Finite Group Databasebeta

← #137×C5⋊D5 #13107304

Subgroups — #137×C5⋊D5 #13107304

Order 1600.

1032 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 1032 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
2C2×Q8×C5⋊D5C211yesmaximal, characteristic, complemented
3C52×#137C211yesmaximal, characteristic, complemented
4#11717271C211yesmaximal, characteristic, complemented
5#11717262C211yesmaximal, characteristic, complemented
6#11717238C212yesmaximal, complemented
7Q16×C5⋊D5C212yesmaximal, complemented
8#11717235C212yesmaximal, complemented
9SD16×C5⋊D5C212yesmaximal, complemented
10#11717229C211yesmaximal, characteristic, complemented
11M16×C5⋊D5C211yesmaximal, characteristic, complemented
12C52×#41C2211yescharacteristic
13C2×C52×Q8C2211yescharacteristic, complemented
14#87904C2211yescharacteristic
15Q8×C5⋊D5—21no—
16Q8×C5⋊D5—21no—
17Q8×C5⋊D5C2212yes—
18Q8×C5⋊D5C2211yescharacteristic
19Q8×C5⋊D5C2212yescomplemented
20#87902—21no—
21#87902C2211yescharacteristic
22D4×C5⋊D5—21no—
23D4×C5⋊D5C2211yescharacteristic
24#87900—21no—
25#87900C2211yescharacteristic
26C2×C4×C5⋊D5—21no—
27C2×C4×C5⋊D5—21no—
28C2×C4×C5⋊D5C2211yescharacteristic
29C2×#6045—21no—
30C2×#6045C2211yescharacteristic, complemented
31C52×Q16C2212yescomplemented
32C52×SD16C2212yescomplemented
33C52×M16C2211yescharacteristic, complemented
34#87812C2212yescomplemented
35#87812C2212yescomplemented
36#87811C2212yescomplemented
37#87810C2212yescomplemented
38#87804C2211yescharacteristic, complemented
39#87802C2212yescomplemented
40#87800C2212yescomplemented
41#87799C2212yescomplemented
42C8×C5⋊D5C2212yes—
43D5×#137—56nomaximal
44C22×C5⋊D5—21no—
45C22×C5⋊D5D411yescharacteristic
46C52×Q8—21no—
47C52×Q8C2312yes—
48C52×Q8C2311yescharacteristic
49C52×D4—21no—
50C52×D4C2311yescharacteristic
51C2×C4×C52—21no—
52C2×C4×C52—21no—
53C2×C4×C52C2311yescharacteristic
54C22⋊(C5⋊D5)—21no—
55C22⋊(C5⋊D5)—21no—
56C22⋊(C5⋊D5)—21no—
57C2×#1039—21no—
58C2×#1039—21no—
59C2×#1039D411yescharacteristic
60C4⋊(C5⋊D5)—21no—
61C4⋊(C5⋊D5)C2311yescharacteristic
62C4×C5⋊D5—21no—
63C4×C5⋊D5—22no—
64C4×C5⋊D5—22no—
65C4×C5⋊D5—21no—
66C4×C5⋊D5—21no—
67C4×C5⋊D5D411yescharacteristic
68C4×C5⋊D5D411yescharacteristic
69C4×C5⋊D5C2311yescharacteristic
70C4×C5⋊D5C2311yescharacteristic
71#6045—21no—
72#6045—22no—
73#6045—22no—
74#6045—21no—
75#6045C2312yes—
76#6045C2311yescharacteristic
77C52×C8C2312yes—
78#6029C2312yes—
79D5×#41—56no—
80C2×Q8×D5—56no—
81C5×#137D516yescomplemented
82#4097—56no—
83#4088—56no—
84#4064—512no—
85D5×Q16—512no—
86#4061—512no—
87D5×SD16—512no—
88#4055—56no—
89D5×M16—56no—
90C22×C52—21no—
91C22×C52C2×D411yescharacteristic
