Finite Group Databasebeta

← C2×#27161231 #38751895

Subgroups — C2×#27161231 #38751895

Order 25216.

996 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15691347C211yesmaximal, characteristic, complemented
3C22×#15691346C211yesmaximal, characteristic, complemented
4C2×#20263306C211yesmaximal, characteristic, complemented
5#27161231C218yesmaximal, direct product factor
6C2×#20263246C211yesmaximal, characteristic, complemented
7C2×#20263238C211yesmaximal, characteristic, complemented
8C2×#102×C197C211yesmaximal, characteristic, complemented
9C24×D197—21no—
10C22×Dic394C2211yescharacteristic, complemented
11C22×D788C2211yescharacteristic
12C22×C22⋊D197—21no—
13C2×D4×D197—24no—
14C2×D4×D197—22no—
15C2×D4×D197—24no—
16C2×D4×D197C2214yes—
17C22×C4×D197C2211yescharacteristic
18C2×#15691347—24no—
19C2×#15691347C2214yescomplemented
20C2×#15691346—24no—
21C2×#15691346C2214yescomplemented
22C22×#13096802C2211yescharacteristic, complemented
23#20263306—22no—
24#20263306C2214yescomplemented
25C2×C4⋊Dic197—21no—
26C2×#15691335—21no—
27#20263246C2214yescomplemented
28#20263238C2214yescomplemented
29C22×D4×C197C2211yescharacteristic
30C22×C8×C197C2211yescharacteristic, complemented
31C2×C4⋊C4×C197C2211yescharacteristic, complemented
32#102×C197C2214yescomplemented
33C23×D197—28no—
34C23×D197—24no—
35C23×D197—21no—
36C23×D197—44no—
37C23×D197—44no—
38C23×D197D411yescharacteristic
39C2×Dic394—21no—
40C2×Dic394—22no—
41C2×Dic394—22no—
42C2×Dic394C2312yescomplemented
43C2×D788—21no—
44C2×D788—22no—
45C2×D788D414yes—
46C2×D788C2312yes—
47C2×C22⋊D197—28no—
48C2×C22⋊D197—24no—
49D4×D197—216no—
50D4×D197—28no—
51D4×D197—28no—
52C22×Dic197—21no—
53C22×Dic197D411yescharacteristic
54C2×C4×D197—22no—
55C2×C4×D197—22no—
56C2×C4×D197D412yes—
57C2×C4×D197C2312yes—
58#15691347—28no—
59#15691346—28no—
60C2×#13096802—22no—
61C2×#13096802C2312yescomplemented
62C4⋊Dic197—24no—
63#15691335—24no—
64C24×C197—21no—
65C2×D4×C197—21no—
66C2×D4×C197—22no—
67C2×D4×C197D414yes—
68C2×D4×C197C2312yes—
69C22×C4×C197—21no—
70C22×C4×C197C2311yescharacteristic
71C2×C8×C197—22no—
72C2×C8×C197C2312yescomplemented
73C4⋊C4×C197—21no—
74C4⋊C4×C197C2312yescomplemented
75C22×D197—22no—
76C22×D197—22no—
77C22×D197—216no—
78C22×D197—24no—
79C22×D197—22no—
80C22×D197—48no—
81C22×D197—432no—
82C22×D197—42no—
83C22×D197—42no—
84C22×D197—48no—
85C22×D197SD1614yescomplemented
86C22×D197C2×D412yes—
87Dic394—22no—
88Dic394—22no—
89Dic394—42no—
90D788—24no—
91D788—22no—
92D788C2×D414yes—
93C22⋊D197—216no—
94C2×Dic197—22no—
95C2×Dic197—22no—
96C2×Dic197—42no—
97C2×Dic197C2×D412yes—
98C4×D197—24no—
99C4×D197—22no—
100C4×D197—22no—
101#13096802—22no—
102C23×C197—24no—
103C23×C197—22no—
104C23×C197—42no—
105C23×C197—42no—
106C23×C197C2×D411yescharacteristic
107D4×C197—24no—
108D4×C197—22no—
109D4×C197C2×D414yes—
110C2×C4×C197—22no—
111C2×C4×C197—42no—
112C2×C4×C197C2×D411yescharacteristic
113C2×C4×C197C2×D412yes—
114C2×C4×C197C2×D412yes—
115C2×C4×C197C2411yescharacteristic
116C8×C197—22no—
117C22×C197—24no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197—48no—
121C22×C197#12012yes—
122C22×C197C2×SD1612yes—
123C22×C197AGL(1,ℤ/8)12yes—
124C22×C197C22×D411yescharacteristic
125C2×D197—22no—
126C2×D197—22no—
127C2×D197—28no—
128C2×D197—24no—
129C2×D197—432no—
130C2×D197—42no—
131C2×D197—42no—
132C2×D197—48no—
133C2×D197C2×SD1614yescomplemented
134C4×C197—42no—
135C4×C197#12012yes—
136C4×C197C22×D411yescharacteristic
137C4×C197C22×D411yescharacteristic
138Dic197—22no—
139Dic197—42no—
140C2×C197—44no—
141C2×C197#45014yescomplemented
142C2×C197C2×#12011yescharacteristic
143C2×C197C22×SD1611yescharacteristic
144C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
145D197—24no—
146D197—44no—
147C197C2×#45011yescharacteristic, complemented, Sylow
148C2×#450—1971nomaximal, Sylow
149C23×D4—1971no—
150C22×SD16—1971no—
151C22×SD16—1971no—
152C2×C22⋊Q8—1971no—
153#450—1978no—
154C2×#102—1971no—
155C2×#102D19711yescharacteristic, complemented
