Finite Group Databasebeta

← C2×#27161228 #38751892

Subgroups — C2×#27161228 #38751892

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×#15691345C211yesmaximal, characteristic, complemented
3C2×#20263302C211yesmaximal, characteristic, complemented
4#27161228C218yesmaximal, direct product factor
5C2×#20263248C211yesmaximal, characteristic, complemented
6C2×#20263247C211yesmaximal, characteristic, complemented
7C2×#20263238C211yesmaximal, characteristic, complemented
8C22×SD16×C197C211yesmaximal, characteristic, complemented
9C24×D197—21no—
10C22×D788C2211yescharacteristic
11C22×C22⋊D197—21no—
12C2×D4×D197—24no—
13C2×D4×D197—22no—
14C2×D4×D197—24no—
15C2×D4×D197C2214yes—
16C22×C4×D197C2211yescharacteristic
17C2×#15691345—24no—
18C2×#15691345C2214yescomplemented
19C22×#13096802C2211yescharacteristic, complemented
20#20263302—22no—
21#20263302C2214yescomplemented
22C2×C4⋊Dic197C2211yescharacteristic, complemented
23C2×#15691337—21no—
24C2×#15691335—21no—
25#20263248C2214yescomplemented
26#20263247C2214yescomplemented
27#20263238C2214yescomplemented
28C22×Q8×C197C2211yescharacteristic, complemented
29C22×D4×C197C2211yescharacteristic
30C2×SD16×C197—24no—
31C2×SD16×C197C2214yescomplemented
32C22×C8×C197C2211yescharacteristic, complemented
33C23×D197—28no—
34C23×D197—24no—
35C23×D197—21no—
36C23×D197—44no—
37C23×D197—44no—
38C23×D197D411yescharacteristic
39C2×D788—21no—
40C2×D788—22no—
41C2×D788D414yes—
42C2×D788C2312yes—
43C2×C22⋊D197—28no—
44C2×C22⋊D197—24no—
45D4×D197—216no—
46D4×D197—28no—
47D4×D197—28no—
48C22×Dic197—21no—
49C22×Dic197D411yescharacteristic
50C2×C4×D197—22no—
51C2×C4×D197—22no—
52C2×C4×D197D412yes—
53C2×C4×D197C2312yes—
54#15691345—28no—
55C2×#13096802—22no—
56C2×#13096802C2312yescomplemented
57C4⋊Dic197—21no—
58C4⋊Dic197C2312yescomplemented
59#15691337—24no—
60#15691335—24no—
61C24×C197—21no—
62C2×Q8×C197—21no—
63C2×Q8×C197—22no—
64C2×Q8×C197—22no—
65C2×Q8×C197C2312yescomplemented
66C2×D4×C197—21no—
67C2×D4×C197—22no—
68C2×D4×C197D414yes—
69C2×D4×C197C2312yes—
70C22×C4×C197—21no—
71C22×C4×C197C2311yescharacteristic
72SD16×C197—28no—
73C2×C8×C197—22no—
74C2×C8×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—
87D788—24no—
88D788—22no—
89D788C2×D414yes—
90C22⋊D197—216no—
91C2×Dic197—22no—
92C2×Dic197—22no—
93C2×Dic197—42no—
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
104Q8×C197—22no—
105Q8×C197—22no—
106Q8×C197—42no—
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×SD16D19711yescharacteristic, complemented
152C2×C22⋊Q8—1971no—
153#450—1978no—
154C2×#102—1971no—
155C2×#102—1971no—
156C2×C22⋊C8—1971no—
157C25—3941no—
158C22×Q8C2×D19711yescharacteristic, complemented
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—24no—
168C2×SD16—3944no—
169C2×SD16—1974no—
170C2×SD16C2×D19714yescomplemented
171C22×C8—1971no—
172C22×C8C2×D19711yescharacteristic, complemented
173C22⋊Q8—3942no—
174C22⋊Q8—1974no—
175C2×C4⋊C4—3941no—
176C2×C4⋊C4—1971no—
177C2×C22⋊C4—3941no—
178#102—1974no—
179#102—1974no—
180C22⋊C8—1974no—
181C24—21no—
182C24—7884no—
183C24—7884no—
184C24—3948no—
185C24—3944no—
186C24—3941no—
187C24—1971no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8C22×D19712yescomplemented
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×D4C22⋊D19714yes—
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—28no—
214SD16—3948no—
215C2×C8—22no—
216C2×C8—3942no—
217C2×C8—1972no—
218C2×C8C22×D19712yescomplemented
219C4⋊C4—3944no—
220C4⋊C4—3941no—
221C4⋊C4—1972no—
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—22no—
241Q8—22no—
242Q8—42no—
243D4—24no—
244D4—22no—
245D4—39416no—
246D4—3944no—
247D4—3942no—
248D4—1974no—
249D4C2×C22⋊D19714yes—
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×C22⋊D19712yes—
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#2026329412yes—
279C22SD16×D19712yes—
280C22#2026335312yes—
281C22C2×D4×D19711yescharacteristic
282C4—42no—
283C4—7882no—
284C4—3942no—
285C4#2026329412yes—
286C4C2×D4×D19711yescharacteristic
287C4C22×C22⋊D19711yescharacteristic
288C2—44no—
289C2—7884no—
290C2—3944no—
291C2#2716122814yesdirect product factor
292C2C2×#2026329411yescharacteristic
293C2C2×SD16×D19711yescharacteristic
294C2C2×#2026335311yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×SD16×C197C2
socleC23×C197C2×D4
prime radicalC22×D4×D197C2
hypercenterC22×SD16D197
hypocenterC197C2×#450
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C22×#15691345C2semidirect
3. C2×#20263302C2semidirect
4. #27161228C2direct (N × H)
5. C2×#20263248C2semidirect
6. C2×#20263247C2semidirect
7. C2×#20263238C2semidirect
8. C22×SD16×C197C2semidirect
18. C2×#15691345C22semidirect
19. C22×#13096802C22semidirect
21. #20263302C22semidirect
22. C2×C4⋊Dic197C22semidirect
25. #20263248C22semidirect
26. #20263247C22semidirect
27. #20263238C22semidirect
28. C22×Q8×C197C22semidirect
31. C2×SD16×C197C22semidirect
32. C22×C8×C197C22semidirect
56. C2×#13096802C23semidirect
58. C4⋊Dic197C23semidirect
65. C2×Q8×C197C23semidirect
74. C2×C8×C197C23semidirect
85. C22×D197SD16semidirect
133. C2×D197C2×SD16semidirect
141. C2×C197#450semidirect
147. C197C2×#450semidirect
151. C22×SD16D197semidirect
158. C22×Q8C2×D197semidirect
170. C2×SD16C2×D197semidirect
172. C22×C8C2×D197semidirect
191. C2×Q8C22×D197semidirect
218. C2×C8C22×D197semidirect
291. C2#27161228direct (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×#156913452
C2×#202633022
#271612282
C2×#202632482
C2×#202632472
C2×#202632382
C22×SD16×C1972
C2×#450197