Finite Group Databasebeta

← #12008677

Subgroups — #12008677

Order 1280.

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

normal subgroups only

This group has 814 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
1C2×C5×#390C211yesmaximal, characteristic
2#10602345C212yesmaximal, complemented
3#10598131C214yesmaximal, complemented
4C23×C5×D4C411yescharacteristic
5C22×C5×C4⋊C4C2211yescharacteristic
6C22×C5×C22⋊C4—22no—
7C22×C42×C5C411yescharacteristic
8C5×#390—22no—
9C5×#390C412yes—
10C5×#390C2212yes—
11#65496—22no—
12#64748—22no—
13#64748C2214yescomplemented
14#11505—51nomaximal, Sylow
15C25×C5—21no—
16C22×C5×D4—24no—
17C22×C5×D4—21no—
18C22×C5×D4—41no—
19C22×C5×D4—41no—
20C22×C5×D4—42no—
21C22×C5×D4C2×C412yes—
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D412yes—
25C2×C5×C4⋊C4—41no—
26C2×C5×C4⋊C4—41no—
27C2×C5×C4⋊C4C2×C411yescharacteristic
28C2×C5×C4⋊C4D412yes—
29C2×C5×C4⋊C4C2311yescharacteristic
30C2×C5×C22⋊C4—24no—
31C2×C5×C22⋊C4—24no—
32C2×C5×C22⋊C4—48no—
33C2×C42×C5—21no—
34C2×C42×C5—42no—
35C2×C42×C5C2×C412yes—
36C2×#681—42no—
37C22⋊#168—24no—
38C22⋊#168—42no—
39C2×#390D511yescharacteristic
40#1463—52no—
41#1318—54no—
42C24×C5—24no—
43C24×C5—22no—
44C24×C5—44no—
45C24×C5—42no—
46C24×C5—44no—
47C24×C5—41no—
48C24×C5—41no—
49C24×C5C22⋊C411yescharacteristic
50C2×C5×D4—24no—
51C2×C5×D4—22no—
52C2×C5×D4—42no—
53C2×C5×D4—42no—
54C2×C5×D4—48no—
55C2×C5×D4—48no—
56C2×C5×D4—42no—
57C2×C5×D4—42no—
58C2×C5×D4M1614yescomplemented
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—24no—
64C22×C4×C5—22no—
65C22×C4×C5—22no—
66C22×C4×C5—22no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5—44no—
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C412yes—
72C22×C4×C5C22×C411yescharacteristic
73C22×C4×C5C2×D412yes—
74C5×C4⋊C4—41no—
75C5×C4⋊C4—42no—
76C5×C4⋊C4—41no—
77C5×C22⋊C4—48no—
78C5×C22⋊C4—48no—
79C42×C5—42no—
80C42×C5—42no—
81#681—84no—
82C2×#168—44no—
83C23×D4Dic511yescharacteristic
84C22×C4⋊C4C2×D511yescharacteristic
85C22×C22⋊C4—22no—
86C22×C42Dic511yescharacteristic
87#407—102no—
88#390—22no—
89#390Dic512yes—
90#390C2×D512yes—
91#324—54no—
92#324—102no—
93C23×C5—22no—
94C23×C5—22no—
95C23×C5—21no—
96C23×C5—21no—
97C23×C5—24no—
98C23×C5—44no—
99C23×C5—44no—
100C23×C5—44no—
101C23×C5—416no—
102C23×C5—44no—
103C23×C5—42no—
104C23×C5—42no—
105C23×C5—88no—
106C23×C5—82no—
107C23×C5—82no—
108C23×C5—82no—
109C23×C5—82no—
110C23×C5#9911yescharacteristic
111C23×C5#9911yescharacteristic
112C23×C5C2×C22⋊C411yescharacteristic
113C5×D4—44no—
114C5×D4—48no—
115C5×D4—44no—
116C2×C4×C5—24no—
117C2×C4×C5—24no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—24no—
123C2×C4×C5—22no—
124C2×C4×C5—22no—
125C2×C4×C5—48no—
126C2×C4×C5—42no—
127C2×C4×C5—44no—
128C2×C4×C5—42no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5—82no—
132C2×C4×C5#10012yes—
133C2×C4×C5C2×M1612yes—
134#168—84no—
135C25—21no—
136C22×D4—24no—
137C22×D4—21no—
138C22×D4—41no—
139C22×D4—41no—
140C22×D4—42no—
141C22×D4C2×Dic512yes—
142C23×C4—21no—
143C23×C4C2×Dic511yescharacteristic
