Finite Group Databasebeta

← #13107301

Subgroups — #13107301

Order 1600.

1048 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 219 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×#87902C211yesmaximal, characteristic, complemented
3C52×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#11717271C211yesmaximal, characteristic, complemented
5#11717252C211yesmaximal, characteristic, complemented
6#11717236C212yesmaximal, complemented
7#11717235C212yesmaximal, complemented
8#11717232C212yesmaximal, complemented
9#11717231C212yesmaximal, complemented
10#11717229C211yesmaximal, characteristic, complemented
11M16×C5⋊D5C211yesmaximal, characteristic, complemented
12C52×#41C2211yescharacteristic
13C2×C52×D4C2211yescharacteristic, complemented
14C2×C22⋊(C5⋊D5)—21no—
15C22×#1039—21no—
16#87904C2211yescharacteristic, complemented
17Q8×C5⋊D5C2211yescharacteristic, complemented
18#87902—21no—
19#87902—21no—
20#87902—21no—
21#87902C2212yes—
22#87902C2211yescharacteristic, complemented
23#87902C2212yescomplemented
24D4×C5⋊D5—21no—
25D4×C5⋊D5C2211yescharacteristic, complemented
26#87900—21no—
27#87900C2211yescharacteristic
28C2×C4×C5⋊D5—21no—
29C2×C4×C5⋊D5C2211yescharacteristic
30C2×#6045C2211yescharacteristic, complemented
31C52×SD16C2212yescomplemented
32C52×D8C2212yescomplemented
33C52×M16C2211yescharacteristic, complemented
34#87812C2212yescomplemented
35#87810C2212yescomplemented
36#87810C2212yescomplemented
37#87809C2212yescomplemented
38#87804C2211yescharacteristic, complemented
39#87802C2212yescomplemented
40#87800C2212yescomplemented
41#87799C2212yescomplemented
42C8×C5⋊D5C2212yescomplemented
43#66107—56nomaximal
44C23×C52—21no—
45C22×C5⋊D5—21no—
46C22×C5⋊D5D411yescharacteristic
47C52×Q8C2311yescharacteristic
48C52×D4—21no—
49C52×D4—21no—
50C52×D4C2312yes—
51C52×D4C2311yescharacteristic
52C2×C4×C52—21no—
53C2×C4×C52C2311yescharacteristic
54C22⋊(C5⋊D5)—21no—
55C22⋊(C5⋊D5)—21no—
56C22⋊(C5⋊D5)—22no—
57C22⋊(C5⋊D5)—22no—
58C22⋊(C5⋊D5)—21no—
59C2×#1039—21no—
60C2×#1039—21no—
61C2×#1039—22no—
62C2×#1039—22no—
63C2×#1039D411yescharacteristic
64C4⋊(C5⋊D5)—21no—
65C4⋊(C5⋊D5)C2311yescharacteristic
66C4×C5⋊D5—21no—
67C4×C5⋊D5—21no—
68C4×C5⋊D5—21no—
69C4×C5⋊D5D411yescharacteristic
70C4×C5⋊D5D411yescharacteristic
71C4×C5⋊D5C2311yescharacteristic
72C4×C5⋊D5C2311yescharacteristic
73#6045—21no—
74#6045—21no—
75#6045C2311yescharacteristic
76#6045C2312yes—
77C52×C8C2312yes—
78#6029C2312yes—
79D5×#41—56no—
80C2×#711—56no—
81C5×AGL(1,ℤ/8)D516yescomplemented
82#4097—56no—
83#4078—56no—
84#4062—512no—
85#4061—512no—
86#4058—512no—
87#4057—512no—
88#4055—56no—
89D5×M16—56no—
90C22×C52—22no—
