Finite Group Databasebeta

← #13107305

Subgroups — #13107305

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×#87904C211yesmaximal, characteristic, complemented
3C52×#137C211yesmaximal, characteristic, complemented
4#11717269C211yesmaximal, characteristic, complemented
5#11717262C211yesmaximal, characteristic, complemented
6#11717239C212yesmaximal, complemented
7#11717238C212yesmaximal, complemented
8#11717236C212yesmaximal, complemented
9#11717234C212yesmaximal, complemented
10#11717228C211yesmaximal, characteristic, complemented
11M16×C5⋊D5C211yesmaximal, characteristic, complemented
12C52×#41C2211yescharacteristic
13C2×C52×Q8C2211yescharacteristic, complemented
14#87904—21no—
15#87904—21no—
16#87904C2212yes—
17#87904C2212yescomplemented
18#87904C2211yescharacteristic, complemented
19Q8×C5⋊D5C2211yescharacteristic, complemented
20#87902—21no—
21#87902C2211yescharacteristic, complemented
22D4×C5⋊D5—21no—
23D4×C5⋊D5C2211yescharacteristic, complemented
24#87900—21no—
25#87900C2211yescharacteristic
26C2×C4⋊(C5⋊D5)—21no—
27C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
28C2×C4×C5⋊D5—21no—
29C2×C4×C5⋊D5—21no—
30C2×C4×C5⋊D5C2211yescharacteristic
31C52×Q16C2212yescomplemented
32C52×SD16C2212yescomplemented
33C52×M16C2211yescharacteristic, complemented
34#87812C2212yescomplemented
35#87811C2212yescomplemented
36#87811C2212yescomplemented
37#87809C2212yescomplemented
38#87804C2211yescharacteristic, complemented
39#87801C2212yescomplemented
40#87800C2212yescomplemented
41#87799C2212yescomplemented
42C8×C5⋊D5C2212yescomplemented
43#66111—56nomaximal
44C22×C5⋊D5—21no—
45C22×C5⋊D5—21no—
46C22×C5⋊D5D411yescharacteristic
47C52×Q8—21no—
48C52×Q8C2311yescharacteristic
49C52×Q8C2312yes—
50C52×D4—21no—
51C52×D4C2311yescharacteristic
52C2×C4×C52—21no—
53C2×C4×C52—21no—
54C2×C4×C52C2311yescharacteristic
55C22⋊(C5⋊D5)—21no—
56C22⋊(C5⋊D5)—21no—
57C22⋊(C5⋊D5)—21no—
58C2×#1039—21no—
59C2×#1039D411yescharacteristic
60C4⋊(C5⋊D5)—22no—
61C4⋊(C5⋊D5)—22no—
62C4⋊(C5⋊D5)—21no—
63C4⋊(C5⋊D5)—21no—
64C4⋊(C5⋊D5)C2312yes—
65C4⋊(C5⋊D5)C2311yescharacteristic
66C4×C5⋊D5—21no—
67C4×C5⋊D5—21no—
68C4×C5⋊D5—21no—
69C4×C5⋊D5—22no—
70C4×C5⋊D5—22no—
71C4×C5⋊D5D411yescharacteristic
72C4×C5⋊D5D411yescharacteristic
73C4×C5⋊D5C2311yescharacteristic
74C4×C5⋊D5C2311yescharacteristic
75#6045—21no—
76#6045C2311yescharacteristic
77C52×C8C2312yes—
78#6029C2312yes—
79D5×#41—56no—
80C2×#713—56no—
81C5×#137D516yescomplemented
82#4095—56no—
83#4088—56no—
84#4065—512no—
85#4064—512no—
86#4062—512no—
87#4060—512no—
88#4054—56no—
89D5×M16—56no—
90C22×C52—21no—
91C22×C52C2×D411yescharacteristic
92C2×C5⋊D5—22no—
93C2×C5⋊D5—21no—
94C2×C5⋊D5—41no—
