Finite Group Databasebeta

← #11996751

Subgroups — #11996751

Order 1280.

644 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2591C211yesmaximal, characteristic
2#10613875C211yesmaximal, characteristic, complemented
3#10602466C212yesmaximal
4#10602345C211yesmaximal, characteristic, complemented
5#10598109C212yesmaximal, complemented
6C2×C5×#121C2211yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C2×#4072—21no—
9C5×#400C412yescomplemented
10C5×#394C412yescomplemented
11C5×#388—22no—
12C5×#386—21no—
13C5×#386—21no—
14C4×C5×C22⋊C4C2211yescharacteristic
15C4×C22⋊Dic5—21no—
16C4×C22⋊Dic5C2212yes—
17#65496C2212yes—
18C4⋊C4×Dic5—24no—
19#64757—24no—
20#64748—22no—
21#9410—51nomaximal, Sylow
22C22×C5×D4—21no—
23C22×C5×D4C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5C2311yescharacteristic
27C5×#121—22no—
28C5×#121—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C4⋊C4D412yes—
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—22no—
38C2×C42×C5—21no—
39C2×C22⋊Dic5—22no—
40#4072—28no—
41C2×C4⋊Dic5—22no—
42C2×C4×Dic5—22no—
43C2×C4×Dic5—22no—
44C2×C4×Dic5—22no—
45C2×#681—22no—
46C5×#95—22no—
47C5×#95—22no—
48C5×#95C2×C412yes—
49#3963—22no—
50#3963D414yes—
51C22⋊#168—22no—
52C22⋊#168—22no—
53#2591D511yescharacteristic
54#2274—51no—
55#1507—52no—
56#1463—51no—
57#1297—52no—
58C24×C5—21no—
59C24×C5C2×D411yescharacteristic
60C2×C5×D4—22no—
61C2×C5×D4—44no—
62C2×C5×D4—41no—
63C2×C5×D4—41no—
64C2×C5×D4—81no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—24no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—22no—
71C22×C4×C5—21no—
72C22×C4×C5—81no—
73C22×C4×C5—81no—
74C22×C4×C5C22⋊C412yes—
75C22×C4×C5C22⋊C412yes—
76C22×C4×C5C22×C411yescharacteristic
77C22×C4×C5C2×D412yes—
78C22×C4×C5C2×D411yescharacteristic
79C22×C4×C5#4112yes—
80C22×Dic5—24no—
81C5×C4⋊C4—24no—
82C5×C4⋊C4—24no—
83C5×C4⋊C4—22no—
84C5×C22⋊C4—42no—
85C5×C22⋊C4—44no—
86C5×C22⋊C4—42no—
87C5×C22⋊C4—82no—
88C42×C5—81no—
89C22⋊Dic5—28no—
90C4⋊Dic5—28no—
91C4×Dic5—28no—
92C4×Dic5—82no—
93C4×Dic5—82no—
94#681—82no—
95C2×#168—42no—
96C2×#121C2×D511yescharacteristic
97C2×#117—101no—
98C22×C4⋊C4C2×D511yescharacteristic
99#407—52no—
100#400Dic512yescomplemented
101#394Dic512yescomplemented
102#388—22no—
103#386—21no—
104#386—21no—
105C4×C4⋊C4—104no—
106C4×C22⋊C4—52no—
107C4×C22⋊C4—101no—
108C4×C22⋊C4C2×D511yescharacteristic
109#327—104no—
110#324—102no—
111C23×C5—22no—
112C23×C5—22no—
113C23×C5—41no—
114C23×C5—42no—
115C23×C5—42no—
116C23×C5—41no—
117C23×C5—42no—
118C23×C5C2×C22⋊C411yescharacteristic
119C23×C5#12011yescharacteristic
120C23×C5#12311yescharacteristic
121C5×D4—82no—
122C5×D4—82no—
123C5×D4—82no—
124C2×C4×C5—22no—
125C2×C4×C5—22no—
126C2×C4×C5—24no—
127C2×C4×C5—22no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—41no—
133C2×C4×C5—41no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—41no—
140C2×C4×C5—42no—
141C2×C4×C5—81no—
142C2×C4×C5#10414yes—
143C2×C4×C5C4×D412yes—
144C2×C4×C5#12112yes—
145C2×Dic5—28no—
146C2×Dic5—44no—
147C2×Dic5—44no—
148#168—82no—
149C22×D4—21no—
150C22×D4C2×Dic511yescharacteristic
