Finite Group Databasebeta

← AGL(1,ℤ/8)×C5⋊D5 #13107300

Subgroups — AGL(1,ℤ/8)×C5⋊D5 #13107300

Order 1600.

1192 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 221 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
1#41×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×D4×C5⋊D5C211yesmaximal, characteristic, complemented
3C52×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#11717269C211yesmaximal, characteristic, complemented
5#11717252C211yesmaximal, characteristic, complemented
6#11717234C212yesmaximal, complemented
7SD16×C5⋊D5C212yesmaximal, complemented
8#11717231C212yesmaximal, complemented
9D8×C5⋊D5C212yesmaximal, complemented
10#11717228C211yesmaximal, characteristic, complemented
11M16×C5⋊D5C211yesmaximal, characteristic, complemented
12C23×C5⋊D5—21no—
13C52×#41C2211yescharacteristic, complemented
14C2×C52×D4C2211yescharacteristic, complemented
15C2×C22⋊(C5⋊D5)—21no—
16#87904C2211yescharacteristic, complemented
17Q8×C5⋊D5C2211yescharacteristic, complemented
18#87902—21no—
19#87902C2211yescharacteristic, complemented
20D4×C5⋊D5—21no—
21D4×C5⋊D5—21no—
22D4×C5⋊D5—21no—
23D4×C5⋊D5C2212yes—
24D4×C5⋊D5C2212yescomplemented
25D4×C5⋊D5C2211yescharacteristic, complemented
26#87900—21no—
27#87900C2211yescharacteristic, complemented
28C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
29C2×C4×C5⋊D5—21no—
30C2×C4×C5⋊D5C2211yescharacteristic
31C52×SD16C2212yescomplemented
32C52×D8C2212yescomplemented
33C52×M16C2211yescharacteristic, complemented
34#87811C2212yescomplemented
35#87810C2212yescomplemented
36#87809C2212yescomplemented
37#87809C2212yescomplemented
38#87804C2211yescharacteristic, complemented
39#87801C2212yescomplemented
40#87800C2212yescomplemented
41#87799C2212yescomplemented
42C8×C5⋊D5C2212yescomplemented
43D5×AGL(1,ℤ/8)—56nomaximal
44C23×C52—21no—
45C22×C5⋊D5—21no—
46C22×C5⋊D5—21no—
47C22×C5⋊D5—22no—
48C22×C5⋊D5—22no—
49C22×C5⋊D5—44no—
50C22×C5⋊D5D411yescharacteristic
51C52×Q8C2311yescharacteristic, complemented
52C52×D4—21no—
53C52×D4—21no—
54C52×D4C2312yes—
55C52×D4C2311yescharacteristic, complemented
56C2×C4×C52—21no—
57C2×C4×C52C2311yescharacteristic
58C22⋊(C5⋊D5)—21no—
59C22⋊(C5⋊D5)—21no—
60C22⋊(C5⋊D5)—21no—
61C22⋊(C5⋊D5)—22no—
62C22⋊(C5⋊D5)—22no—
63C2×#1039—21no—
64C2×#1039D411yescharacteristic
65C4⋊(C5⋊D5)—21no—
66C4⋊(C5⋊D5)—21no—
67C4⋊(C5⋊D5)C2312yes—
68C4⋊(C5⋊D5)C2311yescharacteristic, complemented
69C4×C5⋊D5—21no—
70C4×C5⋊D5—21no—
71C4×C5⋊D5—21no—
72C4×C5⋊D5D411yescharacteristic
73C4×C5⋊D5D411yescharacteristic
74C4×C5⋊D5C2311yescharacteristic
75C4×C5⋊D5C2311yescharacteristic
76#6045—21no—
77#6045C2311yescharacteristic, complemented
78C52×C8C2312yescomplemented
79#6029C2312yescomplemented
80D5×#41—56no—
81C2×D4×D5—56no—
82C5×AGL(1,ℤ/8)D516yescomplemented
83#4095—56no—
84#4078—56no—
85#4060—512no—
86D5×SD16—512no—
87#4057—512no—
88D5×D8—512no—
89#4054—56no—
90D5×M16—56no—
91C22×C52—21no—
92C22×C52—22no—
93C22×C52—42no—
94C22×C52C2×D411yescharacteristic
95C2×C5⋊D5—22no—
96C2×C5⋊D5—21no—
97C2×C5⋊D5—22no—
98C2×C5⋊D5—44no—
99C2×C5⋊D5—42no—
100C2×C5⋊D5—42no—
