Finite Group Databasebeta

← #38752317

Subgroups — #38752317

Order 25216.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15691341C211yesmaximal, characteristic, complemented
2#27161586C212yesmaximal, complemented
3#27161334C212yesmaximal, complemented
4#27161329C212yesmaximal, complemented
5#27161322C214yesmaximal, complemented
6#27161117C211yesmaximal, characteristic, complemented
7#27161075C212yesmaximal, complemented
8C2×#135×C197C211yesmaximal, characteristic, complemented
9C22×C4×D197C2211yescharacteristic
10C2×#15691341—24no—
11C2×#15691341C2212yes—
12C2×#15691341C2212yes—
13C22×#13096802C2211yescharacteristic
14#20263307—22no—
15#20263302C2214yes—
16#20263297C2214yes—
17#20263294—22no—
18C4×C22⋊D197—24no—
19#20263285C2212yes—
20#20263260C2212yescomplemented
21#20263258C2212yescomplemented
22#20263250C2214yescomplemented
23#20263247C2214yescomplemented
24C2×#41×C197C2212yes—
25#135×C197—24no—
26C2×Q16×C197C2211yescharacteristic, complemented
27C2×SD16×C197C2212yescomplemented
28C2×D8×C197C2211yescharacteristic, complemented
29C22×C8×C197C2211yescharacteristic
30C23×D197D411yescharacteristic
31C2×C22⋊D197—24no—
32C22×Dic197D411yescharacteristic
33C2×C4×D197—22no—
34C2×C4×D197—22no—
35C2×C4×D197D412yes—
36C2×C4×D197C2312yes—
37#15691341—24no—
38#15691341—24no—
39C2×#13096802—22no—
40C2×#13096802C2312yes—
41C4⋊Dic197C2314yes—
42#15691337—24no—
43C22⋊Dic197—24no—
44C22⋊Dic197—22no—
45#15691335—24no—
46C4×Dic197—22no—
47#41×C197—24no—
48#41×C197—24no—
49C2×Q8×C197C2312yes—
50C2×D4×C197—22no—
51C2×D4×C197C2312yes—
52C22×C4×C197—22no—
53C22×C4×C197C2311yescharacteristic
54Q16×C197—22no—
55SD16×C197—24no—
56D8×C197—22no—
57C2×C8×C197D414yescomplemented
58C2×C8×C197C2311yescharacteristic
59C2×C8×C197C2311yescharacteristic
60C22×D197—22no—
61C22×D197—42no—
62C22×D197C2×D412yes—
63C22⋊D197—48no—
64C2×Dic197—22no—
65C2×Dic197—24no—
66C2×Dic197C2×D412yes—
67C4×D197—24no—
68C4×D197—22no—
69C4×D197—22no—
70#13096802—22no—
71C23×C197—22no—
72C23×C197C2×D411yescharacteristic
73Q8×C197—22no—
74Q8×C197—22no—
75D4×C197—22no—
76D4×C197—22no—
77D4×C197—44no—
78C2×C4×C197—22no—
79C2×C4×C197—44no—
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D412yes—
82C2×C4×C197#4112yes—
83C2×C4×C197C2411yescharacteristic
84C8×C197C2×D412yes—
85C8×C197C2×D412yes—
86C22×C197—22no—
87C22×C197—42no—
88C22×C197—44no—
89C22×C197#12112yes—
90C22×C197C22×D411yescharacteristic
91C2×D197—22no—
92C2×D197—24no—
93C2×D197—44no—
94C4×C197—42no—
95C4×C197#12112yes—
96C4×C197C22×D411yescharacteristic
97C4×C197C2×#4111yescharacteristic
98Dic197—22no—
99Dic197—44no—
100C2×C197—22no—
101C2×C197—82no—
102C2×C197C2×#12111yescharacteristic
103C2×C197#57512yes—
104D197—42no—
105C197#305411yescharacteristic, complemented, Sylow
106#3054—1971nomaximal, Sylow
107C2×#135D19711yescharacteristic, complemented
108C22×M16—1971no—
109#525—1972no—
110#470—1972no—
111#469—1972no—
