Finite Group Databasebeta

← #469×D101 #27180724

Subgroups — #469×D101 #27180724

Order 12928.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×D8×D101C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#121×D101C212yesmaximal, complemented
4#20265431C211yesmaximal, characteristic, complemented
5#20265430C211yesmaximal, characteristic, complemented
6#20265429C212yesmaximal, complemented
7#20265428C212yesmaximal, complemented
8#20265427C211yesmaximal, characteristic, complemented
9#106×D101C211yesmaximal, characteristic, complemented
10#102×D101C212yesmaximal, complemented
11#469×C101C211yesmaximal, characteristic, complemented
12C2×D4×D101—22no—
13C2×D4×D101C2212yes—
14C22×C4×D101C2211yescharacteristic
15C2×D808C2211yescharacteristic, complemented
16C2×#13108059C2212yescomplemented
17D8×D101—24no—
18C2×#13108058C2211yescharacteristic, complemented
19C2×#13108057C2211yescharacteristic, complemented
20C2×#13108057C2211yescharacteristic, complemented
21M16×D101—24no—
22C2×C8×D101C2211yescharacteristic
23C2×C8×D101C2211yescharacteristic
24#15692230C2212yescomplemented
25#15692229C2212yescomplemented
26#15692228—22no—
27#15692227—22no—
28#15692226C2212yescomplemented
29C4⋊C4×D101C2212yes—
30C22⋊C4×D101—22no—
31#15692187C2211yescharacteristic, complemented
32#15692186C2212yescomplemented
33#15692177C2212yescomplemented
34#15692176C2212yescomplemented
35#15692175C2212yescomplemented
36C2×D8×C101C2211yescharacteristic, complemented
37C2×M16×C101C2211yescharacteristic, complemented
38#121×C101C2212yescomplemented
39#106×C101C2211yescharacteristic, complemented
40#102×C101C2212yescomplemented
41C23×D101—22no—
42C23×D101D411yescharacteristic
43C2×D404—22no—
44C2×D404C2312yes—
45C2×C22⋊D101—22no—
46C2×C22⋊D101—22no—
47C2×C22⋊D101—22no—
48D4×D101—24no—
49D4×D101—24no—
50D4×D101—48no—
51C22×Dic101D411yescharacteristic
52C2×C4×D101—22no—
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55C2×C4×D101D411yescharacteristic
56C2×C4×D101D411yescharacteristic
57C2×C4×D101C2311yescharacteristic
58C2×C4×D101C2311yescharacteristic
59D808—22no—
60#13108059—24no—
61#13108058—22no—
62#13108057—22no—
63#13108057—22no—
64C2×#11717579C2311yescharacteristic
65C2×#11717579C2311yescharacteristic
66C8×D101—22no—
67C8×D101D414yescomplemented
68C4⋊Dic101C2312yes—
69#13108053—22no—
70C22⋊Dic101—22no—
71#13108051—22no—
72#13108051—22no—
73C2×D4×C101—22no—
74C2×D4×C101C2312yes—
75C22×C4×C101C2311yescharacteristic
76D8×C101—22no—
77M16×C101—22no—
78C2×C8×C101C2311yescharacteristic
79C2×C8×C101C2311yescharacteristic
80C4⋊C4×C101C2312yes—
81C22⋊C4×C101—22no—
82C23×C101—22no—
83C23×C101C2×D411yescharacteristic
84C22×D101—22no—
85C22×D101—24no—
86C22×D101—22no—
87C22×D101—44no—
88C22×D101—44no—
89C22×D101—44no—
90C22×D101C2×D411yescharacteristic
91C22×D101C2×D411yescharacteristic
92D4×C101—22no—
93D4×C101—22no—
94D4×C101—44no—
95C2×C4×C101—21no—
96C2×C4×C101—21no—
97C2×C4×C101—22no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2411yescharacteristic
100C22⋊D101—44no—
101C22⋊D101—44no—
102C22⋊D101—44no—
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101C2×D411yescharacteristic
106C2×Dic101C2×D411yescharacteristic
107D404—22no—
108D404—22no—
109D404—44no—
110C4×D101—22no—
111C4×D101—21no—
112C4×D101—21no—
113C4×D101—22no—
114C4×D101—44no—
115C4×D101C2×D412yes—
116C4×D101#4112yes—
117C8×C101—21no—
118C8×C101C2×D412yescomplemented
119#11717579—21no—
120#11717579C2×D412yescomplemented
121C22×C101—21no—
122C22×C101—22no—
123C22×C101—42no—
124C22×C101—42no—
125C22×C101—42no—
126C22×C101C22×D411yescharacteristic
127C2×D101—21no—
128C2×D101—22no—
129C2×D101—44no—
130C2×D101—42no—
131C2×D101—42no—
132C2×D101—42no—
133C2×D101—88no—
134C2×D101#12112yes—
135C2×D101AGL(1,ℤ/8)14yes—
136C4×C101—21no—
137C4×C101—42no—
138C4×C101C22×D411yescharacteristic
139C4×C101C2×#4111yescharacteristic
140Dic101—21no—
141Dic101—42no—
142Dic101#12112yes—
143C2×C101—41no—
144C2×C101—82no—
145C2×C101C2×#12111yescharacteristic
146C2×C101C2×AGL(1,ℤ/8)12yes—
147D101—41no—
148D101—82no—
149D101#46914yesdirect product factor
150C2×#469—1011nomaximal, Sylow
151C101C2×#46911yescharacteristic, complemented, Sylow
