Finite Group Databasebeta

← #13098913

Subgroups — #13098913

Order 1600.

519 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 125 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×C22⋊D25C211yesmaximal, characteristic, complemented
2C2×D4×D25C211yesmaximal, characteristic, complemented
3C25×#120C211yesmaximal, characteristic, complemented
4#11716515C211yesmaximal, characteristic, complemented
5#11716500C212yesmaximal, complemented
6#11716499C212yesmaximal, complemented
7#11716497C211yesmaximal, characteristic, complemented
8#11716448C211yesmaximal, characteristic, complemented
9#11716446C212yesmaximal, complemented
10#11716445C212yesmaximal, complemented
11C22⋊C4×D25C211yesmaximal, characteristic, complemented
12C24×C25C2211yescharacteristic, complemented
13C23×D25C2211yescharacteristic
14C23×D25C2211yescharacteristic, complemented
15C2×D4×C25C2211yescharacteristic
16C2×D4×C25C2212yescomplemented
17C2×C22⋊D25—24no—
18C2×C22⋊D25C2212yes—
19C2×C22⋊D25C2211yescharacteristic
20C2×C22⋊D25C2212yescomplemented
21C2×C22⋊D25C2211yescharacteristic, complemented
22C2×C22⋊D25C2212yescomplemented
23C22×Dic25C2211yescharacteristic, complemented
24D4×D25—24no—
25C2×D100C2211yescharacteristic, complemented
26C2×C4×D25C2211yescharacteristic, complemented
27C2×C4×D25C2212yescomplemented
28C22⋊C4×C25C2211yescharacteristic, complemented
29C22⋊C4×C25C2212yescomplemented
30C22⋊Dic25C2211yescharacteristic, complemented
31C22⋊Dic25C2212yescomplemented
32C22⋊Dic25C2212yescomplemented
33#87720C2212yescomplemented
34#87720C2212yescomplemented
35C4⋊Dic25C2212yescomplemented
36#87718C2212yescomplemented
37#65923—51nomaximal
38C23×C25—22no—
39C23×C25—42no—
40C23×C25C2311yescharacteristic
41C23×C25C2312yes—
42C23×C25C2311yescharacteristic
43C22×D25—22no—
44C22×D25—42no—
45C22×D25—42no—
46C22×D25D414yescomplemented
47C22×D25C2311yescharacteristic
48C22×D25C2311yescharacteristic
49C22×D25C2311yescharacteristic
50D4×C25—22no—
51D4×C25—42no—
52C2×C4×C25C2311yescharacteristic
53C2×C4×C25C2312yes—
54C22⋊D25—22no—
55C22⋊D25—22no—
56C22⋊D25—24no—
57C22⋊D25—24no—
58C22⋊D25—42no—
59C2×Dic25—22no—
60C2×Dic25C2312yes—
61C2×Dic25C2312yes—
62C2×Dic25C2311yescharacteristic
63D100—22no—
64C4×D25—22no—
65C4×D25—42no—
66C22×C22⋊D5—51no—
67C2×D4×D5—51no—
68C5×#120D511yescharacteristic
69#4099—51no—
70#4084—52no—
71#4083—52no—
72#4081—51no—
73#4032—51no—
74#4030—52no—
75#4029—52no—
76D5×C22⋊C4—51no—
77C22×C25—22no—
78C22×C25—22no—
79C22×C25—21no—
80C22×C25—22no—
81C22×C25—22no—
82C22×C25—22no—
83C22×C25—44no—
84C22×C25C2×D412yes—
85C22×C25C2411yescharacteristic
86C2×D25—22no—
87C2×D25—22no—
88C2×D25—21no—
89C2×D25—22no—
90C2×D25—44no—
91C2×D25—44no—
92C2×D25C2×D412yes—
93C2×D25C2×D412yes—
94C4×C25—21no—
95C4×C25—22no—
96Dic25—21no—
97Dic25—22no—
98Dic25—22no—
99C24×C5C2×D511yescharacteristic
100C23×D5—51no—
101C23×D5—51no—
102C2×C5×D4C2×D511yescharacteristic
103C2×C5×D4C2×D512yes—
104C2×C22⋊D5—51no—
105C2×C22⋊D5—51no—
106C2×C22⋊D5—52no—
107C2×C22⋊D5—52no—
108C2×C22⋊D5—52no—
109C2×C22⋊D5—104no—
110C22×Dic5—51no—
111D4×D5—104no—
112C2×D20—51no—
113C2×C4×D5—51no—
114C2×C4×D5—52no—
115C5×C22⋊C4C2×D511yescharacteristic
116C5×C22⋊C4C2×D512yes—
117C22⋊Dic5—51no—
118C22⋊Dic5—52no—
119C22⋊Dic5—52no—
120#685—52no—
121#685—52no—
122C4⋊Dic5—52no—
123#683—52no—
124#534—251noSylow
125C2×C25—22no—
126C2×C25—41no—
127C2×C25—42no—
128C2×C25C22×D411yescharacteristic
129C2×C25H(5,𝔽2)12yes—
130D25—22no—
131D25—22no—
132D25—41no—
133C23×C5—22no—
134C23×C5—42no—
135C23×C5C22×D511yescharacteristic
136C23×C5C22×D512yes—
137C23×C5C22×D511yescharacteristic
138C22×D5—51no—
139C22×D5—51no—
140C22×D5—54no—
141C22×D5—51no—
142C22×D5—102no—
143C22×D5—202no—
144C22×D5—202no—
145C5×D4—22no—
146C5×D4—42no—
147C2×C4×C5C22×D511yescharacteristic
