Finite Group Databasebeta

← #12008776

Subgroups — #12008776

Order 1280.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C4×C5×#124C211yesmaximal, characteristic
2#10602395C212yesmaximal
3#10598134C212yesmaximal, complemented
4#10598123C212yesmaximal, complemented
5C2×C5×#124C411yescharacteristic
6C2×C4×C5×Q8C411yescharacteristic
7C2×C4×C5×D4C2211yescharacteristic
8C5×#391C2211yescharacteristic
9C5×#390C411yescharacteristic
10C5×#388—22no—
11C5×#382C2211yescharacteristic
12C4×C5×C22⋊C4—22no—
13C43×C5C411yescharacteristic
14#65496—22no—
15#64748C2214yes—
16#64745—22no—
17#11531—51nomaximal, Sylow
18C22×C5×Q8C2×C411yescharacteristic
19C22×C5×D4C2×C411yescharacteristic
20C23×C4×C5D412yes—
21C5×#124—22no—
22C5×#124—22no—
23C4×C5×Q8—41no—
24C4×C5×Q8—41no—
25C4×C5×D4—41no—
26C4×C5×D4—41no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C4⋊C4C2311yescharacteristic
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4—22no—
32C2×C5×C22⋊C4D412yes—
33C2×C42×C5—21no—
34C2×C42×C5—21no—
35C2×C42×C5—21no—
36C2×C42×C5C2×C411yescharacteristic
37C2×C42×C5C2×C411yescharacteristic
38C2×C42×C5C2×C411yescharacteristic
39C2×#681—42no—
40C5×#95—22no—
41C5×#95—22no—
42C22⋊#168—24no—
43C22⋊#168—42no—
44C4×#124D511yescharacteristic
45#1465—52no—
46#1324—52no—
47#1319—52no—
48C24×C5C22⋊C412yes—
49C2×C5×Q8—21no—
50C2×C5×Q8—41no—
51C2×C5×Q8—41no—
52C2×C5×Q8M1612yescomplemented
53C2×C5×D4—21no—
54C2×C5×D4—21no—
55C2×C5×D4—41no—
56C2×C5×D4—41no—
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—21no—
63C22×C4×C5—44no—
64C22×C4×C5C22⋊C411yescharacteristic
65C22×C4×C5C22⋊C411yescharacteristic
66C22×C4×C5C22×C411yescharacteristic
67C22×C4×C5C2×D412yes—
68C5×C4⋊C4—41no—
69C5×C4⋊C4—41no—
70C5×C4⋊C4—42no—
71C5×C22⋊C4—28no—
72C5×C22⋊C4—42no—
73C5×C22⋊C4—44no—
74C42×C5—21no—
75C42×C5—21no—
76C42×C5—21no—
77C42×C5—41no—
78C42×C5—42no—
79C42×C5—42no—
80C42×C5—41no—
81C42×C5—41no—
82C42×C5M1612yescomplemented
83#681—84no—
84C2×#168—44no—
85C2×#124Dic511yescharacteristic
86C2×C4×Q8Dic511yescharacteristic
87C2×C4×D4C2×D511yescharacteristic
88#407—102no—
89#391C2×D511yescharacteristic
90#390Dic511yescharacteristic
91#388—22no—
92#382C2×D511yescharacteristic
93C4×C22⋊C4—22no—
94C43Dic511yescharacteristic
95#324—54no—
96#323—102no—
97C23×C5—22no—
98C23×C5—22no—
99C23×C5—22no—
100C23×C5—42no—
101C23×C5—42no—
102C23×C5C2×C22⋊C411yescharacteristic
103C5×Q8—42no—
104C5×Q8—42no—
105C5×D4—42no—
106C5×D4—42no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—21no—
115C2×C4×C5—41no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5—41no—
119C2×C4×C5—41no—
120C2×C4×C5—41no—
121C2×C4×C5—44no—
122C2×C4×C5—41no—
123C2×C4×C5—42no—
124C2×C4×C5—42no—
125C2×C4×C5—41no—
126C2×C4×C5—82no—
127C2×C4×C5#9911yescharacteristic
128C2×C4×C5#9911yescharacteristic
129C2×C4×C5C2×M1611yescharacteristic
130C2×C4×C5C2×M1611yescharacteristic
131#168—84no—
132C22×Q8C2×Dic511yescharacteristic
133C22×D4C2×Dic511yescharacteristic
134C23×C4C22⋊D512yes—
135C2×M16—202no—
136#124—22no—
137#124—22no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×D4—41no—
141C4×D4—41no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4C2×Dic511yescharacteristic
144C2×C4⋊C4C22×D511yescharacteristic
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4C22⋊D512yes—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42C2×Dic511yescharacteristic
152C2×C42C2×Dic511yescharacteristic
153C2×C42C2×Dic511yescharacteristic
154C22⋊C8—104no—
155C22⋊C8—202no—
156#95—22no—
157#95—22no—
158C22×C5—21no—
159C22×C5—21no—
160C22×C5—44no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5—42no—
164C22×C5—82no—
165C22×C5#40711yescharacteristic
166C22×C5C2×#9911yescharacteristic
167C22×C5#41311yescharacteristic
168C4×C5—21no—
169C4×C5—21no—
170C4×C5—21no—
171C4×C5—42no—
172C4×C5—42no—
173C4×C5—42no—
174C4×C5—41no—
175C4×C5—42no—
176C4×C5—81no—
177C4×C5#35412yes—
178C24C22⋊Dic512yes—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8#68112yescomplemented
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—41no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—44no—
194C22×C4C22⋊Dic511yescharacteristic
195C22×C4C22⋊Dic511yescharacteristic
196C22×C4C22×Dic511yescharacteristic
197C22×C4C2×C22⋊D512yes—
198M16—404no—
199C2×C8—204no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C22⋊C4—28no—
204C22⋊C4—42no—
205C22⋊C4—44no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—42no—
211C42—42no—
212C42—41no—
213C42—41no—
214C42#68112yescomplemented
215C2×C5—21no—
216C2×C5—21no—
217C2×C5—82no—
218C2×C5#146511yescharacteristic
219C2×C5C2×#35411yescharacteristic
220C2×C5#213211yescharacteristic
221C23—22no—
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23C2×C22⋊Dic511yescharacteristic
227Q8—42no—
228Q8—42no—
229D4—42no—
230D4—42no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—44no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—82no—
251C2×C4#396611yescharacteristic
252C2×C4#396611yescharacteristic
253C2×C4C2×#68111yescharacteristic
254C2×C4C2×#68111yescharacteristic
255C8—404no—
256C5#1153111yescharacteristic, complemented, Sylow
257C22—21no—
258C22—21no—
259C22—44no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22#6549611yescharacteristic
265C22C2×#396611yescharacteristic
266C22#6552611yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—81no—
276C4#6476112yes—
277C2—21no—
278C2—21no—
279C2—82no—
280C2#1060239511yescharacteristic
281C2C2×#6476111yescharacteristic
282C2#1060261311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC4×C5×#124C2
socleC22×C5C2×#99
prime radicalC22×C5×D4C2×C4
cosocleC2×C5×C4⋊C4C23
hypercenterC4×#124D5
hypocenterC5#11531
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
3. #10598134C2semidirect
4. #10598123C2semidirect
52. C2×C5×Q8M16semidirect
82. C42×C5M16semidirect
182. C2×Q8#681semidirect
214. C42#681semidirect
256. C5#11531semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#1242
#106023952
#105981342
#105981232
#115315