151C23×C4C22⋊D511yescharacteristic
152C23×C4C22⋊D511yescharacteristic
153C23×C4C22×D511yescharacteristic
154C2×M16—102no—
155#121—22no—
156#121—22no—
157#117—108no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—102no—
161C2×C4⋊C4C2×Dic511yescharacteristic
162C2×C4⋊C4C2×Dic512yes—
163C2×C4⋊C4C22⋊D512yes—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—102no—
169C2×C42—21no—
170C2×C42—102no—
171C2×C42—102no—
172C2×C42—102no—
173C22⋊C8—102no—
174C22⋊C8—102no—
175#95—22no—
176#95—22no—
177#95—54no—
178#95—102no—
179#95C2×Dic512yes—
180C22×C5—21no—
181C22×C5—21no—
182C22×C5—21no—
183C22×C5—21no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—21no—
187C22×C5—44no—
188C22×C5—84no—
189C22×C5—81no—
190C22×C5—81no—
191C22×C5—81no—
192C22×C5#38611yescharacteristic
193C22×C5C2×#10412yes—
194C22×C5#45412yes—
195C4×C5—24no—
196C4×C5—22no—
197C4×C5—22no—
198C4×C5—42no—
199C4×C5—82no—
200C4×C5—81no—
201Dic5—48no—
202C24—21no—
203C24C2×C22⋊D511yescharacteristic
204C2×D4—22no—
205C2×D4—44no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—81no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—104no—
219C22×C4C22⋊Dic512yes—
220C22×C4C22⋊Dic512yes—
221C22×C4D4×D512yes—
222C22×C4#71112yes—
223C22×C4C22×Dic511yescharacteristic
224C22×C4C2×C22⋊D511yescharacteristic
225M16—402no—
226C2×C8—202no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—108no—
231C22⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—82no—
235C22⋊C4—108no—
236C42—81no—
237C42—108no—
238C42—402no—
239C42—402no—
240C2×C5—22no—
241C2×C5—22no—
242C2×C5—41no—
243C2×C5—161no—
244C2×C5#188811yescharacteristic
245C2×C5#189611yescharacteristic
246C2×C5#191311yescharacteristic
247C23—22no—
248C23—22no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—42no—
254C23#408111yescharacteristic
255C23#408311yescharacteristic
256C23C2×C22⋊Dic511yescharacteristic
257D4—82no—
258D4—82no—
259D4—82no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—108no—
279C2×C4—204no—
280C2×C4—204no—
281C2×C4#396914yes—
282C2×C4D4×Dic512yes—
283C2×C4#408512yes—
284C8—402no—
285C5#941011yescharacteristic, complemented, Sylow
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#6535112yes—
299C22#6550911yescharacteristic
300C22C2×#396912yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—42no—
305C4—82no—
306C4—81no—
307C4—208no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—161no—
312C2#1060250411yescharacteristic
313C2#1060251711yescharacteristic
314C2#1060264111yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2591C2
socleC22×C5#386
prime radicalC2×C5×#121C22
cosocleC23×C4×C5C23
hypercenter#2591D5
hypocenterC5#9410
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
2. #10613875C2semidirect
4. #10602345C2semidirect
5. #10598109C2semidirect
9. C5×#400C4semidirect
10. C5×#394C4semidirect
100. #400Dic5semidirect
101. #394Dic5semidirect
285. C5#9410semidirect

Maximal subgroups

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

subgroupindex
C5×#25912
#106138752
#106024662
#106023452
#105981092
#94105