Finite Group Databasebeta

← #38752964

Subgroups — #38752964

Order 25216.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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#27161691C211yesmaximal, characteristic, complemented
2#27161420C212yesmaximal, complemented
3#27161379C212yesmaximal, complemented
4#27161284C212yesmaximal, complemented
5#27161230C212yesmaximal
6#27161179C212yesmaximal, complemented
7C4×#15691346C212yesmaximal, complemented
8#27161140C211yesmaximal, characteristic, complemented
9#518×C197C211yesmaximal, characteristic
10C2×#15691346C2212yes—
11C2×#15691346C2212yes—
12#20263326C2211yescharacteristic, complemented
13#20263319—22no—
14#20263318C2211yescharacteristic, complemented
15#20263316—22no—
16#20263305C2212yes—
17C4×Dic394C2212yescomplemented
18#20263259C2212yescomplemented
19#20263257C2212yescomplemented
20C4⋊#13096802C2212yescomplemented
21#20263249C2212yescomplemented
22#20263249C2212yescomplemented
23#20263247C2212yes—
24#20263247C2212yes—
25C22⋊#13096802C2212yes—
26#20263237C2211yescharacteristic, complemented
27C4×#13096802C2211yescharacteristic, complemented
28C22×D4×C197C2211yescharacteristic
29C4×D4×C197—22no—
30C4×D4×C197C2212yes—
31C4×D4×C197C2212yes—
32#117×C197C2211yescharacteristic
33#117×C197C2211yescharacteristic
34C2×C22⋊C4×C197—22no—
35C2×Dic394C2312yes—
36#15691346—24no—
37#15691346—24no—
38C2×#13096802C2312yes—
39C2×#13096802C2312yes—
40C4⋊Dic197—22no—
41C4⋊Dic197C2312yes—
42#15691337—22no—
43#15691337—22no—
44#15691337—22no—
45C22⋊Dic197—22no—
46C4×Dic197—22no—
47C24×C197—21no—
48C2×D4×C197—24no—
49C2×D4×C197—21no—
50C2×D4×C197C2312yes—
51C22×C4×C197—22no—
52C22×C4×C197—22no—
53C22×C4×C197C2311yescharacteristic
54C4⋊C4×C197D412yes—
55C4⋊C4×C197C2312yes—
56C22⋊C4×C197—21no—
57C22⋊C4×C197—22no—
58C22⋊C4×C197—22no—
59C22⋊C4×C197D412yes—
60C42×C197—21no—
61C42×C197C2311yescharacteristic
62C42×C197C2311yescharacteristic
63Dic394—22no—
64Dic394—22no—
65C2×Dic197—22no—
66C2×Dic197—22no—
67#13096802—22no—
68#13096802—22no—
69C23×C197—24no—
70C23×C197—22no—
71C23×C197—82no—
72C23×C197C2×D411yescharacteristic
73D4×C197—24no—
74D4×C197—22no—
75D4×C197#4114yes—
76C2×C4×C197—22no—
77C2×C4×C197—22no—
78C2×C4×C197—82no—
79C2×C4×C197—82no—
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D412yes—
82C2×C4×C197C2×D412yes—
83C2×C4×C197C2411yescharacteristic
84C22×C197—21no—
85C22×C197—24no—
86C22×C197—22no—
87C22×C197—42no—
88C22×C197—42no—
89C22×C197—84no—
90C22×C197C22×D411yescharacteristic
91C4×C197—21no—
92C4×C197—22no—
93C4×C197—22no—
94C4×C197—42no—
95C4×C197C2×#4112yes—
96Dic197—42no—
97Dic197—42no—
98C2×C197—41no—
99C2×C197—44no—
100C2×C197#52511yescharacteristic
101C2×C197#57712yes—
102C197#312311yescharacteristic, complemented, Sylow
103#3123—1971nomaximal, Sylow
104#541—1971no—
105#518D19711yescharacteristic
106#485—1972no—
107#478—1972no—
108#461—1972no—
109#450—1972no—
110#440—1972no—
111C4×SD16—1972no—
112#433—1971no—
113C22×D4C2×D19711yescharacteristic
114C2×SD16—1972no—
115C2×SD16—1972no—
116C4⋊Q8—1971no—
117#126—3942no—
118#125—3942no—
119#125—1971no—
120C22⋊Q8—1972no—
121C4×Q8—1972no—
122C4×D4—22no—
123C4×D4C2×D19712yes—
124C4×D4C2×D19712yes—
125#117C2×D19711yescharacteristic
126#117C2×D19711yescharacteristic
127C2×C22⋊C4—22no—
128#107—1972no—
129#106—1972no—
130C4⋊C8—1972no—
131#103—1972no—
132#103—1972no—
133#102—1972no—
134#102—1972no—
135C22⋊C8—1972no—
136#97—1971no—
137C4×C8—1971no—
138C24—21no—
139C2×Q8—1972no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D19712yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D19711yescharacteristic
146SD16—3944no—
147SD16—3944no—
148C2×C8—1972no—
149C2×C8—1972no—
150C4⋊C4—3942no—
151C4⋊C4—3942no—
152C4⋊C4—3942no—
153C4⋊C4—3942no—
154C4⋊C4—1972no—
155C4⋊C4C22⋊D19712yes—
156C4⋊C4C22×D19712yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—3942no—
161C22⋊C4C22⋊D19712yes—
162C42—21no—
163C42—3942no—
164C42C22×D19711yescharacteristic
165C42C22×D19711yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D19711yescharacteristic
170Q8—3942no—
171Q8—3942no—
172D4—24no—
173D4—22no—
174D4#1569135814yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—3942no—
180C2×C4—3942no—
181C2×C4C2×C22⋊D19711yescharacteristic
182C2×C4C2×C22⋊D19712yes—
183C2×C4C2×C22⋊D19712yes—
184C2×C4C23×D19711yescharacteristic
185C8—3942no—
186C8—3942no—
187C22—22no—
188C22—21no—
189C22—24no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D19711yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—42no—
198C4—7882no—
199C4—7882no—
200C4C2×#1569135812yes—
201C2—41no—
202C2—44no—
203C2#2716159011yescharacteristic
204C2#2716189012yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#518×C197C2
socleC22×C197C22×D4
prime radicalC22×D4×C197C22
hypercenter#518D197
hypocenterC197#3123
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #27161691C2semidirect
2. #27161420C2semidirect
3. #27161379C2semidirect
4. #27161284C2semidirect
6. #27161179C2semidirect
7. C4×#15691346C2semidirect
8. #27161140C2semidirect
12. #20263326C22semidirect
14. #20263318C22semidirect
17. C4×Dic394C22semidirect
18. #20263259C22semidirect
19. #20263257C22semidirect
20. C4⋊#13096802C22semidirect
21. #20263249C22semidirect
22. #20263249C22semidirect
26. #20263237C22semidirect
27. C4×#13096802C22semidirect
102. C197#3123semidirect

Maximal subgroups

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

subgroupindex
#271616912
#271614202
#271613792
#271612842
#271612302
#271611792
C4×#156913462
#271611402
#518×C1972
#3123197