Finite Group Databasebeta

← #13105224

Subgroups — #13105224

Order 1600.

616 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22⋊C4×C5⋊D5C211yesmaximal, characteristic
2C5×#4024C212yesmaximal, complemented
3#11717092C211yesmaximal, characteristic
4#11717082C212yesmaximal, complemented
5C2×#87779C211yesmaximal, characteristic
6#11717009C212yesmaximal, complemented
7#11717008C212yesmaximal, complemented
8#11716990C212yesmaximal, complemented
9#11716984C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C2×C4×C5⋊D5C2212yes—
12C22×C5×Dic5C2212yes—
13C2×C5×Dic10C2212yescomplemented
14C2×#6039C2212yescomplemented
15C2×#6036C2212yes—
16C52×C22⋊C4C2211yescharacteristic
17C22⋊#1039C2211yescharacteristic
18#87808C2212yes—
19C5×C22⋊Dic5C2212yes—
20C5×C4⋊Dic5C2212yescomplemented
21C5×#683C2212yescomplemented
22C5×#683C2212yescomplemented
23#87781C2212yescomplemented
24#87780C2212yescomplemented
25#87780C2212yescomplemented
26#87779C2211yescharacteristic
27#87779C2212yes—
28#87779C2211yescharacteristic
29#87779C2211yescharacteristic
30#87779C2211yescharacteristic
31#65968—52nomaximal
32C23×C52C2311yescharacteristic
33C22×C5⋊D5—42no—
34C22×C5⋊D5Q814yescomplemented
35C22×C5⋊D5C2311yescharacteristic
36C22×C5⋊D5C2311yescharacteristic
37C2×C4×C52C2312yes—
38C2×#1039C2312yes—
39C4×C5⋊D5—42no—
40C2×C5×Dic5—42no—
41C2×C5×Dic5C2312yes—
42C2×C5×Dic5C2312yes—
43C2×C5×Dic5C2312yes—
44C2×C5×Dic5C2312yes—
45C5×Dic10—42no—
46#6039—42no—
47#6036—42no—
48C5×C22⋊Q8—52no—
49#4092—52no—
50#4092—52no—
51C2×#685—52no—
52#4070—52no—
53#4043—52no—
54#4043—52no—
55#4042—52no—
56#4042—52no—
57#4042—52no—
58#4042—52no—
59D5×C22⋊C4—52no—
60D5×C22⋊C4—52no—
61D5×C22⋊C4—102no—
62#4024—52no—
63#4024D512yescomplemented
64C22×C52—22no—
65C22×C52C2×Q812yes—
66C22×C52C2411yescharacteristic
67C2×C5⋊D5—22no—
68C2×C5⋊D5—22no—
69C2×C5⋊D5—44no—
70C2×C5⋊D5C2×Q812yes—
71C2×C5⋊D5C2×Q812yes—
72C4×C52—22no—
73#1039—22no—
74C5×Dic5—22no—
75C5×Dic5—22no—
76C5×Dic5—22no—
77C5×Dic5—22no—
78C23×D5—52no—
79C23×D5—102no—
80C2×C5×Q8—52no—
81C22×C4×C5—52no—
82C22×Dic5C2×D512yes—
83C2×C4×D5—52no—
84C2×C4×D5—52no—
85C2×C4×D5—52no—
86C2×C4×D5—52no—
87C2×C4×D5—104no—
88C2×Dic10—52no—
89C2×Dic10—52no—
90C2×Dic10C2×D512yescomplemented
91C5×C4⋊C4—52no—
92C5×C4⋊C4—52no—
93C5×C4⋊C4—52no—
94C5×C22⋊C4—22no—
95C5×C22⋊C4—52no—
96C5×C22⋊C4C2×D512yes—
97C22⋊Dic5—52no—
98C22⋊Dic5—102no—
99C22⋊Dic5C2×D512yes—
100#685—52no—
101#685—52no—
102#685—52no—
103#685—52no—
104#685—52no—
105#685—52no—
106#685—52no—
107#685—52no—
108#685—104no—
109C4⋊Dic5—52no—
110C4⋊Dic5—52no—
111C4⋊Dic5C2×D512yescomplemented
112#683—52no—
113#683—52no—
114#683—52no—
115#683—52no—
116#683C2×D512yescomplemented
117#683C2×D512yescomplemented
118#543—251noSylow
119C2×C52—22no—
120C2×C52C22×Q811yescharacteristic
121C2×C52H(5,𝔽2)12yes—
122C5⋊D5—22no—
123C5⋊D5—22no—
124C23×C5—22no—
125C23×C5C22×D512yes—
126C22×D5—52no—
127C22×D5—52no—
128C22×D5—58no—
129C22×D5—102no—
130C22×D5—102no—
131C22×D5—108no—
132C22×D5—202no—
133C22×D5—202no—
134C22×D5—208no—
135C5×Q8—202no—
136C2×C4×C5—24no—
137C2×C4×C5—52no—
138C2×C4×C5—52no—
139C2×C4×C5—52no—
140C2×C4×C5—52no—
141C2×C4×C5—202no—
