Finite Group Databasebeta

← #13098910

Subgroups — #13098910

Order 1600.

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

normal subgroups only

This group has 501 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×#87746C211yesmaximal, characteristic, complemented
3C25×#120C211yesmaximal, characteristic, complemented
4#11716515C211yesmaximal, characteristic, complemented
5#11716502C212yesmaximal, complemented
6#11716501C212yesmaximal, complemented
7#11716499C211yesmaximal, characteristic, complemented
8#11716447C212yesmaximal, complemented
9#11716446C212yesmaximal, complemented
10#11716444C211yesmaximal, characteristic, complemented
11#11716439C211yesmaximal, characteristic, complemented
12C24×C25C2211yescharacteristic
13C23×D25C2211yescharacteristic
14C2×D4×C25C2211yescharacteristic
15C2×D4×C25C2212yescomplemented
16C2×C22⋊D25—24no—
17C2×C22⋊D25C2211yescharacteristic
18C2×C22⋊D25C2212yes—
19C2×C22⋊D25C2212yes—
20C2×C22⋊D25C2211yescharacteristic, complemented
21C2×C22⋊D25C2212yescomplemented
22C2×C22⋊D25C2212yescomplemented
23C22×Dic25C2211yescharacteristic
24C22×Dic25C2211yescharacteristic, complemented
25#87746—24no—
26C2×D100C2212yescomplemented
27C2×C4×D25C2211yescharacteristic, complemented
28C2×Dic50C2211yescharacteristic, complemented
29C22⋊C4×C25C2211yescharacteristic
30C22⋊C4×C25C2212yescomplemented
31C22⋊Dic25C2211yescharacteristic
32C22⋊Dic25C2212yescomplemented
33#87720C2212yescomplemented
34#87720C2212yescomplemented
35#87718C2212yescomplemented
36C4×Dic25C2212yescomplemented
37#65926—51nomaximal
38C23×C25—22no—
39C23×C25—42no—
40C23×C25C2312yes—
41C23×C25C2311yescharacteristic
42C23×C25C2311yescharacteristic
43C22×D25—22no—
44C22×D25—42no—
45C22×D25C2312yes—
46C22×D25C2311yescharacteristic
47D4×C25—22no—
48D4×C25—42no—
49C2×C4×C25C2312yes—
50C2×C4×C25C2311yescharacteristic
51C22⋊D25—22no—
52C22⋊D25—22no—
53C22⋊D25—24no—
54C22⋊D25—24no—
55C22⋊D25—42no—
56C22⋊D25—42no—
57C2×Dic25—22no—
58C2×Dic25D414yescomplemented
59C2×Dic25C2312yes—
60C2×Dic25C2311yescharacteristic
61C2×Dic25C2311yescharacteristic
62C2×Dic25C2311yescharacteristic
63D100—42no—
64C4×D25—22no—
65Dic50—22no—
66C22×C22⋊D5—51no—
67C2×#711—51no—
68C5×#120D511yescharacteristic
69#4099—51no—
70#4086—52no—
71#4085—52no—
72#4083—51no—
73#4031—52no—
74#4030—52no—
75#4028—51no—
76#4023—51no—
77C22×C25—22no—
78C22×C25—21no—
79C22×C25—22no—
80C22×C25—22no—
81C22×C25—22no—
82C22×C25—22no—
83C22×C25—44no—
84C22×C25C2×D412yes—
85C22×C25C2411yescharacteristic
86C2×D25—21no—
87C2×D25—22no—
88C2×D25—22no—
89C2×D25—22no—
90C2×D25—22no—
91C2×D25—44no—
92C4×C25—21no—
93C4×C25—22no—
94Dic25—21no—
95Dic25—22no—
96Dic25C2×D412yes—
97Dic25C2×D412yes—
98C24×C5C2×D511yescharacteristic
99C23×D5—51no—
100C2×C5×D4C2×D511yescharacteristic
101C2×C5×D4C2×D512yes—
102C2×C22⋊D5—51no—
103C2×C22⋊D5—51no—
104C2×C22⋊D5—52no—
105C2×C22⋊D5—52no—
106C2×C22⋊D5—52no—
107C2×C22⋊D5—52no—
108C2×C22⋊D5—104no—
109C22×Dic5—51no—
110C22×Dic5—51no—
111#711—104no—
112C2×D20—52no—
113C2×C4×D5—51no—
114C2×Dic10—51no—
115C5×C22⋊C4C2×D511yescharacteristic
116C5×C22⋊C4C2×D512yes—
117C22⋊Dic5—51no—
118C22⋊Dic5—52no—
119#685—52no—
120#685—52no—
121#683—52no—
122C4×Dic5—52no—
123#535—251noSylow
124C2×C25—22no—
125C2×C25—41no—
126C2×C25—42no—
127C2×C25C22×D411yescharacteristic
128C2×C25H(5,𝔽2)12yes—
129D25—41no—
130D25—42no—
131C23×C5—22no—
132C23×C5—42no—
133C23×C5C22×D512yes—
134C23×C5C22×D511yescharacteristic
135C23×C5C22×D511yescharacteristic
136C22×D5—52no—
137C22×D5—51no—
138C22×D5—102no—
139C22×D5—202no—