91C22×C52—21no—
92C22×C52—42no—
93C22×C52C2×D411yescharacteristic
94C2×C5⋊D5—21no—
95C2×C5⋊D5—41no—
96C2×C5⋊D5—42no—
97C2×C5⋊D5C2×D411yescharacteristic
98C2×C5⋊D5C2×D411yescharacteristic
99C4×C52—21no—
100C4×C52C2×D411yescharacteristic
101C4×C52C2411yescharacteristic
102#1039—21no—
103#1039—22no—
104#1039C2×D411yescharacteristic
105#1039C2×D411yescharacteristic
106C5×#41C2×D516yes—
107C2×C5×D4C2×D516yescomplemented
108C2×C22⋊D5—106no—
109C22×Dic5—106no—
110#713—56no—
111Q8×D5—56no—
112#711—512no—
113#711—512no—
114#711—56no—
115#711—106no—
116#711—106no—
117#711—106no—
118D4×D5—56no—
119D4×D5—106no—
120#709—56no—
121#709—106no—
122C2×C4×D5—56no—
123C2×C4×D5—106no—
124C2×Dic10—56no—
125C5×SD16C2×D5112yescomplemented
126C5×D8C2×D5112yescomplemented
127C5×M16C2×D516yescomplemented
128#689—512no—
129#687—512no—
130#687—512no—
131#686—512no—
132#681—56no—
133Dic20—512no—
134#677—512no—
135#676—512no—
136C8×D5—512no—
137#575—251noSylow
138C2×C52—21no—
139C2×C52—41no—
140C2×C52—42no—
141C2×C52C22×D411yescharacteristic
142C5⋊D5—21no—
143C5⋊D5—21no—
144C5⋊D5—41no—
145C23×C5—26no—
146C22×D5—56no—
147C22×D5—106no—
148C5×Q8C22×D516yes—
149C5×D4—26no—
150C5×D4—26no—
151C5×D4C22×D5112yes—
152C5×D4C22×D516yes—
153C2×C4×C5—26no—
154C2×C4×C5C22×D516yes—
155C22⋊D5—106no—
156C22⋊D5—106no—
157C22⋊D5—1012no—
158C22⋊D5—1012no—
159C22⋊D5—106no—
160C2×Dic5—56no—
161C2×Dic5—106no—
162C2×Dic5—1012no—
163C2×Dic5—1012no—
164C2×Dic5—106no—
165D20—56no—
166D20—106no—
167C4×D5—56no—
168C4×D5—56no—
169C4×D5—56no—
170C4×D5—56no—
171C4×D5—106no—
172C4×D5—106no—
173C4×D5—106no—
174Dic10—512no—
175Dic10—56no—
176Dic10—106no—
177Dic10—106no—
178C5×C8C22×D5112yes—
179#168—512no—
180C2×#41—251no—
181C2×#41—251no—
182#137—251no—
183#137—252no—
184#137—251no—
185AGL(1,ℤ/8)—251no—
186AGL(1,ℤ/8)—252no—
187AGL(1,ℤ/8)C5⋊D511yescharacteristic, complemented
188#135—252no—
189#135—252no—
190C2×M16—251no—
191C52#57511yescharacteristic, complemented, Sylow
192C22×C5—212no—
193C22×C5—26no—
194C22×C5—412no—
195C22×C5D4×D516yes—
196C2×D5—56no—
197C2×D5—56no—
198C2×D5—106no—
199C2×D5—206no—
200C2×D5—2012no—
201C4×C5—26no—
202C4×C5D4×D516yes—
203C4×C5C23×D516yes—
204Dic5—56no—
205Dic5—56no—
206Dic5—106no—
207Dic5—1012no—
208#41—251no—
209#41—252no—
210#41—251no—
211#41—252no—
212#41—251no—
213#41—501no—
214#41—501no—
215#41—501no—
216#41—501no—
217#41C2×C5⋊D511yescharacteristic
218C2×Q8—251no—
219C2×Q8—251no—
220C2×D4—251no—
221C2×D4—501no—
222C2×D4—501no—
223C2×D4C2×C5⋊D511yescharacteristic, complemented
224C22×C4—251no—
225C22×C4—501no—
226C22×C4—501no—
227Q16—252no—
228Q16—252no—
229SD16—252no—
230SD16—252no—
231SD16—252no—
