Finite Group Databasebeta

← #38743158

Subgroups — #38743158

Order 25216.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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
3C2×#20263252C212yesmaximal, complemented
4#27160840C212yesmaximal, complemented
5#407×C197C211yesmaximal, characteristic, complemented
6C2×#15691358C2211yescharacteristic
7#20263309—21no—
8#20263305C412yescomplemented
9#20263300C412yescomplemented
10#20263295—21no—
11C4×C22⋊D197—22no—
12#20263285—24no—
13#20263285C2211yescharacteristic
14C2×C4⋊Dic197—21no—
15C2×C22⋊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
24C2×D788C2×C411yescharacteristic
25C2×C22⋊D197—22no—
26C2×C22⋊D197—21no—
27C22×Dic197—22no—
28C2×C4×D197—21no—
29C4⋊Dic197—24no—
30C4⋊Dic197C2×C412yes—
31#15691337—22no—
32C22⋊Dic197—24no—
33C22⋊Dic197—22no—
34C22⋊Dic197—21no—
35#15691335—22no—
36C4×Dic197—22no—
37C4×Dic197—21no—
38C4×Dic197D414yescomplemented
39C24×C197D411yescharacteristic
40C22×C4×C197—22no—
41C22×C4×C197—22no—
42C22×C4×C197C2×C412yes—
43C22×C4×C197D412yes—
44C22×C4×C197D411yescharacteristic
45C22×C4×C197C2311yescharacteristic
46M16×C197—22no—
47M16×C197—22no—
48C2×C8×C197—22no—
49C22×D197—21no—
50Dic394—22no—
51D788—22no—
52C22⋊D197—44no—
53C22⋊D197—41no—
54C22⋊D197—41no—
55C2×Dic197—22no—
56C2×Dic197—21no—
57C2×Dic197—24no—
58C2×Dic197—44no—
59C4×D197—42no—
60C23×C197—21no—
61C23×C197—21no—
62C23×C197—42no—
63C23×C197C22⋊C412yes—
64C23×C197C2×D411yescharacteristic
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—24no—
68C2×C4×C197—24no—
69C2×C4×C197C22⋊C412yes—
70C2×C4×C197C22×C411yescharacteristic
71C2×C4×C197C2×D412yes—
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197#4112yes—
74C8×C197—42no—
75C22×C197—22no—
76C22×C197—22no—
77C22×C197—22no—
78C22×C197—22no—
79C22×C197—42no—
80C22×C197—42no—
81C22×C197C2×C22⋊C411yescharacteristic
82C22×C197#12011yescharacteristic
83C22×C197#12311yescharacteristic
84C2×D197—41no—
85C2×D197—42no—
86C4×C197—22no—
87C4×C197C4×D412yes—
88C4×C197#12112yes—
89Dic197—44no—
90Dic197—41no—
91Dic197—42no—
92C2×C197—21no—
93C2×C197—21no—
94C2×C197—42no—
95C2×C197#38611yescharacteristic
96C2×C197#42112yes—
97D197—81no—
98C197#189711yescharacteristic, complemented, Sylow
99#1897—1971nomaximal, Sylow
100#525—1971no—
101C2×#117—1971no—
102C2×#104—1972no—
103#407D19711yescharacteristic, complemented
104#339—1972no—
105C2×#41—1971no—
106C23×C4C2×D19711yescharacteristic
107C2×M16C2×D19712yes—
108#123—3941no—
109C22⋊Q8—1972no—
110#121—1972no—
111#120—3941no—
112C4×D4—3942no—
113#117—3944no—
114#117—1971no—
115C2×C4⋊C4—3941no—
116C2×C22⋊C4—3941no—
117C2×C42—1972no—
118#104—3944no—
119C22⋊C8—22no—
120C24D78811yescharacteristic
121#41—3942no—
122#41—3942no—
123C2×Q8—1971no—
124C2×D4—3942no—
125C2×D4—3941no—
126C2×D4—1971no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—3942no—
130C22×C4—3941no—
131C22×C4C4×D19712yes—
132C22×C4C22⋊D19712yes—
133C22×C4D78811yescharacteristic
134C22×C4C22×D19711yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—3944no—
139C4⋊C4—3942no—
140C4⋊C4—1972no—
141C22⋊C4—3944no—
142C22⋊C4—3942no—
143C22⋊C4—3941no—
144C22⋊C4—3942no—
145C42—3942no—
146C42—3941no—
147C42—1974no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—3941no—
152C23#1569133512yes—
153C23C2×D78811yescharacteristic
154Q8—3942no—
155D4—7884no—
156D4—7881no—
157D4—7881no—
158D4—3942no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—7884no—
164C2×C4—7882no—
165C2×C4—3942no—
166C2×C4—3941no—
167C2×C4—3944no—
168C2×C4#1569133512yes—
169C2×C4C2×C4×D19711yescharacteristic
170C2×C4D4×D19712yes—
171C2×C4C2×C22⋊D19711yescharacteristic
172C2×C4#1569135712yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—7882no—
181C22—7881no—
182C22C2×#1569133511yescharacteristic
183C22#2026329611yescharacteristic
184C22#2026331011yescharacteristic
185C4—22no—
186C4—7884no—
187C4—7882no—
188C4—7881no—
189C4#2026328612yes—
190C4#2026329812yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—15761no—
195C2#2716095611yescharacteristic
196C2#2716109512yes—
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#1897
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. C2×#20263285C2semidirect
3. C2×#20263252C2semidirect
4. #27160840C2semidirect
5. #407×C197C2semidirect
8. #20263305C4semidirect
9. #20263300C4semidirect
38. C4×Dic197D4semidirect
98. C197#1897semidirect
103. #407D197semidirect

Maximal subgroups

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

subgroupindex
#271615902
C2×#202632852
C2×#202632522
#271608402
#407×C1972
#1897197