Finite Group Databasebeta

← C2×#20265326 #27180312

Subgroups — C2×#20265326 #27180312

Order 12928.

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×D101C211yesmaximal, characteristic
2C22×#13108061C211yesmaximal, characteristic, complemented
3C2×#15692231C211yesmaximal, characteristic, complemented
4#20265326C218yesmaximal, direct product factor
5C2×#15692177C211yesmaximal, characteristic, complemented
6C2×#15692176C211yesmaximal, characteristic, complemented
7C2×#15692167C211yesmaximal, characteristic, complemented
8C22×SD16×C101C211yesmaximal, characteristic, complemented
9C24×D101—21no—
10C22×D404C2211yescharacteristic
11C22×C22⋊D101—21no—
12C2×D4×D101—24no—
13C2×D4×D101—22no—
14C2×D4×D101—24no—
15C2×D4×D101C2214yes—
16C22×C4×D101C2211yescharacteristic
17C2×#13108061—24no—
18C2×#13108061C2214yescomplemented
19C22×#11717579C2211yescharacteristic, complemented
20#15692231—22no—
21#15692231C2214yescomplemented
22C2×C4⋊Dic101C2211yescharacteristic, complemented
23C2×#13108053—21no—
24C2×#13108051—21no—
25#15692177C2214yescomplemented
26#15692176C2214yescomplemented
27#15692167C2214yescomplemented
28C22×Q8×C101C2211yescharacteristic, complemented
29C22×D4×C101C2211yescharacteristic
30C2×SD16×C101—24no—
31C2×SD16×C101C2214yescomplemented
32C22×C8×C101C2211yescharacteristic, complemented
33C23×D101—28no—
34C23×D101—21no—
35C23×D101—24no—
36C23×D101—44no—
37C23×D101—44no—
38C23×D101D411yescharacteristic
39C2×D404—21no—
40C2×D404—22no—
41C2×D404D414yes—
42C2×D404C2312yes—
43C2×C22⋊D101—28no—
44C2×C22⋊D101—24no—
45D4×D101—216no—
46D4×D101—28no—
47D4×D101—28no—
48C22×Dic101—21no—
49C22×Dic101D411yescharacteristic
50C2×C4×D101—22no—
51C2×C4×D101—22no—
52C2×C4×D101D412yes—
53C2×C4×D101C2312yes—
54#13108061—28no—
55C2×#11717579—22no—
56C2×#11717579C2312yescomplemented
57C4⋊Dic101—21no—
58C4⋊Dic101C2312yescomplemented
59#13108053—24no—
60#13108051—24no—
61C24×C101—21no—
62C2×Q8×C101—21no—
63C2×Q8×C101—22no—
64C2×Q8×C101—22no—
65C2×Q8×C101C2312yescomplemented
66C2×D4×C101—21no—
67C2×D4×C101—22no—
68C2×D4×C101D414yes—
69C2×D4×C101C2312yes—
70C22×C4×C101—21no—
71C22×C4×C101C2311yescharacteristic
72SD16×C101—28no—
73C2×C8×C101—22no—
74C2×C8×C101C2312yescomplemented
75C23×C101—24no—
76C23×C101—22no—
77C23×C101—42no—
78C23×C101—42no—
79C23×C101C2×D411yescharacteristic
80C22×D101—22no—
81C22×D101—22no—
82C22×D101—216no—
83C22×D101—24no—
84C22×D101—22no—
85C22×D101—48no—
86C22×D101—432no—
87C22×D101—42no—
88C22×D101—42no—
89C22×D101—48no—
90C22×D101SD1614yescomplemented
91C22×D101C2×D412yes—
92Q8×C101—22no—
93Q8×C101—22no—
94Q8×C101—42no—
95D4×C101—24no—
96D4×C101—22no—
97D4×C101C2×D414yes—
98C2×C4×C101—22no—
99C2×C4×C101—42no—
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101C2×D412yes—
102C2×C4×C101C2×D412yes—
103C2×C4×C101C2411yescharacteristic
104C22⋊D101—216no—
105C2×Dic101—22no—
106C2×Dic101—22no—
107C2×Dic101—42no—
108C2×Dic101C2×D412yes—
109D404—24no—
110D404—22no—
111D404C2×D414yes—
112C4×D101—24no—
113C4×D101—22no—
114C4×D101—22no—
115C8×C101—22no—
116#11717579—22no—
117C22×C101—24no—
118C22×C101—42no—
119C22×C101—42no—
120C22×C101—48no—
121C22×C101#12012yes—
122C22×C101C2×SD1612yes—
123C22×C101AGL(1,ℤ/8)12yes—
124C22×C101C22×D411yescharacteristic
125C2×D101—22no—
126C2×D101—22no—
127C2×D101—28no—
128C2×D101—24no—
129C2×D101—432no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—48no—
133C2×D101C2×SD1614yescomplemented
134C4×C101—42no—
135C4×C101#12012yes—
136C4×C101C22×D411yescharacteristic
137C4×C101C22×D411yescharacteristic
138Dic101—22no—
139Dic101—42no—
140C2×C101—44no—
141C2×C101#45014yescomplemented
142C2×C101C2×#12011yescharacteristic
143C2×C101C22×SD1611yescharacteristic
144C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
145D101—24no—
146D101—44no—
147C2×#450—1011nomaximal, Sylow
148C101C2×#45011yescharacteristic, complemented, Sylow
149C23×D4—1011no—
150C22×SD16—1011no—
151C22×SD16D10111yescharacteristic, complemented
152C2×C22⋊Q8—1011no—
153#450—1018no—
