Finite Group Databasebeta

← #12008784

Subgroups — #12008784

Order 1280.

504 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2328C211yesmaximal, characteristic
2#10602395C212yesmaximal, complemented
3#10598131C212yesmaximal, complemented
4#10598123C212yesmaximal, complemented
5C2×C5×#127C411yescharacteristic
6C2×C4×C5×D4C411yescharacteristic
7C2×C4×C5×D4C2211yescharacteristic
8C5×#390C411yescharacteristic
9C5×#390C2211yescharacteristic
10C5×#386—22no—
11C5×#386—22no—
12C5×#382C2211yescharacteristic
13C5×#376C411yescharacteristic
14#65496—22no—
15#64748C2214yes—
16#64745—22no—
17#11533—51nomaximal, Sylow
18C22×C5×D4—22no—
19C22×C5×D4C2×C411yescharacteristic
20C22×C5×D4C2×C411yescharacteristic
21C23×C4×C5—21no—
22C23×C4×C5D412yes—
23C5×#127—22no—
24C5×#127—22no—
25C4×C5×D4—42no—
26C4×C5×D4—42no—
27C2×C5×C4⋊C4C2×C411yescharacteristic
28C2×C5×C4⋊C4C2311yescharacteristic
29C2×C5×C22⋊C4—21no—
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4—22no—
32C2×C5×C22⋊C4D412yes—
33C2×C42×C5C2×C411yescharacteristic
34C2×C42×C5C2×C411yescharacteristic
35C2×C42×C5C2×C411yescharacteristic
36C2×#681—42no—
37C5×#95—21no—
38C5×#95—21no—
39C5×#95—22no—
40C22⋊#168—24no—
41C22⋊#168—42no—
42#2328D511yescharacteristic
43#1465—52no—
44#1324—52no—
45#1318—52no—
46C24×C5—21no—
47C24×C5C22⋊C412yes—
48C2×C5×D4—28no—
49C2×C5×D4—21no—
50C2×C5×D4—21no—
51C2×C5×D4—21no—
52C2×C5×D4—41no—
53C2×C5×D4—41no—
54C2×C5×D4—41no—
55C2×C5×D4—41no—
56C2×C5×D4—82no—
57C2×C5×D4—82no—
58C2×C5×D4M1612yescomplemented
59C22×C4×C5—21no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—42no—
65C22×C4×C5—82no—
66C22×C4×C5—82no—
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22×C411yescharacteristic
70C22×C4×C5C2×D412yes—
71C5×C4⋊C4—41no—
72C5×C4⋊C4—41no—
73C5×C22⋊C4—42no—
74C5×C22⋊C4—42no—
75C5×C22⋊C4—82no—
76C5×C22⋊C4—82no—
77C42×C5—22no—
78C42×C5—21no—
79C42×C5—41no—
80C42×C5M1612yescomplemented
81#681—84no—
82C2×#168—44no—
83C2×#127Dic511yescharacteristic
84C2×C4×D4Dic511yescharacteristic
85C2×C4×D4C2×D511yescharacteristic
86#407—102no—
87#390Dic511yescharacteristic
88#390C2×D511yescharacteristic
89#386—22no—
90#386—22no—
91#382C2×D511yescharacteristic
92#376Dic511yescharacteristic
93#324—54no—
94#323—102no—
95C23×C5—22no—
96C23×C5—22no—
97C23×C5—22no—
98C23×C5—22no—
99C23×C5—42no—
100C23×C5—42no—
101C23×C5—42no—
102C23×C5—42no—
103C23×C5—41no—
104C23×C5—41no—
105C23×C5C2×C22⋊C411yescharacteristic
106C5×D4—42no—
107C5×D4—42no—
108C5×D4—42no—
109C5×D4—42no—
110C5×D4—88no—
111C2×C4×C5—21no—
112C2×C4×C5—21no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—21no—
116C2×C4×C5—41no—
117C2×C4×C5—44no—
118C2×C4×C5—41no—
119C2×C4×C5—41no—
120C2×C4×C5—42no—
121C2×C4×C5—41no—
122C2×C4×C5—41no—
123C2×C4×C5—41no—
124C2×C4×C5—42no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5—82no—
128C2×C4×C5—84no—
129C2×C4×C5#9912yes—
130C2×C4×C5C2×M1611yescharacteristic
131C2×C4×C5C2×M1611yescharacteristic
132#168—84no—
