Finite Group Databasebeta

← #13107306

Subgroups — #13107306

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#11717522C211yesmaximal, characteristic, complemented
2#11717520C211yesmaximal, characteristic, complemented
3C52×#137C211yesmaximal, characteristic, complemented
4#11717270C211yesmaximal, characteristic, complemented
5C2×#87811C211yesmaximal, characteristic, complemented
6#11717239C212yesmaximal, complemented
7#11717238C212yesmaximal, complemented
8#11717234C212yesmaximal, complemented
9SD16×C5⋊D5C212yesmaximal, complemented
10#11717228C211yesmaximal, characteristic, complemented
11#11717227C211yesmaximal, characteristic, complemented
12C52×#41C2211yescharacteristic
13C2×C52×Q8C2211yescharacteristic, complemented
14#87904—21no—
15#87904C2212yes—
16#87904C2212yescomplemented
17Q8×C5⋊D5—21no—
18Q8×C5⋊D5C2212yescomplemented
19D4×C5⋊D5—22no—
20D4×C5⋊D5C2212yes—
21#87900—22no—
22#87900—21no—
23#87900C2211yescharacteristic
24#87900C2211yescharacteristic, complemented
25C2×C4⋊(C5⋊D5)—21no—
26C2×C4⋊(C5⋊D5)C2211yescharacteristic
27C52×Q16C2212yescomplemented
28C52×SD16C2212yescomplemented
29C52×M16C2211yescharacteristic, complemented
30#87812C2211yescharacteristic, complemented
31#87811C2212yescomplemented
32#87811C2212yescomplemented
33#87811C2211yescharacteristic, complemented
34#87810C2211yescharacteristic, complemented
35#87809C2211yescharacteristic, complemented
36C2×#6029C2211yescharacteristic, complemented
37#87801C2212yescomplemented
38#87800C2212yescomplemented
39#87799C2212yescomplemented
40C8×C5⋊D5C2212yescomplemented
41#66112—56nomaximal
42C22×C5⋊D5—21no—
43C22×C5⋊D5—22no—
44C52×Q8—21no—
45C52×Q8C2312yes—
46C52×Q8C2311yescharacteristic
47C52×D4—21no—
48C52×D4C2311yescharacteristic
49C2×C4×C52—21no—
50C2×C4×C52—21no—
51C2×C4×C52C2311yescharacteristic
52C22⋊(C5⋊D5)—21no—
53C22⋊(C5⋊D5)—22no—
54C22⋊(C5⋊D5)D412yes—
55C4⋊(C5⋊D5)—22no—
56C4⋊(C5⋊D5)—22no—
57C4⋊(C5⋊D5)—21no—
58C4⋊(C5⋊D5)D411yescharacteristic
59C4⋊(C5⋊D5)C2311yescharacteristic
60C4⋊(C5⋊D5)C2312yes—
61C4×C5⋊D5—22no—
62C4×C5⋊D5—21no—
63C4×C5⋊D5—22no—
64C4×C5⋊D5—22no—
65C4×C5⋊D5C2312yes—
66#6045—22no—
67#6045D411yescharacteristic
68#6045C2311yescharacteristic
69C52×C8C2312yes—
70#6029C2311yescharacteristic
71#6029C2311yescharacteristic
72#4149—56no—
73#4147—56no—
74C5×#137D516yescomplemented
75#4096—56no—
76C2×#688—56no—
77#4065—512no—
78#4064—512no—
79#4060—512no—
80D5×SD16—512no—
81#4054—56no—
82#4053—56no—
83C22×C52—21no—
84C22×C52C2×D411yescharacteristic
85C2×C5⋊D5—21no—
86C2×C5⋊D5—22no—
87C2×C5⋊D5—82no—
88C2×C5⋊D5—81no—
89C2×C5⋊D5C2×D412yes—
90C4×C52—22no—
91C4×C52—21no—
92C4×C52C2×D411yescharacteristic
93C4×C52C2411yescharacteristic
94#1039—21no—
95#1039C2×D412yes—
96C5×#41C2×D516yes—
97C2×C5×Q8C2×D516yescomplemented
98#713—512no—
99#713—512no—
100#713—106no—
101Q8×D5—512no—
102Q8×D5—106no—
103D4×D5—512no—
104D4×D5—1012no—
105#709—56no—
106#709—56no—
107#709—106no—
108#709—1012no—
109C2×D20—56no—
110C2×D20—106no—
111C5×Q16C2×D5112yescomplemented
112C5×SD16C2×D5112yescomplemented
113C5×M16C2×D516yescomplemented
114#689—56no—
115#688—56no—
116#688—512no—
117#688—512no—
118#687—56no—
119#686—56no—
120C2×#168—56no—
121D40—512no—
122#677—512no—
123#676—512no—
124C8×D5—512no—
125#577—251noSylow
126C2×C52—21no—
127C2×C52—41no—
128C2×C52C22×D411yescharacteristic
129C5⋊D5—22no—
130C5⋊D5—41no—
131C5⋊D5—42no—
132C22×D5—1012no—
133C22×D5—106no—
134C5×Q8—26no—
135C5×Q8C22×D5112yes—
136C5×Q8C22×D516yes—
137C5×D4—26no—
138C5×D4C22×D516yes—
139C2×C4×C5—26no—
140C2×C4×C5—26no—
141C2×C4×C5C22×D516yes—
142C22⋊D5—512no—
143C22⋊D5—1012no—
144C22⋊D5—106no—
