Finite Group Databasebeta

← #38752075

Subgroups — #38752075

Order 25216.

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

normal subgroups only

This group has 698 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
1C2×#20263374C211yesmaximal, characteristic, complemented
2#27161877C212yesmaximal, complemented
3#27161261C214yesmaximal, complemented
4#27161251C214yesmaximal, complemented
5#27161097C212yesmaximal, complemented
6C2×#20263245C211yesmaximal, characteristic, complemented
7#536×C197C211yesmaximal, characteristic, complemented
8#20263374—24no—
9#20263374—22no—
10#20263374C2214yes—
11C2×#15691359—22no—
12C2×#15691358—22no—
13C2×#15691358C2212yes—
14C2×Q8×D197—22no—
15#20263357C2214yescomplemented
16#20263356—22no—
17#20263352C2214yescomplemented
18#20263351—22no—
19#20263346—24no—
20C2×#15691342C2212yes—
21#20263254C2218yescomplemented
22#20263245C2214yescomplemented
23C2×#41×C197C2211yescharacteristic
24C22×Q8×C197C2211yescharacteristic
25C4⋊Q8×C197C2212yescomplemented
26#124×C197C2212yescomplemented
27#123×C197—22no—
28C22⋊Q8×C197—22no—
29#117×C197C2211yescharacteristic
30#15691359—28no—
31#15691359—28no—
32#15691358—28no—
33#15691358—28no—
34#15691358—22no—
35#15691358—22no—
36#15691358D414yes—
37#15691358C2314yes—
38C2×Dic394—22no—
39C2×Dic394D412yes—
40Q8×D197—28no—
41Q8×D197—28no—
42C2×D788—22no—
43C2×D788D412yes—
44C2×C22⋊D197—22no—
45C2×C4×D197—24no—
46C2×C4×D197—22no—
47#15691348—24no—
48#15691346—24no—
49#15691345—24no—
50#15691343—24no—
51#15691342—22no—
52#15691342C2314yes—
53C2×#13096802—22no—
54#41×C197—22no—
55#41×C197—22no—
56C2×Q8×C197—22no—
57C2×Q8×C197—22no—
58C2×Q8×C197D412yes—
59C2×Q8×C197C2311yescharacteristic
60C2×Q8×C197C2312yes—
61C2×D4×C197—21no—
62C2×D4×C197C2311yescharacteristic
63C22×C4×C197—21no—
64C22×C4×C197D412yes—
65C22×C4×C197C2311yescharacteristic
66C4⋊C4×C197—21no—
67C4⋊C4×C197—24no—
68C22⋊C4×C197—21no—
69C22⋊C4×C197—24no—
70C42×C197C2312yes—
71C22×D197—22no—
72Dic394—28no—
73Dic394—22no—
74Dic394C2×D414yes—
75D788—28no—
76D788—22no—
77D788C2×D414yes—
78C22⋊D197—24no—
79C22⋊D197—24no—
80C2×Dic197—22no—
81C4×D197—216no—
82C4×D197—24no—
83C4×D197—24no—
84#13096802—24no—
85C23×C197—21no—
86C23×C197C2×D411yescharacteristic
87Q8×C197—24no—
88Q8×C197—24no—
89Q8×C197—22no—
90D4×C197—22no—
91D4×C197—41no—
92D4×C197—41no—
93C2×C4×C197—22no—
94C2×C4×C197—22no—
95C2×C4×C197—22no—
96C2×C4×C197—21no—
97C2×C4×C197—42no—
98C2×C4×C197C2×D412yes—
99C2×C4×C197C2×D412yes—
100C2×C4×C197C2×D414yes—
101C2×C4×C197C2×D411yescharacteristic
102C2×C4×C197C2411yescharacteristic
103C22×C197—41no—
104C22×C197—42no—
105C22×C197—41no—
106C22×C197#12011yescharacteristic
107C22×C197#12011yescharacteristic
108C22×C197C22×D411yescharacteristic
109C2×D197—24no—
110C2×D197—44no—
111C4×C197—24no—
112C4×C197—42no—
113C4×C197—41no—
114C4×C197#12012yes—
115C4×C197C22×D412yes—
116Dic197—24no—
117C2×C197—21no—
118C2×C197—21no—
119C2×C197—21no—
120C2×C197—81no—
121C2×C197C2×#12011yescharacteristic
122D197—44no—
123C197#302011yescharacteristic, complemented, Sylow