152C22×D8—1011no—
153C22×M16—1011no—
154C2×#121—1012no—
155#469—1011no—
156#469—1011no—
157#469—1012no—
158#469—1011no—
159#469—1012no—
160#469D10111yescharacteristic, direct product factor
161C2×#106—1011no—
162C2×#102—1012no—
163C22×D4—1012no—
164C22×D4—2022no—
165C23×C4—1011no—
166C2×D8—1012no—
167C2×D8—1011no—
168C2×D8—2024no—
169C2×D8C2×D10111yescharacteristic, complemented
170C2×M16—1011no—
171C2×M16—1011no—
172C2×M16—1011no—
173C2×M16—2024no—
174C2×M16C2×D10111yescharacteristic, complemented
175C22×C8—1011no—
176C22×C8—1011no—
177#121—1012no—
178#121—1012no—
179#121—1012no—
180#121—2022no—
181#121—2022no—
182#121C2×D10112yescomplemented
183C2×C4⋊C4—1012no—
184C2×C22⋊C4—2022no—
185#106—1011no—
186#106—1012no—
187#106C2×D10111yescharacteristic, complemented
188#102—1012no—
189#102—1012no—
190#102—1012no—
191#102C2×D10112yescomplemented
192C24—1011no—
193C24—2022no—
194C2×D4—22no—
195C2×D4—1012no—
196C2×D4—4048no—
197C2×D4—2022no—
198C2×D4—2024no—
199C2×D4—2024no—
200C2×D4—2022no—
201C2×D4—2022no—
202C2×D4—2022no—
203C2×D4C22×D10112yes—
204C22×C4—1011no—
205C22×C4—1011no—
206C22×C4—1011no—
207C22×C4—1011no—
208C22×C4—1011no—
209C22×C4—2022no—
210C22×C4—2022no—
211C22×C4—2022no—
212C22×C4C22×D10111yescharacteristic
213D8—22no—
214D8—2024no—
215D8—2022no—
216M16—22no—
217M16—2022no—
218M16—2022no—
219M16—2022no—
220C2×C8—1011no—
221C2×C8—1014no—
222C2×C8—1011no—
223C2×C8—2022no—
224C2×C8C22×D10111yescharacteristic
225C2×C8C22×D10111yescharacteristic
226C4⋊C4—1012no—
227C4⋊C4—2022no—
228C4⋊C4C22×D10112yes—
229C22⋊C4—22no—
230C22⋊C4—2022no—
231C22⋊C4—2022no—
232C22⋊C4—2022no—
233C23—22no—
234C23—1011no—
235C23—1011no—
236C23—4044no—
237C23—4044no—
238C23—4044no—
239C23—2022no—
240C23—2024no—
241C23—2022no—
242C23D4×D10111yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—4044no—
247D4—4044no—
248D4—4044no—
249D4—4044no—
250D4—2022no—
251D4—2022no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—1011no—
256C2×C4—1012no—
257C2×C4—1012no—
258C2×C4—1011no—
259C2×C4—4044no—
260C2×C4—2022no—
261C2×C4—2022no—
262C2×C4—2022no—
263C2×C4—2022no—
264C2×C4—2021no—
265C2×C4—2021no—
266C2×C4D4×D10111yescharacteristic
267C2×C4C23×D10111yescharacteristic
268C8—21no—
269C8—1012no—
270C8—2021no—
271C8D4×D10112yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—8088no—
278C22—1012no—
279C22—1014no—
280C22—4044no—
281C22—4042no—
282C22—4042no—
283C22—4042no—
284C22—2021no—
285C22—2022no—
286C22C2×D4×D10111yescharacteristic
287C4—21no—
288C4—42no—
289C4—1012no—
290C4—4042no—
291C4—2021no—
292C4C2×D4×D10111yescharacteristic
293C4#41×D10111yescharacteristic
294C2—41no—
295C2—82no—
296C2—8082no—
297C2—1014no—
298C2—4041no—
299C2#121×D10111yescharacteristic
300C2AGL(1,ℤ/8)×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101C2×#469
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D101C2semidirect
2. C2×M16×D101C2semidirect
3. #121×D101C2semidirect
4. #20265431C2semidirect
5. #20265430C2semidirect
6. #20265429C2semidirect
7. #20265428C2semidirect
8. #20265427C2semidirect
9. #106×D101C2semidirect
10. #102×D101C2semidirect
11. #469×C101C2semidirect
15. C2×D808C22semidirect
16. C2×#13108059C22semidirect
18. C2×#13108058C22semidirect
19. C2×#13108057C22semidirect
20. C2×#13108057C22semidirect
24. #15692230C22semidirect
25. #15692229C22semidirect
28. #15692226C22semidirect
31. #15692187C22semidirect
32. #15692186C22semidirect
33. #15692177C22semidirect
34. #15692176C22semidirect
35. #15692175C22semidirect
36. C2×D8×C101C22semidirect
37. C2×M16×C101C22semidirect
38. #121×C101C22semidirect
39. #106×C101C22semidirect
40. #102×C101C22semidirect
67. C8×D101D4semidirect
118. C8×C101C2×D4semidirect
120. #11717579C2×D4semidirect
149. D101#469direct (N × H)
151. C101C2×#469semidirect
160. #469D101direct (N × H)
169. C2×D8C2×D101semidirect
174. C2×M16C2×D101semidirect
182. #121C2×D101semidirect
187. #106C2×D101semidirect
191. #102C2×D101semidirect
271. C8D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1012
C2×M16×D1012
#121×D1012
#202654312
#202654302
#202654292
#202654282
#202654272
#106×D1012
#102×D1012
#469×C1012
C2×#469101