Finite Group Databasebeta

← #13098909

Subgroups — #13098909

Order 1600.

495 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C23×Dic25C211yesmaximal, characteristic, complemented
2C2×#87746C211yesmaximal, characteristic, complemented
3C25×#120C211yesmaximal, characteristic, complemented
4#11716515C211yesmaximal, characteristic, complemented
5#11716501C212yesmaximal, complemented
6#11716497C211yesmaximal, characteristic, complemented
7D4×Dic25C212yesmaximal, complemented
8#11716448C211yesmaximal, characteristic, complemented
9#11716443C212yesmaximal, complemented
10#11716440C212yesmaximal, complemented
11#11716439C211yesmaximal, characteristic, complemented
12C24×C25C2211yescharacteristic
13C2×D4×C25C2212yes—
14C2×D4×C25C2211yescharacteristic
15C2×C22⋊D25C2212yes—
16C2×C22⋊D25C2211yescharacteristic
17C2×C22⋊D25C2211yescharacteristic, complemented
18C22×Dic25—24no—
19C22×Dic25C2212yes—
20C22×Dic25C2212yes—
21C22×Dic25C2211yescharacteristic
22C22×Dic25C2211yescharacteristic
23C22×Dic25C2211yescharacteristic, complemented
24#87746—24no—
25C2×C4×D25C2211yescharacteristic, complemented
26C2×Dic50C2211yescharacteristic, complemented
27C22⋊C4×C25C2211yescharacteristic
28C22⋊C4×C25C2212yes—
29C22⋊Dic25C2211yescharacteristic
30C22⋊Dic25C2212yes—
31C22⋊Dic25C2212yescomplemented
32#87720C2212yescomplemented
33C4⋊Dic25C2212yescomplemented
34#87718C2212yescomplemented
35#87718C2212yescomplemented
36C4×Dic25C2212yescomplemented
37#65920—51nomaximal
38C23×C25—22no—
39C23×C25—24no—
40C23×C25C2311yescharacteristic
41C23×C25C2311yescharacteristic
42C23×C25C2312yes—
43C22×D25C2311yescharacteristic
44D4×C25—22no—
45D4×C25—24no—
46C2×C4×C25C2312yes—
47C2×C4×C25C2311yescharacteristic
48C22⋊D25—22no—
49C22⋊D25—24no—
50C22⋊D25—22no—
51C2×Dic25—22no—
52C2×Dic25—22no—
53C2×Dic25—24no—
54C2×Dic25—24no—
55C2×Dic25D414yescomplemented
56C2×Dic25C2312yes—
57C2×Dic25C2312yes—
58C2×Dic25C2311yescharacteristic
59C2×Dic25C2311yescharacteristic
60C2×Dic25C2311yescharacteristic
61C4×D25—22no—
62Dic50—22no—
63C23×Dic5—51no—
64C2×#711—51no—
65C5×#120D511yescharacteristic
66#4099—51no—
67#4085—52no—
68#4081—51no—
69D4×Dic5—52no—
70#4032—51no—
71#4027—52no—
72#4024—52no—
73#4023—51no—
74C22×C25—28no—
75C22×C25—22no—
76C22×C25—22no—
77C22×C25—22no—
78C22×C25—21no—
79C22×C25—22no—
80C22×C25C2×D412yes—
81C22×C25#4114yescomplemented
82C22×C25C2411yescharacteristic
83C2×D25—21no—
84C2×D25—22no—
85C4×C25—21no—
86C4×C25—22no—
87Dic25—21no—
88Dic25—22no—
89Dic25—22no—
90Dic25C2×D412yes—
91Dic25C2×D412yes—
92C24×C5C2×D511yescharacteristic
93C2×C5×D4C2×D511yescharacteristic
94C2×C5×D4C2×D512yes—
95C2×C22⋊D5—51no—
96C2×C22⋊D5—52no—
97C2×C22⋊D5—51no—
98C22×Dic5—51no—
99C22×Dic5—51no—
100C22×Dic5—52no—
101C22×Dic5—52no—
102C22×Dic5—51no—
103C22×Dic5—104no—
104#711—104no—
105C2×C4×D5—51no—
106C2×Dic10—51no—
107C5×C22⋊C4C2×D511yescharacteristic
108C5×C22⋊C4C2×D512yes—
109C22⋊Dic5—51no—
110C22⋊Dic5—52no—
111C22⋊Dic5—52no—
112#685—52no—
113C4⋊Dic5—52no—
114#683—52no—
115#683—52no—
116C4×Dic5—52no—
117#525—251noSylow
118C2×C25—22no—
119C2×C25—24no—
120C2×C25—41no—
121C2×C25C22×D411yescharacteristic
122C2×C25C2×#4112yes—
123D25—41no—
124C23×C5—22no—
125C23×C5—24no—
126C23×C5C22×D511yescharacteristic
127C23×C5C22×D511yescharacteristic
128C23×C5C22×D512yes—
129C22×D5—51no—
130C5×D4—22no—
131C5×D4—24no—