124#3020—1971nomaximal, Sylow
125C2×2−1+4—1971no—
126#575—1972no—
127#536D19711yescharacteristic, complemented
128#456—1974no—
129#454—1974no—
130#421—1972no—
131C2×#101—1971no—
1322−1+4—3944no—
1332−1+4—3942no—
1342−1+4—1974no—
135C2×#41—3942no—
136C2×#41—3942no—
137C2×#41—1972no—
138C2×#41C2×D19711yescharacteristic
139C22×Q8—3942no—
140C22×Q8C2×D19711yescharacteristic
141#137—3942no—
142#137—1974no—
143AGL(1,ℤ/8)—3942no—
144AGL(1,ℤ/8)—1974no—
145#135—3944no—
146C2×M16—1972no—
147C4⋊Q8C2×D19712yescomplemented
148#124C2×D19712yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D19711yescharacteristic
152#104—1978no—
153#101—1974no—
154#41—22no—
155#41—22no—
156#41—3948no—
157#41—3948no—
158#41—3948no—
159#41—3942no—
160#41—3948no—
161#41—3942no—
162#41—1974no—
163#41—1974no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—3942no—
167C2×Q8—3948no—
168C2×Q8—3948no—
169C2×Q8—1972no—
170C2×Q8C22⋊D19712yes—
171C2×Q8C22×D19711yescharacteristic
172C2×Q8C22×D19712yes—
173C2×D4—21no—
174C2×D4—3942no—
175C2×D4—3942no—
176C2×D4—1972no—
177C2×D4C22×D19711yescharacteristic
178C22×C4—21no—
179C22×C4—3944no—
180C22×C4—3942no—
181C22×C4C22⋊D19712yes—
182C22×C4C22×D19711yescharacteristic
183Q16—3944no—
184SD16—3944no—
185SD16—3944no—
186D8—3944no—
187M16—3942no—
188M16—1974no—
189C2×C8—3942no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D19712yes—
195C23—21no—
196C23—3942no—
197C23C2×C22⋊D19711yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—3948no—
202Q8—3942no—
203Q8—1974no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—3948no—
208D4—3944no—
209D4—3944no—
210D4—3942no—
211D4—1974no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—39416no—
218C2×C4—3942no—
219C2×C4—3944no—
220C2×C4—3944no—
221C2×C4D4×D19712yes—
222C2×C4D4×D19712yes—
223C2×C4C2×C22⋊D19714yes—
224C2×C4C2×C22⋊D19711yescharacteristic
225C2×C4C23×D19711yescharacteristic
226C8—3944no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—7884no—
231C22—3944no—
232C22#2026329411yescharacteristic
233C22#2026329411yescharacteristic
234C22C22×C22⋊D19711yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—3944no—
239C4#2026329412yes—
240C4C2×D4×D19712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—7884no—
246C2C2×#2026329411yescharacteristic
subgroupG/H
centerC2C2×#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#536×C197C2
socleC2×C197C2×#120
hypercenter#536D197
hypocenterC197#3020
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#20263374C2semidirect
2. #27161877C2semidirect
3. #27161261C2semidirect
4. #27161251C2semidirect
5. #27161097C2semidirect
6. C2×#20263245C2semidirect
7. #536×C197C2semidirect
15. #20263357C22semidirect
17. #20263352C22semidirect
21. #20263254C22semidirect
22. #20263245C22semidirect
25. C4⋊Q8×C197C22semidirect
26. #124×C197C22semidirect
123. C197#3020semidirect
127. #536D197semidirect
147. C4⋊Q8C2×D197semidirect
148. #124C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633742
#271618772
#271612612
#271612512
#271610972
C2×#202632452
#536×C1972
#3020197