Finite Group Databasebeta

← #38752090

Subgroups — #38752090

Order 25216.

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

normal subgroups only

This group has 704 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
1#27161915C211yesmaximal, characteristic
2C2×#20263357C212yesmaximal, complemented
3#27161261C214yesmaximal, complemented
4#27161256C214yesmaximal, complemented
5#27161097C212yesmaximal, complemented
6#27161063C211yesmaximal, characteristic, complemented
7#536×C197C211yesmaximal, characteristic, complemented
8#20263374C2212yes—
9#20263373—22no—
10#20263372—22no—
11#20263371—22no—
12#20263371C2212yes—
13C2×#15691358—22no—
14C2×#15691358C2212yes—
15#41×D197—24no—
16#20263357—24no—
17#20263357C2218yescomplemented
18C2×#15691348—22no—
19C2×#15691345—22no—
20C2×#15691342C2212yescomplemented
21#20263254C2218yescomplemented
22#20263245C2212yescomplemented
23#20263243C2212yescomplemented
24C2×#41×C197C2211yescharacteristic
25C22×Q8×C197C2211yescharacteristic, complemented
26C4⋊Q8×C197C2212yescomplemented
27#124×C197C2212yescomplemented
28#123×C197—22no—
29C22⋊Q8×C197—22no—
30#117×C197C2211yescharacteristic, complemented
31#15691359—24no—
32#15691358—24no—
33#15691358—24no—
34#15691358—22no—
35#15691358—24no—
36#15691358—22no—
37#15691358D414yes—
38#15691358C2314yes—
39#15691357—28no—
40#15691357—24no—
41C2×Dic394—22no—
42C2×Dic394D412yes—
43Q8×D197—24no—
44C2×D788—22no—
45C2×D788D412yes—
46C2×C22⋊D197—24no—
47C2×C22⋊D197—22no—
48D4×D197—28no—
49D4×D197—24no—
50C2×C4×D197—24no—
51C2×C4×D197—22no—
52#15691348—28no—
53#15691345—28no—
54#15691342—22no—
55#15691342C2314yes—
56C2×#13096802—22no—
57#41×C197—22no—
58#41×C197—22no—
59C2×Q8×C197—21no—
60C2×Q8×C197—22no—
61C2×Q8×C197—22no—
62C2×Q8×C197C2312yes—
63C2×Q8×C197C2311yescharacteristic
64C2×D4×C197D412yes—
65C2×D4×C197C2311yescharacteristic
66C22×C4×C197—21no—
67C22×C4×C197D412yes—
68C22×C4×C197C2311yescharacteristic
69C4⋊C4×C197—21no—
70C4⋊C4×C197—24no—
71C22⋊C4×C197—21no—
72C22⋊C4×C197—24no—
73C42×C197C2312yes—
74C22×D197—24no—
75C22×D197—22no—
76Dic394—24no—
77Dic394—22no—
78Dic394C2×D414yes—
79D788—24no—
80D788—22no—
81D788C2×D414yes—
82C22⋊D197—28no—
83C22⋊D197—24no—
84C22⋊D197—24no—
85C2×Dic197—24no—
86C2×Dic197—22no—
87C4×D197—28no—
88C4×D197—24no—
89C4×D197—24no—
90#13096802—24no—
91C23×C197C2×D412yes—
92C23×C197C2×D411yescharacteristic
93Q8×C197—22no—
94Q8×C197—24no—
95Q8×C197—41no—
96Q8×C197—41no—
97D4×C197—24no—
98D4×C197—22no—
99C2×C4×C197—22no—
100C2×C4×C197—22no—
101C2×C4×C197—22no—
102C2×C4×C197—22no—
103C2×C4×C197—42no—
104C2×C4×C197C2×D412yes—
105C2×C4×C197C2×D412yes—
106C2×C4×C197C2×D412yes—
107C2×C4×C197C2×D411yescharacteristic
108C2×C4×C197C2411yescharacteristic
109C22×C197—22no—
110C22×C197—41no—
111C22×C197—42no—
112C22×C197#12012yes—
113C22×C197C22×D411yescharacteristic
114C2×D197—24no—
115C2×D197—82no—
116C2×D197—84no—
117C4×C197—22no—
118C4×C197—42no—
119C4×C197—42no—
120C4×C197#12011yescharacteristic
121C4×C197#12011yescharacteristic
122C4×C197C22×D412yes—
123Dic197—24no—
124C2×C197—21no—
125C2×C197—22no—
126C2×C197—42no—
127C2×C197C2×#12011yescharacteristic
128D197—44no—
129C197#302211yescharacteristic, complemented, Sylow
130#3022—1971nomaximal, Sylow
131#585—1971no—
