Finite Group Databasebeta

← #38752289

Subgroups — #38752289

Order 25216.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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×C8×D197C211yesmaximal, characteristic, complemented
2#27161586C212yesmaximal, complemented
3#27161317C212yesmaximal, complemented
4#27161312C212yesmaximal, complemented
5#27161305C214yesmaximal, complemented
6#27161117C211yesmaximal, characteristic, complemented
7#27161075C212yesmaximal, complemented
8C2×#135×C197C211yesmaximal, characteristic, complemented
9C22×C4×D197C2211yescharacteristic
10C22×#13096802C2211yescharacteristic
11C2×C8×D197—24no—
12C2×C8×D197C2212yes—
13C2×C8×D197C2212yes—
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—
37C2×#13096802—22no—
38C2×#13096802C2312yes—
39C8×D197—24no—
40C8×D197—24no—
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—24no—
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—28no—
92C2×D197—22no—
93C2×D197—22no—
94C2×D197#13514yescomplemented
95C4×C197—42no—
96C4×C197#12112yes—
97C4×C197C22×D411yescharacteristic
98C4×C197C2×#4111yescharacteristic
99Dic197—22no—
100Dic197—44no—
101C2×C197—22no—
102C2×C197—82no—
103C2×C197C2×#12111yescharacteristic
104C2×C197C2×#13512yes—
105D197—24no—
106C197#305011yescharacteristic, complemented, Sylow
107#3050—1971nomaximal, Sylow
108C2×#135D19711yescharacteristic, complemented
109C23×C8—1971no—
110#525—1972no—
111#467—1972no—
112#466—1972no—
113#465—1974no—
114#427—1971no—
115#416—1972no—
116C2×#41C2×D19712yes—
117C23×C4—1971no—
118#135—24no—
119C2×Q16C2×D19711yescharacteristic, complemented
120C2×SD16C2×D19712yescomplemented
121C2×D8C2×D19711yescharacteristic, complemented
122C22×C8—3944no—
123C22×C8—1971no—
124C22×C8—1972no—
125C22×C8—1972no—
126C22×C8C2×D19711yescharacteristic
127#123—3942no—
128C22⋊Q8—1974no—
129#121—1974no—
130#120—3942no—
131C4×D4—3944no—
132#117—1972no—
133#107—1972no—
134#106—1972no—
135#103—1974no—
136#102—1974no—
137C24—1971no—
138#41—24no—
139#41—24no—
140C2×Q8C22×D19712yes—
141C2×D4—22no—
142C2×D4—3944no—
143C2×D4C22×D19712yes—
144C22×C4—22no—
145C22×C4—3942no—
146C22×C4—3942no—
147C22×C4—1971no—
148C22×C4—1972no—
149C22×C4—1972no—
150C22×C4C22×D19711yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154C2×C8—3942no—
155C2×C8—3944no—
156C2×C8—3944no—
157C2×C8—1972no—
158C2×C8C22⋊D19714yescomplemented
159C2×C8C22×D19711yescharacteristic
160C2×C8C22×D19711yescharacteristic
161C4⋊C4—3944no—
162C4⋊C4—1974no—
163C22⋊C4—3944no—
164C22⋊C4—3942no—
165C22⋊C4—3944no—
166C42—3942no—
167C23—22no—
168C23—3944no—
169C23—3942no—
170C23—1972no—
171C23D4×D19711yescharacteristic
172Q8—22no—
173Q8—22no—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—7888no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—3944no—
181C2×C4—3942no—
182C2×C4—3944no—
183C2×C4—3942no—
184C2×C4—3942no—
185C2×C4—1972no—
186C2×C4D4×D19711yescharacteristic
187C2×C4C2×C22⋊D19712yes—
188C2×C4#1569135712yes—
189C2×C4C23×D19711yescharacteristic
190C8—3942no—
191C8C2×C22⋊D19712yes—
192C8C2×C22⋊D19712yes—
193C22—22no—
194C22—44no—
195C22—42no—
196C22—3948no—
197C22—3942no—
198C22—3942no—
199C22—1974no—
200C22#2026329712yes—
201C22C2×D4×D19711yescharacteristic
202C4—42no—
203C4—7884no—
204C4—3942no—
205C4#2026329712yes—
206C4C22×C22⋊D19711yescharacteristic
207C4C2×#1569135711yescharacteristic
208C2—22no—
209C2—82no—
210C2—3944no—
211C2C2×#2026329711yescharacteristic
212C2#135×D19712yes—
subgroupG/H
centerC2×C4D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#135×C197C2
socleC22×C197C22×D4
hypercenterC2×#135D197
hypocenterC197#3050
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×C8×D197C2semidirect
2. #27161586C2semidirect
3. #27161317C2semidirect
4. #27161312C2semidirect
5. #27161305C2semidirect
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
94. C2×D197#135semidirect
106. C197#3050semidirect
108. C2×#135D197semidirect
119. C2×Q16C2×D197semidirect
120. C2×SD16C2×D197semidirect
121. C2×D8C2×D197semidirect
158. C2×C8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D1972
#271615862
#271613172
#271613122
#271613052
#271611172
#271610752
C2×#135×C1972
#3050197