Finite Group Databasebeta

← C2×#27161225 #38751914

Subgroups — C2×#27161225 #38751914

Order 25216.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#15691357C211yesmaximal, characteristic, complemented
2C22×#15691347C211yesmaximal, characteristic, complemented
3C22×#15691343C211yesmaximal, characteristic, complemented
4C2×#20263301C211yesmaximal, characteristic, complemented
5#27161225C218yesmaximal, direct product factor
6C2×#20263249C211yesmaximal, characteristic, complemented
7C2×#20263238C211yesmaximal, characteristic, complemented
8C2×#102×C197C211yesmaximal, characteristic, complemented
9C2×#15691357—24no—
10C2×#15691357—22no—
11C2×#15691357—24no—
12C2×#15691357C2214yescomplemented
13C22×Dic394C2211yescharacteristic
14C22×D788—21no—
15C22×D788C2211yescharacteristic
16C22×C22⋊D197—21no—
17C23×Dic197—21no—
18C22×C4×D197C2211yescharacteristic
19C2×#15691347—24no—
20C2×#15691347C2214yescomplemented
21C2×#15691343—24no—
22C2×#15691343C2214yescomplemented
23C22×#13096802C2211yescharacteristic
24#20263301—22no—
25#20263301C2214yescomplemented
26C2×#15691335—21no—
27#20263249C2214yescomplemented
28#20263238C2214yescomplemented
29C22×D4×C197C2211yescharacteristic
30C22×C8×C197C2211yescharacteristic
31C2×C4⋊C4×C197C2211yescharacteristic
32#102×C197C2214yescomplemented
33C23×D197—21no—
34C23×D197D411yescharacteristic
35#15691357—216no—
36#15691357—28no—
37#15691357—28no—
38C2×Dic394—21no—
39C2×Dic394—22no—
40C2×Dic394D414yescomplemented
41C2×Dic394C2312yes—
42C2×D788—24no—
43C2×D788—22no—
44C2×D788—22no—
45C2×D788—21no—
46C2×D788—44no—
47C2×D788C2312yes—
48C2×C22⋊D197—28no—
49C2×C22⋊D197—24no—
50C22×Dic197—28no—
51C22×Dic197—24no—
52C22×Dic197—21no—
53C22×Dic197D411yescharacteristic
54C2×C4×D197—22no—
55C2×C4×D197—22no—
56C2×C4×D197D412yes—
57C2×C4×D197C2312yes—
58#15691347—28no—
59#15691343—28no—
60C2×#13096802—22no—
61C2×#13096802C2312yes—
62#15691335—24no—
63C24×C197—21no—
64C2×D4×C197—21no—
65C2×D4×C197—22no—
66C2×D4×C197D414yescomplemented
67C2×D4×C197C2312yes—
68C22×C4×C197—21no—
69C22×C4×C197C2311yescharacteristic
70C2×C8×C197—22no—
71C2×C8×C197C2312yes—
72C4⋊C4×C197—21no—
73C4⋊C4×C197C2312yes—
74C22×D197—22no—
75C22×D197—22no—
76C22×D197—22no—
77C22×D197—22no—
78C22×D197—42no—
79C22×D197—42no—
80C22×D197—42no—
81C22×D197C2×D412yes—
82Dic394—24no—
83Dic394—22no—
84Dic394C2×D414yescomplemented
85D788—22no—
86D788—22no—
87D788—48no—
88D788—42no—
89C22⋊D197—216no—
90C2×Dic197—216no—
91C2×Dic197—22no—
92C2×Dic197—24no—
93C2×Dic197—22no—
94C2×Dic197C2×D412yes—
95C4×D197—24no—
96C4×D197—22no—
97C4×D197—22no—
98#13096802—22no—
99C23×C197—24no—
100C23×C197—22no—
101C23×C197—42no—
102C23×C197—42no—
103C23×C197C2×D411yescharacteristic
104D4×C197—24no—
105D4×C197—22no—
106D4×C197C2×D414yescomplemented
107C2×C4×C197—22no—
108C2×C4×C197—42no—
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2×D412yes—
111C2×C4×C197C2×D412yes—
112C2×C4×C197C2411yescharacteristic
113C8×C197—22no—
114C22×C197—24no—
115C22×C197—42no—
116C22×C197—42no—
117C22×C197—48no—
118C22×C197#12012yes—
119C22×C197#13512yes—
120C22×C197AGL(1,ℤ/8)12yes—
121C22×C197C22×D411yescharacteristic
122C2×D197—22no—
123C2×D197—22no—
124C2×D197—22no—
125C2×D197—44no—
126C2×D197—42no—
127C2×D197—42no—
128C2×D197—42no—
129C2×D197—84no—
130C4×C197—42no—
131C4×C197#12012yes—
132C4×C197C22×D411yescharacteristic
133C4×C197C22×D411yescharacteristic
134Dic197—22no—
135Dic197—24no—
136C2×C197—44no—
137C2×C197#44914yescomplemented
138C2×C197C2×#12011yescharacteristic
139C2×C197C2×#13511yescharacteristic
140C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
141D197—42no—
142D197—82no—
143C197C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1971nomaximal, Sylow
145C22×#41—1971no—
146C22×SD16—1971no—
