Finite Group Databasebeta

← #13107302

Subgroups — #13107302

Order 1600.

1072 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1072 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#11717517C211yesmaximal, characteristic, complemented
3C52×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#11717270C211yesmaximal, characteristic, complemented
5C2×#87809C211yesmaximal, characteristic, complemented
6#11717236C212yesmaximal, complemented
7#11717234C212yesmaximal, complemented
8#11717231C212yesmaximal, complemented
9D8×C5⋊D5C212yesmaximal, complemented
10#11717228C211yesmaximal, characteristic, complemented
11#11717227C211yesmaximal, characteristic, complemented
12C52×#41C2211yescharacteristic, complemented
13C2×C52×D4C2211yescharacteristic, complemented
14C2×C22⋊(C5⋊D5)—22no—
15#87904C2212yescomplemented
16#87902—21no—
17#87902C2212yescomplemented
18D4×C5⋊D5—22no—
19D4×C5⋊D5—21no—
20D4×C5⋊D5C2212yescomplemented
21D4×C5⋊D5C2212yescomplemented
22#87900—21no—
23#87900C2211yescharacteristic
24#87900C2211yescharacteristic, complemented
25C2×C4⋊(C5⋊D5)—21no—
26C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
27C52×SD16C2212yescomplemented
28C52×D8C2212yescomplemented
29C52×M16C2211yescharacteristic, complemented
30#87812C2211yescharacteristic, complemented
31#87811C2211yescharacteristic, complemented
32#87810C2211yescharacteristic, complemented
33#87809C2211yescharacteristic, complemented
34#87809C2212yescomplemented
35#87809C2212yescomplemented
36C2×#6029C2211yescharacteristic, complemented
37#87801C2212yescomplemented
38#87800C2212yescomplemented
39#87799C2212yescomplemented
40C8×C5⋊D5C2212yescomplemented
41#66108—56nomaximal
42C23×C52—21no—
43C22×C5⋊D5—21no—
44C22×C5⋊D5—22no—
45C22×C5⋊D5—22no—
46C52×Q8C2311yescharacteristic
47C52×D4—21no—
48C52×D4—21no—
49C52×D4C2312yes—
50C52×D4C2311yescharacteristic
51C2×C4×C52—21no—
52C2×C4×C52C2311yescharacteristic
53C22⋊(C5⋊D5)—21no—
54C22⋊(C5⋊D5)—22no—
55C22⋊(C5⋊D5)—22no—
56C22⋊(C5⋊D5)—22no—
57C22⋊(C5⋊D5)D412yes—
58C2×#1039—22no—
59C4⋊(C5⋊D5)—22no—
60C4⋊(C5⋊D5)—21no—
61C4⋊(C5⋊D5)D411yescharacteristic
62C4⋊(C5⋊D5)C2312yes—
63C4⋊(C5⋊D5)C2311yescharacteristic
64C4×C5⋊D5—21no—
65C4×C5⋊D5—22no—
66C4×C5⋊D5C2312yes—
67#6045D411yescharacteristic
68#6045C2311yescharacteristic
69C52×C8C2312yes—
70#6029C2311yescharacteristic
71#6029C2311yescharacteristic
72#4149—56no—
73#4144—56no—
74C5×AGL(1,ℤ/8)D516yescomplemented
75#4096—56no—
76C2×#686—56no—
77#4062—512no—
78#4060—512no—
79#4057—512no—
80D5×D8—512no—
81#4054—56no—
82#4053—56no—
83C22×C52—22no—
84C22×C52—21no—
85C22×C52—81no—
86C22×C52C2×D411yescharacteristic
87C2×C5⋊D5—22no—
88C2×C5⋊D5—21no—
89C2×C5⋊D5—82no—
90C2×C5⋊D5—82no—
91C2×C5⋊D5—81no—
92C2×C5⋊D5C2×D412yes—
93C4×C52—21no—
94C4×C52C2×D411yescharacteristic
95C4×C52C2411yescharacteristic
96#1039—21no—
97#1039C2×D412yes—
98C5×#41C2×D516yescomplemented
99C2×C5×D4C2×D516yescomplemented
100C2×C22⋊D5—1012no—
101#713—512no—
102#711—512no—
103#711—106no—
104D4×D5—512no—
105D4×D5—512no—
106D4×D5—106no—
107D4×D5—1012no—
108#709—56no—
109#709—56no—
110#709—106no—
111C2×D20—56no—
112C2×D20—106no—
113C5×SD16C2×D5112yescomplemented
114C5×D8C2×D5112yescomplemented
115C5×M16C2×D516yescomplemented
116#689—56no—
117#688—56no—
118#687—56no—
119#686—56no—
120#686—512no—
121#686—512no—
122C2×#168—56no—
123D40—512no—
124#677—512no—
125#676—512no—
126C8×D5—512no—
127#576—251noSylow
128C2×C52—21no—
129C2×C52—41no—
130C2×C52—42no—
131C2×C52C22×D411yescharacteristic
132C5⋊D5—22no—
133C5⋊D5—42no—
134C5⋊D5—41no—
135C23×C5—26no—
136C22×D5—1012no—
137C22×D5—1012no—
138C22×D5—106no—
139C5×Q8C22×D516yes—
140C5×D4—26no—
141C5×D4—26no—
142C5×D4C22×D5112yes—
143C5×D4C22×D516yes—
144C2×C4×C5—26no—
145C2×C4×C5C22×D516yes—
146C22⋊D5—512no—
147C22⋊D5—1012no—
148C22⋊D5—1012no—
149C22⋊D5—1012no—
150C22⋊D5—106no—
151C2×Dic5—1012no—
