Finite Group Databasebeta

← #13106096

Subgroups — #13106096

Order 1600.

824 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C2×#87900C211yesmaximal, characteristic, complemented
2C22×C4×C5×D5C211yesmaximal, characteristic, complemented
3C5×#4072C211yesmaximal, characteristic, complemented
4#11717095C212yesmaximal, complemented
5#11717084C212yesmaximal, complemented
6#11717027C212yesmaximal, complemented
7C4×#6038C214yesmaximal, complemented
8#11716975C212yesmaximal, complemented
9C23×C5×D5C2211yescharacteristic
10C22×C4×C52C2211yescharacteristic
11C2×C22⋊(C5⋊D5)C2212yes—
12#87900—24no—
13C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
14C2×C4×C5⋊D5C2212yescomplemented
15C2×#6045C2211yescharacteristic, complemented
16C22×C5×Dic5C2211yescharacteristic
17C2×C4×C5×D5—24no—
18C2×C4×C5×D5C2212yes—
19C2×C4×C5×D5C2212yes—
20C2×#6038C2214yes—
21C5×C22⋊Dic5C2212yes—
22C5×C4⋊Dic5C2212yescomplemented
23C4×C5×Dic5C2212yescomplemented
24#87780C2214yescomplemented
25#87779C2214yescomplemented
26#87778C2214yes—
27#66125—51nomaximal
28#65858—51nomaximal
29C23×C52C2311yescharacteristic
30C22×C5⋊D5C2312yes—
31C22×C5×D5—22no—
32C22×C5×D5—24no—
33C22×C5×D5C2312yes—
34C2×C4×C52D414yescomplemented
35C2×C4×C52C2311yescharacteristic
36C2×C4×C52C2311yescharacteristic
37C22⋊(C5⋊D5)—24no—
38C2×#1039C2312yes—
39C4⋊(C5⋊D5)—22no—
40C4×C5⋊D5—24no—
41#6045—22no—
42C2×C5×Dic5—22no—
43C2×C5×Dic5C2314yes—
44C2×C5×Dic5C2312yes—
45C4×C5×D5—24no—
46C4×C5×D5—24no—
47#6038—28no—
48C23×C4×C5—51no—
49C2×#709—51no—
50C2×#709—51no—
51C2×#709—102no—
52C22×C4×D5D511yescharacteristic, complemented
53C5×#117—51no—
54#4099—52no—
55#4076—52no—
56#4075—52no—
57C4×C22⋊D5—54no—
58#4072D511yescharacteristic, complemented
59#4070—52no—
60#4043—52no—
61#4041—52no—
62#4032—52no—
63#4028—52no—
64C4×D20—54no—
65C22×C52—22no—
66C22×C52C2×D412yes—
67C22×C52C2411yescharacteristic
68C2×C5⋊D5—22no—
69C2×C5⋊D5—24no—
70C2×C5×D5—28no—
71C2×C5×D5—22no—
72C2×C5×D5—22no—
73C2×C5×D5#4114yescomplemented
74C4×C52C2×D412yes—
75C4×C52C2×D412yes—
76#1039—22no—
77C5×Dic5—22no—
78C5×Dic5—24no—
79C24×C5—51no—
80C23×D5C2×D511yescharacteristic
81C22×C4×C5—22no—
82C22×C4×C5—51no—
83C22×C4×C5—52no—
84C22×C4×C5—52no—
85C22×C4×C5—104no—
86C22×C4×C5C2×D511yescharacteristic
87C22×C4×C5C2×D511yescharacteristic
88C2×C22⋊D5—52no—
89C2×C22⋊D5—52no—
90C2×C22⋊D5—54no—
91C2×C22⋊D5—104no—
92C22×Dic5C2×D511yescharacteristic
93#709—1016no—
94#709—104no—
95#709—104no—
96C2×D20—54no—
97C2×D20—51no—
98C2×D20—51no—
99C2×D20—102no—
100C2×C4×D5—24no—
101C2×C4×D5—52no—
102C2×C4×D5—52no—
103C2×C4×D5—104no—
104C2×C4×D5C2×D512yes—
105C2×C4×D5C2×D512yes—
106C2×Dic10—51no—
107C2×Dic10—51no—
108C2×Dic10—102no—
109C5×C4⋊C4—52no—
110C5×C22⋊C4—52no—
111C42×C5—52no—
112C22⋊Dic5—54no—
113C22⋊Dic5C2×D512yes—
114#685—54no—
115#685—54no—
116#685—54no—
117C4⋊Dic5—54no—
118C4⋊Dic5C2×D512yescomplemented
119#683—54no—
120C4×Dic5C2×D512yescomplemented
121#525—251noSylow
122C2×C52—22no—
123C2×C52C22×D411yescharacteristic
124C2×C52C2×#4112yes—
125C5⋊D5—42no—
126C5×D5—24no—
127C23×C5—22no—
128C23×C5—52no—
129C23×C5—104no—
130C23×C5—102no—
131C23×C5C22×D511yescharacteristic
132C23×C5C22×D511yescharacteristic
133C22×D5—22no—
134C22×D5—24no—
135C22×D5—52no—
136C22×D5—52no—
137C22×D5—104no—
138C22×D5C22×D512yes—
139C2×C4×C5—28no—
140C2×C4×C5—22no—
141C2×C4×C5—22no—
142C2×C4×C5—52no—
143C2×C4×C5—54no—
144C2×C4×C5—102no—
145C2×C4×C5—104no—
146C2×C4×C5—104no—
147C2×C4×C5D2014yescomplemented
148C2×C4×C5C22⋊D514yescomplemented
149C2×C4×C5C22×D511yescharacteristic
150C2×C4×C5C22×D511yescharacteristic
151C2×C4×C5C22×D511yescharacteristic
152C2×C4×C5C22×D511yescharacteristic