132C2×C4×C5C22×D512yes—
133C2×C4×C5C22×D511yescharacteristic
134C22⋊D5—102no—
135C22⋊D5—102no—
136C22⋊D5—104no—
137C2×Dic5—54no—
138C2×Dic5—51no—
139C2×Dic5—51no—
140C2×Dic5—52no—
141C2×Dic5—52no—
142C2×Dic5—51no—
143C2×Dic5—102no—
144C2×Dic5—104no—
145C2×Dic5—104no—
146C2×Dic5—102no—
147C4×D5—102no—
148Dic10—102no—
149C2×#41—251no—
150C23×C4—251no—
151#123—251no—
152#123—251no—
153C22⋊Q8—252no—
154#121—252no—
155#120—251no—
156#120D2511yescharacteristic, complemented
157C4×D4—252no—
158C4×D4—252no—
159#117—251no—
160C25#52511yescharacteristic, complemented, Sylow
161C22×C5—28no—
162C22×C5—22no—
163C22×C5—22no—
164C22×C5—22no—
165C22×C5—21no—
166C22×C5—22no—
167C22×C5D4×D512yes—
168C22×C5#71114yes—
169C22×C5C23×D511yescharacteristic
170C2×D5—101no—
171C2×D5—102no—
172C4×C5—21no—
173C4×C5—22no—
174Dic5—52no—
175Dic5—52no—
176Dic5—101no—
177Dic5—102no—
178Dic5—102no—
179C24C2×D2511yescharacteristic
180#41—504no—
181C2×Q8—251no—
182C2×D4—252no—
183C2×D4—251no—
184C2×D4—251no—
185C2×D4C2×D2512yes—
186C2×D4C2×D2511yescharacteristic
187C22×C4—251no—
188C22×C4—252no—
189C22×C4—251no—
190C22×C4—252no—
191C22×C4—251no—
192C22×C4—251no—
193C22×C4—504no—
194C4⋊C4—252no—
195C4⋊C4—252no—
196C4⋊C4—252no—
197C22⋊C4—251no—
198C22⋊C4—252no—
199C22⋊C4—252no—
200C22⋊C4—252no—
201C22⋊C4C2×D2511yescharacteristic
202C22⋊C4C2×D2512yes—
203C42—252no—
204C2×C5—22no—
205C2×C5—24no—
206C2×C5—41no—
207C2×C5C2×D4×D511yescharacteristic
208C2×C5C2×#71112yes—
209D5—201no—
210C23—22no—
211C23—24no—
212C23—251no—
213C23C22×D2511yescharacteristic
214C23C22×D2511yescharacteristic
215C23C22×D2512yes—
216Q8—502no—
217D4—22no—
218D4—24no—
219D4—502no—
220D4—502no—
221D4—504no—
222C2×C4—254no—
223C2×C4—251no—
224C2×C4—251no—
225C2×C4—252no—
226C2×C4—252no—
227C2×C4—251no—
228C2×C4—502no—
229C2×C4—504no—
230C2×C4—504no—
231C2×C4—502no—
232C2×C4—502no—
233C2×C4C22×D2512yes—
234C2×C4C22×D2511yescharacteristic
235C5#6592011yescharacteristic
236C22—28no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—501no—
243C22—502no—
244C22D4×D2512yes—
245C22#8774614yescomplemented
246C22C23×D2511yescharacteristic
247C4—21no—
248C4—22no—
249C4—252no—
250C4—252no—
251C4—502no—
252C4—501no—
253C4—502no—
254C2—22no—
255C2—24no—
256C2—41no—
257C2—1001no—
258C2C2×D4×D2511yescharacteristic
259C2C2×#8774612yes—
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×#120C2
hypercenter#120D25
hypocenterC25#525
LCSG ▷ C22×C25 ▷ C25
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C23×Dic25C2semidirect
2. C2×#87746C2semidirect
3. C25×#120C2semidirect
4. #11716515C2semidirect
5. #11716501C2semidirect
6. #11716497C2semidirect
7. D4×Dic25C2semidirect
8. #11716448C2semidirect
9. #11716443C2semidirect
10. #11716440C2semidirect
11. #11716439C2semidirect
17. C2×C22⋊D25C22semidirect
23. C22×Dic25C22semidirect
25. C2×C4×D25C22semidirect
26. C2×Dic50C22semidirect
31. C22⋊Dic25C22semidirect
32. #87720C22semidirect
33. C4⋊Dic25C22semidirect
34. #87718C22semidirect
35. #87718C22semidirect
36. C4×Dic25C22semidirect
55. C2×Dic25D4semidirect
81. C22×C25#41semidirect
156. #120D25semidirect
160. C25#525semidirect
245. C22#87746semidirect

Maximal subgroups

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

subgroupindex
C23×Dic252
C2×#877462
C25×#1202
#117165152
#117165012
#117164972
D4×Dic252
#117164482
#117164432
#117164402
#117164392
#659205