95C2×C5⋊D5—42no—
96C2×C5⋊D5—42no—
97C2×C5⋊D5C2×D411yescharacteristic
98C2×C5⋊D5C2×D411yescharacteristic
99C4×C52—21no—
100C4×C52—22no—
101C4×C52C2×D411yescharacteristic
102C4×C52C2411yescharacteristic
103#1039—21no—
104#1039C2×D411yescharacteristic
105#1039C2×D411yescharacteristic
106C5×#41C2×D516yes—
107C2×C5×Q8C2×D516yescomplemented
108#713—512no—
109#713—512no—
110#713—56no—
111#713—106no—
112#713—106no—
113Q8×D5—56no—
114#711—56no—
115#711—106no—
116D4×D5—56no—
117D4×D5—106no—
118#709—56no—
119#709—106no—
120C2×D20—56no—
121C2×D20—106no—
122C2×C4×D5—56no—
123C2×C4×D5—106no—
124C2×C4×D5—106no—
125C5×Q16C2×D5112yescomplemented
126C5×SD16C2×D5112yescomplemented
127C5×M16C2×D516yescomplemented
128#689—512no—
129#688—512no—
130#688—512no—
131#686—512no—
132#681—56no—
133D40—512no—
134#677—512no—
135#676—512no—
136C8×D5—512no—
137#575—251noSylow
138C2×C52—21no—
139C2×C52—41no—
140C2×C52C22×D411yescharacteristic
141C5⋊D5—21no—
142C5⋊D5—21no—
143C5⋊D5—42no—
144C5⋊D5—41no—
145C22×D5—56no—
146C22×D5—106no—
147C22×D5—106no—
148C5×Q8—26no—
149C5×Q8C22×D516yes—
150C5×Q8C22×D5112yes—
151C5×D4—26no—
152C5×D4C22×D516yes—
153C2×C4×C5—26no—
154C2×C4×C5—26no—
155C2×C4×C5C22×D516yes—
156C22⋊D5—106no—
157C22⋊D5—106no—
158C22⋊D5—106no—
159C2×Dic5—56no—
160C2×Dic5—106no—
161D20—512no—
162D20—56no—
163D20—1012no—
164D20—1012no—
165D20—106no—
166D20—106no—
167C4×D5—56no—
168C4×D5—56no—
169C4×D5—56no—
170C4×D5—56no—
171C4×D5—106no—
172C4×D5—1012no—
173C4×D5—1012no—
174C4×D5—106no—
175C4×D5—106no—
176Dic10—56no—
177Dic10—106no—
178C5×C8C22×D5112yes—
179#168—512no—
180C2×#41—251no—
181C2×#41—251no—
182#137—251no—
183#137—252no—
184#137C5⋊D511yescharacteristic, complemented
185AGL(1,ℤ/8)—251no—
186AGL(1,ℤ/8)—252no—
187AGL(1,ℤ/8)—251no—
188#135—252no—
189#135—252no—
190C2×M16—251no—
191C52#57511yescharacteristic, complemented, Sylow
192C22×C5—26no—
193C22×C5D4×D516yes—
194C2×D5—56no—
195C2×D5—56no—
196C2×D5—1012no—
197C2×D5—106no—
198C2×D5—206no—
199C2×D5—2012no—
200C2×D5—2012no—
201C4×C5—26no—
202C4×C5—212no—
203C4×C5D4×D516yes—
204C4×C5C23×D516yes—
205Dic5—56no—
206Dic5—56no—
207Dic5—106no—
208#41—251no—
209#41—252no—
210#41—252no—
211#41—251no—
212#41—251no—
213#41—501no—
214#41—501no—
215#41—501no—
216#41—501no—
217#41C2×C5⋊D511yescharacteristic
218C2×Q8—251no—
219C2×Q8C2×C5⋊D511yescharacteristic, complemented
220C2×D4—251no—
221C2×D4—251no—
222C2×D4—501no—
223C2×D4—501no—
224C22×C4—251no—
225C22×C4—501no—
226C22×C4—501no—
227Q16—252no—
228Q16C2×C5⋊D512yescomplemented
229SD16—252no—
230SD16—252no—
231SD16—252no—
232SD16C2×C5⋊D512yescomplemented
233D8—252no—
234D8—252no—
235M16—251no—
236M16—252no—