156C2×C22⋊C8—1971no—
157C25—3941no—
158C22×Q8—1971no—
159C22×D4—3944no—
160C22×D4—3944no—
161C22×D4—3942no—
162C22×D4—3941no—
163C22×D4—1974no—
164C22×D4—1971no—
165C22×D4C2×D19711yescharacteristic
166C23×C4—1971no—
167C2×SD16—3944no—
168C2×SD16—3944no—
169C2×SD16—1974no—
170C2×SD16—1974no—
171C22×C8—1971no—
172C22×C8C2×D19711yescharacteristic, complemented
173C22⋊Q8—3942no—
174C22⋊Q8—1974no—
175C2×C4⋊C4—3941no—
176C2×C4⋊C4C2×D19711yescharacteristic, complemented
177C2×C22⋊C4—3941no—
178#102—1974no—
179#102C2×D19714yescomplemented
180C22⋊C8—1974no—
181C24—21no—
182C24—7884no—
183C24—7884no—
184C24—3948no—
185C24—3944no—
186C24—3941no—
187C24—1971no—
188C2×Q8—3942no—
189C2×Q8—3942no—
190C2×Q8—3941no—
191C2×Q8—1972no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—39416no—
195C2×D4—3948no—
196C2×D4—3948no—
197C2×D4—3948no—
198C2×D4—3944no—
199C2×D4—3941no—
200C2×D4—3942no—
201C2×D4—1974no—
202C2×D4—1972no—
203C2×D4D78814yes—
204C2×D4C22×D19712yes—
205C22×C4—21no—
206C22×C4—3941no—
207C22×C4—3942no—
208C22×C4—3942no—
209C22×C4—1971no—
210C22×C4—1972no—
211C22×C4—1972no—
212C22×C4C22×D19711yescharacteristic
213SD16—3948no—
214SD16—3948no—
215C2×C8—22no—
216C2×C8—3942no—
217C2×C8—1972no—
218C2×C8C22×D19712yescomplemented
219C4⋊C4—21no—
220C4⋊C4—3944no—
221C4⋊C4C22×D19712yescomplemented
222C22⋊C4—3944no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—7888no—
228C23—78832no—
229C23—7888no—
230C23—7882no—
231C23—7882no—
232C23—39416no—
233C23—3944no—
234C23—3942no—
235C23—3942no—
236C23—3942no—
237C23—1974no—
238C23—1972no—
239C23D4×D19711yescharacteristic
240Q8—7882no—
241Q8—3942no—
242Q8—3942no—
243D4—24no—
244D4—22no—
245D4—39416no—
246D4—3944no—
247D4—3942no—
248D4—1974no—
249D4C2×D78814yes—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—7882no—
253C2×C4—3942no—
254C2×C4—3942no—
255C2×C4—3944no—
256C2×C4—3942no—
257C2×C4—3942no—
258C2×C4—1972no—
259C2×C4D4×D19711yescharacteristic
260C2×C4D4×D19712yes—
261C2×C4C2×D78812yes—
262C2×C4C23×D19711yescharacteristic
263C8—22no—
264C8—3942no—
265C22—24no—
266C22—42no—
267C22—48no—
268C22—42no—
269C22—78832no—
270C22—7882no—
271C22—7888no—
272C22—7882no—
273C22—3944no—
274C22—3942no—
275C22—3948no—
276C22—3942no—
277C22—1974no—
278C22#2026329612yes—
279C22SD16×D19712yes—
280C22#2026335012yes—
281C22C2×D4×D19711yescharacteristic
282C4—42no—
283C4—7882no—
284C4—3942no—
285C4#2026329612yes—
286C4C2×D4×D19711yescharacteristic
287C4C22×D78811yescharacteristic
288C2—44no—
289C2—7884no—
290C2—3944no—
291C2#2716123114yesdirect product factor
292C2C2×#2026329611yescharacteristic
293C2C2×SD16×D19711yescharacteristic
294C2C2×#2026335011yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#102×C197C2
socleC23×C197C2×D4
prime radicalC22×D4×D197C2
hypercenterC2×#102D197
hypocenterC197C2×#450
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
2. C22×#15691347C2semidirect
3. C22×#15691346C2semidirect
4. C2×#20263306C2semidirect
5. #27161231C2direct (N × H)
6. C2×#20263246C2semidirect
7. C2×#20263238C2semidirect
8. C2×#102×C197C2semidirect
10. C22×Dic394C22semidirect
19. C2×#15691347C22semidirect
21. C2×#15691346C22semidirect
22. C22×#13096802C22semidirect
24. #20263306C22semidirect
27. #20263246C22semidirect
28. #20263238C22semidirect
30. C22×C8×C197C22semidirect
31. C2×C4⋊C4×C197C22semidirect
32. #102×C197C22semidirect
42. C2×Dic394C23semidirect
61. C2×#13096802C23semidirect
72. C2×C8×C197C23semidirect
74. C4⋊C4×C197C23semidirect
85. C22×D197SD16semidirect
133. C2×D197C2×SD16semidirect
141. C2×C197#450semidirect
147. C197C2×#450semidirect
155. C2×#102D197semidirect
172. C22×C8C2×D197semidirect
176. C2×C4⋊C4C2×D197semidirect
179. #102C2×D197semidirect
218. C2×C8C22×D197semidirect
221. C4⋊C4C22×D197semidirect
291. C2#27161231direct (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×#156913472
C22×#156913462
C2×#202633062
#271612312
C2×#202632462
C2×#202632382
C2×#102×C1972
C2×#450197