Finite Group Databasebeta

← #12010259

Subgroups — #12010259

Order 1280.

336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3168C211yesmaximal, characteristic
2#10600555C211yesmaximal, characteristic, complemented
3#10600538C211yesmaximal, characteristic, complemented
4#10598267C212yesmaximal, complemented
5#10598265C212yesmaximal
6C5×#529—21no—
7C5×#499C411yescharacteristic, complemented
8C5×#495C411yescharacteristic, complemented
9C5×#488—21no—
10C5×#467C2211yescharacteristic
11C5×#466C2211yescharacteristic
12C5×#465—21no—
13C4×C5×Q16C411yescharacteristic, complemented
14C4×C5×SD16—21no—
15C4×C5×D8C411yescharacteristic, complemented
16C2×C4×C5×C8C2211yescharacteristic
17#65224—21no—
18#64771—21no—
19#64771C2212yes—
20C22⋊#672C2212yes—
21C4⋊#672—21no—
22#64681—21no—
23#11860—51nomaximal, Sylow
24C2×C5×Q16C2×C411yescharacteristic
25C2×C5×SD16—21no—
26C2×C5×D8C2×C411yescharacteristic
27C22×C5×C8D411yescharacteristic
28C22×C5×C8C2311yescharacteristic
29C5×#126—41no—
30C5×#125—21no—
31C5×#124—21no—
32C5×#123—41no—
33C5×C22⋊Q8—21no—
34C5×#121—21no—
35C4×C5×Q8—21no—
36C4×C5×D4—21no—
37C4×C5×D4—41no—
38C5×#117—41no—
39C2×C42×C5D411yescharacteristic
40C2×C4⋊Dic5—22no—
41C2×C4⋊Dic5—41no—
42C2×C4⋊Dic5—41no—
43C5×#107C2×C411yescharacteristic
44C5×#107C2×C411yescharacteristic
45C5×#106—21no—
46C5×#103—21no—
47C5×#103—21no—
48C5×#102—21no—
49C5×#102—21no—
50C4×C5×C8C2×C411yescharacteristic
51C4×C5×C8C2×C411yescharacteristic
52C4×C5×C8D411yescharacteristic
53C4×C5×C8D411yescharacteristic
54C2×#672—22no—
55#3168D511yescharacteristic
56#2163—51no—
57#1834—51no—
58#1348—52no—
59#1347—52no—
60C2×C5×Q8—21no—
61C2×C5×D4—21no—
62C2×C5×D4—41no—
63C22×C4×C5—41no—
64C22×C4×C5D811yescharacteristic
65C22×C4×C5D811yescharacteristic
66C22×C4×C5C2×D411yescharacteristic
67C22×Dic5—42no—
68C5×Q16—41no—
69C5×SD16—42no—
70C5×D8—41no—
71C2×C5×C8—41no—
72C2×C5×C8—41no—
73C2×C5×C8C22⋊C411yescharacteristic
74C2×C5×C8C22⋊C411yescharacteristic
75C2×C5×C8C22×C411yescharacteristic
76C2×C5×C8C2×D411yescharacteristic
77C5×C4⋊C4—21no—
78C5×C4⋊C4—21no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C22⋊C4—41no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4—41no—
86C5×C22⋊C4—41no—
87C5×C22⋊C4—41no—
88C42×C5—41no—
89C42×C5C22⋊C411yescharacteristic
90C42×C5C22⋊C411yescharacteristic
91C42×C5SD1611yescharacteristic
92C42×C5SD1611yescharacteristic
93C4⋊Dic5—42no—
94C4⋊Dic5—42no—
95C4⋊Dic5—82no—
96C4⋊Dic5—82no—
97#672—42no—
98#529—21no—
99#499Dic511yescharacteristic, complemented
100#495Dic511yescharacteristic, complemented
101#488—21no—
102#467C2×D511yescharacteristic
103#466C2×D511yescharacteristic
104#465—21no—
105C4×Q16Dic511yescharacteristic, complemented
106C4×SD16—21no—
107C4×D8Dic511yescharacteristic, complemented
108C2×C4×C8C2×D511yescharacteristic
109#384—101no—
110C4⋊C16—101no—
111C22⋊C16—52no—
112#346—101no—
113#340—52no—
114#340—101no—
115C23×C5—41no—
116C23×C5C2×D811yescharacteristic
117C5×Q8—41no—
118C5×Q8—41no—
119C5×D4—41no—
120C5×D4—41no—
121C5×D4—81no—
122C5×D4—81no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—22no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5—41no—
131C2×C4×C5—82no—
132C2×C4×C5#10211yescharacteristic
133C2×C4×C5#10211yescharacteristic
134C2×C4×C5#10211yescharacteristic
135C2×C4×C5#10211yescharacteristic
136C2×C4×C5C2×C22⋊C411yescharacteristic
137C2×C4×C5C2×SD1611yescharacteristic
138C2×Dic5—82no—
139C2×Dic5—82no—
140C2×Dic5—82no—
141C5×C8—21no—
142C5×C8—21no—
143C5×C8—21no—
144C5×C8—21no—
145C2×Q16C2×Dic511yescharacteristic
146C2×SD16—21no—
147C2×D8C2×Dic511yescharacteristic
148C22×C8C22⋊D511yescharacteristic
