Finite Group Databasebeta

← C2×#27161082 #38751211

Subgroups — C2×#27161082 #38751211

Order 25216.

1008 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 229 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×D4×D197C211yesmaximal, characteristic
2C22×#15691341C211yesmaximal, characteristic, complemented
3C2×#20263284C211yesmaximal, characteristic, complemented
4#27161082C218yesmaximal, direct product factor
5C2×#20263248C211yesmaximal, characteristic, complemented
6C2×#20263247C211yesmaximal, characteristic, complemented
7C2×#20263246C211yesmaximal, characteristic, complemented
8C2×#102×C197C211yesmaximal, characteristic, complemented
9C24×D197—21no—
10C22×D788C2211yescharacteristic
11C22×C22⋊D197—21no—
12C2×D4×D197—22no—
13C2×D4×D197—24no—
14C2×D4×D197C418yescomplemented
15C2×D4×D197C2214yes—
16C22×C4×D197C2211yescharacteristic
17C2×#15691341—24no—
18C2×#15691341C2214yescomplemented
19C22×#13096802C2211yescharacteristic, complemented
20#20263284—22no—
21#20263284C2214yescomplemented
22C2×C4⋊Dic197C2211yescharacteristic, complemented
23C2×#15691335—21no—
24C2×C4×Dic197—21no—
25#20263248C2214yescomplemented
26#20263247C2214yescomplemented
27#20263246C2214yescomplemented
28C22×D4×C197C2211yescharacteristic
29C22×C8×C197C2211yescharacteristic, complemented
30C2×C4⋊C4×C197C2211yescharacteristic, complemented
31#102×C197C2214yescomplemented
32C23×D197—28no—
33C23×D197—24no—
34C23×D197—21no—
35C23×D197—44no—
36C23×D197—44no—
37C23×D197D411yescharacteristic
38C2×D788—21no—
39C2×D788—22no—
40C2×D788C2×C414yes—
41C2×D788C2312yes—
42C2×C22⋊D197—28no—
43C2×C22⋊D197—24no—
44D4×D197—216no—
45D4×D197—28no—
46D4×D197C2×C4116yescomplemented
47C22×Dic197—21no—
48C22×Dic197D411yescharacteristic
49C2×C4×D197C2×C414yes—
50C2×C4×D197D412yes—
51C2×C4×D197D414yes—
52C2×C4×D197C2312yes—
53#15691341—28no—
54C2×#13096802—22no—
55C2×#13096802C2312yescomplemented
56C4⋊Dic197—21no—
57C4⋊Dic197C2312yescomplemented
58#15691335—24no—
59C4×Dic197—24no—
60C24×C197—21no—
61C2×D4×C197—21no—
62C2×D4×C197—22no—
63C2×D4×C197C2×C414yes—
64C2×D4×C197C2312yes—
65C22×C4×C197—21no—
66C22×C4×C197C2311yescharacteristic
67C2×C8×C197—22no—
68C2×C8×C197C2312yescomplemented
69C4⋊C4×C197—21no—
70C4⋊C4×C197C2312yescomplemented
71C22×D197—22no—
72C22×D197—22no—
73C22×D197—24no—
74C22×D197—22no—
75C22×D197—216no—
76C22×D197—48no—
77C22×D197—48no—
78C22×D197—432no—
79C22×D197—42no—
80C22×D197—42no—
81C22×D197C22⋊C414yes—
82C22×D197C2×D412yes—
83D788—24no—
84D788—22no—
85D788C22×C414yes—
86C22⋊D197—216no—
87C2×Dic197—22no—
88C2×Dic197—42no—
89C2×Dic197C22⋊C414yes—
90C2×Dic197C2×D412yes—
91C4×D197C22⋊C418yes—
92C4×D197C22×C414yes—
93C4×D197C2×D414yes—
94#13096802—22no—
95C23×C197—24no—
96C23×C197—22no—
97C23×C197—42no—
98C23×C197—42no—
99C23×C197C2×D411yescharacteristic
100D4×C197—24no—
101D4×C197—22no—
102D4×C197C22×C414yes—
103C2×C4×C197—22no—
104C2×C4×C197—42no—
105C2×C4×C197C22×C412yes—
106C2×C4×C197C2×D411yescharacteristic
107C2×C4×C197C2×D412yes—
108C2×C4×C197C2411yescharacteristic
109C8×C197—22no—
110C22×C197—24no—
111C22×C197—42no—
112C22×C197—42no—
113C22×C197—48no—
114C22×C197C2×C22⋊C412yes—
115C22×C197AGL(1,ℤ/8)12yes—
116C22×C197AGL(1,ℤ/8)12yes—
117C22×C197C22×D411yescharacteristic
118C2×D197—22no—
119C2×D197—22no—
120C2×D197—28no—
121C2×D197—24no—
122C2×D197—432no—
123C2×D197—42no—
124C2×D197—42no—
125C2×D197—48no—
126C2×D197C2×C22⋊C414yes—
127C4×C197—42no—
128C4×C197C2×C22⋊C412yes—
129C4×C197C23×C411yescharacteristic
130C4×C197C22×D411yescharacteristic
131Dic197—42no—
132Dic197C2×C22⋊C414yes—
133C2×C197—44no—
134C2×C197#41814yescomplemented
135C2×C197C22×C22⋊C411yescharacteristic
136C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
138D197—24no—
139D197—44no—
140C197C2×#41811yescharacteristic, complemented, Sylow
141C2×#418—1971nomaximal, Sylow
142C23×D4—1971no—
143C22×M16—1971no—
144C2×#117—1971no—
145#418—1978no—
146C2×#102—1971no—
147C2×#102—1971no—