133C22×D4—22no—
134C22×D4C2×Dic511yescharacteristic
135C22×D4C2×Dic511yescharacteristic
136C23×C4—21no—
137C23×C4C22⋊D512yes—
138C2×M16—202no—
139#127—22no—
140#127—22no—
141C4×D4—42no—
142C4×D4—42no—
143C2×C4⋊C4C2×Dic511yescharacteristic
144C2×C4⋊C4C22×D511yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4C22⋊D512yes—
149C2×C42C2×Dic511yescharacteristic
150C2×C42C2×Dic511yescharacteristic
151C2×C42C2×Dic511yescharacteristic
152C22⋊C8—104no—
153C22⋊C8—202no—
154#95—21no—
155#95—21no—
156#95—22no—
157C22×C5—21no—
158C22×C5—21no—
159C22×C5—44no—
160C22×C5—44no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5—42no—
164C22×C5—42no—
165C22×C5—82no—
166C22×C5—82no—
167C22×C5—81no—
168C22×C5—81no—
169C22×C5#40711yescharacteristic
170C22×C5C2×#9911yescharacteristic
171C22×C5#41311yescharacteristic
172C4×C5—22no—
173C4×C5—21no—
174C4×C5—21no—
175C4×C5—42no—
176C4×C5—42no—
177C4×C5—41no—
178C4×C5—81no—
179C4×C5—81no—
180C24—21no—
181C24C22⋊Dic512yes—
182C2×D4—28no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—82no—
191C2×D4—82no—
192C2×D4#68112yescomplemented
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4C22⋊Dic511yescharacteristic
202C22×C4C22⋊Dic511yescharacteristic
203C22×C4C22×Dic511yescharacteristic
204C22×C4C2×C22⋊D512yes—
205M16—404no—
206C2×C8—204no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—22no—
214C42—21no—
215C42—41no—
216C42#68112yescomplemented
217C2×C5—21no—
218C2×C5—21no—
219C2×C5—82no—
220C2×C5—82no—
221C2×C5#146511yescharacteristic
222C2×C5#212711yescharacteristic
223C2×C5#213411yescharacteristic
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23C2×C22⋊Dic511yescharacteristic
235D4—42no—
236D4—42no—
237D4—42no—
238D4—42no—
239D4—88no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—84no—
258C2×C4#396612yes—
259C2×C4C2×#68111yescharacteristic
260C2×C4C2×#68111yescharacteristic
261C8—404no—
262C5#1153311yescharacteristic, complemented, Sylow
263C22—21no—
264C22—21no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—82no—
273C22—81no—
274C22—81no—
275C22#6549611yescharacteristic
276C22C2×#396611yescharacteristic
277C22#6552611yescharacteristic
278C4—21no—
279C4—21no—
280C4—22no—
281C4—42no—
282C4—42no—
283C4—41no—
284C4—81no—
285C4—81no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—82no—
290C2#1060239511yescharacteristic
291C2#1060265311yescharacteristic
292C2#1060265511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2328C2
socleC22×C5C2×#99
prime radicalC2×C5×#127C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2328D5
hypocenterC5#11533
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #10602395C2semidirect
3. #10598131C2semidirect
4. #10598123C2semidirect
58. C2×C5×D4M16semidirect
80. C42×C5M16semidirect
192. C2×D4#681semidirect
216. C42#681semidirect
262. C5#11533semidirect

Maximal subgroups

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

subgroupindex
C5×#23282
#106023952
#105981312
#105981232
#115335