Finite Group Databasebeta

← #38752233

Subgroups — #38752233

Order 25216.

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

normal subgroups only

This group has 490 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×#20263357C211yesmaximal, characteristic, complemented
2C22×#15691345C211yesmaximal, characteristic, complemented
3#27161648C211yesmaximal, characteristic
4#27161301C214yesmaximal, complemented
5#27161279C214yesmaximal, complemented
6#27161106C211yesmaximal, characteristic, complemented
7#27161083C211yesmaximal, characteristic, complemented
8#27161073C211yesmaximal, characteristic, complemented
9#519×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11C22×D788C2211yescharacteristic
12#20263357—24no—
13C2×#15691348C2212yescomplemented
14C2×#15691345—24no—
15C2×#15691345C2214yes—
16C2×#15691345C2212yescomplemented
17C2×#15691342C2211yescharacteristic
18C22×#13096802C2211yescharacteristic
19#20263323C2212yes—
20#20263314C2212yes—
21#20263301—22no—
22#20263296—22no—
23C4⋊#13096802C2214yescomplemented
24#20263249C2212yescomplemented
25#20263246C2214yes—
26#20263246C2212yescomplemented
27C22×Q8×C197C2211yescharacteristic
28C4×Q8×C197—22no—
29C4×Q8×C197C2214yescomplemented
30#117×C197—22no—
31#117×C197C2211yescharacteristic
32#117×C197C2211yescharacteristic
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#15691348—24no—
46#15691345—28no—
47#15691345—24no—
48#15691342—22no—
49C2×#13096802—22no—
50C2×#13096802C2312yes—
51C2×#13096802C2312yes—
52#15691335—24no—
53C2×Q8×C197—21no—
54C2×Q8×C197—22no—
55C2×Q8×C197D414yescomplemented
56C2×Q8×C197C2312yes—
57C22×C4×C197—21no—
58C22×C4×C197C2311yescharacteristic
59C4⋊C4×C197—24no—
60C4⋊C4×C197D412yes—
61C4⋊C4×C197C2312yes—
62C22⋊C4×C197—21no—
63C22⋊C4×C197D412yes—
64C42×C197—24no—
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×C197C2×D411yescharacteristic
85Q8×C197—24no—
86Q8×C197—22no—
87Q8×C197C2×D414yes—
88C2×C4×C197—24no—
89C2×C4×C197—22no—
90C2×C4×C197—22no—
91C2×C4×C197C2×D414yes—
92C2×C4×C197C2×D411yescharacteristic
93C2×C4×C197C2×D412yes—
94C2×C4×C197#4112yes—
95C2×C4×C197C2411yescharacteristic
96C22×C197—22no—
97C22×C197#12112yes—
98C22×C197C22×D411yescharacteristic
99C2×D197—41no—
100C2×D197—42no—
101C2×D197—42no—
102C2×D197—44no—
103C2×D197—84no—
104C4×C197—24no—
105C4×C197—24no—
106C4×C197—42no—
107C4×C197#12112yes—
108C4×C197C22×D411yescharacteristic
109C4×C197C2×#4111yescharacteristic
110Dic197—41no—
111C2×C197—22no—
112C2×C197C2×#12111yescharacteristic
113C2×C197#57712yes—
114D197—81no—
115D197—82no—
116C197#304311yescharacteristic, complemented, Sylow
117#3043—1971nomaximal, Sylow
118C2×#137—1971no—
119C22×SD16—1971no—
120#535—1971no—
121#519D19711yescharacteristic, complemented
122#464—1974no—
123#460—1974no—
124#424—1971no—
125#418—1971no—
126#416—1971no—
127C2×#41—1971no—
128C22×Q8C2×D19711yescharacteristic
129C22×D4—1971no—
130#137—3944no—
131C2×Q16—1972no—
132C2×SD16—3944no—
133C2×SD16—1974no—
134C2×SD16—1972no—
135C2×M16—1971no—
136C22×C8—1971no—
137#127—1972no—
138#124—1972no—
139#121—3942no—
140#120—3942no—
141C4×Q8—22no—
142C4×Q8C2×D19714yescomplemented
143#117—22no—
144#117C2×D19711yescharacteristic
145#117C2×D19711yescharacteristic
146C4⋊C8—1974no—
147#103—1972no—
148#102—1974no—
149#102—1972no—
150C24—3941no—
151#41—3942no—
152#41—3942no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—1971no—
156C2×Q8D78814yescomplemented
157C2×Q8C22×D19712yes—
158C2×D4—3941no—
159C2×D4—3941no—
160C2×D4—3944no—
161C2×D4—3942no—
162C2×D4—3942no—
163C2×D4—1972no—
164C2×D4—1971no—
165C22×C4—21no—
166C22×C4—3941no—
167C22×C4C22×D19711yescharacteristic
168Q16—3944no—
169SD16—3948no—
170SD16—3944no—
171M16—3942no—
172C2×C8—3942no—
173C2×C8—1972no—
174C2×C8—1972no—
175C4⋊C4—24no—
176C4⋊C4C22⋊D19712yes—
177C4⋊C4C22×D19712yes—
178C22⋊C4—21no—
179C22⋊C4—3944no—
180C22⋊C4C22⋊D19712yes—
181C42—24no—
182C42—21no—
183C42C22×D19712yes—
184C23—7882no—
185C23—15762no—
186C23—3942no—
187C23—3941no—
188C23C2×C22⋊D19711yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—3941no—
192Q8—3941no—
193Q8C2×D78814yes—
194D4—7882no—
195D4—7882no—
196D4—15764no—
197D4—3942no—
198D4—3941no—
199D4—3942no—
200D4—3941no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—7882no—
205C2×C4—3941no—
206C2×C4C2×C22⋊D19714yes—
207C2×C4C2×C22⋊D19711yescharacteristic
208C2×C4C2×D78812yes—
209C2×C4#1569135812yes—
210C2×C4C23×D19711yescharacteristic
211C8—3942no—
212C8—3942no—
213C22—22no—
214C22—7882no—
215C22—7884no—
216C22—7881no—
217C22—7882no—
218C22—15764no—
219C22#2026330012yes—
220C22C22×C22⋊D19711yescharacteristic
221C4—24no—
222C4—24no—
223C4—42no—
224C4—7881no—
225C4#2026330012yes—
226C4C22×D78811yescharacteristic
227C4C2×#1569135811yescharacteristic
228C2—22no—
229C2—15762no—
230C2—15761no—
231C2C2×#2026330011yescharacteristic
232C2#2716188712yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#519×C197C2
socleC22×C197C22×D4
prime radical#27161648C2
hypercenter#519D197
hypocenterC197#3043
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C2×#20263357C2semidirect
2. C22×#15691345C2semidirect
4. #27161301C2semidirect
5. #27161279C2semidirect
6. #27161106C2semidirect
7. #27161083C2semidirect
8. #27161073C2semidirect
9. #519×C197C2semidirect
13. C2×#15691348C22semidirect
16. C2×#15691345C22semidirect
23. C4⋊#13096802C22semidirect
24. #20263249C22semidirect
26. #20263246C22semidirect
29. C4×Q8×C197C22semidirect
55. C2×Q8×C197D4semidirect
116. C197#3043semidirect
121. #519D197semidirect
142. C4×Q8C2×D197semidirect
156. C2×Q8D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
C22×#156913452
#271616482
#271613012
#271612792
#271611062
#271610832
#271610732
#519×C1972
#3043197