Finite Group Databasebeta

← #13106527

Subgroups — #13106527

Order 1600.

572 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 128 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#11717267C211yesmaximal, characteristic, complemented
2C5×#4092C212yesmaximal, complemented
3#11717041C211yesmaximal, characteristic
4D5×#685C212yesmaximal, complemented
5#11717022C212yesmaximal, complemented
6#11717016C211yesmaximal, characteristic, complemented
7#11717011C212yesmaximal, complemented
8#11716996C212yesmaximal, complemented
9#11716966C212yesmaximal, complemented
10C22×D52C2211yescharacteristic
11C2×C52×Q8C2211yescharacteristic, complemented
12C2×C4⋊(C5⋊D5)C2211yescharacteristic
13C2×C4×C5×D5C2212yes—
14C2×#6038C2212yes—
15C2×#6036C2211yescharacteristic, complemented
16C2×D5×Dic5C2212yes—
17#87808C2212yes—
18#87808C2212yescomplemented
19C4×#1039C2211yescharacteristic, complemented
20C5×#685C2212yes—
21C5×#685C2212yes—
22C5×C4⋊Dic5C2212yescomplemented
23C5×#683C2212yescomplemented
24C5×#683C2212yescomplemented
25#87780C2212yescomplemented
26#87779C2211yescharacteristic
27#87779C2212yes—
28#87779C2211yescharacteristic, complemented
29#87778C2212yes—
30Dic52C2212yescomplemented
31#65971—52nomaximal
32C22×C5⋊D5C2311yescharacteristic
33C22×C5⋊D5C2311yescharacteristic
34C22×C5×D5C2312yes—
35C2×D52—24no—
36C2×D52—41no—
37C52×Q8—41no—
38C2×C4×C52C2312yes—
39C2×C4×C52C2311yescharacteristic
40C2×#1039C2312yes—
41C4⋊(C5⋊D5)—41no—
42C2×C5×Dic5C2312yes—
43C2×C5×Dic5C2312yes—
44C2×C5×Dic5C2312yes—
45C4×C5×D5—42no—
46#6038—24no—
47#6036—41no—
48D5×Dic5—24no—
49C5×C22⋊Q8—52no—
50#4093—52no—
51#4093—102no—
52#4092D512yescomplemented
53C4×C22⋊D5—52no—
54C2×#685—52no—
55#4044—52no—
56#4044—52no—
57#4040—52no—
58#4040—52no—
59#4039—52no—
60#4038—52no—
61#4032—52no—
62#4028—52no—
63D5×C22⋊C4—52no—
64#4023—52no—
65C22×C52C2411yescharacteristic
66C2×C5⋊D5—22no—
67C2×C5⋊D5—22no—
68C2×C5⋊D5—21no—
69C2×C5⋊D5—21no—
70C2×C5×D5—22no—
71C2×C5×D5—22no—
72C2×C5×D5#4114yes—
73D52—44no—
74C4×C52—21no—
75C4×C52—22no—
76#1039—22no—
77C5×Dic5—22no—
78C5×Dic5—22no—
79C5×Dic5—22no—
80C23×D5—52no—
81C2×C5×Q8—22no—
82C2×C5×Q8C2×D512yescomplemented
83C22×C4×C5—52no—
84C2×C22⋊D5—52no—
85C22×Dic5—52no—
86C2×D20—52no—
87C2×D20—52no—
88C2×D20—102no—
89C2×C4×D5—52no—
90C2×C4×D5—52no—
91C2×C4×D5C2×D512yes—
92C5×C4⋊C4—52no—
93C5×C4⋊C4—52no—
94C5×C4⋊C4—52no—
95C5×C22⋊C4—52no—
96C5×C22⋊C4—52no—
97C22⋊Dic5—52no—
98#685—52no—
99#685—52no—
100#685—52no—
101#685—52no—
102#685—52no—
103#685—52no—
104#685—52no—
105#685—52no—
106#685—52no—
107#685—104no—
108#685—104no—
109#685C2×D512yes—
110#685C2×D512yes—
111C4⋊Dic5—52no—
112C4⋊Dic5C2×D512yescomplemented
113#683—52no—
114#683C2×D512yescomplemented
115#683C2×D512yescomplemented
116C4×Dic5—52no—
117C4×Dic5—52no—
118C4×Dic5—52no—
119C4×Dic5—102no—
120#551—251noSylow
121C2×C52C2×#4112yes—
122C2×C52H(5,𝔽2)11yescharacteristic
123C5⋊D5—41no—
124C5⋊D5—41no—
125C5×D5—24no—
126C23×C5—52no—
127C22×D5—52no—
128C22×D5—52no—
129C22×D5—102no—
130C22×D5—102no—
131C22×D5—104no—
132C22×D5—104no—
133C22×D5—202no—
134C22×D5C22×D512yes—
135C5×Q8—42no—
136C5×Q8—44no—
137C2×C4×C5—22no—
138C2×C4×C5—24no—
139C2×C4×C5—52no—
140C2×C4×C5—52no—
141C2×C4×C5—52no—