101C2×C5⋊D5—44no—
102C2×C5⋊D5C2×D411yescharacteristic
103C2×C5⋊D5C2×D411yescharacteristic
104C4×C52—21no—
105C4×C52C2×D411yescharacteristic
106C4×C52C2411yescharacteristic
107#1039—21no—
108#1039C2×D411yescharacteristic
109#1039C2×D411yescharacteristic
110C23×D5—106no—
111C5×#41C2×D516yescomplemented
112C2×C5×D4C2×D516yescomplemented
113C2×C22⋊D5—106no—
114#713—56no—
115Q8×D5—56no—
116#711—56no—
117#711—106no—
118D4×D5—512no—
119D4×D5—512no—
120D4×D5—56no—
121D4×D5—106no—
122D4×D5—106no—
123D4×D5—106no—
124#709—56no—
125#709—106no—
126C2×D20—56no—
127C2×C4×D5—56no—
128C2×C4×D5—106no—
129C5×SD16C2×D5112yescomplemented
130C5×D8C2×D5112yescomplemented
131C5×M16C2×D516yescomplemented
132#688—512no—
133#687—512no—
134#686—512no—
135#686—512no—
136#681—56no—
137D40—512no—
138#677—512no—
139#676—512no—
140C8×D5—512no—
141C2×AGL(1,ℤ/8)—251noSylow
142C2×C52—21no—
143C2×C52—42no—
144C2×C52—41no—
145C2×C52C22×D411yescharacteristic
146C5⋊D5—21no—
147C5⋊D5—41no—
148C5⋊D5—42no—
149C5⋊D5AGL(1,ℤ/8)12yesdirect product factor
150C23×C5—26no—
151C22×D5—56no—
152C22×D5—106no—
153C22×D5—1012no—
154C22×D5—1012no—
155C22×D5—106no—
156C22×D5—2024no—
157C5×Q8C22×D516yescomplemented
158C5×D4—26no—
159C5×D4—26no—
160C5×D4C22×D5112yes—
161C5×D4C22×D516yescomplemented
162C2×C4×C5—26no—
163C2×C4×C5C22×D516yes—
164C22⋊D5—106no—
165C22⋊D5—1012no—
166C22⋊D5—1012no—
167C22⋊D5—106no—
168C22⋊D5—106no—
169C2×Dic5—56no—
170C2×Dic5—106no—
171D20—512no—
172D20—56no—
173D20—106no—
174D20—106no—
175C4×D5—56no—
176C4×D5—56no—
177C4×D5—56no—
178C4×D5—56no—
179C4×D5—106no—
180C4×D5—106no—
181C4×D5—106no—
182Dic10—56no—
183Dic10—106no—
184C5×C8C22×D5112yescomplemented
185#168—512no—
186C2×#41—251no—
187C22×D4—251no—
188AGL(1,ℤ/8)—251no—
189AGL(1,ℤ/8)—251no—
190AGL(1,ℤ/8)—252no—
191AGL(1,ℤ/8)—252no—
192AGL(1,ℤ/8)—251no—
193AGL(1,ℤ/8)C5⋊D511yescharacteristic, direct product factor
194C2×SD16—252no—
195C2×D8—252no—
196C2×M16—251no—
197C52C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
198C22×C5—26no—
199C22×C5—212no—
200C22×C5—412no—
201C22×C5D4×D516yes—
202C2×D5—56no—
203C2×D5—56no—
204C2×D5—1012no—
205C2×D5—106no—
206C2×D5—1012no—
207C2×D5—2012no—
208C2×D5—2024no—
209C2×D5—2024no—
210C2×D5—2012no—
211C4×C5—26no—
212C4×C5D4×D516yes—
213C4×C5C23×D516yes—
214Dic5—56no—
215Dic5—56no—
216Dic5—106no—
217C24—501no—
218#41—251no—
219#41—251no—
220#41—251no—
221#41—501no—
222#41—501no—
223#41C2×C5⋊D511yescharacteristic, complemented
224C2×Q8—251no—
225C2×D4—252no—
226C2×D4—252no—
227C2×D4—251no—
228C2×D4—251no—
229C2×D4—501no—
230C2×D4—501no—
231C2×D4—501no—
232C2×D4—501no—
233C2×D4C2×C5⋊D511yescharacteristic, complemented
234C22×C4—251no—
235C22×C4—501no—
236SD16—252no—
237SD16—252no—
238SD16—252no—
239SD16C2×C5⋊D512yescomplemented
240D8—252no—
241D8—252no—
242D8—252no—
243D8C2×C5⋊D512yescomplemented
244M16—251no—
245M16—252no—
246M16C2×C5⋊D511yescharacteristic, complemented
247C2×C8—252no—
248C2×C5—26no—
249C2×C5—412no—
250C2×C5—46no—
251C2×C5C2×D4×D516yes—
252D5—512no—
253D5—106no—
254D5—206no—
255D5—2012no—
256C23—21no—