149C22×C8C22×D511yescharacteristic
150#126—41no—
151#125—21no—
152#124—21no—
153#123—41no—
154C22⋊Q8—21no—
155#121—21no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—41no—
159#117—41no—
160C2×C4⋊C4—102no—
161C2×C4⋊C4—201no—
162C2×C4⋊C4—201no—
163C2×C42C22⋊D511yescharacteristic
164C2×C16—102no—
165#107C2×Dic511yescharacteristic
166#107C2×Dic511yescharacteristic
167#106—21no—
168#103—21no—
169#103—21no—
170#102—21no—
171#102—21no—
172C4×C8C2×Dic511yescharacteristic
173C4×C8C2×Dic511yescharacteristic
174C4×C8C22⋊D511yescharacteristic
175C4×C8C22⋊D511yescharacteristic
176C22×C5—21no—
177C22×C5—21no—
178C22×C5—81no—
179C22×C5—81no—
180C22×C5—81no—
181C22×C5C2×#10211yescharacteristic
182C22×C5#50911yescharacteristic
183C22×C5#51011yescharacteristic
184C4×C5—21no—
185C4×C5—21no—
186C4×C5—21no—
187C4×C5—21no—
188C4×C5—81no—
189C4×C5—81no—
190C4×C5—81no—
191C4×C5#36012yes—
192C4×C5#41011yescharacteristic
193C4×C5#42111yescharacteristic
194Dic5—162no—
195C2×Q8—21no—
196C2×D4—21no—
197C2×D4—41no—
198C22×C4—41no—
199C22×C4—202no—
200C22×C4#68611yescharacteristic
201C22×C4#68611yescharacteristic
202C22×C4C2×C22⋊D511yescharacteristic
203Q16—41no—
204SD16—42no—
205D8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8C22⋊Dic511yescharacteristic
209C2×C8C22⋊Dic511yescharacteristic
210C2×C8C22×Dic511yescharacteristic
211C2×C8C2×C22⋊D511yescharacteristic
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—202no—
219C4⋊C4—202no—
220C4⋊C4—402no—
221C4⋊C4—402no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C42—41no—
228C42#68711yescharacteristic
229C42#68711yescharacteristic
230C42C22⋊Dic511yescharacteristic
231C42C22⋊Dic511yescharacteristic
232C16—202no—
233C2×C5—21no—
234C2×C5—21no—
235C2×C5—161no—
236C2×C5#151711yescharacteristic
237C2×C5#213911yescharacteristic
238C2×C5C2×#36011yescharacteristic
239C23—41no—
240C23C2×#68611yescharacteristic
241Q8—41no—
242Q8—41no—
243D4—41no—
244D4—41no—
245D4—81no—
246D4—81no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—82no—
256C2×C4—402no—
257C2×C4—402no—
258C2×C4—402no—
259C2×C4#396411yescharacteristic
260C2×C4#396411yescharacteristic
261C2×C4#396411yescharacteristic
262C2×C4#396411yescharacteristic
263C2×C4C2×#68711yescharacteristic
264C2×C4C2×C22⋊Dic511yescharacteristic
265C8—21no—
266C8—21no—
267C8—21no—
268C8—21no—
269C5#1186011yescharacteristic, complemented, Sylow
270C22—21no—
271C22—21no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22#6548211yescharacteristic
276C22#6548411yescharacteristic
277C22C2×#396411yescharacteristic
278C4—21no—
279C4—21no—
280C4—21no—
281C4—21no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—802no—
286C4#6478612yes—
287C4#6552511yescharacteristic
288C4#6552911yescharacteristic
289C2—21no—
290C2—21no—
291C2—161no—
292C2#1060260711yescharacteristic
293C2C2×#6478611yescharacteristic
294C2#1060287111yescharacteristic
subgroupG/H
centerC22C2×#3964
commutatorC2×C5×C8C22×C4
FrattiniC22×C8C22×D5
FittingC5×#3168C2
socleC22×C5C2×#102
prime radicalC5×#466C22
cosocleC22×C5×C8C23
hypercenter#3168D5
hypocenterC5#11860
LCSG ▷ C2×C5×C8 ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
2. #10600555C2semidirect
3. #10600538C2semidirect
4. #10598267C2semidirect
7. C5×#499C4semidirect
8. C5×#495C4semidirect
13. C4×C5×Q16C4semidirect
15. C4×C5×D8C4semidirect
99. #499Dic5semidirect
100. #495Dic5semidirect
105. C4×Q16Dic5semidirect
107. C4×D8Dic5semidirect
269. C5#11860semidirect

Maximal subgroups

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

subgroupindex
C5×#31682
#106005552
#106005382
#105982672
#105982652
#118605