Finite Group Databasebeta

← #46847979

Subgroups — #46847979

Order 34688.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22576187C212yesmaximal, complemented
2#31774569C211yesmaximal, characteristic, complemented
3#31774532C211yesmaximal, characteristic, complemented
4#31774273C211yesmaximal, characteristic, complemented
5#31774149C212yesmaximal, complemented
6#31774093C211yesmaximal, characteristic, complemented
7#31774041C212yesmaximal, complemented
8#31774032C212yesmaximal, complemented
9C4×#16864641C211yesmaximal, characteristic, complemented
10#31773974C211yesmaximal, characteristic, complemented
11#460×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2212yescomplemented
13C2×D4×D271—21no—
14C2×D4×D271C2212yes—
15#22576187—28no—
16C2×#16864641C2211yescharacteristic
17C2×D2168C2212yes—
18C2×#16864639C2212yescomplemented
19C2×#16864639C2212yescomplemented
20C2×#16864637C2212yescomplemented
21#22576157—21no—
22#22576157C2211yescharacteristic, complemented
23#22576135—22no—
24#22576135C2212yescomplemented
25#22576131—21no—
26#22576131C2212yes—
27#22576128—22no—
28C4×D1084—21no—
29C4×D1084C2211yescharacteristic
30C4×D1084C2211yescharacteristic
31#22576091C2211yescharacteristic, complemented
32#22576084C2211yescharacteristic, complemented
33#22576082C2212yescomplemented
34#22576080C2211yescharacteristic
35(C2×D271)⋊C8C2212yescomplemented
36C4×C271⋊C8C2211yescharacteristic, complemented
37C2×SD16×C271C2212yescomplemented
38#127×C271C2211yescharacteristic
39C4×Q8×C271C2211yescharacteristic, complemented
40C4⋊C8×C271C2211yescharacteristic, complemented
41#102×C271C2212yescomplemented
42C23×D271—22no—
43#16864655—24no—
44#16864655—42no—
45C2×D1084—22no—
46C2×D1084—22no—
47C2×D1084—21no—
48C2×D1084—21no—
49C2×D1084D411yescharacteristic
50C2×D1084C2312yes—
51C2×D1084C2311yescharacteristic
52C2×C22⋊D271—22no—
53C2×C22⋊D271—22no—
54D4×D271—24no—
55D4×D271—42no—
56D4×D271—44no—
57C2×C4×D271—22no—
58C2×C4×D271—21no—
59C2×C4×D271C2312yes—
60#16864641D414yescomplemented
61D2168—24no—
62#16864639—24no—
63#16864639—42no—
64#16864637—24no—
65C2×C271⋊C8C2311yescharacteristic
66C2×C271⋊C8C2311yescharacteristic
67C4⋊Dic271D411yescharacteristic
68C4⋊Dic271C2311yescharacteristic
69C22⋊Dic271—22no—
70(C2×D271)⋊C4—22no—
71(C2×D271)⋊C4—21no—
72(C2×D271)⋊C4D412yes—
73C2×Q8×C271C2311yescharacteristic
74C2×D4×C271—21no—
75C2×D4×C271—21no—
76C2×D4×C271C2312yes—
77SD16×C271—24no—
78C2×C8×C271C2312yes—
79C4⋊C4×C271—21no—
80C4⋊C4×C271C2311yescharacteristic
81C42×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—22no—
85C22×D271—44no—
86C22×D271—44no—
87C22×D271—82no—
88C22×D271C2×D412yes—
89D1084—22no—
90D1084—22no—
91D1084—21no—
92D1084—22no—
93D1084—44no—
94D1084—42no—
95D1084—81no—
96D1084—82no—
97D1084C2×D412yes—
98C22⋊D271—44no—
99C22⋊D271—44no—
100C2×Dic271—21no—
101C2×Dic271C2×D412yes—
102C4×D271—22no—
103C4×D271—22no—
104C4×D271—44no—
105C4×D271—42no—
106C271⋊C8—21no—
107C271⋊C8C2×D412yescomplemented
108C23×C271—22no—
109Q8×C271—21no—
110Q8×C271C2×D412yescomplemented
111D4×C271—22no—
112D4×C271—22no—
113D4×C271—42no—
114D4×C271—81no—
115C2×C4×C271—21no—
116C2×C4×C271—21no—
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2×D411yescharacteristic
119C2×C4×C271C2411yescharacteristic
120C8×C271—22no—
121C22×C271—42no—
122C22×C271—42no—
123C22×C271—42no—
124C22×C271C22×D411yescharacteristic
125C2×D271—22no—
126C2×D271—22no—
127C2×D271—22no—
128C2×D271—22no—
129C2×D271—41no—
130C2×D271—41no—
131C2×D271—42no—
132C2×D271—42no—
133C2×D271—42no—
134C2×D271—88no—
135C4×C271—22no—
136C4×C271—21no—
137C4×C271—41no—
138C4×C271AGL(1,ℤ/8)12yescomplemented
139C4×C271C22×D411yescharacteristic
140C4×C271H(5,𝔽2)11yescharacteristic
141Dic271—22no—
142Dic271—41no—
143C2×C271—82no—
144C2×C271D4211yescharacteristic
145C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C271#57611yescharacteristic
147D271—42no—
148D271—42no—
149D271—82no—
150C271#327411yescharacteristic, complemented, Sylow
151#3274—2711nomaximal, Sylow
152C2×AGL(1,ℤ/8)—2712no—
153#550—2711no—
154D42—2711no—
155#493—2711no—
