Finite Group Databasebeta

← #13106525

Subgroups — #13106525

Order 1600.

558 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 134 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#11717266C211yesmaximal, characteristic, complemented
2C5×#4092C212yesmaximal, complemented
3#11717039C211yesmaximal, characteristic
4D5×#685C212yesmaximal
5#11717014C212yesmaximal, complemented
6#11717009C212yesmaximal, complemented
7#11717007C212yesmaximal, complemented
8#11716984C211yesmaximal, characteristic, complemented
9#11716969C212yesmaximal, complemented
10C22×D52C2211yescharacteristic
11C2×C52×Q8C2211yescharacteristic, complemented
12C2×C4×C5⋊D5C2211yescharacteristic
13C2×C4×C5×D5C2212yes—
14C2×#6039C2211yescharacteristic, complemented
15C2×#6039C2212yescomplemented
16C2×#6036C2211yescharacteristic
17C2×D5×Dic5C2212yes—
18#87808C2212yes—
19C4⋊#1039C2211yescharacteristic, complemented
20#87806C2212yescomplemented
21C5×#685C2212yes—
22C5×#685C2212yes—
23C5×C4⋊Dic5C2212yescomplemented
24C5×#683C2212yescomplemented
25C5×#683C2212yescomplemented
26#87781C2211yescharacteristic, complemented
27#87780C2212yescomplemented
28#87779C2212yes—
29#87778C2212yes—
30#87778C2212yes—
31#65968—52nomaximal
32C22×C5⋊D5C2311yescharacteristic
33C22×C5×D5C2312yes—
34C2×D52—41no—
35C2×D52—41no—
36C2×D52Q814yescomplemented
37C52×Q8—41no—
38C2×C4×C52C2312yes—
39C2×C4×C52C2311yescharacteristic
40C2×#1039C2312yes—
41C2×#1039C2311yescharacteristic
42C4×C5⋊D5—41no—
43C2×C5×Dic5C2312yes—
44C2×C5×Dic5C2312yes—
45C2×C5×Dic5C2312yes—
46C4×C5×D5—42no—
47#6039—42no—
48#6039—41no—
49#6036—41no—
50D5×Dic5—42no—
51C5×C22⋊Q8—52no—
52#4092—52no—
53#4092—102no—
54#4092D512yescomplemented
55C2×#685—52no—
56#4070—52no—
57#4043—52no—
58#4043—52no—
59#4042—52no—
60#4042—52no—
61#4042—52no—
62#4042—52no—
63D5×C22⋊C4—52no—
64D5×C22⋊C4—52no—
65#4024—52no—
66#4024—52no—
67C22×C52C2411yescharacteristic
68C2×C5⋊D5—21no—
69C2×C5⋊D5—21no—
70C2×C5⋊D5C2×Q812yes—
71C2×C5×D5—22no—
72C2×C5×D5—22no—
73C2×C5×D5C2×Q814yes—
74D52—44no—
75C4×C52—21no—
76C4×C52—22no—
77#1039—22no—
78#1039—21no—
79C5×Dic5—22no—
80C5×Dic5—22no—
81C5×Dic5—22no—
82C23×D5—52no—
83C2×C5×Q8—22no—
84C2×C5×Q8C2×D512yescomplemented
85C22×C4×C5—52no—
86C22×Dic5—52no—
87C2×C4×D5—52no—
88C2×C4×D5—52no—
89C2×C4×D5—52no—
90C2×C4×D5—102no—
91C2×C4×D5C2×D512yes—
92C2×Dic10—52no—
93C2×Dic10—52no—
94C2×Dic10—52no—
95C5×C4⋊C4—52no—
96C5×C4⋊C4—52no—
97C5×C4⋊C4—52no—
98C5×C22⋊C4—52no—
99C5×C22⋊C4—52no—
100C22⋊Dic5—52no—
101C22⋊Dic5—52no—
102#685—52no—
103#685—52no—
104#685—52no—
105#685—52no—
106#685—52no—
107#685—52no—
108#685—104no—
109#685C2×D512yes—
110#685C2×D512yes—
111C4⋊Dic5—52no—
112C4⋊Dic5—52no—
113C4⋊Dic5—102no—
114C4⋊Dic5C2×D512yescomplemented
115#683—52no—
116#683—52no—
117#683—52no—
118#683—52no—
119#683—104no—
120#683C2×D512yescomplemented
121#683C2×D512yescomplemented
122#543—251noSylow
123C2×C52C22×Q811yescharacteristic
124C2×C52H(5,𝔽2)11yescharacteristic
125C2×C52H(5,𝔽2)11yescharacteristic
126C5⋊D5—22no—
127C5×D5—24no—
128C23×C5—52no—
129C22×D5—52no—
130C22×D5—58no—
131C22×D5—102no—
132C22×D5—202no—
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—
146C2×Dic5—52no—
147C2×Dic5—52no—
