Finite Group Databasebeta

← #38742629

Subgroups — #38742629

Order 25216.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#15691358C211yesmaximal, characteristic
2C2×#20263240C214yesmaximal, complemented
3#27160942C211yesmaximal, characteristic, complemented
4#381×C197C211yesmaximal, characteristic, complemented
5C2×#15691358—28no—
6C2×#15691358—22no—
7C2×#15691358—22no—
8C22×Dic394C411yescharacteristic, complemented
9C22×D788C411yescharacteristic, complemented
10C22×C22⋊D197C2212yes—
11C22×C4×D197C412yescomplemented
12C2×C22⋊Dic197C2212yes—
13#20263240—28no—
14#20263233—22no—
15C23×C4×C197C2211yescharacteristic
16C2×C22⋊C4×C197C2212yes—
17#95×C197—22no—
18C23×D197C2×C412yes—
19#15691358—216no—
20#15691358—216no—
21C2×Dic394—24no—
22C2×Dic394—22no—
23C2×D788—24no—
24C2×D788—22no—
25C2×C22⋊D197—24no—
26C2×C22⋊D197—22no—
27C2×C22⋊D197—22no—
28C2×C22⋊D197D418yes—
29C22×Dic197—22no—
30C22×Dic197C2×C412yes—
31C2×C4×D197—24no—
32C2×C4×D197—24no—
33C2×C4×D197—24no—
34C22⋊Dic197—24no—
35C24×C197C2311yescharacteristic
36C22×C4×C197—24no—
37C22×C4×C197—22no—
38C22×C4×C197C2×C411yescharacteristic
39C22×C4×C197C2×C411yescharacteristic
40C22×C4×C197D412yes—
41C22×C4×C197D412yes—
42C22⋊C4×C197—24no—
43C22×D197—22no—
44C22×D197—22no—
45C22×D197—44no—
46C22×D197C22⋊C414yes—
47Dic394—28no—
48D788—28no—
49C22⋊D197—28no—
50C22⋊D197—28no—
51C2×Dic197—22no—
52C2×Dic197—22no—
53C2×Dic197—44no—
54C2×Dic197—42no—
55C2×Dic197C22⋊C414yes—
56C4×D197—216no—
57C23×C197—42no—
58C23×C197C22×C411yescharacteristic
59C23×C197C2×D414yes—
60C23×C197C2×D411yescharacteristic
61C23×C197C2×D411yescharacteristic
62C2×C4×C197—22no—
63C2×C4×C197—22no—
64C2×C4×C197—24no—
65C2×C4×C197—24no—
66C2×C4×C197—44no—
67C2×C4×C197—42no—
68C2×C4×C197C22⋊C412yes—
69C2×C4×C197C22⋊C412yes—
70C22×C197—22no—
71C22×C197—22no—
72C22×C197—22no—
73C22×C197—44no—
74C22×C197C2×C22⋊C412yes—
75C22×C197C2×C22⋊C411yescharacteristic
76C22×C197#12012yes—
77C22×C197#12012yes—
78C2×D197—24no—
79C2×D197—44no—
80C2×D197—44no—
81C2×D197—44no—
82C4×C197—22no—
83C4×C197—22no—
84C4×C197—82no—
85Dic197—24no—
86Dic197—82no—
87C2×C197—22no—
88C2×C197—22no—
89C2×C197—22no—
90C2×C197#37911yescharacteristic
91C2×C197#41012yes—
92D197—44no—
93C197#178211yescharacteristic, complemented, Sylow
94#1782—1971nomaximal, Sylow
95C22×#41—1971no—
96C2×#99—1974no—
97#381—1971no—
98#381D19711yescharacteristic, complemented
99C2×#41—3948no—
100C2×#41—3942no—
101C2×#41—3942no—
102C22×Q8—1971no—
103C22×D4—1972no—
104C22×D4—1971no—
105C23×C4—1972no—
106C23×C4C2×D19711yescharacteristic
107C2×C22⋊C4—1972no—
108C2×C22⋊C4C2×D19712yes—
109#99—3948no—
110#95—22no—
111#95—3942no—
112C24—1972no—
113C24C22×D19711yescharacteristic
114#41—39416no—
115#41—39416no—
116C2×Q8—3944no—
117C2×Q8—3942no—
118C2×D4—3944no—
119C2×D4—3944no—
120C2×D4—3942no—
121C2×D4—3942no—
122C2×D4—3942no—
123C2×D4—1978no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—3944no—
127C22×C4—3944no—
128C22×C4—3942no—
129C22×C4—3944no—
130C22×C4—1972no—
131C22×C4C4×D19711yescharacteristic
132C22×C4C4×D19711yescharacteristic
133C22×C4C22⋊D19712yes—
134C22×C4D78812yes—
135C22⋊C4—24no—
136C22⋊C4—3944no—
137C23—42no—
138C23—7884no—
139C23—3942no—
140C23—3942no—
141C23—1974no—
142C23C2×C4×D19711yescharacteristic
143C23D4×D19714yes—
144C23C2×C22⋊D19711yescharacteristic
145C23C2×D78811yescharacteristic
146Q8—3948no—
147D4—3948no—
148D4—3948no—
149D4—3948no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—7884no—
157C2×C4—7882no—
158C2×C4—39416no—
159C2×C4—3942no—
160C2×C4—3942no—
161C2×C4—1974no—
162C2×C4#1569133512yes—
163C2×C4#1569133512yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—7884no—
169C22—7884no—
170C22—7884no—
171C22—3944no—
172C22C22⋊C4×D19712yes—
173C22C2×#1569133511yescharacteristic
174C22#2026329412yes—
175C22#2026329612yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—15762no—
180C4—3944no—
181C2—22no—
182C2—22no—
183C2—22no—
184C2—7884no—
185C2#2716093411yescharacteristic
186C2#2716105212yes—
subgroupG/H
centerC22C2×#15691335
commutatorC23×C197C22×C4
FrattiniC24C22×D197
Fitting#381×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC22×#15691358C2
cosocleC24×C197C23
hypercenter#381D197
hypocenterC197#1782
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#20263240C2semidirect
3. #27160942C2semidirect
4. #381×C197C2semidirect
8. C22×Dic394C4semidirect
9. C22×D788C4semidirect
11. C22×C4×D197C4semidirect
93. C197#1782semidirect
98. #381D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913582
C2×#202632402
#271609422
#381×C1972
#1782197