257C23—251no—
258C23—501no—
259C23—502no—
260C23—502no—
261C23—501no—
262C23—1004no—
263Q8—251no—
264Q8—501no—
265Q8C22×C5⋊D511yescharacteristic, complemented
266D4—21no—
267D4—21no—
268D4—251no—
269D4—252no—
270D4—501no—
271D4—502no—
272D4—502no—
273D4—501no—
274D4—501no—
275D4—501no—
276D4—501no—
277D4C22×C5⋊D512yes—
278D4C22×C5⋊D511yescharacteristic, complemented
279C2×C4—21no—
280C2×C4—251no—
281C2×C4—251no—
282C2×C4—251no—
283C2×C4—251no—
284C2×C4—251no—
285C2×C4—501no—
286C2×C4—501no—
287C2×C4—501no—
288C2×C4—501no—
289C2×C4C22×C5⋊D511yescharacteristic
290C8—252no—
291C8C22×C5⋊D512yescomplemented
292C5D5×AGL(1,ℤ/8)16yescomplemented
293C22—21no—
294C22—22no—
295C22—42no—
296C22—251no—
297C22—251no—
298C22—502no—
299C22—501no—
300C22—502no—
301C22—1002no—
302C22—1004no—
303C22—1002no—
304C22—1004no—
305C22D4×C5⋊D511yescharacteristic
306C4—21no—
307C4—251no—
308C4—251no—
309C4—501no—
310C4D4×C5⋊D511yescharacteristic
311C4C23×C5⋊D511yescharacteristic
312C2—21no—
313C2—42no—
314C2—41no—
315C2—252no—
316C2—501no—
317C2—1002no—
318C2—1001no—
319C2C2×D4×C5⋊D511yescharacteristic
subgroupG/H
centerC2C2×D4×C5⋊D5
commutator, cosocleC4×C52C24
FrattiniC4C23×C5⋊D5
FittingC52×AGL(1,ℤ/8)C2
socleC2×C52C22×D4
hypercenterAGL(1,ℤ/8)C5⋊D5
hypocenterC52C2×AGL(1,ℤ/8)
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×C5⋊D5C2semidirect
2. C2×D4×C5⋊D5C2semidirect
3. C52×AGL(1,ℤ/8)C2semidirect
4. #11717269C2semidirect
5. #11717252C2semidirect
6. #11717234C2semidirect
7. SD16×C5⋊D5C2semidirect
8. #11717231C2semidirect
9. D8×C5⋊D5C2semidirect
10. #11717228C2semidirect
11. M16×C5⋊D5C2semidirect
13. C52×#41C22semidirect
14. C2×C52×D4C22semidirect
16. #87904C22semidirect
17. Q8×C5⋊D5C22semidirect
19. #87902C22semidirect
24. D4×C5⋊D5C22semidirect
25. D4×C5⋊D5C22semidirect
27. #87900C22semidirect
28. C2×C4⋊(C5⋊D5)C22semidirect
31. C52×SD16C22semidirect
32. C52×D8C22semidirect
33. C52×M16C22semidirect
34. #87811C22semidirect
35. #87810C22semidirect
36. #87809C22semidirect
37. #87809C22semidirect
38. #87804C22semidirect
39. #87801C22semidirect
40. #87800C22semidirect
41. #87799C22semidirect
42. C8×C5⋊D5C22semidirect
51. C52×Q8C23semidirect
55. C52×D4C23semidirect
68. C4⋊(C5⋊D5)C23semidirect
77. #6045C23semidirect
78. C52×C8C23semidirect
79. #6029C23semidirect
82. C5×AGL(1,ℤ/8)D5semidirect
111. C5×#41C2×D5semidirect
112. C2×C5×D4C2×D5semidirect
129. C5×SD16C2×D5semidirect
130. C5×D8C2×D5semidirect
131. C5×M16C2×D5semidirect
149. C5⋊D5AGL(1,ℤ/8)direct (N × H)
157. C5×Q8C22×D5semidirect
161. C5×D4C22×D5semidirect
184. C5×C8C22×D5semidirect
193. AGL(1,ℤ/8)C5⋊D5direct (N × H)
197. C52C2×AGL(1,ℤ/8)semidirect
223. #41C2×C5⋊D5semidirect
233. C2×D4C2×C5⋊D5semidirect
239. SD16C2×C5⋊D5semidirect
243. D8C2×C5⋊D5semidirect
246. M16C2×C5⋊D5semidirect
265. Q8C22×C5⋊D5semidirect
278. D4C22×C5⋊D5semidirect
291. C8C22×C5⋊D5semidirect
292. C5D5×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#41×C5⋊D52
C2×D4×C5⋊D52
C52×AGL(1,ℤ/8)2
#117172692
#117172522
#117172342
SD16×C5⋊D52
#117172312
D8×C5⋊D52
#117172282
M16×C5⋊D52
D5×AGL(1,ℤ/8)5