144C23×C4C22⋊D512yes—
145C2×M16—202no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4C2×Dic511yescharacteristic
149C2×C4⋊C4C22⋊D512yes—
150C2×C4⋊C4C22×D511yescharacteristic
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—48no—
154C2×C42—21no—
155C2×C42—42no—
156C2×C42C2×Dic512yes—
157C22⋊C8—104no—
158C22⋊C8—202no—
159C22×C5—24no—
160C22×C5—24no—
161C22×C5—21no—
162C22×C5—21no—
163C22×C5—21no—
164C22×C5—21no—
165C22×C5—21no—
166C22×C5—21no—
167C22×C5—48no—
168C22×C5—44no—
169C22×C5—84no—
170C22×C5—88no—
171C22×C5—88no—
172C22×C5—82no—
173C22×C5—82no—
174C22×C5#35312yes—
175C22×C5#35512yes—
176C22×C5#40711yescharacteristic
177C22×C5C2×#9911yescharacteristic
178C22×C5C2×#10011yescharacteristic
179C4×C5—22no—
180C4×C5—22no—
181C4×C5—44no—
182C4×C5—42no—
183C4×C5—42no—
184C24—24no—
185C24—22no—
186C24—44no—
187C24—42no—
188C24—44no—
189C24—41no—
190C24—41no—
191C24C22⋊Dic511yescharacteristic
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—48no—
197C2×D4—48no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4#68114yescomplemented
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4C22⋊Dic511yescharacteristic
213C22×C4C22⋊Dic512yes—
214C22×C4C22×Dic511yescharacteristic
215C22×C4C2×C22⋊D512yes—
216M16—404no—
217C2×C8—204no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C42—42no—
224C42—42no—
225C2×C5—21no—
226C2×C5—21no—
227C2×C5—22no—
228C2×C5—22no—
229C2×C5—84no—
230C2×C5#146211yescharacteristic
231C2×C5C2×#35311yescharacteristic
232C2×C5C2×#35511yescharacteristic
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—24no—
238C23—44no—
239C23—44no—
240C23—44no—
241C23—416no—
242C23—44no—
243C23—42no—
244C23—42no—
245C23—88no—
246C23—82no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23#396611yescharacteristic
251C23#396611yescharacteristic
252C23C2×C22⋊Dic511yescharacteristic
253D4—44no—
254D4—48no—
255D4—44no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—48no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4#396512yes—
273C2×C4C2×#68112yes—
274C8—404no—
275C5#1150511yescharacteristic, complemented, Sylow
276C22—24no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—48no—
285C22—44no—
286C22—84no—
287C22—88no—
288C22—88no—
289C22—82no—
290C22—82no—
291C22#6476212yes—
292C22#6476512yes—
293C22#6549611yescharacteristic
294C22C2×#396511yescharacteristic
295C22C2×#396611yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—42no—
300C4—42no—
301C2—21no—
302C2—21no—
303C2—22no—
304C2—22no—
305C2—84no—
306C2#1060233211yescharacteristic
307C2C2×#6476211yescharacteristic
308C2C2×#6476511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC2×C5×#390C2
socleC22×C5C2×#99
prime radicalC23×C5×D4C4
cosocleC2×C5×C4⋊C4C23
hypercenterC2×#390D5
hypocenterC5#11505
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. #10602345C2semidirect
3. #10598131C2semidirect
13. #64748C22semidirect
58. C2×C5×D4M16semidirect
200. C2×D4#681semidirect
275. C5#11505semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3902
#106023452
#105981312
#115055