Finite Group Databasebeta

← #38752215

Subgroups — #38752215

Order 25216.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20263352C211yesmaximal, characteristic, complemented
2C22×#15691343C211yesmaximal, characteristic, complemented
3#27161648C211yesmaximal, characteristic, complemented
4#27161294C214yesmaximal, complemented
5#27161274C214yesmaximal, complemented
6#27161106C211yesmaximal, characteristic, complemented
7#27161083C211yesmaximal, characteristic, complemented
8#27161073C211yesmaximal, characteristic, complemented
9#518×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11C22×D788C2211yescharacteristic
12#20263352—24no—
13C2×#15691346C2212yescomplemented
14C2×#15691343—24no—
15C2×#15691343C2214yes—
16C2×#15691343C2212yescomplemented
17C2×#15691342C2211yescharacteristic
18C22×#13096802C2211yescharacteristic
19#20263323C2212yescomplemented
20#20263314C2212yescomplemented
21#20263301—22no—
22#20263296—22no—
23C4⋊#13096802C2214yescomplemented
24#20263249C2212yescomplemented
25#20263246C2212yescomplemented
26#20263246C2214yescomplemented
27C22×D4×C197C2211yescharacteristic
28C4×D4×C197—22no—
29C4×D4×C197C2214yescomplemented
30#117×C197C2211yescharacteristic
31#117×C197C2211yescharacteristic
32C2×C22⋊C4×C197—22no—
33C23×D197—21no—
34#15691358—22no—
35#15691358—22no—
36C2×Dic394C2311yescharacteristic
37C2×D788—22no—
38C2×D788—21no—
39C2×D788—24no—
40C2×D788—22no—
41C2×D788C2312yes—
42C2×D788C2311yescharacteristic
43C2×C22⋊D197—21no—
44C2×C4×D197—21no—
45#15691346—24no—
46#15691343—28no—
47#15691343—24no—
48#15691342—22no—
49C2×#13096802—22no—
50C2×#13096802C2312yes—
51C2×#13096802C2312yes—
52#15691335—24no—
53C24×C197—21no—
54C2×D4×C197—21no—
55C2×D4×C197—22no—
56C2×D4×C197D414yescomplemented
57C2×D4×C197C2312yes—
58C22×C4×C197—24no—
59C22×C4×C197C2311yescharacteristic
60C4⋊C4×C197D412yes—
61C4⋊C4×C197C2312yes—
62C22⋊C4×C197—21no—
63C22⋊C4×C197—24no—
64C22⋊C4×C197D412yes—
65C42×C197—21no—
66C42×C197C2312yes—
67C22×D197—22no—
68C22×D197—21no—
69C22×D197—42no—
70C22×D197—82no—
71Dic394—21no—
72Dic394—21no—
73D788—22no—
74D788—21no—
75D788—22no—
76D788—21no—
77D788—42no—
78D788—84no—
79C22⋊D197—42no—
80C2×Dic197—21no—
81C4×D197—42no—
82#13096802—22no—
83#13096802—22no—
84C23×C197—22no—
85C23×C197—24no—
86C23×C197—82no—
87C23×C197C2×D411yescharacteristic
88D4×C197—24no—
89D4×C197—22no—
90D4×C197C2×D414yes—
91C2×C4×C197—22no—
92C2×C4×C197—84no—
93C2×C4×C197C2×D414yes—
94C2×C4×C197C2×D411yescharacteristic
95C2×C4×C197C2×D412yes—
96C2×C4×C197#4112yes—
97C2×C4×C197C2411yescharacteristic
98C22×C197—22no—
99C22×C197—24no—
100C22×C197—44no—
101C22×C197—84no—
102C22×C197#12112yes—
103C22×C197C22×D411yescharacteristic
104C2×D197—41no—
105C2×D197—42no—
106C2×D197—42no—
107C2×D197—44no—
108C2×D197—84no—
109C4×C197—24no—
110C4×C197—42no—
111C4×C197#12112yes—
112C4×C197C22×D411yescharacteristic
113C4×C197C2×#4111yescharacteristic
114Dic197—41no—
115C2×C197—22no—
116C2×C197—44no—
117C2×C197C2×#12111yescharacteristic
118C2×C197#57612yes—
119D197—81no—
120D197—82no—
121C197#304111yescharacteristic, complemented, Sylow
122#3041—1971nomaximal, Sylow
123C2×AGL(1,ℤ/8)—1971no—
124C22×D8—1971no—
