Finite Group Databasebeta

← #38743150

Subgroups — #38743150

Order 25216.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#20263300C211yesmaximal, characteristic
2C2×#20263248C212yesmaximal, complemented
3#27160939C211yesmaximal, characteristic, complemented
4#27160842C212yesmaximal, complemented
5C22×M16×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7C22×C22⋊D197—21no—
8#20263300—22no—
9#20263300C414yescomplemented
10C2×C4⋊Dic197C2212yes—
11C2×C4⋊Dic197C2211yescharacteristic
12C2×C22⋊Dic197—21no—
13C2×#15691335—21no—
14#20263248—24no—
15#20263233—21no—
16C23×C4×C197C2211yescharacteristic
17C2×M16×C197—22no—
18C2×M16×C197—24no—
19C22×C8×C197C2212yes—
20C23×D197—21no—
21C2×D788—22no—
22C2×D788—21no—
23C2×D788—22no—
24C2×D788C2×C412yes—
25C2×C22⋊D197—24no—
26C2×C22⋊D197—44no—
27C22×Dic197—21no—
28C22×Dic197—22no—
29C4⋊Dic197—21no—
30C4⋊Dic197—22no—
31C4⋊Dic197—22no—
32C4⋊Dic197C2×C412yes—
33C22⋊Dic197—42no—
34#15691335—24no—
35C24×C197D411yescharacteristic
36C22×C4×C197—22no—
37C22×C4×C197—22no—
38C22×C4×C197C2×C412yes—
39C22×C4×C197D412yes—
40C22×C4×C197D411yescharacteristic
41C22×C4×C197C2311yescharacteristic
42M16×C197—28no—
43C2×C8×C197—22no—
44C2×C8×C197—22no—
45C2×C8×C197D414yescomplemented
46C22×D197—22no—
47C22×D197—42no—
48C22×D197—42no—
49C22×D197—41no—
50C22×D197—41no—
51D788—24no—
52D788—41no—
53D788—41no—
54C22⋊D197—48no—
55C2×Dic197—22no—
56C2×Dic197—42no—
57C2×Dic197—42no—
58C2×Dic197—42no—
59C2×Dic197—42no—
60C23×C197—22no—
61C23×C197—21no—
62C23×C197—22no—
63C23×C197—21no—
64C23×C197C22⋊C412yes—
65C23×C197C2×D411yescharacteristic
66C2×C4×C197—22no—
67C2×C4×C197—24no—
68C2×C4×C197—22no—
69C2×C4×C197—22no—
70C2×C4×C197C22⋊C412yes—
71C2×C4×C197C22×C411yescharacteristic
72C2×C4×C197C2×D412yes—
73C2×C4×C197C2×D411yescharacteristic
74C2×C4×C197#4112yes—
75C8×C197—24no—
76C22×C197—22no—
77C22×C197—22no—
78C22×C197—22no—
79C22×C197—42no—
80C22×C197—42no—
81C22×C197C2×C22⋊C411yescharacteristic
82C22×C197#12112yes—
83C22×C197AGL(1,ℤ/8)11yescharacteristic
84C22×C197AGL(1,ℤ/8)12yes—
85C22×C197#13711yescharacteristic
86C2×D197—42no—
87C2×D197—42no—
88C2×D197—42no—
89C2×D197—82no—
90C2×D197—82no—
91C4×C197—22no—
92C4×C197C4×D412yes—
93C4×C197#12011yescharacteristic
94C4×C197#12311yescharacteristic
95Dic197—42no—
96Dic197—82no—
97C2×C197—42no—
98C2×C197#38611yescharacteristic
99C2×C197#41711yescharacteristic
100C2×C197#41811yescharacteristic
101C2×C197#46812yes—
102C2×C197#46912yes—
103D197—82no—
104C197#189511yescharacteristic, complemented, Sylow
105#1895—1971nomaximal, Sylow
106C22×M16D19711yescharacteristic, complemented
107C2×#121—1971no—
108C2×#102—1972no—
109#380—1971no—
110#340—1972no—
111C22×D4—3941no—
112C22×D4—1971no—
113C23×C4C2×D19711yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D19712yes—
117#121—3942no—
118#121—1974no—
119C2×C4⋊C4—1972no—
120C2×C4⋊C4—1971no—
121C2×C22⋊C4—3941no—
122C2×C22⋊C4—3941no—
123#102—3944no—
124#95—3941no—
125C24—3941no—
126C24D78811yescharacteristic
127C2×D4—7884no—
128C2×D4—3944no—
129C2×D4—3941no—
130C2×D4—3942no—
131C2×D4—3942no—
132C2×D4—1972no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—3942no—
136C22×C4—3941no—
137C22×C4C4×D19712yes—
138C22×C4C22⋊D19712yes—
139C22×C4D78811yescharacteristic
140C22×C4C22×D19711yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D19714yescomplemented
145C4⋊C4—3942no—
146C4⋊C4—3942no—
147C4⋊C4—3941no—
148C4⋊C4—1972no—
149C22⋊C4—7882no—
150C22⋊C4—3944no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—7882no—
156C23—7882no—
157C23—7881no—
158C23—7881no—
159C23—3942no—
160C23#1569133512yes—
161C23C2×D78811yescharacteristic
162D4—7888no—
163D4—7881no—
164D4—7881no—
165D4—3944no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—7882no—
171C2×C4—7882no—
172C2×C4—7882no—
173C2×C4—7882no—
174C2×C4—3942no—
175C2×C4#1569133512yes—
176C2×C4C2×C4×D19711yescharacteristic
177C2×C4C2×C22⋊D19712yes—
178C2×C4C2×C22⋊D19711yescharacteristic
179C2×C4#1569135812yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—7882no—
187C22—7882no—
188C22—7882no—
189C22—15762no—
190C22—15762no—
191C22C2×#1569133511yescharacteristic
192C22#2026330012yes—
193C22#2026335412yes—
194C22#2026335411yescharacteristic
195C22#2026335911yescharacteristic
196C4—22no—
197C4—7882no—
198C4—15762no—
199C4C4×C22⋊D19712yes—
200C4#2026329511yescharacteristic
201C4#2026330911yescharacteristic
202C2—42no—
203C2—15762no—
204C2#2716095811yescharacteristic
205C2#2716108111yescharacteristic
206C2#2716108511yescharacteristic
207C2#2716132512yes—
208C2#2716133212yes—
subgroupG/H
centerC23C2×D788
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
FittingC22×M16×C197C2
socleC23×C197C2×D4
prime radicalC2×#20263300C2
cosocleC22×C4×C197C23
hypercenterC22×M16D197
hypocenterC197#1895
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#20263248C2semidirect
3. #27160939C2semidirect
4. #27160842C2semidirect
5. C22×M16×C197C2semidirect
9. #20263300C4semidirect
45. C2×C8×C197D4semidirect
104. C197#1895semidirect
106. C22×M16D197semidirect
144. C2×C8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633002
C2×#202632482
#271609392
#271608422
C22×M16×C1972
#1895197