Finite Group Databasebeta

← #470×D271 #46845962

Subgroups — #470×D271 #46845962

Order 34688.

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×D271C211yesmaximal, characteristic, complemented
2C2×M16×D271C211yesmaximal, characteristic
3C22⋊Q8×D271C212yesmaximal
4#31774156C211yesmaximal, characteristic, complemented
5#31774155C211yesmaximal, characteristic, complemented
6#31774154C212yesmaximal, complemented
7#31774153C212yesmaximal, complemented
8#31774152C211yesmaximal, characteristic, complemented
9#106×D271C211yesmaximal, characteristic, complemented
10#103×D271C212yesmaximal, complemented
11#470×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271C2212yes—
13C22×C4×D271C2211yescharacteristic
14C2×Dic1084C2211yescharacteristic, complemented
15C2×#16864644C2212yescomplemented
16Q16×D271—24no—
17C2×#16864638C2211yescharacteristic
18C2×#16864637C2211yescharacteristic
19C2×#16864637C2211yescharacteristic
20M16×D271—24no—
21C2×C8×D271C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic
23#22576140C2212yes—
24#22576139C2212yes—
25#22576138—22no—
26#22576137—22no—
27#22576136C2212yes—
28C4⋊C4×D271—22no—
29C4⋊C4×D271C2212yes—
30C22⋊C4×D271—22no—
31#22576092C2211yescharacteristic, complemented
32#22576091C2212yescomplemented
33#22576085C2212yescomplemented
34#22576084C2212yescomplemented
35#22576083C2212yescomplemented
36C2×Q16×C271C2211yescharacteristic, complemented
37C2×M16×C271C2211yescharacteristic
38C22⋊Q8×C271C2212yes—
39#106×C271C2211yescharacteristic, complemented
40#103×C271C2212yescomplemented
41C23×D271D411yescharacteristic
42C2×Dic542—22no—
43C2×Dic542C2312yes—
44Q8×D271—24no—
45Q8×D271—24no—
46C22×Dic271D411yescharacteristic
47C2×C4×D271—22no—
48C2×C4×D271—22no—
49C2×C4×D271—22no—
50C2×C4×D271—22no—
51C2×C4×D271D411yescharacteristic
52C2×C4×D271D411yescharacteristic
53C2×C4×D271C2311yescharacteristic
54C2×C4×D271C2311yescharacteristic
55Dic1084—22no—
56#16864644—24no—
57#16864638—22no—
58#16864637—22no—
59#16864637—22no—
60C2×C271⋊C8C2311yescharacteristic
61C2×C271⋊C8C2311yescharacteristic
62C8×D271—22no—
63C8×D271D414yes—
64C4⋊Dic271—22no—
65C4⋊Dic271C2312yes—
66Dic271⋊C4—22no—
67Dic271⋊C4—22no—
68Dic271⋊C4—22no—
69C22⋊Dic271—22no—
70(C2×D271)⋊C4—22no—
71(C2×D271)⋊C4—22no—
72C2×Q8×C271C2312yes—
73C22×C4×C271C2311yescharacteristic
74Q16×C271—22no—
75M16×C271—22no—
76C2×C8×C271C2311yescharacteristic
77C2×C8×C271C2311yescharacteristic
78C4⋊C4×C271—22no—
79C4⋊C4×C271C2312yes—
80C22⋊C4×C271—22no—
81C22×D271—22no—
82C22×D271—24no—
83C22×D271C2×D411yescharacteristic
84C22×D271C2×D411yescharacteristic
85Dic542—22no—
86Dic542—22no—
87Dic542—44no—
88C2×Dic271—22no—
89C2×Dic271—22no—
90C2×Dic271—22no—
91C2×Dic271C2×D411yescharacteristic
92C2×Dic271C2×D411yescharacteristic
93C4×D271—22no—
94C4×D271—21no—
95C4×D271—21no—
96C4×D271—22no—
97C4×D271—44no—
98C4×D271—44no—
99C4×D271C2×D412yes—
100C4×D271#4112yes—
101C271⋊C8—21no—
102C271⋊C8C2×D412yes—
103C23×C271C2×D411yescharacteristic
104Q8×C271—22no—
105Q8×C271—22no—
106C2×C4×C271—21no—
107C2×C4×C271—21no—
108C2×C4×C271—22no—
109C2×C4×C271—22no—
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2411yescharacteristic
112C8×C271—21no—
113C8×C271C2×D412yes—
114C22×C271—21no—
115C22×C271—22no—
116C22×C271C22×D411yescharacteristic
117C2×D271—21no—
118C2×D271—22no—
119C2×D271—44no—
120C2×D271#12112yes—
121C2×D271#13714yes—
122C4×C271—21no—
123C4×C271—42no—
124C4×C271—42no—
125C4×C271C22×D411yescharacteristic
126C4×C271C2×#4111yescharacteristic
127Dic271—21no—
128Dic271—42no—
129Dic271—42no—
130Dic271#12112yes—
131C2×C271—41no—
132C2×C271C2×#12111yescharacteristic
133C2×C271C2×#13712yes—
134D271—41no—
135D271#47014yesdirect product factor