148C2×C4×C5C22×D512yes—
149C22⋊D5—102no—
150C22⋊D5—104no—
151C22⋊D5—104no—
152C22⋊D5—102no—
153C22⋊D5—202no—
154C2×Dic5—52no—
155C2×Dic5—52no—
156C2×Dic5—51no—
157C2×Dic5—102no—
158D20—102no—
159C4×D5—102no—
160C4×D5—202no—
161C22×D4—251no—
162C22×D4—251no—
163#123—251no—
164#123—252no—
165#123—251no—
166#121—252no—
167#121—252no—
168#120—251no—
169#120—252no—
170#120D2511yescharacteristic, complemented
171C2×C22⋊C4—251no—
172C25#53411yescharacteristic, complemented, Sylow
173C22×C5—22no—
174C22×C5—22no—
175C22×C5—21no—
176C22×C5—22no—
177C22×C5—22no—
178C22×C5—22no—
179C22×C5—44no—
180C22×C5D4×D512yes—
181C22×C5C23×D511yescharacteristic
182C2×D5—52no—
183C2×D5—52no—
184C2×D5—102no—
185C2×D5—102no—
186C2×D5—101no—
187C2×D5—102no—
188C2×D5—204no—
189C2×D5—204no—
190C4×C5—21no—
191C4×C5—22no—
192Dic5—101no—
193Dic5—102no—
194Dic5—102no—
195C24—251no—
196C24—251no—
197C24C2×D2511yescharacteristic, complemented
198C2×D4—252no—
199C2×D4—251no—
200C2×D4—251no—
201C2×D4—252no—
202C2×D4—252no—
203C2×D4—251no—
204C2×D4—504no—
205C2×D4—504no—
206C2×D4C2×D2511yescharacteristic
207C2×D4C2×D2512yescomplemented
208C22×C4—251no—
209C22×C4—251no—
210C22×C4—252no—
211C4⋊C4—252no—
212C4⋊C4—252no—
213C22⋊C4—251no—
214C22⋊C4—252no—
215C22⋊C4—252no—
216C22⋊C4—252no—
217C22⋊C4—252no—
218C22⋊C4C2×D2511yescharacteristic, complemented
219C22⋊C4C2×D2512yescomplemented
220C2×C5—22no—
221C2×C5—41no—
222C2×C5—42no—
223C2×C5C2×D4×D511yescharacteristic
224C2×C5#414412yes—
225D5—102no—
226D5—102no—
227D5—201no—
228C23—22no—
229C23—42no—
230C23—251no—
231C23—251no—
232C23—254no—
233C23—251no—
234C23—502no—
235C23—1002no—
236C23—1002no—
237C23C22×D2511yescharacteristic
238C23C22×D2512yes—
239C23C22×D2511yescharacteristic
240D4—22no—
241D4—42no—
242D4—502no—
243D4—502no—
244D4—504no—
245D4—504no—
246D4—502no—
247D4—1002no—
248C2×C4—252no—
249C2×C4—251no—
250C2×C4—252no—
251C2×C4—502no—
252C2×C4—502no—
253C2×C4—1002no—
254C2×C4C22×D2512yes—
255C2×C4C22×D2511yescharacteristic
256C5#6592311yescharacteristic
257C22—22no—
258C22—21no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—44no—
264C22—252no—
265C22—252no—
266C22—502no—
267C22—502no—
268C22—501no—
269C22—502no—
270C22—1004no—
271C22—1004no—
272C22D4×D2512yes—
273C22C23×D2511yescharacteristic
274C4—21no—
275C4—22no—
276C4—501no—
277C4—502no—
278C4—502no—
279C2—22no—
280C2—42no—
281C2—41no—
282C2—502no—
283C2—502no—
284C2—1001no—
285C2C2×D4×D2511yescharacteristic
286C2#1171656312yes—
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×#120C2
hypercenter#120D25
hypocenterC25#534
LCSG ▷ C22×C25 ▷ C25
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D25C2semidirect
2. C2×D4×D25C2semidirect
3. C25×#120C2semidirect
4. #11716515C2semidirect
5. #11716500C2semidirect
6. #11716499C2semidirect
7. #11716497C2semidirect
8. #11716448C2semidirect
9. #11716446C2semidirect
10. #11716445C2semidirect
11. C22⋊C4×D25C2semidirect
12. C24×C25C22semidirect
14. C23×D25C22semidirect
16. C2×D4×C25C22semidirect
20. C2×C22⋊D25C22semidirect
21. C2×C22⋊D25C22semidirect
22. C2×C22⋊D25C22semidirect
23. C22×Dic25C22semidirect
25. C2×D100C22semidirect
26. C2×C4×D25C22semidirect
27. C2×C4×D25C22semidirect
28. C22⋊C4×C25C22semidirect
29. C22⋊C4×C25C22semidirect
30. C22⋊Dic25C22semidirect
31. C22⋊Dic25C22semidirect
32. C22⋊Dic25C22semidirect
33. #87720C22semidirect
34. #87720C22semidirect
35. C4⋊Dic25C22semidirect
36. #87718C22semidirect
46. C22×D25D4semidirect
170. #120D25semidirect
172. C25#534semidirect
197. C24C2×D25semidirect
207. C2×D4C2×D25semidirect
218. C22⋊C4C2×D25semidirect
219. C22⋊C4C2×D25semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D252
C2×D4×D252
C25×#1202
#117165152
#117165002
#117164992
#117164972
#117164482
#117164462
#117164452
C22⋊C4×D252
#659235