Finite Group Databasebeta

← #470×D101 #27180730

Subgroups — #470×D101 #27180730

Order 12928.

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

normal subgroups only

This group has 492 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×Q16×D101C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic
3C22⋊Q8×D101C212yesmaximal
4#20265436C211yesmaximal, characteristic, complemented
5#20265435C211yesmaximal, characteristic, complemented
6#20265434C212yesmaximal, complemented
7#20265433C212yesmaximal, complemented
8#20265432C211yesmaximal, characteristic, complemented
9#106×D101C211yesmaximal, characteristic, complemented
10#103×D101C212yesmaximal, complemented
11#470×C101C211yesmaximal, characteristic, complemented
12C2×Q8×D101C2212yes—
13C22×C4×D101C2211yescharacteristic
14C2×Dic404C2211yescharacteristic, complemented
15C2×#13108064C2212yescomplemented
16Q16×D101—24no—
17C2×#13108058C2211yescharacteristic
18C2×#13108057C2211yescharacteristic
19C2×#13108057C2211yescharacteristic
20M16×D101—24no—
21C2×C8×D101C2211yescharacteristic
22C2×C8×D101C2211yescharacteristic
23#15692235C2212yes—
24#15692234C2212yes—
25#15692233—22no—
26#15692232—22no—
27#15692231C2212yes—
28C4⋊C4×D101—22no—
29C4⋊C4×D101C2212yes—
30C22⋊C4×D101—22no—
31#15692187C2211yescharacteristic, complemented
32#15692186C2212yescomplemented
33#15692180C2212yescomplemented
34#15692179C2212yescomplemented
35#15692178C2212yescomplemented
36C2×Q16×C101C2211yescharacteristic, complemented
37C2×M16×C101C2211yescharacteristic
38C22⋊Q8×C101C2212yes—
39#106×C101C2211yescharacteristic, complemented
40#103×C101C2212yescomplemented
41C23×D101D411yescharacteristic
42C2×Dic202—22no—
43C2×Dic202C2312yes—
44Q8×D101—24no—
45Q8×D101—24no—
46C22×Dic101D411yescharacteristic
47C2×C4×D101—22no—
48C2×C4×D101—22no—
49C2×C4×D101—22no—
50C2×C4×D101—22no—
51C2×C4×D101D411yescharacteristic
52C2×C4×D101D411yescharacteristic
53C2×C4×D101C2311yescharacteristic
54C2×C4×D101C2311yescharacteristic
55Dic404—22no—
56#13108064—24no—
57#13108058—22no—
58#13108057—22no—
59#13108057—22no—
60C2×#11717579C2311yescharacteristic
61C2×#11717579C2311yescharacteristic
62C8×D101—22no—
63C8×D101D414yes—
64C4⋊Dic101—22no—
65C4⋊Dic101C2312yes—
66#13108053—22no—
67#13108053—22no—
68#13108053—22no—
69C22⋊Dic101—22no—
70#13108051—22no—
71#13108051—22no—
72C2×Q8×C101C2312yes—
73C22×C4×C101C2311yescharacteristic
74Q16×C101—22no—
75M16×C101—22no—
76C2×C8×C101C2311yescharacteristic
77C2×C8×C101C2311yescharacteristic
78C4⋊C4×C101—22no—
79C4⋊C4×C101C2312yes—
80C22⋊C4×C101—22no—
81C23×C101C2×D411yescharacteristic
82C22×D101—22no—
83C22×D101—24no—
84C22×D101C2×D411yescharacteristic
85C22×D101C2×D411yescharacteristic
86Q8×C101—22no—
87Q8×C101—22no—
88C2×C4×C101—21no—
89C2×C4×C101—21no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2411yescharacteristic
94C2×Dic101—22no—
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101C2×D411yescharacteristic
98C2×Dic101C2×D411yescharacteristic
99C4×D101—22no—
100C4×D101—21no—
101C4×D101—21no—
102C4×D101—22no—
103C4×D101—44no—
104C4×D101—44no—
105C4×D101C2×D412yes—
106C4×D101#4112yes—
107Dic202—22no—
108Dic202—22no—
109Dic202—44no—
110C8×C101—21no—
111C8×C101C2×D412yes—
112#11717579—21no—
113#11717579C2×D412yes—
114C22×C101—21no—
115C22×C101—22no—
116C22×C101C22×D411yescharacteristic
117C2×D101—21no—
118C2×D101—22no—
119C2×D101—44no—
120C2×D101#12112yes—
121C2×D101#13714yes—
122C4×C101—21no—
123C4×C101—42no—
124C4×C101—42no—
125C4×C101C22×D411yescharacteristic
126C4×C101C2×#4111yescharacteristic
127Dic101—21no—
128Dic101—42no—
129Dic101—42no—
130Dic101#12112yes—
131C2×C101—41no—
132C2×C101C2×#12111yescharacteristic
133C2×C101C2×#13712yes—
134D101—41no—