92C2×C5⋊D5—22no—
93C2×C5⋊D5—21no—
94C2×C5⋊D5—42no—
95C2×C5⋊D5C2×D411yescharacteristic
96C2×C5⋊D5C2×D411yescharacteristic
97C4×C52—22no—
98C4×C52—21no—
99C4×C52C2×D411yescharacteristic
100C4×C52C2411yescharacteristic
101#1039—22no—
102#1039—21no—
103#1039C2×D411yescharacteristic
104#1039C2×D411yescharacteristic
105C5×#41C2×D516yes—
106C2×C5×Q8C2×D516yescomplemented
107#713—56no—
108Q8×D5—512no—
109Q8×D5—512no—
110Q8×D5—56no—
111Q8×D5—106no—
112Q8×D5—106no—
113#711—56no—
114#711—106no—
115D4×D5—56no—
116D4×D5—106no—
117#709—56no—
118#709—106no—
119C2×C4×D5—56no—
120C2×C4×D5—106no—
121C2×C4×D5—106no—
122C2×Dic10—56no—
123C2×Dic10—106no—
124C5×Q16C2×D5112yescomplemented
125C5×SD16C2×D5112yescomplemented
126C5×M16C2×D516yescomplemented
127#689—512no—
128#689—512no—
129#688—512no—
130#687—512no—
131#681—56no—
132Dic20—512no—
133#677—512no—
134#676—512no—
135C8×D5—512no—
136C2×#137—251noSylow
137C2×C52—21no—
138C2×C52—41no—
139C2×C52C22×D411yescharacteristic
140C5⋊D5—21no—
141C5⋊D5—41no—
142C5⋊D5#13712yesdirect product factor
143C22×D5—56no—
144C22×D5—106no—
145C5×Q8—26no—
146C5×Q8C22×D5112yes—
147C5×Q8C22×D516yes—
148C5×D4—26no—
149C5×D4C22×D516yes—
150C2×C4×C5—26no—
151C2×C4×C5—26no—
152C2×C4×C5C22×D516yes—
153C22⋊D5—106no—
154C22⋊D5—106no—
155C22⋊D5—106no—
156C2×Dic5—56no—
157C2×Dic5—106no—
158C2×Dic5—106no—
159D20—56no—
160D20—106no—
161C4×D5—56no—
162C4×D5—56no—
163C4×D5—56no—
164C4×D5—56no—
165C4×D5—106no—
166C4×D5—106no—
167C4×D5—1012no—
168C4×D5—1012no—
169C4×D5—106no—
170Dic10—512no—
171Dic10—56no—
172Dic10—1012no—
173Dic10—1012no—
174Dic10—106no—
175Dic10—106no—
176C5×C8C22×D5112yes—
177#168—512no—
178C2×#41—251no—
179C22×Q8—251no—
180#137—251no—
181#137—251no—
182#137—252no—
183#137—252no—
184#137—251no—
185#137C5⋊D511yescharacteristic, direct product factor
186C2×Q16—252no—
187C2×SD16—252no—
188C2×M16—251no—
189C52C2×#13711yescharacteristic, complemented, Sylow
190C22×C5—26no—
191C22×C5D4×D516yes—
192C2×D5—56no—
193C2×D5—56no—
194C2×D5—1012no—
195C2×D5—106no—
196C2×D5—2012no—
197C4×C5—212no—
198C4×C5—26no—
199C4×C5D4×D516yes—
200C4×C5C23×D516yes—
201Dic5—56no—
202Dic5—56no—
203Dic5—1012no—
204Dic5—106no—
205#41—251no—
206#41—251no—
207#41—251no—
208#41—501no—
209#41—501no—
210#41C2×C5⋊D511yescharacteristic
211C2×Q8—252no—
212C2×Q8—251no—
213C2×Q8—252no—
214C2×Q8—251no—
215C2×Q8—501no—
216C2×Q8—501no—
217C2×Q8—501no—
218C2×Q8C2×C5⋊D511yescharacteristic, complemented
219C2×D4—251no—
220C2×D4—501no—
221C22×C4—251no—
222C22×C4—501no—
223C22×C4—501no—
224Q16—252no—
225Q16—252no—
226Q16—252no—
227Q16C2×C5⋊D512yescomplemented
228SD16—252no—
229SD16—252no—
