Finite Group Databasebeta

← C2×#27161244 #38751924

Subgroups — C2×#27161244 #38751924

Order 25216.

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

normal subgroups only

This group has 760 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×#15691359C211yesmaximal, characteristic
2C22×#15691348C211yesmaximal, characteristic, complemented
3C22×#15691347C211yesmaximal, characteristic, complemented
4C2×#20263306C211yesmaximal, characteristic, complemented
5#27161244C218yesmaximal, direct product factor
6C2×#20263246C211yesmaximal, characteristic, complemented
7C2×#20263238C211yesmaximal, characteristic, complemented
8C2×#103×C197C211yesmaximal, characteristic, complemented
9C2×#15691359—24no—
10C2×#15691359—22no—
11C2×#15691359—24no—
12C2×#15691359C2214yes—
13C22×Dic394C2211yescharacteristic
14C22×D788—21no—
15C22×D788C2211yescharacteristic
16C22×C4×D197—21no—
17C22×C4×D197C2211yescharacteristic
18C2×#15691348—24no—
19C2×#15691348C2214yescomplemented
20C2×#15691347—24no—
21C2×#15691347C2214yescomplemented
22C22×#13096802C2211yescharacteristic
23#20263306—22no—
24#20263306C2214yescomplemented
25C2×C4⋊Dic197—21no—
26C2×#15691335—21no—
27#20263246C2214yescomplemented
28#20263238C2214yescomplemented
29C22×Q8×C197C2211yescharacteristic
30C22×C8×C197C2211yescharacteristic
31C2×C4⋊C4×C197C2211yescharacteristic
32#103×C197C2214yescomplemented
33C23×D197—21no—
34C23×D197D411yescharacteristic
35#15691359—216no—
36#15691359—28no—
37#15691359—28no—
38C2×Dic394—21no—
39C2×Dic394—22no—
40C2×Dic394—22no—
41C2×Dic394C2312yes—
42C2×D788—28no—
43C2×D788—24no—
44C2×D788—21no—
45C2×D788—22no—
46C2×D788D414yes—
47C2×D788C2312yes—
48C22×Dic197—21no—
49C22×Dic197D411yescharacteristic
50C2×C4×D197—28no—
51C2×C4×D197—24no—
52C2×C4×D197—22no—
53C2×C4×D197—22no—
54C2×C4×D197D412yes—
55C2×C4×D197C2312yes—
56#15691348—28no—
57#15691347—28no—
58C2×#13096802—22no—
59C2×#13096802C2312yes—
60C4⋊Dic197—24no—
61#15691335—24no—
62C2×Q8×C197—22no—
63C2×Q8×C197—21no—
64C2×Q8×C197D414yes—
65C2×Q8×C197C2312yes—
66C22×C4×C197—21no—
67C22×C4×C197—21no—
68C22×C4×C197C2311yescharacteristic
69C2×C8×C197—22no—
70C2×C8×C197C2312yes—
71C4⋊C4×C197—21no—
72C4⋊C4×C197C2312yes—
73C22×D197—22no—
74C22×D197—24no—
75C22×D197—22no—
76C22×D197—22no—
77C22×D197—22no—
78C22×D197—42no—
79C22×D197—42no—
80C22×D197C2×D412yes—
81Dic394—22no—
82Dic394—22no—
83Dic394—42no—
84D788—216no—
85D788—24no—
86D788—22no—
87D788C2×D414yes—
88C2×Dic197—22no—
89C2×Dic197—22no—
90C2×Dic197—42no—
91C2×Dic197C2×D412yes—
92C4×D197—216no—
93C4×D197—24no—
94C4×D197—22no—
95C4×D197—22no—
96#13096802—22no—
97C23×C197C2×D411yescharacteristic
98Q8×C197—24no—
99Q8×C197—22no—
100Q8×C197C2×D414yes—
101C2×C4×C197—24no—
102C2×C4×C197—22no—
103C2×C4×C197—22no—
104C2×C4×C197—42no—
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197C2×D412yes—
107C2×C4×C197C2×D412yes—
108C2×C4×C197C2411yescharacteristic
109C8×C197—22no—
110C22×C197#12012yes—
111C22×C197#13512yes—
112C22×C197#13712yes—
113C22×C197C22×D411yescharacteristic
114C2×D197—22no—
115C2×D197—24no—
116C2×D197—22no—
117C2×D197—22no—
118C2×D197—42no—
119C2×D197—42no—
120C2×D197—44no—
121C2×D197—48no—
122C4×C197—24no—
123C4×C197—42no—
124C4×C197#12012yes—
125C4×C197C22×D411yescharacteristic
126C4×C197C22×D411yescharacteristic
127Dic197—22no—
128Dic197—42no—
129C2×C197#45214yescomplemented
130C2×C197C2×#12011yescharacteristic
131C2×C197C2×#13511yescharacteristic
132C2×C197C2×#13711yescharacteristic
133D197—42no—
134D197—44no—
135C197C2×#45211yescharacteristic, complemented, Sylow