154C2×#102—1011no—
155C2×#102—1011no—
156C2×C22⋊C8—1011no—
157C25—2021no—
158C22×Q8C2×D10111yescharacteristic, complemented
159C22×D4—1014no—
160C22×D4—1011no—
161C22×D4—2022no—
162C22×D4—2021no—
163C22×D4—2024no—
164C22×D4—2024no—
165C22×D4C2×D10111yescharacteristic
166C23×C4—1011no—
167C2×SD16—24no—
168C2×SD16—1014no—
169C2×SD16—2024no—
170C2×SD16C2×D10114yescomplemented
171C22×C8—1011no—
172C22×C8C2×D10111yescharacteristic, complemented
173C22⋊Q8—1014no—
174C22⋊Q8—2022no—
175C2×C4⋊C4—1011no—
176C2×C4⋊C4—2021no—
177C2×C22⋊C4—2021no—
178#102—1014no—
179#102—1014no—
180C22⋊C8—1014no—
181C24—21no—
182C24—1011no—
183C24—4044no—
184C24—4044no—
185C24—2024no—
186C24—2028no—
187C24—2021no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8C22×D10112yescomplemented
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—1014no—
195C2×D4—1012no—
196C2×D4—2024no—
197C2×D4—2028no—
198C2×D4—20216no—
199C2×D4—2028no—
200C2×D4—2028no—
201C2×D4—2021no—
202C2×D4—2022no—
203C2×D4C22⋊D10114yes—
204C2×D4C22×D10112yes—
205C22×C4—21no—
206C22×C4—1011no—
207C22×C4—1012no—
208C22×C4—1012no—
209C22×C4—2021no—
210C22×C4—2022no—
211C22×C4—2022no—
212C22×C4C22×D10111yescharacteristic
213SD16—28no—
214SD16—2028no—
215C2×C8—22no—
216C2×C8—1012no—
217C2×C8—2022no—
218C2×C8C22×D10112yescomplemented
219C4⋊C4—1012no—
220C4⋊C4—2024no—
221C4⋊C4—2021no—
222C22⋊C4—2024no—
223C23—22no—
224C23—24no—
225C23—42no—
226C23—42no—
227C23—1012no—
228C23—1014no—
229C23—4048no—
230C23—40432no—
231C23—4048no—
232C23—4042no—
233C23—4042no—
234C23—20216no—
235C23—2022no—
236C23—2024no—
237C23—2022no—
238C23—2022no—
239C23D4×D10111yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—42no—
243D4—22no—
244D4—24no—
245D4—1014no—
246D4—20216no—
247D4—2022no—
248D4—2024no—
249D4C2×C22⋊D10114yes—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—1012no—
253C2×C4—4042no—
254C2×C4—2022no—
255C2×C4—2022no—
256C2×C4—2022no—
257C2×C4—2022no—
258C2×C4—2024no—
259C2×C4D4×D10112yes—
260C2×C4D4×D10111yescharacteristic
261C2×C4C2×C22⋊D10112yes—
262C2×C4C23×D10111yescharacteristic
263C8—22no—
264C8—2022no—
265C22—24no—
266C22—48no—
267C22—42no—
268C22—42no—
269C22—1014no—
270C22—40432no—
271C22—4048no—
272C22—4042no—
273C22—4042no—
274C22—2024no—
275C22—2028no—
276C22—2022no—
277C22—2022no—
278C22#1569222312yes—
279C22SD16×D10112yes—
280C22#1569228212yes—
281C22C2×D4×D10111yescharacteristic
282C4—42no—
283C4—4042no—
284C4—2022no—
285C4#1569222312yes—
286C4C2×D4×D10111yescharacteristic
287C4C22×C22⋊D10111yescharacteristic
288C2—44no—
289C2—4044no—
290C2—2024no—
291C2#2026532614yesdirect product factor
292C2C2×#1569222311yescharacteristic
293C2C2×SD16×D10111yescharacteristic
294C2C2×#1569228211yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×SD16×C101C2
socleC23×C101C2×D4
prime radicalC22×D4×D101C2
hypercenterC22×SD16D101
hypocenterC101C2×#450
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C22×#13108061C2semidirect
3. C2×#15692231C2semidirect
4. #20265326C2direct (N × H)
5. C2×#15692177C2semidirect
6. C2×#15692176C2semidirect
7. C2×#15692167C2semidirect
8. C22×SD16×C101C2semidirect
18. C2×#13108061C22semidirect
19. C22×#11717579C22semidirect
21. #15692231C22semidirect
22. C2×C4⋊Dic101C22semidirect
25. #15692177C22semidirect
26. #15692176C22semidirect
27. #15692167C22semidirect
28. C22×Q8×C101C22semidirect
31. C2×SD16×C101C22semidirect
32. C22×C8×C101C22semidirect
56. C2×#11717579C23semidirect
58. C4⋊Dic101C23semidirect
65. C2×Q8×C101C23semidirect
74. C2×C8×C101C23semidirect
90. C22×D101SD16semidirect
133. C2×D101C2×SD16semidirect
141. C2×C101#450semidirect
148. C101C2×#450semidirect
151. C22×SD16D101semidirect
158. C22×Q8C2×D101semidirect
170. C2×SD16C2×D101semidirect
172. C22×C8C2×D101semidirect
191. C2×Q8C22×D101semidirect
218. C2×C8C22×D101semidirect
291. C2#20265326direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1012
C22×#131080612
C2×#156922312
#202653262
C2×#156921772
C2×#156921762
C2×#156921672
C22×SD16×C1012
C2×#450101