Finite Group Databasebeta

← #38743844

Subgroups — #38743844

Order 25216.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#27161641C211yesmaximal, characteristic
2C2×#20263310C212yesmaximal, complemented
3C2×#20263240C212yesmaximal, complemented
4#27160940C211yesmaximal, characteristic, complemented
5#342×C197C211yesmaximal, characteristic, complemented
6C22×C22⋊D197C2212yes—
7C23×Dic197C2211yescharacteristic
8#20263310—22no—
9#20263310—24no—
10#20263310—24no—
11#20263300—22no—
12#20263296—22no—
13C2×C4⋊Dic197—22no—
14C2×C22⋊Dic197C2211yescharacteristic
15C2×#15691335—22no—
16#20263240—24no—
17#20263233—22no—
18C2×C22⋊C4×C197C2211yescharacteristic
19C2×C22⋊C4×C197C2212yes—
20C23×D197D412yes—
21C2×D788—22no—
22C2×C22⋊D197—22no—
23C2×C22⋊D197—22no—
24C2×C22⋊D197—22no—
25C2×C22⋊D197—22no—
26C2×C22⋊D197—22no—
27C2×C22⋊D197—22no—
28C22×Dic197—21no—
29C22×Dic197—22no—
30C22×Dic197—22no—
31C22×Dic197—21no—
32C22×Dic197—21no—
33C22×Dic197D412yes—
34C22×Dic197D412yescomplemented
35C4⋊Dic197—28no—
36C4⋊Dic197—22no—
37C22⋊Dic197—22no—
38#15691335—28no—
39#15691335—24no—
40C24×C197C2311yescharacteristic
41C22×C4×C197—22no—
42C22×C4×C197D412yes—
43C22⋊C4×C197—22no—
44C22⋊C4×C197—22no—
45C22⋊C4×C197D414yescomplemented
46C22×D197—22no—
47C22×D197—22no—
48C22×D197—22no—
49C22×D197—44no—
50D788—82no—
51C22⋊D197—42no—
52C22⋊D197—42no—
53C22⋊D197—42no—
54C22⋊D197—42no—
55C2×Dic197—24no—
56C2×Dic197—22no—
57C2×Dic197—22no—
58C2×Dic197—22no—
59C2×Dic197—22no—
60C2×Dic197—41no—
61C2×Dic197—42no—
62C2×Dic197—41no—
63C2×Dic197—42no—
64C2×Dic197—41no—
65C23×C197—42no—
66C23×C197C2×D412yes—
67C23×C197C2×D411yescharacteristic
68C23×C197C2×D411yescharacteristic
69C23×C197C2×D412yes—
70C23×C197#4111yescharacteristic
71C2×C4×C197—22no—
72C2×C4×C197—24no—
73C2×C4×C197—44no—
74C2×C4×C197—42no—
75C22×C197—22no—
76C22×C197—24no—
77C22×C197—41no—
78C22×C197—42no—
79C22×C197—41no—
80C22×C197#12012yes—
81C22×C197#12011yescharacteristic
82C22×C197#12111yescharacteristic
83C22×C197#12112yes—
84C22×C197#12711yescharacteristic
85C2×D197—44no—
86C2×D197—42no—
87C2×D197—44no—
88C2×D197—82no—
89C4×C197—44no—
90C4×C197—42no—
91Dic197—42no—
92Dic197—42no—
93Dic197—81no—
94C2×C197—21no—
95C2×C197—22no—
96C2×C197—22no—
97C2×C197—21no—
98C2×C197#39211yescharacteristic
99C2×C197#45812yes—
100D197—82no—
101C197#202211yescharacteristic, complemented, Sylow
102#2022—1971nomaximal, Sylow
103#534—1971no—
104C2×#123—1972no—
105C2×#99—1972no—
106#381—1971no—
107#342D19711yescharacteristic, complemented
108C22×D4—1972no—
109C23×C4—1971no—
110#123—3944no—
111#123—3942no—
112#123—3944no—
113#121—3942no—
114#120—3942no—
115C2×C4⋊C4—3942no—
116C2×C22⋊C4—3942no—
117C2×C22⋊C4—1971no—
118C2×C22⋊C4C2×D19711yescharacteristic
119C2×C22⋊C4C2×D19712yes—
120#99—3944no—
121#95—3942no—
122C24—1972no—
123C24C22×D19711yescharacteristic
124C2×D4—3942no—
125C2×D4—3942no—
126C2×D4—3942no—
127C2×D4—3942no—
128C2×D4—3942no—
129C2×D4—3942no—
130C2×D4—3942no—
131C22×C4—22no—
132C22×C4—3941no—
133C22×C4—3942no—
134C22×C4—3941no—
135C22×C4—3942no—
136C22×C4—3941no—
137C22×C4—1972no—
138C22×C4—1972no—
139C22×C4D78812yes—
140C4⋊C4—3948no—
141C4⋊C4—3942no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3948no—
145C22⋊C4—3944no—
146C22⋊C4—3942no—
147C22⋊C4D78814yescomplemented
148C23—42no—
149C23—7884no—
150C23—3942no—
151C23—3942no—
152C23—3942no—
153C23D4×D19712yes—
154C23D4×D19711yescharacteristic
155C23C2×D78811yescharacteristic
156C23C2×D78812yes—
157C23#1569135911yescharacteristic
158D4—7882no—
159D4—7882no—
160D4—7882no—
161D4—7882no—
162D4—15762no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—7881no—
168C2×C4—7882no—
169C2×C4—7881no—
170C2×C4—7882no—
171C2×C4—7881no—
172C2×C4—3944no—
173C2×C4—3942no—
174C2×C4—3942no—
175C2×C4—3942no—
176C2×C4—3942no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—7884no—
183C22—7882no—
184C22—7884no—
185C22—15762no—
186C22#2026329612yes—
187C22#2026329611yescharacteristic
188C22#2026330112yes—
189C22#2026330111yescharacteristic
190C22#2026332311yescharacteristic
191C4—44no—
192C4—42no—
193C4—7882no—
194C4—7882no—
195C4—15761no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—15762no—
201C2#2716098911yescharacteristic
202C2#2716127012yes—
subgroupG/H
centerC22#20263296
commutator, cosocleC24×C197C23
FrattiniC24C22×D197
Fitting#342×C197C2
socleC22×C197#120
prime radical#27161641C2
hypercenter#342D197
hypocenterC197#2022
LCSG ▷ C24×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#20263310C2semidirect
3. C2×#20263240C2semidirect
4. #27160940C2semidirect
5. #342×C197C2semidirect
34. C22×Dic197D4semidirect
45. C22⋊C4×C197D4semidirect
101. C197#2022semidirect
107. #342D197semidirect
147. C22⋊C4D788semidirect

Maximal subgroups

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

subgroupindex
#271616412
C2×#202633102
C2×#202632402
#271609402
#342×C1972
#2022197