136C271C2×#47011yescharacteristic, complemented, Sylow
137C2×#470—2711nomaximal, Sylow
138C22×Q16—2711no—
139C22×M16—2711no—
140C2×C22⋊Q8—2712no—
141#470—2711no—
142#470—2711no—
143#470—2712no—
144#470—2711no—
145#470—2712no—
146#470D27111yescharacteristic, direct product factor
147C2×#106—2711no—
148C2×#103—2712no—
149C22×Q8—2712no—
150C23×C4—2711no—
151C2×Q16—2712no—
152C2×Q16—2711no—
153C2×Q16—5424no—
154C2×Q16C2×D27111yescharacteristic, complemented
155C2×M16—2711no—
156C2×M16—2711no—
157C2×M16—2711no—
158C2×M16—5424no—
159C2×M16C2×D27111yescharacteristic
160C22×C8—2711no—
161C22×C8—2711no—
162C22⋊Q8—2712no—
163C22⋊Q8—2712no—
164C22⋊Q8—2712no—
165C22⋊Q8—5422no—
166C22⋊Q8—5422no—
167C22⋊Q8C2×D27112yes—
168C2×C4⋊C4—2712no—
169C2×C4⋊C4—5422no—
170C2×C22⋊C4—5422no—
171#106—2711no—
172#106—2712no—
173#106C2×D27111yescharacteristic, complemented
174#103—2712no—
175#103—2712no—
176#103—2712no—
177#103C2×D27112yescomplemented
178C24—2711no—
179C2×Q8—2712no—
180C2×Q8—5422no—
181C2×Q8—5424no—
182C2×Q8—5424no—
183C2×Q8C22×D27112yes—
184C22×C4—2711no—
185C22×C4—2711no—
186C22×C4—2711no—
187C22×C4—2711no—
188C22×C4—2711no—
189C22×C4—5422no—
190C22×C4—5422no—
191C22×C4—5422no—
192C22×C4—5422no—
193C22×C4C22×D27111yescharacteristic
194Q16—22no—
195Q16—5424no—
196Q16—5422no—
197M16—22no—
198M16—5422no—
199M16—5422no—
200M16—5422no—
201C2×C8—2711no—
202C2×C8—2714no—
203C2×C8—2711no—
204C2×C8—5422no—
205C2×C8C22×D27111yescharacteristic
206C2×C8C22×D27111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—2712no—
209C4⋊C4—5422no—
210C4⋊C4—5422no—
211C4⋊C4—5422no—
212C4⋊C4—5422no—
213C4⋊C4C22×D27112yes—
214C22⋊C4—22no—
215C22⋊C4—5422no—
216C22⋊C4—5422no—
217C22⋊C4—5422no—
218C23—2711no—
219C23—2711no—
220C23—5422no—
221C23—5424no—
222C23D4×D27111yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—5422no—
226Q8—5422no—
227Q8—10844no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—2711no—
233C2×C4—2712no—
234C2×C4—2712no—
235C2×C4—2711no—
236C2×C4—5422no—
237C2×C4—5422no—
238C2×C4—5422no—
239C2×C4—5421no—
240C2×C4—5421no—
241C2×C4—5422no—
242C2×C4—5422no—
243C2×C4—10844no—
244C2×C4—10844no—
245C2×C4D4×D27111yescharacteristic
246C2×C4C23×D27111yescharacteristic
247C8—21no—
248C8—2712no—
249C8—5421no—
250C8D4×D27112yes—
251C22—21no—
252C22—22no—
253C22—2712no—
254C22—2714no—
255C22—5421no—
256C22—5422no—
257C22—10844no—
258C22C2×D4×D27111yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—2712no—
263C4—5421no—
264C4—10842no—
265C4—10842no—
266C4C2×D4×D27111yescharacteristic
267C4#41×D27111yescharacteristic
268C2—41no—
269C2—2714no—
270C2—10841no—
271C2#121×D27111yescharacteristic
272C2#137×D27112yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#470×C271C2
socleC22×C271C22×D4
prime radicalC23×D271D4
hypercenter#470D271
hypocenterC271C2×#470
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D271C2semidirect
4. #31774156C2semidirect
5. #31774155C2semidirect
6. #31774154C2semidirect
7. #31774153C2semidirect
8. #31774152C2semidirect
9. #106×D271C2semidirect
10. #103×D271C2semidirect
11. #470×C271C2semidirect
14. C2×Dic1084C22semidirect
15. C2×#16864644C22semidirect
31. #22576092C22semidirect
32. #22576091C22semidirect
33. #22576085C22semidirect
34. #22576084C22semidirect
35. #22576083C22semidirect
36. C2×Q16×C271C22semidirect
39. #106×C271C22semidirect
40. #103×C271C22semidirect
135. D271#470direct (N × H)
136. C271C2×#470semidirect
146. #470D271direct (N × H)
154. C2×Q16C2×D271semidirect
173. #106C2×D271semidirect
177. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D2712
C2×M16×D2712
C22⋊Q8×D2712
#317741562
#317741552
#317741542
#317741532
#317741522
#106×D2712
#103×D2712
#470×C2712
C2×#470271