D197semidirect
185. C22#20263252semidirect

Maximal subgroups

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

subgroupindex
#271615902
C22×C4×Dic1972
C2×#202632522
#271608402
#407×C1972
#1896197