148C2×Dic5—52no—
149C2×Dic5—102no—
150C2×Dic5—104no—
151C2×Dic5—202no—
152C2×Dic5C22×D512yes—
153C2×Dic5C22×D512yes—
154C2×Dic5C22×D512yes—
155C4×D5—42no—
156C4×D5—202no—
157C4×D5—202no—
158C4×D5—202no—
159C4×D5—204no—
160Dic10—202no—
161Dic10—202no—
162Dic10—202no—
163C22⋊Q8—52no—
164C22⋊Q8—252no—
165C22⋊Q8—252no—
166C22⋊Q8—251no—
167C22⋊Q8—252no—
168C22⋊Q8—252no—
169C22⋊Q8—251no—
170C2×C22⋊C4—251no—
171C2×C22⋊C4—252no—
172C52#54311yescharacteristic, complemented, Sylow
173C22×C5—22no—
174C22×C5—54no—
175C22×C5—102no—
176C22×C5—102no—
177C22×C5C23×D512yes—
178C2×D5—22no—
179C2×D5—22no—
180C2×D5—54no—
181C2×D5—104no—
182C2×D5—102no—
183C2×D5—102no—
184C2×D5—202no—
185C2×D5—202no—
186C2×D5—208no—
187C2×D5Q8×D514yes—
188C4×C5—24no—
189C4×C5—22no—
190C4×C5—22no—
191C4×C5—22no—
192C4×C5—44no—
193C4×C5—102no—
194C4×C5—102no—
195C4×C5—102no—
196Dic5—22no—
197Dic5—22no—
198Dic5—22no—
199Dic5—102no—
200Dic5—102no—
201Dic5—102no—
202Dic5—202no—
203Dic5—204no—
204C24—251no—
205C2×Q8—252no—
206C2×Q8—251no—
207C2×Q8D5211yescharacteristic, complemented
208C22×C4—52no—
209C22×C4—251no—
210C22×C4—252no—
211C22×C4—251no—
212C4⋊C4—52no—
213C4⋊C4—52no—
214C4⋊C4—52no—
215C4⋊C4—251no—
216C4⋊C4—252no—
217C4⋊C4—252no—
218C4⋊C4—251no—
219C22⋊C4—52no—
220C22⋊C4—52no—
221C22⋊C4—252no—
222C22⋊C4—252no—
223C22⋊C4—252no—
224C22⋊C4—252no—
225C2×C5—22no—
226C2×C5—22no—
227C2×C5—22no—
228C2×C5—104no—
229C2×C5#414412yes—
230C2×C5C2×Q8×D512yes—
231C2×C5#414912yes—
232D5—24no—
233D5—104no—
234D5—204no—
235C23—52no—
236C23—254no—
237C23—251no—
238C23—1001no—
239C23—1001no—
240Q8—41no—
241Q8—1002no—
242Q8—1001no—
243C2×C4—52no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—202no—
247C2×C4—252no—
248C2×C4—251no—
249C2×C4—1001no—
250C2×C4—1001no—
251C2×C4—1002no—
252C2×C4C2×D5211yescharacteristic
253C2×C4C2×D5212yes—
254C5—22no—
255C5#6596812yescomplemented
256C22—54no—
257C22—102no—
258C22—102no—
259C22—252no—
260C22—501no—
261C22—501no—
262C22—1004no—
263C22C22×D5211yescharacteristic
264C4—22no—
265C4—21no—
266C4—102no—
267C4—102no—
268C4—102no—
269C4—502no—
270C4—501no—
271C2—104no—
272C2—502no—
273C2Q8×D5211yescharacteristic
274C2#1171748511yescharacteristic
275C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radicalC22×D52C22
hypercenterC2×Q8D52
hypocenterC52#543
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11717266C2semidirect
2. C5×#4092C2semidirect
5. #11717014C2semidirect
6. #11717009C2semidirect
7. #11717007C2semidirect
8. #11716984C2semidirect
9. #11716969C2semidirect
11. C2×C52×Q8C22semidirect
14. C2×#6039C22semidirect
15. C2×#6039C22semidirect
19. C4⋊#1039C22semidirect
20. #87806C22semidirect
23. C5×C4⋊Dic5C22semidirect
24. C5×#683C22semidirect
25. C5×#683C22semidirect
26. #87781C22semidirect
27. #87780C22semidirect
36. C2×D52Q8semidirect
54. #4092D5semidirect
84. C2×C5×Q8C2×D5semidirect
114. C4⋊Dic5C2×D5semidirect
120. #683C2×D5semidirect
121. #683C2×D5semidirect
172. C52#543semidirect
207. C2×Q8D52semidirect
255. C5#65968semidirect

Maximal subgroups

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

subgroupindex
#117172662
C5×#40922
#117170392
D5×#6852
#117170142
#117170092
#117170072
#117169842
#117169692
#659685