136C2×#452—1971nomaximal, Sylow
137C22×#41—1971no—
138C22×Q16—1971no—
139C22×SD16—1971no—
140C2×C22⋊Q8—1971no—
141#452—1978no—
142C2×#103D19711yescharacteristic, complemented
143C2×#102—1971no—
144C2×C22⋊C8—1971no—
145C2×#41—3944no—
146C2×#41—3944no—
147C2×#41—3942no—
148C2×#41—1974no—
149C22×Q8—1971no—
150C22×Q8C2×D19711yescharacteristic
151C22×D4—3941no—
152C22×D4—1971no—
153C23×C4—3941no—
154C23×C4—1971no—
155C2×Q16—3944no—
156C2×Q16—1974no—
157C2×SD16—3944no—
158C2×SD16—1974no—
159C22×C8—1971no—
160C22×C8C2×D19711yescharacteristic
161C22⋊Q8—3942no—
162C22⋊Q8—1974no—
163C2×C4⋊C4—3941no—
164C2×C4⋊C4C2×D19711yescharacteristic
165C2×C22⋊C4—3941no—
166#103C2×D19714yescomplemented
167#102—1974no—
168C22⋊C8—1974no—
169C24—3941no—
170C24—1971no—
171#41—39416no—
172#41—3948no—
173#41—3948no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—3942no—
177C2×Q8—3942no—
178C2×Q8—3941no—
179C2×Q8—1972no—
180C2×Q8D78814yes—
181C2×Q8C22×D19712yes—
182C2×D4—3948no—
183C2×D4—3944no—
184C2×D4—3941no—
185C2×D4—3942no—
186C2×D4—1974no—
187C2×D4—1972no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—3948no—
191C22×C4—3944no—
192C22×C4—3941no—
193C22×C4—3942no—
194C22×C4—3942no—
195C22×C4—1971no—
196C22×C4—1972no—
197C22×C4—1972no—
198C22×C4C22×D19711yescharacteristic
199Q16—3948no—
200SD16—3948no—
201C2×C8—22no—
202C2×C8—3942no—
203C2×C8—1972no—
204C2×C8C22×D19712yes—
205C4⋊C4—21no—
206C4⋊C4—3944no—
207C4⋊C4C22×D19712yes—
208C22⋊C4—3944no—
209C23—7882no—
210C23—7882no—
211C23—3942no—
212C23—3944no—
213C23—3942no—
214C23—3942no—
215C23—3942no—
216C23—1972no—
217C23D4×D19711yescharacteristic
218Q8—24no—
219Q8—22no—
220Q8—7882no—
221Q8—3942no—
222Q8—3942no—
223Q8C2×D78814yes—
224D4—39416no—
225D4—3944no—
226D4—3942no—
227D4—1974no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—7882no—
233C2×C4—39416no—
234C2×C4—3942no—
235C2×C4—3942no—
236C2×C4—3944no—
237C2×C4—3942no—
238C2×C4—3942no—
239C2×C4—1972no—
240C2×C4D4×D19711yescharacteristic
241C2×C4D4×D19712yes—
242C2×C4C2×D78812yes—
243C2×C4C23×D19711yescharacteristic
244C8—22no—
245C8—3942no—
246C22—7882no—
247C22—7882no—
248C22—7888no—
249C22—7884no—
250C22—3942no—
251C22—3942no—
252C22—3942no—
253C22—3944no—
254C22#2026329612yes—
255C22#2026334712yes—
256C22#2026335812yes—
257C22C2×D4×D19711yescharacteristic
258C4—24no—
259C4—42no—
260C4—7882no—
261C4—3942no—
262C4#2026329612yes—
263C4C2×D4×D19711yescharacteristic
264C4C22×D78811yescharacteristic
265C2—7884no—
266C2—7882no—
267C2#2716124414yesdirect product factor
268C2C2×#2026329611yescharacteristic
269C2C2×#2026334711yescharacteristic
270C2C2×#2026335811yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#103×C197C2
socleC23×C197C2×D4
prime radicalC22×#15691359C2
hypercenterC2×#103D197
hypocenterC197C2×#452
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C22×#15691348C2semidirect
3. C22×#15691347C2semidirect
4. C2×#20263306C2semidirect
5. #27161244C2direct (N × H)
6. C2×#20263246C2semidirect
7. C2×#20263238C2semidirect
8. C2×#103×C197C2semidirect
19. C2×#15691348C22semidirect
21. C2×#15691347C22semidirect
24. #20263306C22semidirect
27. #20263246C22semidirect
28. #20263238C22semidirect
32. #103×C197C22semidirect
129. C2×C197#452semidirect
135. C197C2×#452semidirect
142. C2×#103D197semidirect
166. #103C2×D197semidirect
267. C2#27161244direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913592
C22×#156913482
C22×#156913472
C2×#202633062
#271612442
C2×#202632462
C2×#202632382
C2×#103×C1972
C2×#452197