135D101#47014yesdirect product factor
136C2×#470—1011nomaximal, Sylow
137C101C2×#47011yescharacteristic, complemented, Sylow
138C22×Q16—1011no—
139C22×M16—1011no—
140C2×C22⋊Q8—1012no—
141#470—1011no—
142#470—1011no—
143#470—1012no—
144#470—1011no—
145#470—1012no—
146#470D10111yescharacteristic, direct product factor
147C2×#106—1011no—
148C2×#103—1012no—
149C22×Q8—1012no—
150C23×C4—1011no—
151C2×Q16—1012no—
152C2×Q16—1011no—
153C2×Q16—2024no—
154C2×Q16C2×D10111yescharacteristic, complemented
155C2×M16—1011no—
156C2×M16—1011no—
157C2×M16—1011no—
158C2×M16—2024no—
159C2×M16C2×D10111yescharacteristic
160C22×C8—1011no—
161C22×C8—1011no—
162C22⋊Q8—1012no—
163C22⋊Q8—1012no—
164C22⋊Q8—1012no—
165C22⋊Q8—2022no—
166C22⋊Q8—2022no—
167C22⋊Q8C2×D10112yes—
168C2×C4⋊C4—1012no—
169C2×C4⋊C4—2022no—
170C2×C22⋊C4—2022no—
171#106—1011no—
172#106—1012no—
173#106C2×D10111yescharacteristic, complemented
174#103—1012no—
175#103—1012no—
176#103—1012no—
177#103C2×D10112yescomplemented
178C24—1011no—
179C2×Q8—1012no—
180C2×Q8—2022no—
181C2×Q8—2024no—
182C2×Q8—2024no—
183C2×Q8C22×D10112yes—
184C22×C4—1011no—
185C22×C4—1011no—
186C22×C4—1011no—
187C22×C4—1011no—
188C22×C4—1011no—
189C22×C4—2022no—
190C22×C4—2022no—
191C22×C4—2022no—
192C22×C4—2022no—
193C22×C4C22×D10111yescharacteristic
194Q16—22no—
195Q16—2024no—
196Q16—2022no—
197M16—22no—
198M16—2022no—
199M16—2022no—
200M16—2022no—
201C2×C8—1011no—
202C2×C8—1014no—
203C2×C8—1011no—
204C2×C8—2022no—
205C2×C8C22×D10111yescharacteristic
206C2×C8C22×D10111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—1012no—
209C4⋊C4—2022no—
210C4⋊C4—2022no—
211C4⋊C4—2022no—
212C4⋊C4—2022no—
213C4⋊C4C22×D10112yes—
214C22⋊C4—22no—
215C22⋊C4—2022no—
216C22⋊C4—2022no—
217C22⋊C4—2022no—
218C23—1011no—
219C23—1011no—
220C23—2022no—
221C23—2024no—
222C23D4×D10111yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—4044no—
226Q8—2022no—
227Q8—2022no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—1011no—
233C2×C4—1012no—
234C2×C4—1012no—
235C2×C4—1011no—
236C2×C4—4044no—
237C2×C4—4044no—
238C2×C4—2022no—
239C2×C4—2022no—
240C2×C4—2022no—
241C2×C4—2022no—
242C2×C4—2022no—
243C2×C4—2021no—
244C2×C4—2021no—
245C2×C4D4×D10111yescharacteristic
246C2×C4C23×D10111yescharacteristic
247C8—21no—
248C8—1012no—
249C8—2021no—
250C8D4×D10112yes—
251C22—21no—
252C22—22no—
253C22—1012no—
254C22—1014no—
255C22—4044no—
256C22—2021no—
257C22—2022no—
258C22C2×D4×D10111yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—1012no—
263C4—4042no—
264C4—4042no—
265C4—2021no—
266C4C2×D4×D10111yescharacteristic
267C4#41×D10111yescharacteristic
268C2—41no—
269C2—1014no—
270C2—4041no—
271C2#121×D10111yescharacteristic
272C2#137×D10112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#470×C101C2
socleC22×C101C22×D4
prime radicalC23×D101D4
hypercenter#470D101
hypocenterC101C2×#470
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D101C2semidirect
4. #20265436C2semidirect
5. #20265435C2semidirect
6. #20265434C2semidirect
7. #20265433C2semidirect
8. #20265432C2semidirect
9. #106×D101C2semidirect
10. #103×D101C2semidirect
11. #470×C101C2semidirect
14. C2×Dic404C22semidirect
15. C2×#13108064C22semidirect
31. #15692187C22semidirect
32. #15692186C22semidirect
33. #15692180C22semidirect
34. #15692179C22semidirect
35. #15692178C22semidirect
36. C2×Q16×C101C22semidirect
39. #106×C101C22semidirect
40. #103×C101C22semidirect
135. D101#470direct (N × H)
137. C101C2×#470semidirect
146. #470D101direct (N × H)
154. C2×Q16C2×D101semidirect
173. #106C2×D101semidirect
177. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1012
C2×M16×D1012
C22⋊Q8×D1012
#202654362
#202654352
#202654342
#202654332
#202654322
#106×D1012
#103×D1012
#470×C1012
C2×#470101