112#468—1974no—
113#427—1971no—
114#416—1972no—
115C2×#41C2×D19712yes—
116C23×C4—1971no—
117#135—24no—
118C2×Q16C2×D19711yescharacteristic, complemented
119C2×SD16C2×D19712yescomplemented
120C2×D8C2×D19711yescharacteristic, complemented
121C2×M16—3944no—
122C2×M16—1972no—
123C2×M16—1972no—
124C22×C8—1971no—
125C22×C8C2×D19711yescharacteristic
126#123—3942no—
127C22⋊Q8—1974no—
128#121—1974no—
129#120—3942no—
130C4×D4—3944no—
131#117—1972no—
132#107—1972no—
133#106—1972no—
134#103—1974no—
135#102—1974no—
136C24—1971no—
137#41—24no—
138#41—24no—
139C2×Q8C22×D19712yes—
140C2×D4—22no—
141C2×D4—3944no—
142C2×D4C22×D19712yes—
143C22×C4—22no—
144C22×C4—3942no—
145C22×C4—3942no—
146C22×C4—1971no—
147C22×C4—1972no—
148C22×C4—1972no—
149C22×C4C22×D19711yescharacteristic
150Q16—22no—
151SD16—24no—
152D8—22no—
153M16—3944no—
154M16—3944no—
155C2×C8—3942no—
156C2×C8—1972no—
157C2×C8C22⋊D19714yescomplemented
158C2×C8C22×D19711yescharacteristic
159C2×C8C22×D19711yescharacteristic
160C4⋊C4—3944no—
161C4⋊C4—1974no—
162C22⋊C4—3944no—
163C22⋊C4—3942no—
164C22⋊C4—3944no—
165C42—3942no—
166C23—22no—
167C23—7882no—
168C23—3942no—
169C23—1972no—
170C23D4×D19711yescharacteristic
171Q8—22no—
172Q8—22no—
173D4—22no—
174D4—22no—
175D4—44no—
176D4—7888no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—3944no—
180C2×C4—3942no—
181C2×C4—3944no—
182C2×C4—3942no—
183C2×C4—3942no—
184C2×C4—1972no—
185C2×C4D4×D19711yescharacteristic
186C2×C4C2×C22⋊D19712yes—
187C2×C4#1569135712yes—
188C2×C4C23×D19711yescharacteristic
189C8—3942no—
190C8C2×C22⋊D19712yes—
191C8C2×C22⋊D19712yes—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—7884no—
196C22—3944no—
197C22—3942no—
198C22#2026329712yes—
199C22C2×D4×D19711yescharacteristic
200C4—42no—
201C4—7884no—
202C4—3942no—
203C4#2026329712yes—
204C4C22×C22⋊D19711yescharacteristic
205C4C2×#1569135711yescharacteristic
206C2—22no—
207C2—82no—
208C2—7882no—
209C2C2×#2026329711yescharacteristic
210C2#2716187612yes—
subgroupG/H
centerC2×C4D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#135×C197C2
socleC22×C197C22×D4
hypercenterC2×#135D197
hypocenterC197#3054
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×#15691341C2semidirect
2. #27161586C2semidirect
3. #27161334C2semidirect
4. #27161329C2semidirect
5. #27161322C2semidirect
6. #27161117C2semidirect
7. #27161075C2semidirect
8. C2×#135×C197C2semidirect
20. #20263260C22semidirect
21. #20263258C22semidirect
22. #20263250C22semidirect
23. #20263247C22semidirect
26. C2×Q16×C197C22semidirect
27. C2×SD16×C197C22semidirect
28. C2×D8×C197C22semidirect
57. C2×C8×C197D4semidirect
105. C197#3054semidirect
107. C2×#135D197semidirect
118. C2×Q16C2×D197semidirect
119. C2×SD16C2×D197semidirect
120. C2×D8C2×D197semidirect
157. C2×C8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913412
#271615862
#271613342
#271613292
#271613222
#271611172
#271610752
C2×#135×C1972
#3054197