147C22×D8—1971no—
148C2×#121—1971no—
149#449—1978no—
150C2×#103—1971no—
151C2×#102D19711yescharacteristic, complemented
152C2×C22⋊C8—1971no—
153C2×#41—3944no—
154C2×#41—3944no—
155C2×#41—3942no—
156C2×#41—1974no—
157C22×Q8—1971no—
158C22×D4—3941no—
159C22×D4—3941no—
160C22×D4—1971no—
161C22×D4C2×D19711yescharacteristic
162C23×C4—3941no—
163C23×C4—1971no—
164C2×SD16—3944no—
165C2×SD16—1974no—
166C2×D8—3944no—
167C2×D8—1974no—
168C22×C8—1971no—
169C22×C8C2×D19711yescharacteristic
170#121—3942no—
171#121—1974no—
172C2×C4⋊C4C2×D19711yescharacteristic
173C2×C22⋊C4—3941no—
174#103—1974no—
175#102C2×D19714yescomplemented
176C22⋊C8—1974no—
177C24—21no—
178C24—3941no—
179C24—1971no—
180#41—39416no—
181#41—3948no—
182#41—3948no—
183C2×Q8—3941no—
184C2×Q8—3942no—
185C2×Q8—1974no—
186C2×Q8—1972no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—7884no—
190C2×D4—3944no—
191C2×D4—3948no—
192C2×D4—3944no—
193C2×D4—3941no—
194C2×D4—3942no—
195C2×D4—3942no—
196C2×D4—1972no—
197C2×D4D78814yescomplemented
198C2×D4C22×D19712yes—
199C22×C4—21no—
200C22×C4—3948no—
201C22×C4—3941no—
202C22×C4—3944no—
203C22×C4—3942no—
204C22×C4—3942no—
205C22×C4—1971no—
206C22×C4—1972no—
207C22×C4—1972no—
208C22×C4C22×D19711yescharacteristic
209SD16—3948no—
210D8—3948no—
211C2×C8—22no—
212C2×C8—3942no—
213C2×C8—1972no—
214C2×C8C22×D19712yes—
215C4⋊C4—21no—
216C4⋊C4C22×D19712yes—
217C22⋊C4—3944no—
218C23—24no—
219C23—22no—
220C23—42no—
221C23—42no—
222C23—7882no—
223C23—7882no—
224C23—7882no—
225C23—3942no—
226C23—3942no—
227C23—3942no—
228C23—3942no—
229C23—1972no—
230C23D4×D19711yescharacteristic
231Q8—3944no—
232Q8—3942no—
233Q8—1974no—
234D4—24no—
235D4—22no—
236D4—7888no—
237D4—7882no—
238D4—39416no—
239D4—3942no—
240D4—3942no—
241D4C2×D78814yescomplemented
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—39416no—
245C2×C4—3942no—
246C2×C4—3944no—
247C2×C4—3942no—
248C2×C4—3944no—
249C2×C4—3942no—
250C2×C4—3942no—
251C2×C4—1972no—
252C2×C4D4×D19711yescharacteristic
253C2×C4D4×D19712yes—
254C2×C4C2×D78812yes—
255C2×C4C23×D19711yescharacteristic
256C8—22no—
257C8—3942no—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—48no—
262C22—7884no—
263C22—7882no—
264C22—7882no—
265C22—7882no—
266C22—15764no—
267C22—3942no—
268C22—3942no—
269C22—3942no—
270C22#2026329612yes—
271C22#2026334712yes—
272C22#2026335012yes—
273C22C2×D4×D19711yescharacteristic
274C4—42no—
275C4—3942no—
276C4—3944no—
277C4#2026329612yes—
278C4C2×D4×D19711yescharacteristic
279C4C22×D78811yescharacteristic
280C2—44no—
281C2—7882no—
282C2—15762no—
283C2#2716122514yesdirect product factor
284C2C2×#2026329611yescharacteristic
285C2C2×#2026334711yescharacteristic
286C2C2×#2026335011yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#102×C197C2
socleC23×C197C2×D4
hypercenterC2×#102D197
hypocenterC197C2×#449
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#15691357C2semidirect
2. C22×#15691347C2semidirect
3. C22×#15691343C2semidirect
4. C2×#20263301C2semidirect
5. #27161225C2direct (N × H)
6. C2×#20263249C2semidirect
7. C2×#20263238C2semidirect
8. C2×#102×C197C2semidirect
12. C2×#15691357C22semidirect
20. C2×#15691347C22semidirect
22. C2×#15691343C22semidirect
25. #20263301C22semidirect
27. #20263249C22semidirect
28. #20263238C22semidirect
32. #102×C197C22semidirect
40. C2×Dic394D4semidirect
66. C2×D4×C197D4semidirect
84. Dic394C2×D4semidirect
106. D4×C197C2×D4semidirect
137. C2×C197#449semidirect
143. C197C2×#449semidirect
151. C2×#102D197semidirect
175. #102C2×D197semidirect
197. C2×D4D788semidirect
241. D4C2×D788semidirect
283. C2#27161225direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913572
C22×#156913472
C22×#156913432
C2×#202633012
#271612252
C2×#202632492
C2×#202632382
C2×#102×C1972
C2×#449197