145D20—56no—
146D20—512no—
147D20—56no—
148D20—1012no—
149D20—1012no—
150D20—106no—
151C4×D5—512no—
152C4×D5—1012no—
153C4×D5—1012no—
154C4×D5—1012no—
155C4×D5—106no—
156Dic10—56no—
157Dic10—56no—
158Dic10—1012no—
159C5×C8C22×D5112yes—
160#168—56no—
161#168—56no—
1622−1+4—251no—
163H(5,𝔽2)—251no—
164#137—252no—
165#137C5⋊D511yescharacteristic, complemented
166AGL(1,ℤ/8)—252no—
167AGL(1,ℤ/8)—251no—
168#135—251no—
169#135—252no—
170C2×SD16—251no—
171C2×SD16—252no—
172#131—251no—
173C52#57711yescharacteristic, complemented, Sylow
174C22×C5—26no—
175C22×C5D4×D516yes—
176C2×D5—512no—
177C2×D5—106no—
178C2×D5—1012no—
179C2×D5—4012no—
180C2×D5—406no—
181C4×C5—212no—
182C4×C5—26no—
183C4×C5D4×D516yes—
184C4×C5C23×D516yes—
185Dic5—512no—
186Dic5—106no—
187#41—251no—
188#41—252no—
189#41—252no—
190#41—251no—
191#41—502no—
192#41—501no—
193#41—501no—
194#41C2×C5⋊D511yescharacteristic
195C2×Q8—252no—
196C2×Q8—501no—
197C2×Q8C2×C5⋊D511yescharacteristic, complemented
198C2×D4—252no—
199C2×D4—251no—
200C2×D4—501no—
201C2×D4—502no—
202Q16—251no—
203Q16C2×C5⋊D512yescomplemented
204SD16—252no—
205SD16—252no—
206SD16—251no—
207SD16—251no—
208SD16—252no—
209SD16C2×C5⋊D512yescomplemented
210D8—251no—
211D8—252no—
212M16—252no—
213M16C2×C5⋊D511yescharacteristic, complemented
214C2×C8—251no—
215C2×C8—252no—
216C2×C5—26no—
217C2×C5—46no—
218C2×C5C2×D4×D516yes—
219D5—1012no—
220D5—206no—
221D5—2012no—
222C23—502no—
223C23—501no—
224Q8—21no—
225Q8—251no—
226Q8—251no—
227Q8—502no—
228Q8C22×C5⋊D512yes—
229Q8C22×C5⋊D511yescharacteristic
230D4—21no—
231D4—252no—
232D4—251no—
233D4—251no—
234D4—252no—
235D4—502no—
236D4—502no—
237D4—502no—
238D4—501no—
239D4—501no—
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—251no—
250C8—251no—
251C8C22×C5⋊D512yes—
252C5#6611216yescomplemented
253C22—21no—
254C22—252no—
255C22—501no—
256C22—502no—
257C22—2001no—
258C22—2002no—
259C22D4×C5⋊D511yescharacteristic
260C4—21no—
261C4—22no—
262C4—252no—
263C4—501no—
264C4D4×C5⋊D511yescharacteristic
265C4C23×C5⋊D511yescharacteristic
266C2—21no—
267C2—41no—
268C2—502no—
269C2—1002no—
270C2—1001no—
271C2C2×D4×C5⋊D511yescharacteristic
subgroupG/H
centerC2C2×D4×C5⋊D5
commutator, cosocleC4×C52C24
FrattiniC4C23×C5⋊D5
FittingC52×#137C2
socleC2×C52C22×D4
hypercenter#137C5⋊D5
hypocenterC52#577
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. #11717522C2semidirect
2. #11717520C2semidirect
3. C52×#137C2semidirect
4. #11717270C2semidirect
5. C2×#87811C2semidirect
6. #11717239C2semidirect
7. #11717238C2semidirect
8. #11717234C2semidirect
9. SD16×C5⋊D5C2semidirect
10. #11717228C2semidirect
11. #11717227C2semidirect
13. C2×C52×Q8C22semidirect
16. #87904C22semidirect
18. Q8×C5⋊D5C22semidirect
24. #87900C22semidirect
27. C52×Q16C22semidirect
28. C52×SD16C22semidirect
29. C52×M16C22semidirect
30. #87812C22semidirect
31. #87811C22semidirect
32. #87811C22semidirect
33. #87811C22semidirect
34. #87810C22semidirect
35. #87809C22semidirect
36. C2×#6029C22semidirect
37. #87801C22semidirect
38. #87800C22semidirect
39. #87799C22semidirect
40. C8×C5⋊D5C22semidirect
74. C5×#137D5semidirect
97. C2×C5×Q8C2×D5semidirect
111. C5×Q16C2×D5semidirect
112. C5×SD16C2×D5semidirect
113. C5×M16C2×D5semidirect
165. #137C5⋊D5semidirect
173. C52#577semidirect
197. C2×Q8C2×C5⋊D5semidirect
203. Q16C2×C5⋊D5semidirect
209. SD16C2×C5⋊D5semidirect
213. M16C2×C5⋊D5semidirect
252. C5#66112semidirect

Maximal subgroups

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

subgroupindex
#117175222
#117175202
C52×#1372
#117172702
C2×#878112
#117172392
#117172382
#117172342
SD16×C5⋊D52
#117172282
#117172272
#661125