Finite Group Databasebeta

← #11997030

Subgroups — #11997030

Order 1280.

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

normal subgroups only

This group has 528 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×#2502C211yesmaximal, characteristic
2#10613886C211yesmaximal, characteristic
3#10602482C212yesmaximal
4#10602355C211yesmaximal, characteristic
5#10598107C212yesmaximal
6C2×C5×C22⋊Q8C2211yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#398—22no—
9C5×#391C412yescomplemented
10C5×#385—21no—
11C5×#385—21no—
12C5×#384C412yescomplemented
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#9461—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⋊C4—22no—
32C2×C5×C4⋊C4—22no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C4⋊C4C2×C412yes—
35C2×C5×C4⋊C4D412yes—
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C42×C5C2×C412yes—
39C2×C22⋊Dic5—22no—
40C2×C4⋊Dic5—22no—
41C2×C4×Dic5—22no—
42C2×#681—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#2502D511yescharacteristic
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—22no—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—42no—
68C22×C4×C5—81no—
69C22×C4×C5—81no—
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×C4⋊C4—82no—
82C5×C4⋊C4—82no—
83C5×C22⋊C4—24no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4—41no—
86C42×C5—42no—
87C22⋊Dic5—44no—
88C4⋊Dic5—82no—
89C4×Dic5—82no—
90#681—44no—
91C2×#168—42no—
92C2×C22⋊Q8C2×D511yescharacteristic
93C22×C4⋊C4C2×D511yescharacteristic
94#407—52no—
95#398—22no—
96#391Dic512yescomplemented
97#387—104no—
98#385—21no—
99#385—21no—
100#384Dic512yescomplemented
101#381—52no—
102#380—101no—
103C4×C22⋊C4—101no—
104C4×C22⋊C4C2×D511yescharacteristic
105#343—102no—
106#328—104no—
107C23×C5—22no—
108C23×C5—22no—
109C23×C5—41no—
110C23×C5C2×C22⋊C411yescharacteristic
111C23×C5#12011yescharacteristic
112C23×C5#12311yescharacteristic
113C5×Q8—82no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—24no—
117C2×C4×C5—22no—
118C2×C4×C5—22no—
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—42no—
126C2×C4×C5—42no—
127C2×C4×C5—42no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5—42no—
131C2×C4×C5—81no—
132C2×C4×C5—81no—
133C2×C4×C5C4×D412yes—
134C2×C4×C5#12112yes—
135C2×Dic5—44no—
136C2×Dic5—44no—
137C2×Dic5—44no—
138#168—82no—
139C22×Q8C2×Dic511yescharacteristic
140C23×C4C22⋊D511yescharacteristic
141C23×C4C22⋊D511yescharacteristic
142C23×C4C22×D511yescharacteristic
143C2×M16—102no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—102no—
152C2×C4⋊C4C2×Dic511yescharacteristic
153C2×C4⋊C4C2×Dic512yes—
154C2×C4⋊C4C22⋊D512yes—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—102no—
158C2×C42—102no—
159C2×C42C2×Dic512yes—
160C22⋊C8—102no—
161C22⋊C8—102no—
162#95—22no—
163#95—22no—
164#95—21no—
165#95—54no—
166#95—102no—
167#95—102no—
168#95—102no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—21no—
176C22×C5—44no—
177C22×C5#38611yescharacteristic
178C22×C5#41012yes—
179C22×C5#45612yes—
180C4×C5—22no—
181C4×C5—22no—
182C4×C5—44no—
183C4×C5—42no—
184C4×C5—42no—
185C4×C5—81no—
186C4×C5—81no—
187Dic5—84no—
188C24C2×C22⋊D511yescharacteristic
189C2×Q8—22no—
190C2×Q8—42no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—104no—
202C22×C4C22⋊Dic512yes—
203C22×C4C22⋊Dic512yes—
204C22×C4D4×D512yes—
205C22×C4#71112yes—
206C22×C4C22×Dic511yescharacteristic
207C22×C4C2×C22⋊D511yescharacteristic
208M16—204no—
209C2×C8—202no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—402no—
217C22⋊C4—24no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—204no—
221C42—42no—
222C42—402no—
223C2×C5—22no—
224C2×C5—22no—
225C2×C5—41no—
226C2×C5#188611yescharacteristic
227C2×C5#189911yescharacteristic
228C2×C5#191111yescharacteristic
229C23—22no—
230C23—22no—
231C23—41no—
232C23#408111yescharacteristic
233C23#408311yescharacteristic
234C23C2×C22⋊Dic511yescharacteristic
235Q8—82no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—204no—
256C2×C4—204no—
257C2×C4—204no—
258C2×C4D4×Dic512yes—
259C2×C4#408512yes—
260C8—402no—
261C5#946111yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#6538412yes—
271C22#6550911yescharacteristic
272C22#6552912yes—
273C4—22no—
274C4—22no—
275C4—44no—
276C4—42no—
277C4—42no—
278C4—81no—
279C4—81no—
280C4—404no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#1060249811yescharacteristic
285C2#1060257711yescharacteristic
286C2#1060264611yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2502C2
socleC22×C5#386
prime radicalC24×C5C2×D4
cosocleC23×C4×C5C23
hypercenter#2502D5
hypocenterC5#9461
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
9. C5×#391C4semidirect
12. C5×#384C4semidirect
96. #391Dic5semidirect
100. #384Dic5semidirect
261. C5#9461semidirect

Maximal subgroups

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

subgroupindex
C5×#25022
#106138862
#106024822
#106023552
#105981072
#94615