153C22⋊D5—108no—
154C22⋊D5—1016no—
155C22⋊D5—104no—
156C22⋊D5—104no—
157C2×Dic5—22no—
158C2×Dic5—52no—
159C2×Dic5—52no—
160C2×Dic5—104no—
161C2×Dic5C22×D514yes—
162C2×Dic5C22×D512yes—
163D20—108no—
164D20—108no—
165D20—102no—
166D20—102no—
167C4×D5—24no—
168C4×D5—24no—
169C4×D5—1016no—
170C4×D5—104no—
171C4×D5—104no—
172Dic10—108no—
173Dic10—102no—
174Dic10—102no—
175C2×#41—251no—
176C23×C4—51no—
177#123—252no—
178C22⋊Q8—252no—
179#121—252no—
180#120—252no—
181C4×D4—254no—
182#117—51no—
183C52#52511yescharacteristic, complemented, Sylow
184C22×C5—22no—
185C22×C5—22no—
186C22×C5—24no—
187C22×C5—22no—
188C22×C5—44no—
189C22×C5—54no—
190C22×C5—108no—
191C22×C5—102no—
192C22×C5—102no—
193C22×C5C2×D2012yes—
194C22×C5C2×C22⋊D512yes—
195C22×C5C23×D511yescharacteristic
196C22×C5C23×D511yescharacteristic
197C2×D5—28no—
198C2×D5—22no—
199C2×D5—22no—
200C2×D5—108no—
201C2×D5—102no—
202C2×D5—104no—
203C2×D5—104no—
204C2×D5—102no—
205C2×D5—208no—
206C2×D5#70914yescomplemented
207C4×C5—24no—
208C4×C5—24no—
209C4×C5—102no—
210C4×C5—104no—
211C4×C5C2×D2012yes—
212C4×C5C2×D2012yes—
213C4×C5C2×C22⋊D512yes—
214C4×C5C2×C22⋊D512yes—
215Dic5—24no—
216Dic5—22no—
217Dic5—108no—
218Dic5—102no—
219Dic5—102no—
220C24—51no—
221#41—504no—
222C2×Q8—251no—
223C2×D4—254no—
224C2×D4—252no—
225C2×D4—251no—
226C22×C4—51no—
227C22×C4—52no—
228C22×C4—52no—
229C22×C4—104no—
230C22×C4—252no—
231C22×C4D5211yescharacteristic
232C4⋊C4—52no—
233C4⋊C4—254no—
234C22⋊C4—52no—
235C22⋊C4—254no—
236C22⋊C4—254no—
237C42—52no—
238C2×C5—22no—
239C2×C5—22no—
240C2×C5—24no—
241C2×C5—22no—
242C2×C5—44no—
243C2×C5—104no—
244C2×C5C22×D2011yescharacteristic
245C2×C5C2×#70912yes—
246C2×C5D5×#4112yes—
247C2×C5C22×C22⋊D511yescharacteristic
248D5—24no—
249D5—208no—
250D5—202no—
251D5—202no—
252C23—52no—
253C23—104no—
254C23—102no—
255C23—252no—
256C23C2×D5211yescharacteristic
257Q8—502no—
258D4—508no—
259D4—504no—
260D4—502no—
261C2×C4—54no—
262C2×C4—52no—
263C2×C4—102no—
264C2×C4—104no—
265C2×C4—104no—
266C2×C4—252no—
267C2×C4—504no—
268C2×C4#603814yescomplemented
269C2×C4C2×D5211yescharacteristic
270C2×C4C2×D5211yescharacteristic
271C5—22no—
272C5#6585811yescharacteristic, complemented
273C5#6612511yescharacteristic, complemented
274C22—22no—
275C22—54no—
276C22—108no—
277C22—102no—
278C22—102no—
279C22—504no—
280C22—502no—
281C22C2×#603812yes—
282C22C22×D5211yescharacteristic
283C4—104no—
284C4—102no—
285C4—502no—
286C4C2×#603812yes—
287C4C2×#603812yes—
288C2—22no—
289C2—104no—
290C2—1002no—
291C2D5×#70912yes—
292C2C22×#603811yescharacteristic
subgroupG/H
centerC2×C4C2×D52
commutator, socle, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC22×C4×C52C22
hypercenterC22×C4D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#87900C2semidirect
2. C22×C4×C5×D5C2semidirect
3. C5×#4072C2semidirect
4. #11717095C2semidirect
5. #11717084C2semidirect
6. #11717027C2semidirect
7. C4×#6038C2semidirect
8. #11716975C2semidirect
13. C2×C4⋊(C5⋊D5)C22semidirect
14. C2×C4×C5⋊D5C22semidirect
15. C2×#6045C22semidirect
22. C5×C4⋊Dic5C22semidirect
23. C4×C5×Dic5C22semidirect
24. #87780C22semidirect
25. #87779C22semidirect
34. C2×C4×C52D4semidirect
52. C22×C4×D5D5semidirect
58. #4072D5semidirect
73. C2×C5×D5#41semidirect
118. C4⋊Dic5C2×D5semidirect
120. C4×Dic5C2×D5semidirect
147. C2×C4×C5D20semidirect
148. C2×C4×C5C22⋊D5semidirect
183. C52#525semidirect
206. C2×D5#709semidirect
268. C2×C4#6038semidirect
272. C5#65858semidirect
273. C5#66125semidirect

Maximal subgroups

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

subgroupindex
C2×#879002
C22×C4×C5×D52
C5×#40722
#117170952
#117170842
#117170272
C4×#60382
#117169752
#661255
#658585