140C5×D4—22no—
141C5×D4—42no—
142C2×C4×C5C22×D512yes—
143C2×C4×C5C22×D511yescharacteristic
144C22⋊D5—102no—
145C22⋊D5—104no—
146C22⋊D5—104no—
147C22⋊D5—102no—
148C22⋊D5—202no—
149C22⋊D5—202no—
150C2×Dic5—54no—
151C2×Dic5—51no—
152C2×Dic5—51no—
153C2×Dic5—52no—
154C2×Dic5—51no—
155C2×Dic5—102no—
156D20—202no—
157C4×D5—102no—
158Dic10—102no—
159C2×#41—251no—
160C22×D4—251no—
161#127—252no—
162#124—252no—
163#121—252no—
164#121—252no—
165#120—251no—
166#120—251no—
167#120—251no—
168#120D2511yescharacteristic, complemented
169#117—251no—
170C25#53511yescharacteristic, complemented, Sylow
171C22×C5—22no—
172C22×C5—21no—
173C22×C5—22no—
174C22×C5—22no—
175C22×C5—22no—
176C22×C5—22no—
177C22×C5—44no—
178C22×C5D4×D512yes—
179C22×C5C23×D511yescharacteristic
180C2×D5—101no—
181C2×D5—102no—
182C2×D5—102no—
183C2×D5—102no—
184C2×D5—102no—
185C2×D5—204no—
186C4×C5—21no—
187C4×C5—22no—
188Dic5—52no—
189Dic5—52no—
190Dic5—101no—
191Dic5—102no—
192C24—251no—
193C24C2×D2511yescharacteristic
194#41—504no—
195C2×Q8—251no—
196C2×D4—251no—
197C2×D4—252no—
198C2×D4—252no—
199C2×D4—251no—
200C2×D4—252no—
201C2×D4—252no—
202C2×D4—252no—
203C2×D4—504no—
204C2×D4C2×D2511yescharacteristic
205C2×D4C2×D2512yescomplemented
206C22×C4—251no—
207C22×C4—251no—
208C22×C4—251no—
209C4⋊C4—252no—
210C22⋊C4—251no—
211C22⋊C4—252no—
212C22⋊C4—252no—
213C22⋊C4—252no—
214C22⋊C4C2×D2511yescharacteristic
215C22⋊C4C2×D2512yescomplemented
216C42—252no—
217C2×C5—22no—
218C2×C5—42no—
219C2×C5—41no—
220C2×C5C2×D4×D511yescharacteristic
221C2×C5#414412yes—
222D5—202no—
223D5—201no—
224C23—22no—
225C23—42no—
226C23—252no—
227C23—251no—
228C23—502no—
229C23—1002no—
230C23C22×D2512yes—
231C23C22×D2511yescharacteristic
232C23C22×D2511yescharacteristic
233Q8—502no—
234D4—22no—
235D4—42no—
236D4—504no—
237D4—504no—
238D4—502no—
239D4—502no—
240D4—1002no—
241D4—1002no—
242D4—1002no—
243C2×C4—254no—
244C2×C4—251no—
245C2×C4—251no—
246C2×C4—252no—
247C2×C4—251no—
248C2×C4—502no—
249C2×C4—502no—
250C2×C4C22×D2512yes—
251C2×C4C22×D2511yescharacteristic
252C5#6592611yescharacteristic
253C22—22no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—44no—
260C22—501no—
261C22—502no—
262C22—502no—
263C22—502no—
264C22—502no—
265C22—1004no—
266C22D4×D2512yes—
267C22C23×D2511yescharacteristic
268C4—21no—
269C4—22no—
270C4—252no—
271C4—252no—
272C4—501no—
273C4—502no—
274C2—22no—
275C2—42no—
276C2—41no—
277C2—1002no—
278C2—1001no—
279C2C2×D4×D2511yescharacteristic
280C2#1171656312yes—
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×#120C2
hypercenter#120D25
hypocenterC25#535
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×#87746C2semidirect
3. C25×#120C2semidirect
4. #11716515C2semidirect
5. #11716502C2semidirect
6. #11716501C2semidirect
7. #11716499C2semidirect
8. #11716447C2semidirect
9. #11716446C2semidirect
10. #11716444C2semidirect
11. #11716439C2semidirect
15. C2×D4×C25C22semidirect
20. C2×C22⋊D25C22semidirect
21. C2×C22⋊D25C22semidirect
22. C2×C22⋊D25C22semidirect
24. C22×Dic25C22semidirect
26. C2×D100C22semidirect
27. C2×C4×D25C22semidirect
28. C2×Dic50C22semidirect
30. C22⋊C4×C25C22semidirect
32. C22⋊Dic25C22semidirect
33. #87720C22semidirect
34. #87720C22semidirect
35. #87718C22semidirect
36. C4×Dic25C22semidirect
58. C2×Dic25D4semidirect
168. #120D25semidirect
170. C25#535semidirect
205. C2×D4C2×D25semidirect
215. 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×#877462
C25×#1202
#117165152
#117165022
#117165012
#117164992
#117164472
#117164462
#117164442
#117164392
#659265