142C2×C4×C5C22×D512yes—
143C2×C4×C5C22×D512yes—
144C2×Dic5—42no—
145C2×Dic5—52no—
146C2×Dic5—52no—
147C2×Dic5—104no—
148C2×Dic5C22×D512yes—
149C2×Dic5C22×D512yes—
150C2×Dic5C22×D512yes—
151C2×Dic5C22×D512yes—
152C4×D5—202no—
153C4×D5—202no—
154C4×D5—202no—
155C4×D5—202no—
156C4×D5—208no—
157Dic10—42no—
158Dic10—202no—
159Dic10—202no—
160C22⋊Q8—52no—
161C22⋊Q8—252no—
162C22⋊Q8—252no—
163C22⋊Q8—252no—
164C22⋊Q8—252no—
165C22⋊Q8—252no—
166C2×C22⋊C4—251no—
167C2×C22⋊C4—251no—
168C2×C22⋊C4—251no—
169C52#54311yescharacteristic, complemented, Sylow
170C22×C5—22no—
171C22×C5—22no—
172C22×C5—24no—
173C22×C5—22no—
174C22×C5—44no—
175C22×C5Q8×D514yes—
176C22×C5C23×D512yes—
177C2×D5—54no—
178C2×D5—54no—
179C2×D5—104no—
180C2×D5—104no—
181C2×D5—108no—
182C2×D5—102no—
183C2×D5—102no—
184C2×D5—102no—
185C2×D5—102no—
186C2×D5—204no—
187C2×D5—208no—
188C2×D5—2016no—
189C4×C5—22no—
190C4×C5—22no—
191C4×C5—44no—
192C4×C5—102no—
193C4×C5—102no—
194C4×C5—102no—
195C4×C5—102no—
196Dic5—22no—
197Dic5—22no—
198Dic5—22no—
199Dic5—22no—
200Dic5—102no—
201Dic5—102no—
202Dic5—204no—
203C24—251no—
204C2×Q8—52no—
205C2×Q8—252no—
206C22×C4—52no—
207C22×C4—252no—
208C22×C4—252no—
209C4⋊C4—52no—
210C4⋊C4—52no—
211C4⋊C4—52no—
212C4⋊C4—252no—
213C4⋊C4—252no—
214C4⋊C4—252no—
215C22⋊C4—52no—
216C22⋊C4—251no—
217C22⋊C4—251no—
218C22⋊C4—251no—
219C22⋊C4—252no—
220C22⋊C4—251no—
221C22⋊C4—251no—
222C22⋊C4—252no—
223C22⋊C4D5211yescharacteristic
224C2×C5—24no—
225C2×C5—22no—
226C2×C5—24no—
227C2×C5—44no—
228C2×C5#414412yes—
229C2×C5C2×Q8×D512yes—
230C2×C5#414912yes—
231D5—108no—
232D5—104no—
233D5—104no—
234D5—204no—
235C23—251no—
236C23—251no—
237C23—254no—
238C23—1002no—
239C23C2×D5211yescharacteristic
240Q8—202no—
241Q8—1002no—
242C2×C4—52no—
243C2×C4—52no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—202no—
247C2×C4—252no—
248C2×C4—1002no—
249C2×C4—1002no—
250C2×C4C2×D5212yes—
251C5—22no—
252C5#6596812yescomplemented
253C22—22no—
254C22—252no—
255C22—252no—
256C22—502no—
257C22—502no—
258C22—1004no—
259C22#8787212yes—
260C22C22×D5211yescharacteristic
261C4—22no—
262C4—102no—
263C4—102no—
264C4—102no—
265C4—102no—
266C4—502no—
267C2—22no—
268C2—502no—
269C2—502no—
270C2C2×#8787211yescharacteristic
271C2#1171747912yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC23×C5⋊D5C22
hypercenterC22⋊C4D52
hypocenterC52#543
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#4024C2semidirect
4. #11717082C2semidirect
6. #11717009C2semidirect
7. #11717008C2semidirect
8. #11716990C2semidirect
9. #11716984C2semidirect
13. C2×C5×Dic10C22semidirect
14. C2×#6039C22semidirect
20. C5×C4⋊Dic5C22semidirect
21. C5×#683C22semidirect
22. C5×#683C22semidirect
23. #87781C22semidirect
24. #87780C22semidirect
25. #87780C22semidirect
34. C22×C5⋊D5Q8semidirect
63. #4024D5semidirect
90. C2×Dic10C2×D5semidirect
111. C4⋊Dic5C2×D5semidirect
116. #683C2×D5semidirect
117. #683C2×D5semidirect
169. C52#543semidirect
252. C5#65968semidirect

Maximal subgroups

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

subgroupindex
C22⋊C4×C5⋊D52
C5×#40242
#117170922
#117170822
C2×#877792
#117170092
#117170082
#117169902
#117169842
#659685