232SD16C2×C5⋊D512yescomplemented
233D8—252no—
234D8C2×C5⋊D512yescomplemented
235M16—251no—
236M16—252no—
237M16C2×C5⋊D511yescharacteristic, complemented
238C2×C8—252no—
239C2×C5—26no—
240C2×C5—46no—
241C2×C5—412no—
242C2×C5C2×D4×D516yes—
243D5—106no—
244D5—106no—
245D5—206no—
246C23—21no—
247C23—251no—
248C23—501no—
249Q8—251no—
250Q8—252no—
251Q8—501no—
252Q8—501no—
253Q8C22×C5⋊D511yescharacteristic
254D4—21no—
255D4—21no—
256D4—251no—
257D4—501no—
258D4—501no—
259D4—501no—
260D4—502no—
261D4—502no—
262D4—501no—
263D4C22×C5⋊D512yes—
264D4C22×C5⋊D511yescharacteristic
265C2×C4—21no—
266C2×C4—251no—
267C2×C4—251no—
268C2×C4—251no—
269C2×C4—251no—
270C2×C4—251no—
271C2×C4—501no—
272C2×C4—501no—
273C2×C4—501no—
274C2×C4—502no—
275C2×C4—502no—
276C2×C4—501no—
277C2×C4—501no—
278C2×C4C22×C5⋊D511yescharacteristic
279C8—252no—
280C8C22×C5⋊D512yes—
281C5#6610716yescomplemented
282C22—21no—
283C22—22no—
284C22—42no—
285C22—251no—
286C22—251no—
287C22—501no—
288C22—1001no—
289C22—1002no—
290C22D4×C5⋊D511yescharacteristic
291C4—21no—
292C4—251no—
293C4—251no—
294C4—501no—
295C4—502no—
296C4D4×C5⋊D511yescharacteristic
297C4C23×C5⋊D511yescharacteristic
298C2—21no—
299C2—42no—
300C2—41no—
301C2—501no—
302C2—501no—
303C2—1001no—
304C2C2×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
hypocenterC52#575
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×#87902C2semidirect
3. C52×AGL(1,ℤ/8)C2semidirect
4. #11717271C2semidirect
5. #11717252C2semidirect
6. #11717236C2semidirect
7. #11717235C2semidirect
8. #11717232C2semidirect
9. #11717231C2semidirect
10. #11717229C2semidirect
11. M16×C5⋊D5C2semidirect
13. C2×C52×D4C22semidirect
16. #87904C22semidirect
17. Q8×C5⋊D5C22semidirect
22. #87902C22semidirect
23. #87902C22semidirect
25. D4×C5⋊D5C22semidirect
30. C2×#6045C22semidirect
31. C52×SD16C22semidirect
32. C52×D8C22semidirect
33. C52×M16C22semidirect
34. #87812C22semidirect
35. #87810C22semidirect
36. #87810C22semidirect
37. #87809C22semidirect
38. #87804C22semidirect
39. #87802C22semidirect
40. #87800C22semidirect
41. #87799C22semidirect
42. C8×C5⋊D5C22semidirect
81. C5×AGL(1,ℤ/8)D5semidirect
107. C2×C5×D4C2×D5semidirect
125. C5×SD16C2×D5semidirect
126. C5×D8C2×D5semidirect
127. C5×M16C2×D5semidirect
187. AGL(1,ℤ/8)C5⋊D5semidirect
191. C52#575semidirect
223. C2×D4C2×C5⋊D5semidirect
232. SD16C2×C5⋊D5semidirect
234. D8C2×C5⋊D5semidirect
237. M16C2×C5⋊D5semidirect
281. C5#66107semidirect

Maximal subgroups

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

subgroupindex
#41×C5⋊D52
C2×#879022
C52×AGL(1,ℤ/8)2
#117172712
#117172522
#117172362
#117172352
#117172322
#117172312
#117172292
M16×C5⋊D52
#661075