152D20—56no—
153D20—512no—
154D20—56no—
155D20—1012no—
156D20—106no—
157C4×D5—512no—
158C4×D5—1012no—
159C4×D5—106no—
160Dic10—56no—
161Dic10—56no—
162C5×C8C22×D5112yes—
163#168—56no—
164#168—56no—
165H(5,𝔽2)—251no—
166H(5,𝔽2)—251no—
167AGL(1,ℤ/8)—252no—
168AGL(1,ℤ/8)—252no—
169AGL(1,ℤ/8)—251no—
170AGL(1,ℤ/8)C5⋊D511yescharacteristic, complemented
171#135—251no—
172#135—252no—
173C2×D8—251no—
174C2×D8—252no—
175#131—251no—
176C52#57611yescharacteristic, complemented, Sylow
177C22×C5—212no—
178C22×C5—26no—
179C22×C5—86no—
180C22×C5D4×D516yes—
181C2×D5—512no—
182C2×D5—1012no—
183C2×D5—106no—
184C2×D5—4012no—
185C2×D5—4012no—
186C2×D5—406no—
187C4×C5—26no—
188C4×C5D4×D516yes—
189C4×C5C23×D516yes—
190Dic5—512no—
191Dic5—106no—
192#41—251no—
193#41—252no—
194#41—252no—
195#41—251no—
196#41—501no—
197#41—501no—
198#41C2×C5⋊D511yescharacteristic, complemented
199C2×D4—252no—
200C2×D4—252no—
201C2×D4—251no—
202C2×D4—501no—
203C2×D4—502no—
204C2×D4—501no—
205C2×D4—502no—
206C2×D4C2×C5⋊D511yescharacteristic, complemented
207Q16—251no—
208SD16—251no—
209SD16—251no—
210SD16—252no—
211SD16C2×C5⋊D512yescomplemented
212D8—252no—
213D8—251no—
214D8—252no—
215D8—252no—
216D8C2×C5⋊D512yescomplemented
217M16—252no—
218M16C2×C5⋊D511yescharacteristic, complemented
219C2×C8—251no—
220C2×C8—252no—
221C2×C5—26no—
222C2×C5—46no—
223C2×C5—412no—
224C2×C5C2×D4×D516yes—
225D5—1012no—
226D5—2012no—
227D5—206no—
228C23—21no—
229C23—502no—
230C23—502no—
231C23—501no—
232Q8—251no—
233Q8—251no—
234Q8C22×C5⋊D511yescharacteristic
235D4—21no—
236D4—21no—
237D4—252no—
238D4—251no—
239D4—252no—
240D4—251no—
241D4—502no—
242D4—502no—
243D4—502no—
244D4—502no—
245D4—501no—
246D4—501no—
247D4C22×C5⋊D512yes—
248D4C22×C5⋊D511yescharacteristic
249C2×C4—21no—
250C2×C4—252no—
251C2×C4—502no—
252C2×C4—502no—
253C2×C4—501no—
254C2×C4C22×C5⋊D511yescharacteristic
255C8—251no—
256C8—251no—
257C8C22×C5⋊D512yes—
258C5#6610816yescomplemented
259C22—21no—
260C22—22no—
261C22—81no—
262C22—252no—
263C22—502no—
264C22—501no—
265C22—2001no—
266C22—2002no—
267C22—2002no—
268C22D4×C5⋊D511yescharacteristic
269C4—21no—
270C4—252no—
271C4—501no—
272C4D4×C5⋊D511yescharacteristic
273C4C23×C5⋊D511yescharacteristic
274C2—21no—
275C2—42no—
276C2—41no—
277C2—502no—
278C2—1002no—
279C2—1001no—
280C2C2×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#576
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. #11717522C2semidirect
2. #11717517C2semidirect
3. C52×AGL(1,ℤ/8)C2semidirect
4. #11717270C2semidirect
5. C2×#87809C2semidirect
6. #11717236C2semidirect
7. #11717234C2semidirect
8. #11717231C2semidirect
9. D8×C5⋊D5C2semidirect
10. #11717228C2semidirect
11. #11717227C2semidirect
12. C52×#41C22semidirect
13. C2×C52×D4C22semidirect
15. #87904C22semidirect
17. #87902C22semidirect
20. D4×C5⋊D5C22semidirect
21. D4×C5⋊D5C22semidirect
24. #87900C22semidirect
26. C2×C4⋊(C5⋊D5)C22semidirect
27. C52×SD16C22semidirect
28. C52×D8C22semidirect
29. C52×M16C22semidirect
30. #87812C22semidirect
31. #87811C22semidirect
32. #87810C22semidirect
33. #87809C22semidirect
34. #87809C22semidirect
35. #87809C22semidirect
36. C2×#6029C22semidirect
37. #87801C22semidirect
38. #87800C22semidirect
39. #87799C22semidirect
40. C8×C5⋊D5C22semidirect
74. C5×AGL(1,ℤ/8)D5semidirect
98. C5×#41C2×D5semidirect
99. C2×C5×D4C2×D5semidirect
113. C5×SD16C2×D5semidirect
114. C5×D8C2×D5semidirect
115. C5×M16C2×D5semidirect
170. AGL(1,ℤ/8)C5⋊D5semidirect
176. C52#576semidirect
198. #41C2×C5⋊D5semidirect
206. C2×D4C2×C5⋊D5semidirect
211. SD16C2×C5⋊D5semidirect
216. D8C2×C5⋊D5semidirect
218. M16C2×C5⋊D5semidirect
258. C5#66108semidirect

Maximal subgroups

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

subgroupindex
#117175222
#117175172
C52×AGL(1,ℤ/8)2
#117172702
C2×#878092
#117172362
#117172342
#117172312
D8×C5⋊D52
#117172282
#117172272
#661085