Finite Group Databasebeta

← #38741666

Subgroups — #38741666

Order 25216.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#27161220C212yesmaximal, complemented
3#27160991C211yesmaximal, characteristic, complemented
4#27160814C212yesmaximal, complemented
5#323×C197C211yesmaximal, characteristic, complemented
6C22×Dic394C2211yescharacteristic, complemented
7C22×C22⋊D197—21no—
8C2×#15691347—22no—
9#20263310—22no—
10#20263300—21no—
11#20263300C2212yes—
12#20263296—22no—
13C2×C22⋊Dic197—21no—
14#20263251—22no—
15#20263233—21no—
16#20263233C2211yescharacteristic, complemented
17C2×C22⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yescomplemented
19C23×D197—21no—
20C2×Dic394—22no—
21C2×Dic394—42no—
22C2×D788—21no—
23C2×D788D412yes—
24C2×C22⋊D197—22no—
25C2×C22⋊D197—22no—
26C2×C22⋊D197—42no—
27C2×C22⋊D197—42no—
28C22×Dic197—21no—
29C22×Dic197—21no—
30C22×Dic197—21no—
31#15691347—44no—
32C4⋊Dic197—21no—
33C4⋊Dic197D412yes—
34C22⋊Dic197—42no—
35#15691335—22no—
36#15691335—22no—
37C24×C197D411yescharacteristic
38C22×C4×C197D411yescharacteristic
39C22×C4×C197C2311yescharacteristic
40C2×C8×C197—22no—
41C22⋊C4×C197—22no—
42C22×D197—22no—
43C22×D197—42no—
44C22×D197—81no—
45C22×D197—81no—
46Dic394—24no—
47Dic394—42no—
48D788—42no—
49D788—81no—
50C22⋊D197—44no—
51C22⋊D197—42no—
52C22⋊D197—42no—
53C2×Dic197—22no—
54C2×Dic197—22no—
55C2×Dic197—42no—
56C2×Dic197—41no—
57C2×Dic197—41no—
58C2×Dic197—41no—
59C2×Dic197—41no—
60C2×Dic197—41no—
61C23×C197—21no—
62C23×C197—21no—
63C23×C197—41no—
64C23×C197—41no—
65C23×C197SD1612yescomplemented
66C23×C197C2×D411yescharacteristic
67C2×C4×C197—21no—
68C2×C4×C197—41no—
69C2×C4×C197—41no—
70C2×C4×C197C2×D412yes—
71C8×C197—42no—
72C22×C197—21no—
73C22×C197—21no—
74C22×C197—22no—
75C22×C197—42no—
76C22×C197—41no—
77C22×C197—41no—
78C22×C197—81no—
79C22×C197#12011yescharacteristic
80C22×C197C2×SD1611yescharacteristic
81C22×C197AGL(1,ℤ/8)11yescharacteristic
82C2×D197—42no—
83C2×D197—42no—
84C2×D197—42no—
85C2×D197—84no—
86C4×C197—22no—
87C4×C197—41no—
88Dic197—42no—
89Dic197—42no—
90Dic197—81no—
91C2×C197—21no—
92C2×C197—21no—
93C2×C197—42no—
94C2×C197#45011yescharacteristic
95C2×C197#45511yescharacteristic
96C2×C197B(4,𝔽2)11yescharacteristic
97D197—82no—
98C197#160111yescharacteristic, complemented, Sylow
99#1601—1971nomaximal, Sylow
100#534—1971no—
101#448—1972no—
102#393—1971no—
103#327—1972no—
104#323D19711yescharacteristic, complemented
105C22×Q8—1971no—
106C22×D4—3941no—
107C2×SD16—3942no—
108#123—3942no—
109#121—3941no—
110#121—1972no—
111#120—3942no—
112C2×C22⋊C4—3941no—
113C2×C22⋊C4C2×D19711yescharacteristic
114#103—3942no—
115C22⋊C8C2×D19712yescomplemented
116#95—3941no—
117#95—1971no—
118C24—3941no—
119C24D78811yescharacteristic
120C2×Q8—7882no—
121C2×Q8—3942no—
122C2×D4—7882no—
123C2×D4—7882no—
124C2×D4—3941no—
125C2×D4—3942no—
126C2×D4—3942no—
127C2×D4—1972no—
128C22×C4—3941no—
129C22×C4—3941no—
130C22×C4—3941no—
131C22×C4D78811yescharacteristic
132C22×C4C22×D19711yescharacteristic
133SD16—7884no—
134C2×C8—22no—
135C4⋊C4—3941no—
136C4⋊C4—1972no—
137C22⋊C4—22no—
138C22⋊C4—7882no—
139C22⋊C4—3942no—
140C22⋊C4—3942no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—7882no—
146C23—15761no—
147C23—15761no—
148C23—3942no—
149C23#1569134712yescomplemented
150C23C2×D78811yescharacteristic
151Q8—7882no—
152Q8—3944no—
153D4—7882no—
154D4—7884no—
155D4—7882no—
156D4—7882no—
157D4—15761no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—7882no—
162C2×C4—7881no—
163C2×C4—7881no—
164C2×C4—7881no—
165C2×C4—7881no—
166C2×C4—7881no—
167C2×C4—3942no—
168C2×C4—3942no—
169C2×C4D4×D19712yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—7882no—
179C22—7882no—
180C22—7882no—
181C22—15764no—
182C22#2026329611yescharacteristic
183C22C2×#1569134711yescharacteristic
184C22#2026335411yescharacteristic
185C4—22no—
186C4—41no—
187C4—7882no—
188C4—7882no—
189C4—15761no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—15762no—
194C2#2716123211yescharacteristic
195C2#2716125911yescharacteristic
196C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197#120
prime radical#27161641C2
hypercenter#323D197
hypocenterC197#1601
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27161220C2semidirect
3. #27160991C2semidirect
4. #27160814C2semidirect
5. #323×C197C2semidirect
6. C22×Dic394C22semidirect
16. #20263233C22semidirect
18. C22⋊C8×C197C22semidirect
65. C23×C197SD16semidirect
98. C197#1601semidirect
104. #323D197semidirect
115. C22⋊C8C2×D197semidirect
149. C23#15691347semidirect

Maximal subgroups

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

subgroupindex
#271616412
#271612202
#271609912
#271608142
#323×C1972
#1601197