Finite Group Databasebeta

← #38741630

Subgroups — #38741630

Order 25216.

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

normal subgroups only

This group has 368 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#27161212C212yesmaximal, complemented
2#27160988C211yesmaximal, characteristic, complemented
3#27160813C212yesmaximal, complemented
4#27160803C211yesmaximal, characteristic, complemented
5#534×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic, complemented
7C22×C22⋊D197—21no—
8C2×#15691343—22no—
9C2×#15691335—21no—
10#20263246—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
19C23×D197—21no—
20C2×D788—22no—
21C2×D788—42no—
22C2×C22⋊D197—42no—
23C2×C22⋊D197—42no—
24C2×C22⋊D197—42no—
25C22×Dic197—21no—
26#15691343—44no—
27C2×#13096802—22no—
28#15691335—42no—
29C24×C197—21no—
30C24×C197D411yescharacteristic
31C2×D4×C197—22no—
32C2×D4×C197—22no—
33C2×D4×C197—21no—
34C2×D4×C197—42no—
35C2×D4×C197—42no—
36C2×D4×C197D412yescomplemented
37C22×C4×C197—21no—
38C22×C4×C197D411yescharacteristic
39C22×C4×C197C2311yescharacteristic
40C4⋊C4×C197—21no—
41C4⋊C4×C197D412yescomplemented
42C22⋊C4×C197—22no—
43C22⋊C4×C197—22no—
44C22⋊C4×C197—22no—
45C22×D197—22no—
46C22×D197—41no—
47C22×D197—41no—
48C22×D197—41no—
49C22×D197—41no—
50C22×D197—41no—
51C22×D197—41no—
52D788—24no—
53D788—42no—
54C22⋊D197—48no—
55C2×Dic197—22no—
56C2×Dic197—41no—
57C2×Dic197—41no—
58#13096802—42no—
59C23×C197—21no—
60C23×C197—21no—
61C23×C197—22no—
62C23×C197—41no—
63C23×C197—41no—
64C23×C197—42no—
65C23×C197—81no—
66C23×C197—81no—
67C23×C197D812yescomplemented
68C23×C197C2×D411yescharacteristic
69D4×C197—42no—
70D4×C197—44no—
71D4×C197—42no—
72D4×C197—42no—
73D4×C197—81no—
74C2×C4×C197—21no—
75C2×C4×C197—22no—
76C2×C4×C197—41no—
77C2×C4×C197—42no—
78C2×C4×C197—41no—
79C2×C4×C197C2×D412yes—
80C22×C197—21no—
81C22×C197—21no—
82C22×C197—22no—
83C22×C197—42no—
84C22×C197—41no—
85C22×C197—41no—
86C22×C197—42no—
87C22×C197—42no—
88C22×C197—42no—
89C22×C197—81no—
90C22×C197—84no—
91C22×C197#12011yescharacteristic
92C22×C197C2×D811yescharacteristic
93C22×C197AGL(1,ℤ/8)11yescharacteristic
94C2×D197—42no—
95C2×D197—42no—
96C2×D197—42no—
97C2×D197—81no—
98C2×D197—81no—
99C2×D197—81no—
100C2×D197—81no—
101C4×C197—22no—
102C4×C197—42no—
103C4×C197—41no—
104Dic197—42no—
105C2×C197—21no—
106C2×C197—21no—
107C2×C197—42no—
108C2×C197—82no—
109C2×C197#44711yescharacteristic
110C2×C197#45311yescharacteristic
111C2×C197B(4,𝔽2)11yescharacteristic
112D197—82no—
113C197#159511yescharacteristic, complemented, Sylow
114#1595—1971nomaximal, Sylow
115#534D19711yescharacteristic, complemented
116#447—1972no—
117#392—1971no—
118#327—1972no—
119#323—1971no—
120C22×D4—21no—
121C22×D4—3941no—
122C22×D4—1971no—
123C2×D8—3942no—
124#123—22no—
125#121—21no—
126#121C2×D19712yes—
127#120—22no—
128C2×C22⋊C4—3941no—
129C2×C22⋊C4C2×D19711yescharacteristic
130#102—3942no—
131C22⋊C8—1972no—
132#95—1971no—
133C24—21no—
134C24—3941no—
135C24C22⋊D19711yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—7882no—
142C2×D4—7882no—
143C2×D4—7882no—
144C2×D4—7882no—
145C2×D4—3942no—
146C2×D4C22⋊D19712yescomplemented
147C22×C4—21no—
148C22×C4—3941no—
149C22×C4C22⋊D19711yescharacteristic
150C22×C4C22×D19711yescharacteristic
151D8—7884no—
152C2×C8—3942no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D19712yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—7882no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—42no—
165C23—81no—
166C23—81no—
167C23—7881no—
168C23—7881no—
169C23—7881no—
170C23—7881no—
171C23—7881no—
172C23—7881no—
173C23—3942no—
174C23#1569134312yescomplemented
175C23C2×C22⋊D19711yescharacteristic
176D4—42no—
177D4—44no—
178D4—42no—
179D4—42no—
180D4—81no—
181D4—7882no—
182D4—7888no—
183D4—3944no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—7881no—
190C2×C4—7881no—
191C2×C4—3942no—
192C2×C4C2×C22⋊D19712yes—
193C8—7882no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—84no—
205C22—7882no—
206C22—7882no—
207C22—7882no—
208C22—15761no—
209C22—15761no—
210C22—15761no—
211C22—15761no—
212C22#2026329511yescharacteristic
213C22C2×#1569134311yescharacteristic
214C22#2026335211yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—7882no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—15762no—
224C2#2716121311yescharacteristic
225C2#2716124711yescharacteristic
226C2#2716126611yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#534×C197C2
socleC22×C197#120
hypercenter#534D197
hypocenterC197#1595
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. #27161212C2semidirect
2. #27160988C2semidirect
3. #27160813C2semidirect
4. #27160803C2semidirect
5. #534×C197C2semidirect
6. C22×D788C22semidirect
11. C22⋊#13096802C22semidirect
12. #20263233C22semidirect
36. C2×D4×C197D4semidirect
41. C4⋊C4×C197D4semidirect
67. C23×C197D8semidirect
113. C197#1595semidirect
115. #534D197semidirect
146. C2×D4C22⋊D197semidirect
154. C4⋊C4C22⋊D197semidirect
174. C23#15691343semidirect

Maximal subgroups

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

subgroupindex
#271612122
#271609882
#271608132
#271608032
#534×C1972
#1595197