142C2×C4×C5—202no—
143C2×C4×C5C22×D512yes—
144C2×C4×C5C22×D512yes—
145C2×C4×C5C22×D512yes—
146C22⋊D5—104no—
147C2×Dic5—52no—
148C2×Dic5—52no—
149C2×Dic5—104no—
150C2×Dic5—104no—
151C2×Dic5C22×D512yes—
152C2×Dic5C22×D512yes—
153C2×Dic5C22×D512yes—
154D20—104no—
155D20—202no—
156D20—204no—
157C4×D5—42no—
158C4×D5—104no—
159C4×D5—202no—
160#126—252no—
161#124—251no—
162#123—252no—
163#123—251no—
164C22⋊Q8—52no—
165#120—251no—
166C4×D4—252no—
167#117—252no—
168C2×C22⋊C4—252no—
169C52#55111yescharacteristic, complemented, Sylow
170C22×C5—22no—
171C22×C5—54no—
172C22×C5—102no—
173C22×C5—102no—
174C22×C5C23×D512yes—
175C2×D5—22no—
176C2×D5—22no—
177C2×D5—102no—
178C2×D5—102no—
179C2×D5—102no—
180C2×D5—102no—
181C2×D5—102no—
182C2×D5—102no—
183C2×D5—202no—
184C2×D5—202no—
185C2×D5—204no—
186C2×D5—204no—
187C2×D5—208no—
188C2×D5#71314yes—
189C4×C5—24no—
190C4×C5—22no—
191C4×C5—22no—
192C4×C5—22no—
193C4×C5—44no—
194C4×C5—102no—
195C4×C5—102no—
196C4×C5—102no—
197Dic5—22no—
198Dic5—22no—
199Dic5—22no—
200Dic5—102no—
201Dic5—102no—
202Dic5—204no—
203C24—251no—
204C2×Q8D5211yescharacteristic, complemented
205C2×D4—252no—
206C2×D4—251no—
207C22×C4—52no—
208C22×C4—251no—
209C22×C4—252no—
210C4⋊C4—52no—
211C4⋊C4—52no—
212C4⋊C4—52no—
213C4⋊C4—252no—
214C22⋊C4—52no—
215C22⋊C4—52no—
216C22⋊C4—252no—
217C22⋊C4—251no—
218C22⋊C4—251no—
219C22⋊C4—252no—
220C22⋊C4—252no—
221C22⋊C4—252no—
222C42—252no—
223C42—251no—
224C2×C5—24no—
225C2×C5—22no—
226C2×C5—104no—
227C2×C5#414412yes—
228C2×C5C2×#71312yes—
229C2×C5D5×#4112yes—
230D5—24no—
231D5—202no—
232D5—202no—
233D5—402no—
234D5—402no—
235C23—52no—
236C23—251no—
237C23—251no—
238C23—504no—
239C23—1001no—
240Q8—41no—
241D4—504no—
242D4—1001no—
243C2×C4—52no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—202no—
247C2×C4—252no—
248C2×C4—504no—
249C2×C4—1001no—
250C2×C4C2×D5211yescharacteristic
251C2×C4C2×D5212yes—
252C5—22no—
253C5#6597112yescomplemented
254C22—54no—
255C22—102no—
256C22—102no—
257C22—502no—
258C22—502no—
259C22—501no—
260C22—501no—
261C22—1004no—
262C22C22×D5211yescharacteristic
263C4—22no—
264C4—21no—
265C4—102no—
266C4—102no—
267C4—102no—
268C4—502no—
269C2—104no—
270C2—1001no—
271C2—1001no—
272C2D5×#71312yes—
273C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radical#11717041C2
hypercenterC2×Q8D52
hypocenterC52#551
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11717267C2semidirect
2. C5×#4092C2semidirect
4. D5×#685C2semidirect
5. #11717022C2semidirect
6. #11717016C2semidirect
7. #11717011C2semidirect
8. #11716996C2semidirect
9. #11716966C2semidirect
11. C2×C52×Q8C22semidirect
15. C2×#6036C22semidirect
18. #87808C22semidirect
19. C4×#1039C22semidirect
22. C5×C4⋊Dic5C22semidirect
23. C5×#683C22semidirect
24. C5×#683C22semidirect
25. #87780C22semidirect
28. #87779C22semidirect
30. Dic52C22semidirect
52. #4092D5semidirect
82. C2×C5×Q8C2×D5semidirect
112. C4⋊Dic5C2×D5semidirect
114. #683C2×D5semidirect
115. #683C2×D5semidirect
169. C52#551semidirect
204. C2×Q8D52semidirect
253. C5#65971semidirect

Maximal subgroups

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

subgroupindex
#117172672
C5×#40922
#117170412
D5×#6852
#117170222
#117170162
#117170112
#117169962
#117169662
#659715