Finite Group Databasebeta

← #38751949

Subgroups — #38751949

Order 25216.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×Dic394C211yesmaximal, characteristic, complemented
2C2×#20263359C212yesmaximal, complemented
3#27161590C211yesmaximal, characteristic
4#27161237C214yesmaximal, complemented
5#27161220C214yesmaximal, complemented
6#27161089C212yesmaximal, complemented
7#407×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2211yescharacteristic
9C22×Dic394—28no—
10C22×Dic394—22no—
11C22×Dic394—21no—
12C22×Dic394C2214yes—
13C22×Dic394C2212yes—
14C23×Dic197—21no—
15#20263359—28no—
16C2×Dic788C2214yescomplemented
17C2×#15691347C2214yescomplemented
18#20263309—21no—
19#20263305C2212yes—
20#20263300C2212yes—
21#20263295—21no—
22C4×C22⋊D197—22no—
23#20263285C2211yescharacteristic
24#20263251C2218yescomplemented
25C23×C4×C197C2211yescharacteristic
26C2×M16×C197C2212yes—
27C22⋊C8×C197C2214yes—
28#15691358—24no—
29C2×Dic394—28no—
30C2×Dic394—216no—
31C2×Dic394—216no—
32C2×Dic394—24no—
33C2×Dic394—24no—
34C2×Dic394—22no—
35C2×Dic394D418yescomplemented
36C2×Dic394C2311yescharacteristic
37C2×Dic394C2314yes—
38C2×D788C2311yescharacteristic
39C2×C22⋊D197—22no—
40C2×C22⋊D197—21no—
41C22×Dic197—28no—
42C22×Dic197—22no—
43C22×Dic197—24no—
44C2×C4×D197—21no—
45Dic788—28no—
46#15691347—28no—
47C4⋊Dic197C2312yes—
48#15691337—22no—
49C22⋊Dic197—21no—
50C22⋊Dic197—22no—
51#15691335—22no—
52C4×Dic197—21no—
53C24×C197D411yescharacteristic
54C22×C4×C197—24no—
55C22×C4×C197D412yes—
56C22×C4×C197D411yescharacteristic
57C22×C4×C197C2312yes—
58C22×C4×C197C2311yescharacteristic
59M16×C197—24no—
60C2×C8×C197C2314yes—
61C22×D197—21no—
62Dic394—216no—
63Dic394—28no—
64Dic394—24no—
65Dic394—22no—
66Dic394C2×D418yes—
67D788—22no—
68C22⋊D197—42no—
69C22⋊D197—44no—
70C2×Dic197—216no—
71C2×Dic197—28no—
72C2×Dic197—24no—
73C2×Dic197—22no—
74C2×Dic197—21no—
75C4×D197—42no—
76C23×C197—22no—
77C23×C197—24no—
78C23×C197C2×D411yescharacteristic
79C23×C197C2×D412yes—
80C2×C4×C197—22no—
81C2×C4×C197—24no—
82C2×C4×C197—24no—
83C2×C4×C197C2×D414yes—
84C2×C4×C197C2×D412yes—
85C2×C4×C197C2×D411yescharacteristic
86C2×C4×C197C2411yescharacteristic
87C8×C197—24no—
88C22×C197—28no—
89C22×C197—22no—
90C22×C197—22no—
91C22×C197—22no—
92C22×C197#12012yes—
93C22×C197#13714yescomplemented
94C22×C197C22×D411yescharacteristic
95C2×D197—41no—
96C2×D197—42no—
97C4×C197—22no—
98C4×C197#12012yes—
99C4×C197C22×D412yes—
100Dic197—28no—
101Dic197—41no—
102Dic197—42no—
103C2×C197—22no—
104C2×C197—24no—
105C2×C197C2×#12011yescharacteristic
106C2×C197C2×#13712yes—
107D197—81no—
108C197#300411yescharacteristic, complemented, Sylow
109#3004—1971nomaximal, Sylow
110C23×Q8—1971no—
111C2×#137—1972no—
112#525—1971no—
113#451—1974no—
114#448—1974no—
115#419—1972no—
116#407D19711yescharacteristic, complemented
117C2×#41—1971no—
118C22×Q8—3948no—
119C22×Q8—3942no—
120C22×Q8—3941no—
121C22×Q8—1974no—
122C22×Q8—1972no—
123C23×C4—3941no—
124C23×C4C2×D19711yescharacteristic
125#137—3948no—
126C2×Q16—1974no—
127C2×SD16—1974no—
128C2×M16C2×D19712yes—
129#123—3941no—
130C22⋊Q8—1972no—
131#121—1972no—
132#120—3941no—
133C4×D4—3942no—
134#117—1971no—
135#103—1978no—
136C22⋊C8C2×D19714yes—
137C24D78811yescharacteristic
138#41—3944no—
139C2×Q8—3948no—
140C2×Q8—39416no—
141C2×Q8—39416no—
142C2×Q8—3944no—
143C2×Q8—3944no—
144C2×Q8—3942no—
145C2×Q8—1978no—
146C2×Q8—1971no—
147C2×Q8—1974no—
148C2×D4—3942no—
149C2×D4—3941no—
150C2×D4—1971no—
151C22×C4—24no—
152C22×C4—3941no—
153C22×C4—3948no—
154C22×C4—3944no—
155C22×C4—3942no—
156C22×C4D78812yes—
157C22×C4D78811yescharacteristic
158C22×C4C22×D19712yes—
159C22×C4C22×D19711yescharacteristic
160Q16—3948no—
161SD16—3948no—
162M16—24no—
163C2×C8C22×D19714yes—
164C4⋊C4—3942no—
165C4⋊C4—1972no—
166C22⋊C4—3941no—
167C22⋊C4—3942no—
168C22⋊C4—3942no—
169C42—3941no—
170C23—22no—
171C23—24no—
172C23—3941no—
173C23C2×D78811yescharacteristic
174C23C2×D78812yes—
175Q8—39416no—
176Q8—3942no—
177Q8—3948no—
178Q8—3944no—
179Q8—1978no—
180D4—7882no—
181D4—7884no—
182D4—3942no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—7882no—
187C2×C4—3942no—
188C2×C4—3941no—
189C2×C4—39416no—
190C2×C4—3948no—
191C2×C4—3944no—
192C2×C4D4×D19714yes—
193C2×C4C2×D78812yes—
194C2×C4C2×D78811yescharacteristic
195C2×C4C23×D19711yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—7882no—
202C22—7881no—
203C22#2026329612yes—
204C22#2026335914yescomplemented
205C22C22×D78811yescharacteristic
206C4—22no—
207C4—7882no—
208C4—7881no—
209C4—3948no—
210C4#2026329612yes—
211C4C2×D4×D19712yes—
212C2—22no—
213C2—24no—
214C2—15761no—
215C2C2×#2026329611yescharacteristic
216C2C2×#2026335912yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#407×C197C2
socleC22×C197C22×D4
prime radical#27161590C2
hypercenter#407D197
hypocenterC197#3004
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic394C2semidirect
2. C2×#20263359C2semidirect
4. #27161237C2semidirect
5. #27161220C2semidirect
6. #27161089C2semidirect
7. #407×C197C2semidirect
16. C2×Dic788C22semidirect
17. C2×#15691347C22semidirect
24. #20263251C22semidirect
35. C2×Dic394D4semidirect
93. C22×C197#137semidirect
108. C197#3004semidirect
116. #407D197semidirect
204. C22#20263359semidirect

Maximal subgroups

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

subgroupindex
C23×Dic3942
C2×#202633592
#271615902
#271612372
#271612202
#271610892
#407×C1972
#3004197