Finite Group Databasebeta

← #13107298

Subgroups — #13107298

Order 1600.

1032 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1032 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#11717523C211yesmaximal, characteristic, complemented
2#11717522C211yesmaximal, characteristic, complemented
3C52×#135C211yesmaximal, characteristic, complemented
4#11717271C211yesmaximal, characteristic, complemented
5#11717269C211yesmaximal, characteristic, complemented
6#11717238C212yesmaximal, complemented
7#11717236C212yesmaximal, complemented
8SD16×C5⋊D5C212yesmaximal, complemented
9#11717231C212yesmaximal, complemented
10C2×#87800C211yesmaximal, characteristic, complemented
11#11717221C211yesmaximal, characteristic, complemented
12C52×#41C2211yescharacteristic
13C52×#41C2211yescharacteristic, complemented
14#87904C2212yes—
15Q8×C5⋊D5C2212yescomplemented
16#87902—22no—
17#87902C2212yescomplemented
18D4×C5⋊D5—22no—
19D4×C5⋊D5C2212yes—
20#87900—21no—
21#87900—21no—
22#87900C2211yescharacteristic
23C2×C4⋊(C5⋊D5)—21no—
24C2×C4⋊(C5⋊D5)C2211yescharacteristic
25C2×#6045—21no—
26C2×#6045C2211yescharacteristic, complemented
27C52×Q16C2211yescharacteristic, complemented
28C52×SD16C2211yescharacteristic, complemented
29C52×SD16C2211yescharacteristic, complemented
30C52×D8C2211yescharacteristic, complemented
31C2×C52×C8C2211yescharacteristic, complemented
32#87812C2212yescomplemented
33#87811C2212yescomplemented
34#87810C2212yescomplemented
35#87809C2212yescomplemented
36#87804C2211yescharacteristic, complemented
37#87800C2212yescomplemented
38#87800C2212yescomplemented
39#87799C2212yescomplemented
40C8×C5⋊D5C2212yescomplemented
41#66104—56nomaximal
42C22×C5⋊D5—21no—
43C22×C5⋊D5—22no—
44C52×Q8C2311yescharacteristic
45C52×Q8C2311yescharacteristic
46C52×D4—21no—
47C52×D4—21no—
48C52×D4C2311yescharacteristic
49C52×D4C2311yescharacteristic
50C2×C4×C52—21no—
51C2×C4×C52—21no—
52C2×C4×C52C2311yescharacteristic
53C22⋊(C5⋊D5)—22no—
54C22⋊(C5⋊D5)—22no—
55C22⋊(C5⋊D5)D412yes—
56C2×#1039—21no—
57C2×#1039—22no—
58C4⋊(C5⋊D5)—22no—
59C4⋊(C5⋊D5)—21no—
60C4⋊(C5⋊D5)D411yescharacteristic
61C4⋊(C5⋊D5)C2312yes—
62C4×C5⋊D5—22no—
63C4×C5⋊D5—22no—
64C4×C5⋊D5C2312yes—
65#6045—21no—
66#6045—22no—
67#6045D411yescharacteristic
68#6045C2312yes—
69C52×C8C2311yescharacteristic
70C52×C8C2311yescharacteristic
71#6029C2312yes—
72#4150—56no—
73#4149—56no—
74C5×#135D516yescomplemented
75#4097—56no—
76#4095—56no—
77#4064—512no—
78#4062—512no—
79D5×SD16—512no—
80#4057—512no—
81C2×#677—56no—
82#4047—56no—
83C22×C52—21no—
84C22×C52—21no—
85C22×C52C2×D411yescharacteristic
86C2×C5⋊D5—22no—
87C2×C5⋊D5—82no—
88C2×C5⋊D5—81no—
89C2×C5⋊D5C2×D412yes—
90C4×C52—21no—
91C4×C52—21no—
92C4×C52C2×D411yescharacteristic
93C4×C52C2411yescharacteristic
94#1039—22no—
95#1039C2×D412yes—
96C5×#41C2×D516yes—
97C5×#41C2×D516yescomplemented
98#713—512no—
99Q8×D5—512no—
100#711—512no—
101#711—1012no—
102D4×D5—512no—
103D4×D5—1012no—
104#709—56no—
105#709—106no—
106#709—106no—
107C2×D20—56no—
108C2×D20—106no—
109C2×Dic10—56no—
110C2×Dic10—106no—
111C5×Q16C2×D516yescomplemented
112C5×SD16C2×D516yescomplemented
113C5×SD16C2×D516yescomplemented
114C5×D8C2×D516yescomplemented
115C2×C5×C8C2×D516yescomplemented
116#689—512no—
117#688—512no—
118#687—512no—
119#686—512no—
120#681—56no—
121#677—512no—
122#677—512no—
123#676—512no—
124C8×D5—512no—
125#577—251noSylow
126C2×C52—21no—
127C2×C52—41no—
128C2×C52—41no—
129C2×C52C22×D411yescharacteristic
130C5⋊D5—22no—
131C5⋊D5—42no—
132C22×D5—1012no—
133C22×D5—106no—
134C5×Q8C22×D516yes—
135C5×Q8C22×D516yes—
136C5×D4—26no—
137C5×D4—26no—
138C5×D4C22×D516yes—
139C5×D4C22×D516yes—
140C2×C4×C5—26no—
141C2×C4×C5—26no—
142C2×C4×C5C22×D516yes—
143C22⋊D5—512no—
144C22⋊D5—1012no—
145C22⋊D5—1012no—
146C2×Dic5—1012no—
147C2×Dic5—106no—
148D20—56no—
149D20—512no—
150D20—1012no—
151D20—106no—
152C4×D5—512no—
153C4×D5—1012no—
154C4×D5—1012no—