237M16C2×C5⋊D511yescharacteristic, complemented
238C2×C8—252no—
239C2×C5—26no—
240C2×C5—46no—
241C2×C5C2×D4×D516yes—
242D5—106no—
243D5—106no—
244D5—2012no—
245D5—206no—
246C23—251no—
247C23—501no—
248C23—501no—
249Q8—21no—
250Q8—251no—
251Q8—501no—
252Q8C22×C5⋊D511yescharacteristic
253Q8C22×C5⋊D512yes—
254D4—21no—
255D4—252no—
256D4—251no—
257D4—501no—
258D4—502no—
259D4—502no—
260D4—501no—
261D4—501no—
262D4—501no—
263D4—501no—
264D4C22×C5⋊D511yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—251no—
268C2×C4—251no—
269C2×C4—251no—
270C2×C4—251no—
271C2×C4—251no—
272C2×C4—501no—
273C2×C4—502no—
274C2×C4—502no—
275C2×C4—501no—
276C2×C4—501no—
277C2×C4—501no—
278C2×C4C22×C5⋊D511yescharacteristic
279C8—252no—
280C8C22×C5⋊D512yes—
281C5#6611116yescomplemented
282C22—21no—
283C22—251no—
284C22—251no—
285C22—502no—
286C22—501no—
287C22—1001no—
288C22—1002no—
289C22—1002no—
290C22D4×C5⋊D511yescharacteristic
291C4—21no—
292C4—22no—
293C4—251no—
294C4—251no—
295C4—501no—
296C4D4×C5⋊D511yescharacteristic
297C4C23×C5⋊D511yescharacteristic
298C2—21no—
299C2—41no—
300C2—501no—
301C2—501no—
302C2—1002no—
303C2—1001no—
304C2C2×D4×C5⋊D511yescharacteristic
subgroupG/H
centerC2C2×D4×C5⋊D5
commutator, cosocleC4×C52C24
FrattiniC4C23×C5⋊D5
FittingC52×#137C2
socleC2×C52C22×D4
hypercenter#137C5⋊D5
hypocenterC52#575
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
1. #41×C5⋊D5C2semidirect
2. C2×#87904C2semidirect
3. C52×#137C2semidirect
4. #11717269C2semidirect
5. #11717262C2semidirect
6. #11717239C2semidirect
7. #11717238C2semidirect
8. #11717236C2semidirect
9. #11717234C2semidirect
10. #11717228C2semidirect
11. M16×C5⋊D5C2semidirect
13. C2×C52×Q8C22semidirect
17. #87904C22semidirect
18. #87904C22semidirect
19. Q8×C5⋊D5C22semidirect
21. #87902C22semidirect
23. D4×C5⋊D5C22semidirect
27. C2×C4⋊(C5⋊D5)C22semidirect
31. C52×Q16C22semidirect
32. C52×SD16C22semidirect
33. C52×M16C22semidirect
34. #87812C22semidirect
35. #87811C22semidirect
36. #87811C22semidirect
37. #87809C22semidirect
38. #87804C22semidirect
39. #87801C22semidirect
40. #87800C22semidirect
41. #87799C22semidirect
42. C8×C5⋊D5C22semidirect
81. C5×#137D5semidirect
107. C2×C5×Q8C2×D5semidirect
125. C5×Q16C2×D5semidirect
126. C5×SD16C2×D5semidirect
127. C5×M16C2×D5semidirect
184. #137C5⋊D5semidirect
191. C52#575semidirect
219. C2×Q8C2×C5⋊D5semidirect
228. Q16C2×C5⋊D5semidirect
232. SD16C2×C5⋊D5semidirect
237. M16C2×C5⋊D5semidirect
281. C5#66111semidirect

Maximal subgroups

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

subgroupindex
#41×C5⋊D52
C2×#879042
C52×#1372
#117172692
#117172622
#117172392
#117172382
#117172362
#117172342
#117172282
M16×C5⋊D52
#661115