125#535—1971no—
126#518D19711yescharacteristic, complemented
127#463—1974no—
128#459—1974no—
129#424—1971no—
130#418—1971no—
131#416—1971no—
132C2×#41—1971no—
133C22×D4—1971no—
134C22×D4C2×D19711yescharacteristic
135AGL(1,ℤ/8)—3944no—
136C2×SD16—1972no—
137C2×D8—3944no—
138C2×D8—1974no—
139C2×D8—1972no—
140C2×M16—1971no—
141C22×C8—1971no—
142#127—1972no—
143#124—1972no—
144#121—3942no—
145#120—3942no—
146C4×D4—22no—
147C4×D4C2×D19714yescomplemented
148#117C2×D19711yescharacteristic
149#117C2×D19711yescharacteristic
150C2×C22⋊C4—22no—
151C4⋊C8—1974no—
152#103—1972no—
153#102—1972no—
154#102—1974no—
155C24—21no—
156C24—3941no—
157#41—3942no—
158#41—3942no—
159C2×Q8—1971no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—3941no—
163C2×D4—3941no—
164C2×D4—3944no—
165C2×D4—3942no—
166C2×D4—3942no—
167C2×D4—1972no—
168C2×D4—1971no—
169C2×D4D78814yescomplemented
170C2×D4C22×D19712yes—
171C22×C4—24no—
172C22×C4—3941no—
173C22×C4C22×D19711yescharacteristic
174SD16—3944no—
175D8—3948no—
176D8—3944no—
177M16—3942no—
178C2×C8—3942no—
179C2×C8—1972no—
180C2×C8—1972no—
181C4⋊C4C22⋊D19712yes—
182C4⋊C4C22×D19712yes—
183C22⋊C4—21no—
184C22⋊C4—24no—
185C22⋊C4—3944no—
186C22⋊C4C22⋊D19712yes—
187C42—21no—
188C42C22×D19712yes—
189C23—22no—
190C23—24no—
191C23—82no—
192C23—7882no—
193C23—15762no—
194C23—3942no—
195C23—3941no—
196C23C2×C22⋊D19711yescharacteristic
197Q8—3941no—
198Q8—3941no—
199D4—24no—
200D4—22no—
201D4—7882no—
202D4—7882no—
203D4—15764no—
204D4—3942no—
205D4—3941no—
206D4—3942no—
207D4—3941no—
208D4C2×D78814yes—
209C2×C4—22no—
210C2×C4—84no—
211C2×C4—7882no—
212C2×C4—3941no—
213C2×C4C2×C22⋊D19714yes—
214C2×C4C2×C22⋊D19711yescharacteristic
215C2×C4C2×D78812yes—
216C2×C4#1569135812yes—
217C2×C4C23×D19711yescharacteristic
218C8—3942no—
219C8—3942no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—84no—
224C22—7882no—
225C22—7884no—
226C22—7881no—
227C22—7882no—
228C22—15764no—
229C22#2026330012yes—
230C22C22×C22⋊D19711yescharacteristic
231C4—24no—
232C4—42no—
233C4—7881no—
234C4#2026330012yes—
235C4C22×D78811yescharacteristic
236C4C2×#1569135811yescharacteristic
237C2—22no—
238C2—44no—
239C2—15762no—
240C2—15761no—
241C2C2×#2026330011yescharacteristic
242C2#2716188212yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#518×C197C2
socleC22×C197C22×D4
hypercenter#518D197
hypocenterC197#3041
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. C2×#20263352C2semidirect
2. C22×#15691343C2semidirect
3. #27161648C2semidirect
4. #27161294C2semidirect
5. #27161274C2semidirect
6. #27161106C2semidirect
7. #27161083C2semidirect
8. #27161073C2semidirect
9. #518×C197C2semidirect
13. C2×#15691346C22semidirect
16. C2×#15691343C22semidirect
19. #20263323C22semidirect
20. #20263314C22semidirect
23. C4⋊#13096802C22semidirect
24. #20263249C22semidirect
25. #20263246C22semidirect
26. #20263246C22semidirect
29. C4×D4×C197C22semidirect
56. C2×D4×C197D4semidirect
121. C197#3041semidirect
126. #518D197semidirect
147. C4×D4C2×D197semidirect
169. C2×D4D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
C22×#156913432
#271616482
#271612942
#271612742
#271611062
#271610832
#271610732
#518×C1972
#3041197