148C2×#102—1971no—
149C2×#102D19711yescharacteristic, complemented
150C25—3941no—
151C22×D4—3942no—
152C22×D4—3944no—
153C22×D4—3941no—
154C22×D4—1978no—
155C22×D4—1974no—
156C22×D4—1971no—
157C22×D4C2×D19711yescharacteristic
158C23×C4—1971no—
159C2×M16—3944no—
160C2×M16—1974no—
161C22×C8—1971no—
162C22×C8C2×D19711yescharacteristic, complemented
163#117—3942no—
164#117—1974no—
165C2×C4⋊C4—1971no—
166C2×C4⋊C4C2×D19711yescharacteristic, complemented
167C2×C22⋊C4—3941no—
168C2×C42—3941no—
169#102—1974no—
170#102—1974no—
171#102—1974no—
172#102C2×D19714yescomplemented
173C24—21no—
174C24—7884no—
175C24—7884no—
176C24—3948no—
177C24—3944no—
178C24—3941no—
179C24—1971no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—39416no—
183C2×D4—3948no—
184C2×D4—3948no—
185C2×D4—3944no—
186C2×D4—3941no—
187C2×D4—3942no—
188C2×D4—19716no—
189C2×D4—1974no—
190C2×D4—1972no—
191C2×D4C4×D19714yes—
192C2×D4C22×D19712yes—
193C22×C4—21no—
194C22×C4—3941no—
195C22×C4—1971no—
196C22×C4—1974no—
197C22×C4—1972no—
198C22×C4—1974no—
199C22×C4—1972no—
200C22×C4C22×D19711yescharacteristic
201M16—3948no—
202C2×C8—22no—
203C2×C8—3942no—
204C2×C8—1972no—
205C2×C8C22×D19712yescomplemented
206C4⋊C4—21no—
207C4⋊C4—3941no—
208C4⋊C4—1972no—
209C4⋊C4C22×D19712yescomplemented
210C22⋊C4—3944no—
211C42—3944no—
212C23—24no—
213C23—22no—
214C23—42no—
215C23—42no—
216C23—7888no—
217C23—7888no—
218C23—78832no—
219C23—7882no—
220C23—7882no—
221C23—39416no—
222C23—3944no—
223C23—3942no—
224C23—3942no—
225C23—3942no—
226C23—1974no—
227C23—1972no—
228C23D4×D19711yescharacteristic
229D4—24no—
230D4—22no—
231D4—39416no—
232D4—3944no—
233D4—3942no—
234D4—1974no—
235D4C2×C4×D19714yes—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—7882no—
239C2×C4—3942no—
240C2×C4—1974no—
241C2×C4—1972no—
242C2×C4—1978no—
243C2×C4—1974no—
244C2×C4—1974no—
245C2×C4C2×C4×D19712yes—
246C2×C4D4×D19711yescharacteristic
247C2×C4D4×D19712yes—
248C2×C4C23×D19711yescharacteristic
249C8—22no—
250C8—3942no—
251C22—24no—
252C22—42no—
253C22—42no—
254C22—48no—
255C22—78832no—
256C22—7882no—
257C22—7882no—
258C22—7888no—
259C22—3944no—
260C22—3942no—
261C22—3942no—
262C22—3948no—
263C22—1974no—
264C22C22⋊C4×D19712yes—
265C22#2026335012yes—
266C22#2026335312yes—
267C22C2×D4×D19711yescharacteristic
268C4—42no—
269C4—7882no—
270C4—1974no—
271C4C22⋊C4×D19712yes—
272C4C22×C4×D19711yescharacteristic
273C4C2×D4×D19711yescharacteristic
274C2—44no—
275C2—7884no—
276C2—3944no—
277C2#2716108214yesdirect product factor
278C2C2×C22⋊C4×D19711yescharacteristic
279C2C2×#2026335011yescharacteristic
280C2C2×#2026335311yescharacteristic
subgroupG/H
centerC23D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
FittingC2×#102×C197C2
socleC23×C197C2×D4
prime radicalC22×D4×D197C2
cosocleC2×C4×C197C24
hypercenterC2×#102D197
hypocenterC197C2×#418
LCSG ▷ 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
2. C22×#15691341C2semidirect
3. C2×#20263284C2semidirect
4. #27161082C2direct (N × H)
5. C2×#20263248C2semidirect
6. C2×#20263247C2semidirect
7. C2×#20263246C2semidirect
8. C2×#102×C197C2semidirect
14. C2×D4×D197C4semidirect
18. C2×#15691341C22semidirect
19. C22×#13096802C22semidirect
21. #20263284C22semidirect
22. C2×C4⋊Dic197C22semidirect
25. #20263248C22semidirect
26. #20263247C22semidirect
27. #20263246C22semidirect
29. C22×C8×C197C22semidirect
30. C2×C4⋊C4×C197C22semidirect
31. #102×C197C22semidirect
46. D4×D197C2×C4semidirect
55. C2×#13096802C23semidirect
57. C4⋊Dic197C23semidirect
68. C2×C8×C197C23semidirect
70. C4⋊C4×C197C23semidirect
134. C2×C197#418semidirect
140. C197C2×#418semidirect
149. C2×#102D197semidirect
162. C22×C8C2×D197semidirect
166. C2×C4⋊C4C2×D197semidirect
172. #102C2×D197semidirect
205. C2×C8C22×D197semidirect
209. C4⋊C4C22×D197semidirect
277. C2#27161082direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C22×#156913412
C2×#202632842
#271610822
C2×#202632482
C2×#202632472
C2×#202632462
C2×#102×C1972
C2×#418197