Finite Group Databasebeta

← #38741732

Subgroups — #38741732

Order 25216.

288 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 57 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#27161395C212yesmaximal, complemented
2#27161015C211yesmaximal, characteristic, complemented
3#27160813C212yesmaximal, complemented
4#27160803C211yesmaximal, characteristic, complemented
5#534×C197C211yesmaximal, characteristic
6C2×C4⋊Dic197C2211yescharacteristic, complemented
7C2×#15691337—21no—
8C2×C22⋊Dic197—21no—
9#20263257—22no—
10#20263247—22no—
11C22⋊#13096802C2212yescomplemented
12#20263233C2211yescharacteristic, complemented
13C22×D4×C197—21no—
14#123×C197—22no—
15#121×C197—21no—
16#121×C197C2212yes—
17#120×C197—22no—
18C2×C22⋊C4×C197C2211yescharacteristic
19C22×Dic197—21no—
20C22×Dic197—21no—
21C2×#13096802—22no—
22C4⋊Dic197—22no—
23#15691337—42no—
24C22⋊Dic197—42no—
25C24×C197—21no—
26C24×C197D411yescharacteristic
27C2×D4×C197—22no—
28C2×D4×C197—22no—
29C2×D4×C197—21no—
30C2×D4×C197—42no—
31C2×D4×C197—42no—
32C2×D4×C197D412yes—
33C22×C4×C197—21no—
34C22×C4×C197D411yescharacteristic
35C22×C4×C197C2311yescharacteristic
36C4⋊C4×C197—21no—
37C4⋊C4×C197D412yes—
38C22⋊C4×C197—22no—
39C22⋊C4×C197—22no—
40C22⋊C4×C197—22no—
41C2×Dic197—22no—
42C2×Dic197—41no—
43C2×Dic197—41no—
44C2×Dic197—41no—
45C2×Dic197—41no—
46C2×Dic197—41no—
47#13096802—42no—
48C23×C197—21no—
49C23×C197—21no—
50C23×C197—22no—
51C23×C197—41no—
52C23×C197—41no—
53C23×C197—42no—
54C23×C197—81no—
55C23×C197—81no—
56C23×C197SD1612yes—
57C23×C197C2×D411yescharacteristic
58D4×C197—42no—
59D4×C197—44no—
60D4×C197—42no—
61D4×C197—42no—
62D4×C197—81no—
63C2×C4×C197—21no—
64C2×C4×C197—22no—
65C2×C4×C197—41no—
66C2×C4×C197—42no—
67C2×C4×C197—41no—
68C2×C4×C197C2×D412yes—
69C22×C197—21no—
70C22×C197—21no—
71C22×C197—22no—
72C22×C197—42no—
73C22×C197—41no—
74C22×C197—41no—
75C22×C197—42no—
76C22×C197—42no—
77C22×C197—42no—
78C22×C197—81no—
79C22×C197—84no—
80C22×C197#12011yescharacteristic
81C22×C197C2×SD1611yescharacteristic
82C22×C197AGL(1,ℤ/8)11yescharacteristic
83C4×C197—22no—
84C4×C197—42no—
85C4×C197—41no—
86Dic197—42no—
87Dic197—81no—
88C2×C197—21no—
89C2×C197—21no—
90C2×C197—42no—
91C2×C197—82no—
92C2×C197#45011yescharacteristic
93C2×C197#45311yescharacteristic
94C2×C197#45811yescharacteristic
95C197#161311yescharacteristic, complemented, Sylow
96#1613—1971nomaximal, Sylow
97#534D19711yescharacteristic
98#481—1972no—
99#399—1971no—
100#327—1972no—
101#323—1971no—
102C22×D4—21no—
103#123—22no—
104#121—21no—
105#121C2×D19712yes—
106#120—22no—
107C2×C4⋊C4—3941no—
108C2×C4⋊C4—1971no—
109C2×C22⋊C4—3941no—
110C2×C22⋊C4C2×D19711yescharacteristic
111#106—3942no—
112#102—3942no—
113C22⋊C8—1972no—
114#95—1971no—
115C24—21no—
116C24C22⋊D19711yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C22⋊D19712yes—
123C22×C4—21no—
124C22×C4—3941no—
125C22×C4—3941no—
126C22×C4C22⋊D19711yescharacteristic
127C22×C4C22×D19711yescharacteristic
128C2×C8—3942no—
129C4⋊C4—21no—
130C4⋊C4—7882no—
131C4⋊C4—3942no—
132C4⋊C4C22⋊D19712yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—7882no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#1569134612yes—
146C23C2×C22⋊D19711yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—7881no—
158C2×C4—7881no—
159C2×C4—7881no—
160C2×C4—7881no—
161C2×C4—7881no—
162C2×C4—3942no—
163C2×C4C2×C22⋊D19712yes—
164C8—7882no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#2026329511yescharacteristic
177C22C2×#1569134611yescharacteristic
178C22#2026335211yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—7882no—
183C4—15761no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#2716123011yescharacteristic
189C2#2716124711yescharacteristic
190C2#2716126911yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting, prime radical#534×C197C2
socleC22×C197#120
hypercenter#534D197
hypocenterC197#1613
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #27161395C2semidirect
2. #27161015C2semidirect
3. #27160813C2semidirect
4. #27160803C2semidirect
6. C2×C4⋊Dic197C22semidirect
11. C22⋊#13096802C22semidirect
12. #20263233C22semidirect
95. C197#1613semidirect

Maximal subgroups

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

subgroupindex
#271613952
#271610152
#271608132
#271608032
#534×C1972
#1613197