Finite Group Databasebeta

← #11997025

Subgroups — #11997025

Order 1280.

512 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C5×#2503C211yesmaximal, characteristic
2#10613886C211yesmaximal, characteristic
3#10602482C212yesmaximal
4#10602355C211yesmaximal, characteristic
5#10598107C212yesmaximal
6C2×C5×C22⋊Q8C2211yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#400—22no—
9C5×#391C412yescomplemented
10C5×#382C412yescomplemented
11C5×#381—21no—
12C5×#381—21no—
13C4×C5×C22⋊C4C2211yescharacteristic
14#65501—21no—
15#65500C2212yes—
16C4×C22⋊Dic5—21no—
17#65496C2212yes—
18#65270—24no—
19#64758—24no—
20#64750—22no—
21#9460—51nomaximal, Sylow
22C22×C5×Q8C2×C411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5C2311yescharacteristic
26C5×C22⋊Q8—22no—
27C5×C22⋊Q8—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C4⋊C4D412yes—
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—21no—
36C2×C42×C5C2×C412yes—
37C2×C22⋊Dic5—22no—
38C2×C4⋊Dic5—22no—
39C2×C4×Dic5—22no—
40C2×#681—22no—
41C5×#95—22no—
42C5×#95—22no—
43C5×#95—22no—
44C5×#95—22no—
45C5×#95—21no—
46#3963—22no—
47#3963—22no—
48#3963—22no—
49#3963D414yes—
50C22⋊#168—22no—
51C22⋊#168—22no—
52#2503D511yescharacteristic
53#2313—51no—
54#1821—51no—
55#1511—52no—
56#1284—52no—
57C24×C5C2×D411yescharacteristic
58C2×C5×Q8—22no—
59C2×C5×Q8—42no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—22no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5—42no—
70C22×C4×C5C22⋊C412yes—
71C22×C4×C5C22⋊C412yes—
72C22×C4×C5C22×C411yescharacteristic
73C22×C4×C5C2×D412yes—
74C22×C4×C5C2×D411yescharacteristic
75C22×C4×C5#4112yes—
76C22×Dic5—24no—
77C5×C4⋊C4—24no—
78C5×C4⋊C4—24no—
79C5×C4⋊C4—22no—
80C5×C4⋊C4—42no—
81C5×C22⋊C4—42no—
82C5×C22⋊C4—41no—
83C5×C22⋊C4—41no—
84C42×C5—42no—
85C22⋊Dic5—44no—
86C4⋊Dic5—82no—
87C4×Dic5—82no—
88#681—44no—
89C2×#168—42no—
90C2×C22⋊Q8C2×D511yescharacteristic
91C22×C4⋊C4C2×D511yescharacteristic
92#407—52no—
93#400—22no—
94#391Dic512yescomplemented
95#387—104no—
96#382Dic512yescomplemented
97#381—21no—
98#381—21no—
99#381—52no—
100#380—101no—
101C4×C22⋊C4—101no—
102C4×C22⋊C4C2×D511yescharacteristic
103#343—102no—
104#328—104no—
105C23×C5—22no—
106C23×C5—22no—
107C23×C5—41no—
108C23×C5C2×C22⋊C411yescharacteristic
109C23×C5#12011yescharacteristic
110C23×C5#12311yescharacteristic
111C5×Q8—82no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—42no—
117C2×C4×C5—42no—
118C2×C4×C5—42no—
119C2×C4×C5—42no—
120C2×C4×C5—42no—
121C2×C4×C5—42no—
122C2×C4×C5—42no—
123C2×C4×C5—42no—
124C2×C4×C5—42no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—81no—
130C2×C4×C5—81no—
131C2×C4×C5C4×D412yes—
132C2×C4×C5#12112yes—
133C2×Dic5—44no—
134C2×Dic5—44no—
135C2×Dic5—44no—
136#168—82no—
137C22×Q8C2×Dic511yescharacteristic
138C23×C4C22⋊D511yescharacteristic
139C23×C4C22⋊D511yescharacteristic
140C23×C4C22×D511yescharacteristic
141C2×M16—102no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—102no—
148C2×C4⋊C4C2×Dic511yescharacteristic
149C2×C4⋊C4C2×Dic512yes—
150C2×C4⋊C4C22⋊D512yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—102no—
154C2×C42—102no—
155C2×C42C2×Dic512yes—
156C22⋊C8—102no—
157C22⋊C8—102no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—21no—
163#95—54no—
164#95—102no—
165#95—102no—
166#95—102no—
167C22×C5—21no—
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5—44no—
175C22×C5#38611yescharacteristic
176C22×C5#41012yes—
177C22×C5#45512yes—
178C4×C5—22no—
179C4×C5—22no—
180C4×C5—42no—
181C4×C5—42no—
182C4×C5—82no—
183C4×C5—81no—
184C4×C5—81no—
185Dic5—84no—
186C24C2×C22⋊D511yescharacteristic
187C2×Q8—22no—
188C2×Q8—42no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—104no—
200C22×C4C22⋊Dic512yes—
201C22×C4C22⋊Dic512yes—
202C22×C4D4×D512yes—
203C22×C4#71112yes—
204C22×C4C22×Dic511yescharacteristic
205C22×C4C2×C22⋊D511yescharacteristic
206M16—204no—
207C2×C8—202no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—402no—
213C22⋊C4—42no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—204no—
217C42—42no—
218C42—402no—
219C2×C5—22no—
220C2×C5—22no—
221C2×C5—41no—
222C2×C5#188511yescharacteristic
223C2×C5#189911yescharacteristic
224C2×C5#191211yescharacteristic
225C23—22no—
226C23—22no—
227C23—41no—
228C23#408111yescharacteristic
229C23#408311yescharacteristic
230C23C2×C22⋊Dic511yescharacteristic
231Q8—82no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—204no—
252C2×C4—204no—
253C2×C4—204no—
254C2×C4D4×Dic512yes—
255C2×C4#408512yes—
256C8—402no—
257C5#946011yescharacteristic, complemented, Sylow
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#6535012yes—
267C22#6550911yescharacteristic
268C22#6552912yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—81no—
275C4—81no—
276C4—404no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#1060249811yescharacteristic
281C2#1060251611yescharacteristic
282C2#1060265711yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2503C2
socleC22×C5#386
prime radicalC24×C5C2×D4
cosocleC23×C4×C5C23
hypercenter#2503D5
hypocenterC5#9460
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
9. C5×#391C4semidirect
10. C5×#382C4semidirect
94. #391Dic5semidirect
96. #382Dic5semidirect
257. C5#9460semidirect

Maximal subgroups

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

subgroupindex
C5×#25032
#106138862
#106024822
#106023552
#105981072
#94605