Finite Group Databasebeta

← #38741632

Subgroups — #38741632

Order 25216.

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

normal subgroups only

This group has 368 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, complemented
2#27161215C212yesmaximal, complemented
3#27160988C211yesmaximal, characteristic, complemented
4#27160814C212yesmaximal, complemented
5#323×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic, complemented
7C22×C22⋊D197—21no—
8C22×C22⋊D197—21no—
9C2×D1576—22no—
10#20263310—22no—
11#20263300—21no—
12#20263300C2212yes—
13#20263296—22no—
14C2×#15691335—21no—
15#20263248—22no—
16#20263233C2211yescharacteristic, complemented
17C2×C22⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yescomplemented
19C23×D197—21no—
20C23×D197—21no—
21C2×D788—21no—
22C2×D788—22no—
23C2×D788—42no—
24C2×D788D412yescomplemented
25C2×C22⋊D197—22no—
26C2×C22⋊D197—22no—
27C2×C22⋊D197—42no—
28C2×C22⋊D197—42no—
29C2×C22⋊D197—42no—
30C2×C22⋊D197—42no—
31C2×C22⋊D197—42no—
32C22×Dic197—21no—
33C22×Dic197—21no—
34D1576—44no—
35C4⋊Dic197—21no—
36C4⋊Dic197D412yescomplemented
37#15691335—22no—
38#15691335—22no—
39#15691335—42no—
40C24×C197D411yescharacteristic
41C22×C4×C197D411yescharacteristic
42C22×C4×C197C2311yescharacteristic
43C2×C8×C197—22no—
44C22⋊C4×C197—22no—
45C22×D197—22no—
46C22×D197—22no—
47C22×D197—41no—
48C22×D197—41no—
49C22×D197—42no—
50C22×D197—41no—
51C22×D197—41no—
52C22×D197—41no—
53C22×D197—41no—
54C22×D197—81no—
55C22×D197—81no—
56D788—24no—
57D788—42no—
58D788—42no—
59D788—81no—
60C22⋊D197—44no—
61C22⋊D197—48no—
62C22⋊D197—42no—
63C22⋊D197—42no—
64C2×Dic197—22no—
65C2×Dic197—22no—
66C2×Dic197—41no—
67C2×Dic197—42no—
68C2×Dic197—41no—
69C23×C197—21no—
70C23×C197—21no—
71C23×C197—41no—
72C23×C197—41no—
73C23×C197D812yescomplemented
74C23×C197C2×D411yescharacteristic
75C2×C4×C197—21no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197C2×D412yes—
79C8×C197—42no—
80C22×C197—21no—
81C22×C197—21no—
82C22×C197—22no—
83C22×C197—42no—
84C22×C197—41no—
85C22×C197—41no—
86C22×C197—81no—
87C22×C197#12011yescharacteristic
88C22×C197C2×D811yescharacteristic
89C22×C197AGL(1,ℤ/8)11yescharacteristic
90C2×D197—42no—
91C2×D197—42no—
92C2×D197—42no—
93C2×D197—42no—
94C2×D197—42no—
95C2×D197—42no—
96C2×D197—81no—
97C2×D197—81no—
98C2×D197—84no—
99C2×D197—81no—
100C2×D197—81no—
101C4×C197—22no—
102C4×C197—41no—
103Dic197—42no—
104Dic197—42no—
105C2×C197—21no—
106C2×C197—21no—
107C2×C197—42no—
108C2×C197#44711yescharacteristic
109C2×C197#45311yescharacteristic
110C2×C197B(4,𝔽2)11yescharacteristic
111D197—82no—
112D197—82no—
113C197#159511yescharacteristic, complemented, Sylow
114#1595—1971nomaximal, Sylow
115#534—1971no—
116#447—1972no—
117#392—1971no—
118#327—1972no—
119#323D19711yescharacteristic, complemented
120C22×D4—3941no—
121C22×D4—3941no—
122C22×D4—1971no—
123C2×D8—3942no—
124#123—3942no—
125#121—3941no—
126#121—1972no—
127#120—3942no—
128C2×C22⋊C4—3941no—
129C2×C22⋊C4C2×D19711yescharacteristic
130#102—3942no—
131C22⋊C8C2×D19712yescomplemented
132#95—1971no—
133C24—3941no—
134C24—3941no—
135C24D78811yescharacteristic
136C2×D4—7882no—
137C2×D4—7882no—
138C2×D4—7882no—
139C2×D4—7882no—
140C2×D4—7882no—
141C2×D4—7882no—
142C2×D4—3941no—
143C2×D4—3942no—
144C2×D4—3942no—
145C2×D4—3942no—
146C2×D4—1972no—
147C22×C4—3941no—
148C22×C4—3941no—
149C22×C4D78811yescharacteristic
150C22×C4C22×D19711yescharacteristic
151D8—7884no—
152C2×C8—22no—
153C4⋊C4—3941no—
154C4⋊C4—1972no—
155C22⋊C4—22no—
156C22⋊C4—7882no—
157C22⋊C4—3942no—
158C22⋊C4—3942no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—7881no—
164C23—7881no—
165C23—7881no—
166C23—7882no—
167C23—7881no—
168C23—7881no—
169C23—7881no—
170C23—15761no—
171C23—15761no—
172C23—3942no—
173C23—3942no—
174C23D157612yescomplemented
175C23C2×D78811yescharacteristic
176D4—7882no—
177D4—7882no—
178D4—7884no—
179D4—7888no—
180D4—7882no—
181D4—7882no—
182D4—15761no—
183D4—3944no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—7881no—
188C2×C4—7882no—
189C2×C4—7881no—
190C2×C4—3942no—
191C2×C4—3942no—
192C2×C4D4×D19712yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—7882no—
202C22—7882no—
203C22—7882no—
204C22—7882no—
205C22—7882no—
206C22—7882no—
207C22—15761no—
208C22—15761no—
209C22—15764no—
210C22—15761no—
211C22—15761no—
212C22#2026329611yescharacteristic
213C22C2×D157611yescharacteristic
214C22#2026335411yescharacteristic
215C4—22no—
216C4—41no—
217C4—7882no—
218C4—7882no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—15762no—
223C2—15762no—
224C2#2716121511yescharacteristic
225C2#2716124911yescharacteristic
226C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197#120
hypercenter#323D197
hypocenterC197#1595
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
1. #27161641C2semidirect
2. #27161215C2semidirect
3. #27160988C2semidirect
4. #27160814C2semidirect
5. #323×C197C2semidirect
6. C22×D788C22semidirect
16. #20263233C22semidirect
18. C22⋊C8×C197C22semidirect
24. C2×D788D4semidirect
36. C4⋊Dic197D4semidirect
73. C23×C197D8semidirect
113. C197#1595semidirect
119. #323D197semidirect
131. C22⋊C8C2×D197semidirect
174. C23D1576semidirect

Maximal subgroups

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

subgroupindex
#271616412
#271612152
#271609882
#271608142
#323×C1972
#1595197