156#469—2712no—
157#460D27111yescharacteristic, complemented
158#459—2711no—
159#449—2712no—
160#447—2712no—
161C4×SD16—2711no—
162#436—2711no—
163C2×#41—2712no—
164C22×D4—2712no—
165C22×D4—5421no—
166AGL(1,ℤ/8)—5428no—
167C2×SD16—2711no—
168C2×SD16C2×D27112yescomplemented
169C2×D8—2712no—
170C2×D8—2712no—
171C2×D8—2712no—
172C2×M16—2712no—
173#127—2711no—
174#127—5421no—
175#127C2×D27111yescharacteristic
176#121—2712no—
177#121—2712no—
178#121—5422no—
179#121—5421no—
180#120—5422no—
181C4×Q8C2×D27111yescharacteristic, complemented
182C4×D4—2711no—
183C4×D4—2711no—
184C4×D4—5421no—
185#106—2711no—
186C4⋊C8C2×D27111yescharacteristic, complemented
187#103—2711no—
188#102—2712no—
189#102—2711no—
190#102C2×D27112yescomplemented
191C22⋊C8—2712no—
192C4×C8—2711no—
193C24—5422no—
194#41—5424no—
195#41—10842no—
196C2×Q8C22×D27111yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—2711no—
200C2×D4—2711no—
201C2×D4—2712no—
202C2×D4—5422no—
203C2×D4—5422no—
204C2×D4—5422no—
205C2×D4—5424no—
206C2×D4—5422no—
207C2×D4—5421no—
208C2×D4—5421no—
209C2×D4—10842no—
210C2×D4—10844no—
211C2×D4C22×D27112yes—
212C22×C4—2712no—
213C22×C4—5422no—
214C22×C4—5421no—
215SD16—24no—
216SD16—2714no—
217D8—5424no—
218D8—5424no—
219D8—10842no—
220M16—5424no—
221C2×C8—2711no—
222C2×C8—2711no—
223C2×C8C22×D27112yes—
224C4⋊C4—21no—
225C4⋊C4—2711no—
226C4⋊C4—2711no—
227C4⋊C4C22×D27111yescharacteristic
228C22⋊C4—2712no—
229C22⋊C4—5422no—
230C22⋊C4—5422no—
231C22⋊C4—5421no—
232C42—21no—
233C42C22×D27111yescharacteristic
234C23—22no—
235C23—2712no—
236C23—5421no—
237C23—5422no—
238C23—10844no—
239C23—10844no—
240C23—21682no—
241Q8—21no—
242Q8D4×D27112yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—2712no—
248D4—5422no—
249D4—5422no—
250D4—5421no—
251D4—5422no—
252D4—10844no—
253D4—10844no—
254D4—10844no—
255D4—10842no—
256D4—21681no—
257D4—21682no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—2712no—
261C2×C4—5422no—
262C2×C4—5422no—
263C2×C4—5421no—
264C2×C4—10844no—
265C2×C4—10842no—
266C2×C4D4×D27111yescharacteristic
267C2×C4D4×D27111yescharacteristic
268C2×C4C23×D27111yescharacteristic
269C8—22no—
270C8—2712no—
271C8—5421no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—5422no—
276C22—5422no—
277C22—5422no—
278C22—5422no—
279C22—10841no—
280C22—10841no—
281C22—10842no—
282C22—10842no—
283C22—10842no—
284C22—21688no—
285C22C2×D4×D27111yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—5422no—
290C4—10841no—
291C4#2257618712yescomplemented
292C4C2×D4×D27111yescharacteristic
293C4#2257620611yescharacteristic
294C2—82no—
295C2—10842no—
296C2—10842no—
297C2—21682no—
298C2#3177453111yescharacteristic
299C2C2×#2257618711yescharacteristic
300C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#460×C271C2
socleC22×C271C22×D4
hypercenter#460D271
hypocenterC271#3274
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. #31774569C2semidirect
3. #31774532C2semidirect
4. #31774273C2semidirect
5. #31774149C2semidirect
6. #31774093C2semidirect
7. #31774041C2semidirect
8. #31774032C2semidirect
9. C4×#16864641C2semidirect
10. #31773974C2semidirect
11. #460×C271C2semidirect
12. C2×#16864655C22semidirect
18. C2×#16864639C22semidirect
19. C2×#16864639C22semidirect
20. C2×#16864637C22semidirect
22. #22576157C22semidirect
24. #22576135C22semidirect
31. #22576091C22semidirect
32. #22576084C22semidirect
33. #22576082C22semidirect
35. (C2×D271)⋊C8C22semidirect
36. C4×C271⋊C8C22semidirect
37. C2×SD16×C271C22semidirect
39. C4×Q8×C271C22semidirect
40. C4⋊C8×C271C22semidirect
41. #102×C271C22semidirect
60. #16864641D4semidirect
107. C271⋊C8C2×D4semidirect
110. Q8×C271C2×D4semidirect
138. C4×C271AGL(1,ℤ/8)semidirect
150. C271#3274semidirect
157. #460D271semidirect
168. C2×SD16C2×D271semidirect
181. C4×Q8C2×D271semidirect
186. C4⋊C8C2×D271semidirect
190. #102C2×D271semidirect
242. Q8D4×D271semidirect
291. C4#22576187semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
#317745692
#317745322
#317742732
#317741492
#317740932
#317740412
#317740322
C4×#168646412
#317739742
#460×C2712
#3274271