Finite Group Databasebeta

← #38752652

Subgroups — #38752652

Order 25216.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#20263300C211yesmaximal, characteristic
2#27161403C214yesmaximal, complemented
3#27161397C212yesmaximal, complemented
4#27161392C212yesmaximal, complemented
5#27161122C212yesmaximal, complemented
6#27161083C212yesmaximal, complemented
7#27161043C211yesmaximal, characteristic, complemented
8C2×#117×C197C211yesmaximal, characteristic, complemented
9C22×D788C2211yescharacteristic
10C22×C22⋊D197—21no—
11C2×#15691342C2212yes—
12#20263300—21no—
13#20263300—21no—
14#20263300C2212yes—
15#20263300C2212yes—
16C2×C4⋊Dic197C2211yescharacteristic
17C2×#15691335—21no—
18#20263259C2214yescomplemented
19#20263257C2214yescomplemented
20#20263246C2214yescomplemented
21#20263246C2214yescomplemented
22C22⋊#13096802C2212yescomplemented
23C22⋊#13096802C2212yescomplemented
24C23×C4×C197C2211yescharacteristic
25#117×C197—21no—
26#117×C197—21no—
27#117×C197C2212yes—
28#117×C197C2212yescomplemented
29C2×C4⋊C4×C197C2211yescharacteristic, complemented
30C2×C4⋊C4×C197C2211yescharacteristic, complemented
31C2×C22⋊C4×C197—21no—
32C2×C42×C197—21no—
33C23×D197—21no—
34C2×D788—21no—
35C2×D788—24no—
36C2×D788C2312yes—
37C2×C22⋊D197—22no—
38C2×C22⋊D197—22no—
39C2×C22⋊D197—44no—
40C22×Dic197—21no—
41#15691342—24no—
42C2×#13096802C2314yes—
43C4⋊Dic197—21no—
44C4⋊Dic197C2312yes—
45#15691335—22no—
46#15691335—22no—
47C24×C197D411yescharacteristic
48C22×C4×C197—24no—
49C22×C4×C197—21no—
50C22×C4×C197—21no—
51C22×C4×C197D411yescharacteristic
52C22×C4×C197D411yescharacteristic
53C22×C4×C197D411yescharacteristic
54C22×C4×C197C2311yescharacteristic
55C22×C4×C197C2311yescharacteristic
56C22×C4×C197C2311yescharacteristic
57C4⋊C4×C197—21no—
58C4⋊C4×C197—21no—
59C4⋊C4×C197C2312yes—
60C4⋊C4×C197C2312yes—
61C22⋊C4×C197—22no—
62C22⋊C4×C197—22no—
63C42×C197—22no—
64C42×C197—22no—
65C22×D197—22no—
66C22×D197—42no—
67C22×D197—42no—
68C22×D197—42no—
69D788—24no—
70D788—42no—
71C22⋊D197—44no—
72C22⋊D197—44no—
73C2×Dic197—22no—
74C2×Dic197—42no—
75#13096802—24no—
76C23×C197—22no—
77C23×C197—22no—
78C23×C197—41no—
79C23×C197C2×D411yescharacteristic
80C23×C197C2×D411yescharacteristic
81C23×C197C2×D411yescharacteristic
82C2×C4×C197—22no—
83C2×C4×C197—22no—
84C2×C4×C197—22no—
85C2×C4×C197—22no—
86C2×C4×C197—22no—
87C2×C4×C197—22no—
88C2×C4×C197—22no—
89C2×C4×C197—42no—
90C2×C4×C197—42no—
91C2×C4×C197C2×D411yescharacteristic
92C2×C4×C197C2×D411yescharacteristic
93C2×C4×C197C2×D411yescharacteristic
94C2×C4×C197#4114yes—
95C2×C4×C197C2411yescharacteristic
96C22×C197—21no—
97C22×C197—21no—
98C22×C197—21no—
99C22×C197—21no—
100C22×C197—21no—
101C22×C197—21no—
102C22×C197—21no—
103C22×C197—44no—
104C22×C197#12312yes—
105C22×C197AGL(1,ℤ/8)12yes—
106C22×C197C22×D411yescharacteristic
107C2×D197—42no—
108C2×D197—42no—
109C2×D197—42no—
110C2×D197—84no—
111C4×C197—21no—
112C4×C197—21no—
113C4×C197—42no—
114C4×C197—42no—
115C4×C197#12312yes—
116C4×C197C2×#4112yes—
117Dic197—42no—
118C2×C197—22no—
119C2×C197—22no—
120C2×C197—41no—
121C2×C197C2×#12311yescharacteristic
122C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C197#57511yescharacteristic
124D197—82no—
125C197#309411yescharacteristic, complemented, Sylow
126#3094—1971nomaximal, Sylow