155Dic10—56no—
156Dic10—512no—
157Dic10—1012no—
158Dic10—106no—
159C5×C8C22×D516yes—
160C5×C8C22×D516yes—
161#168—512no—
1622−1+4—251no—
163H(5,𝔽2)—251no—
164#137—251no—
165#137—252no—
166AGL(1,ℤ/8)—251no—
167AGL(1,ℤ/8)—252no—
168#135—252no—
169#135C5⋊D511yescharacteristic, complemented
170C2×SD16—252no—
171C2×SD16—251no—
172#131—251no—
173C52#57711yescharacteristic, complemented, Sylow
174C22×C5—26no—
175C22×C5—26no—
176C22×C5D4×D516yes—
177C2×D5—512no—
178C2×D5—1012no—
179C2×D5—4012no—
180C2×D5—406no—
181C4×C5—26no—
182C4×C5—26no—
183C4×C5D4×D516yes—
184C4×C5C23×D516yes—
185Dic5—512no—
186Dic5—1012no—
187#41—251no—
188#41—252no—
189#41—252no—
190#41—501no—
191#41—501no—
192#41—502no—
193#41C2×C5⋊D511yescharacteristic
194#41C2×C5⋊D511yescharacteristic, complemented
195C2×Q8—252no—
196C2×Q8—251no—
197C2×Q8—501no—
198C2×D4—252no—
199C2×D4—251no—
200C2×D4—501no—
201C2×D4—502no—
202Q16—252no—
203Q16C2×C5⋊D511yescharacteristic, complemented
204SD16—252no—
205SD16—252no—
206SD16—252no—
207SD16—252no—
208SD16C2×C5⋊D511yescharacteristic, complemented
209SD16C2×C5⋊D511yescharacteristic, complemented
210D8—252no—
211D8C2×C5⋊D511yescharacteristic, complemented
212M16—251no—
213M16—252no—
214C2×C8—252no—
215C2×C8C2×C5⋊D511yescharacteristic, complemented
216C2×C5—26no—
217C2×C5—46no—
218C2×C5—46no—
219C2×C5C2×D4×D516yes—
220D5—1012no—
221D5—2012no—
222C23—502no—
223C23—501no—
224Q8—251no—
225Q8—252no—
226Q8—502no—
227Q8—501no—
228Q8C22×C5⋊D511yescharacteristic
229Q8C22×C5⋊D511yescharacteristic
230D4—21no—
231D4—21no—
232D4—252no—
233D4—251no—
234D4—252no—
235D4—502no—
236D4—502no—
237D4—502no—
238D4—501no—
239D4C22×C5⋊D511yescharacteristic
240D4C22×C5⋊D511yescharacteristic
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—252no—
244C2×C4—502no—
245C2×C4—502no—
246C2×C4—502no—
247C2×C4—501no—
248C2×C4C22×C5⋊D511yescharacteristic
249C8—252no—
250C8C22×C5⋊D511yescharacteristic
251C8C22×C5⋊D511yescharacteristic
252C5#6610416yescomplemented
253C22—21no—
254C22—21no—
255C22—252no—
256C22—502no—
257C22—2001no—
258C22—2002no—
259C22D4×C5⋊D511yescharacteristic
260C4—21no—
261C4—21no—
262C4—252no—
263C4—502no—
264C4D4×C5⋊D511yescharacteristic
265C4C23×C5⋊D511yescharacteristic
266C2—21no—
267C2—41no—
268C2—41no—
269C2—502no—
270C2—1002no—
271C2C2×D4×C5⋊D511yescharacteristic
subgroupG/H
centerC2C2×D4×C5⋊D5
commutator, cosocleC4×C52C24
FrattiniC4C23×C5⋊D5
FittingC52×#135C2
socleC2×C52C22×D4
hypercenter#135C5⋊D5
hypocenterC52#577
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #11717523C2semidirect
2. #11717522C2semidirect
3. C52×#135C2semidirect
4. #11717271C2semidirect
5. #11717269C2semidirect
6. #11717238C2semidirect
7. #11717236C2semidirect
8. SD16×C5⋊D5C2semidirect
9. #11717231C2semidirect
10. C2×#87800C2semidirect
11. #11717221C2semidirect
13. C52×#41C22semidirect
15. Q8×C5⋊D5C22semidirect
17. #87902C22semidirect
26. C2×#6045C22semidirect
27. C52×Q16C22semidirect
28. C52×SD16C22semidirect
29. C52×SD16C22semidirect
30. C52×D8C22semidirect
31. C2×C52×C8C22semidirect
32. #87812C22semidirect
33. #87811C22semidirect
34. #87810C22semidirect
35. #87809C22semidirect
36. #87804C22semidirect
37. #87800C22semidirect
38. #87800C22semidirect
39. #87799C22semidirect
40. C8×C5⋊D5C22semidirect
74. C5×#135D5semidirect
97. C5×#41C2×D5semidirect
111. C5×Q16C2×D5semidirect
112. C5×SD16C2×D5semidirect
113. C5×SD16C2×D5semidirect
114. C5×D8C2×D5semidirect
115. C2×C5×C8C2×D5semidirect
169. #135C5⋊D5semidirect
173. C52#577semidirect
194. #41C2×C5⋊D5semidirect
203. Q16C2×C5⋊D5semidirect
208. SD16C2×C5⋊D5semidirect
209. SD16C2×C5⋊D5semidirect
211. D8C2×C5⋊D5semidirect
215. C2×C8C2×C5⋊D5semidirect
252. C5#66104semidirect

Maximal subgroups

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

subgroupindex
#117175232
#117175222
C52×#1352
#117172712
#117172692
#117172382
#117172362
SD16×C5⋊D52
#117172312
C2×#878002
#117172212
#661045