230SD16—252no—
231SD16C2×C5⋊D512yescomplemented
232M16—251no—
233M16—252no—
234M16C2×C5⋊D511yescharacteristic, complemented
235C2×C8—252no—
236C2×C5—26no—
237C2×C5—46no—
238C2×C5C2×D4×D516yes—
239D5—512no—
240D5—106no—
241D5—206no—
242C23—251no—
243C23—501no—
244Q8—21no—
245Q8—252no—
246Q8—251no—
247Q8—502no—
248Q8—502no—
249Q8—501no—
250Q8—501no—
251Q8C22×C5⋊D512yes—
252Q8C22×C5⋊D511yescharacteristic
253D4—21no—
254D4—251no—
255D4—501no—
256D4—501no—
257D4—501no—
258D4—501no—
259D4C22×C5⋊D511yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—251no—
263C2×C4—251no—
264C2×C4—251no—
265C2×C4—251no—
266C2×C4—251no—
267C2×C4—501no—
268C2×C4—501no—
269C2×C4—502no—
270C2×C4—502no—
271C2×C4—501no—
272C2×C4—501no—
273C2×C4—501no—
274C2×C4C22×C5⋊D511yescharacteristic
275C8—252no—
276C8C22×C5⋊D512yes—
277C5D5×#13716yescomplemented
278C22—21no—
279C22—251no—
280C22—251no—
281C22—502no—
282C22—501no—
283C22—1002no—
284C22D4×C5⋊D511yescharacteristic
285C4—21no—
286C4—22no—
287C4—251no—
288C4—251no—
289C4—501no—
290C4—502no—
291C4D4×C5⋊D511yescharacteristic
292C4C23×C5⋊D511yescharacteristic
293C2—21no—
294C2—41no—
295C2—252no—
296C2—501no—
297C2—1001no—
298C2C2×D4×C5⋊D511yescharacteristic
subgroupG/H
centerC2C2×D4×C5⋊D5
commutator, cosocleC4×C52C24
FrattiniC4C23×C5⋊D5
FittingC52×#137C2
socleC2×C52C22×D4
prime radical#41×C5⋊D5C2
hypercenter#137C5⋊D5
hypocenterC52C2×#137
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C5⋊D5C2semidirect
3. C52×#137C2semidirect
4. #11717271C2semidirect
5. #11717262C2semidirect
6. #11717238C2semidirect
7. Q16×C5⋊D5C2semidirect
8. #11717235C2semidirect
9. SD16×C5⋊D5C2semidirect
10. #11717229C2semidirect
11. M16×C5⋊D5C2semidirect
13. C2×C52×Q8C22semidirect
19. Q8×C5⋊D5C22semidirect
30. C2×#6045C22semidirect
31. C52×Q16C22semidirect
32. C52×SD16C22semidirect
33. C52×M16C22semidirect
34. #87812C22semidirect
35. #87812C22semidirect
36. #87811C22semidirect
37. #87810C22semidirect
38. #87804C22semidirect
39. #87802C22semidirect
40. #87800C22semidirect
41. #87799C22semidirect
81. C5×#137D5semidirect
106. C2×C5×Q8C2×D5semidirect
124. C5×Q16C2×D5semidirect
125. C5×SD16C2×D5semidirect
126. C5×M16C2×D5semidirect
142. C5⋊D5#137direct (N × H)
185. #137C5⋊D5direct (N × H)
189. C52C2×#137semidirect
218. C2×Q8C2×C5⋊D5semidirect
227. Q16C2×C5⋊D5semidirect
231. SD16C2×C5⋊D5semidirect
234. M16C2×C5⋊D5semidirect
277. C5D5×#137semidirect

Maximal subgroups

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

subgroupindex
#41×C5⋊D52
C2×Q8×C5⋊D52
C52×#1372
#117172712
#117172622
#117172382
Q16×C5⋊D52
#117172352
SD16×C5⋊D52
#117172292
M16×C5⋊D52
D5×#1375