127C2×#121—1971no—
128C2×#117D19711yescharacteristic, complemented
129#482—1974no—
130#481—1972no—
131#480—1972no—
132#428—1972no—
133#418—1972no—
134#407—1971no—
135C22×D4—3941no—
136C22×D4—1971no—
137C23×C4C2×D19711yescharacteristic
138C2×M16—1972no—
139#121—3941no—
140#121—3941no—
141#121—1972no—
142#121—1972no—
143#117—21no—
144#117—21no—
145#117C2×D19712yes—
146#117C2×D19712yescomplemented
147C2×C4⋊C4—1971no—
148C2×C4⋊C4C2×D19711yescharacteristic, complemented
149C2×C4⋊C4C2×D19711yescharacteristic, complemented
150C2×C22⋊C4—21no—
151C2×C22⋊C4—3941no—
152C2×C42—21no—
153#107—1974no—
154#106—1974no—
155#102—1974no—
156#102—1974no—
157C22⋊C8—1972no—
158C22⋊C8—1972no—
159C24—3941no—
160C24C22⋊D19711yescharacteristic
161C2×D4—7884no—
162C2×D4—3942no—
163C2×D4—3942no—
164C2×D4—3941no—
165C2×D4—3944no—
166C2×D4—1972no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—3941no—
171C22×C4C22⋊D19711yescharacteristic
172C22×C4C22⋊D19711yescharacteristic
173C22×C4C22⋊D19711yescharacteristic
174C22×C4C22×D19711yescharacteristic
175C22×C4C22×D19711yescharacteristic
176C22×C4C22×D19711yescharacteristic
177M16—3944no—
178C2×C8—1974no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—3941no—
182C4⋊C4—1972no—
183C4⋊C4C22×D19712yes—
184C4⋊C4C22×D19712yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—3942no—
188C22⋊C4—3942no—
189C42—22no—
190C42—22no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—7882no—
195C23—7882no—
196C23—7882no—
197C23—3942no—
198C23C2×C22⋊D19711yescharacteristic
199C23C2×C22⋊D19711yescharacteristic
200C23C2×C22⋊D19711yescharacteristic
201D4—7884no—
202D4—7884no—
203D4—7882no—
204D4—3944no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—7882no—
215C2×C4—3942no—
216C2×C4C2×C22⋊D19711yescharacteristic
217C2×C4C2×C22⋊D19711yescharacteristic
218C2×C4C2×C22⋊D19711yescharacteristic
219C2×C4#1569135814yes—
220C2×C4C23×D19711yescharacteristic
221C8—3944no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—7882no—
231C22—7882no—
232C22—7882no—
233C22—15764no—
234C22#2026330912yes—
235C22#2026335112yes—
236C22C22×C22⋊D19711yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—7882no—
242C4#2026330912yes—
243C4C2×#1569135812yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—15762no—
248C2C2×#2026330911yescharacteristic
249C2C2×#2026335111yescharacteristic
250C2#2716187711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#117×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263300C2
hypercenterC2×#117D197
hypocenterC197#3094
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #27161403C2semidirect
3. #27161397C2semidirect
4. #27161392C2semidirect
5. #27161122C2semidirect
6. #27161083C2semidirect
7. #27161043C2semidirect
8. C2×#117×C197C2semidirect
18. #20263259C22semidirect
19. #20263257C22semidirect
20. #20263246C22semidirect
21. #20263246C22semidirect
22. C22⋊#13096802C22semidirect
23. C22⋊#13096802C22semidirect
28. #117×C197C22semidirect
29. C2×C4⋊C4×C197C22semidirect
30. C2×C4⋊C4×C197C22semidirect
125. C197#3094semidirect
128. C2×#117D197semidirect
146. #117C2×D197semidirect
148. C2×C4⋊C4C2×D197semidirect
149. C2×C4⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633002
#271614032
#271613972
#271613922
#271611222
#271610832
#271610432
C2×#117×C1972
#3094197