132C2×#137—1972no—
133#536D19711yescharacteristic, complemented
134#456—1974no—
135#455—1974no—
136#421—1972no—
137#413—1971no—
1382−1+4—3942no—
1392−1+4—1972no—
140H(5,𝔽2)—3942no—
141H(5,𝔽2)—3942no—
142H(5,𝔽2)—1972no—
143C2×#41—3944no—
144C2×#41—3942no—
145C2×#41—1972no—
146C2×#41C2×D19711yescharacteristic
147C22×Q8C2×D19711yescharacteristic, complemented
148#137—3944no—
149#137—1978no—
150C2×Q16—3942no—
151C2×SD16—3942no—
152C2×M16—1972no—
153C4⋊Q8C2×D19712yescomplemented
154#124C2×D19712yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D19711yescharacteristic, complemented
158#104—1978no—
159#101—1972no—
160#100—1972no—
161#41—22no—
162#41—22no—
163#41—3944no—
164#41—3944no—
165#41—3948no—
166#41—3944no—
167#41—3944no—
168#41—3942no—
169#41—3942no—
170#41—3944no—
171#41—1974no—
172#41—1974no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—3942no—
177C2×Q8—3944no—
178C2×Q8—1972no—
179C2×Q8C22×D19712yes—
180C2×Q8C22×D19711yescharacteristic
181C2×D4—3942no—
182C2×D4—3948no—
183C2×D4—3944no—
184C2×D4—3944no—
185C2×D4—3942no—
186C2×D4—1972no—
187C2×D4C22⋊D19712yes—
188C2×D4C22×D19711yescharacteristic
189C22×C4—21no—
190C22×C4—3944no—
191C22×C4—3942no—
192C22×C4C22⋊D19712yes—
193C22×C4C22×D19711yescharacteristic
194Q16—3948no—
195SD16—3948no—
196M16—3942no—
197M16—1974no—
198C2×C8—3942no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D19712yes—
204C23—3944no—
205C23—3942no—
206C23C2×C22⋊D19712yes—
207C23C2×C22⋊D19711yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—3944no—
213Q8—3942no—
214Q8—1974no—
215D4—24no—
216D4—22no—
217D4—3944no—
218D4—3948no—
219D4—3944no—
220D4—3944no—
221D4—3942no—
222D4—1974no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—3948no—
229C2×C4—3944no—
230C2×C4—3942no—
231C2×C4—3944no—
232C2×C4—3944no—
233C2×C4D4×D19712yes—
234C2×C4D4×D19712yes—
235C2×C4C2×C22⋊D19712yes—
236C2×C4C2×C22⋊D19711yescharacteristic
237C2×C4C23×D19711yescharacteristic
238C8—3944no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—15762no—
243C22—15764no—
244C22—3944no—
245C22#2026329412yes—
246C22C22×C22⋊D19711yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—3944no—
251C4#2026329411yescharacteristic
252C4#2026329411yescharacteristic
253C4C2×D4×D19712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—7884no—
258C2C2×#2026329411yescharacteristic
subgroupG/H
centerC2C2×#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#536×C197C2
socleC2×C197C2×#120
prime radical#27161915C2
hypercenter#536D197
hypocenterC197#3022
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
2. C2×#20263357C2semidirect
3. #27161261C2semidirect
4. #27161256C2semidirect
5. #27161097C2semidirect
6. #27161063C2semidirect
7. #536×C197C2semidirect
17. #20263357C22semidirect
20. C2×#15691342C22semidirect
21. #20263254C22semidirect
22. #20263245C22semidirect
23. #20263243C22semidirect
25. C22×Q8×C197C22semidirect
26. C4⋊Q8×C197C22semidirect
27. #124×C197C22semidirect
30. #117×C197C22semidirect
129. C197#3022semidirect
133. #536D197semidirect
147. C22×Q8C2×D197semidirect
153. C4⋊Q8C2×D197semidirect
154. #124C2×D197semidirect
157. #117C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
C2×#202633572
#271612612
#271612562
#271610972
#271610632
#536×C1972
#3022197