Finite Group Databasebeta

← #12010193

Subgroups — #12010193

Order 1280.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2611C211yesmaximal, characteristic
2#10602387C211yesmaximal, characteristic
3#10602364C211yesmaximal, characteristic
4#10598142C212yesmaximal
5#10598140C212yesmaximal
6C2×C5×#125C411yescharacteristic, complemented
7C2×C5×#117C2211yescharacteristic
8C5×#400C411yescharacteristic, complemented
9C5×#399C2211yescharacteristic
10C5×#398C411yescharacteristic, complemented
11C5×#397C2211yescharacteristic
12C5×#389—21no—
13C5×#388—21no—
14C5×#387—21no—
15C5×#385—21no—
16C5×#376C411yescharacteristic, complemented
17#65500—21no—
18#65214—21no—
19#64748—21no—
20#64746—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11846—51nomaximal, Sylow
24C23×C4×C5D411yescharacteristic
25C5×#125—41no—
26C5×#125—41no—
27C5×#117—41no—
28C5×#117—41no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C4⋊C4C2×C411yescharacteristic
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C5×C22⋊C4C2311yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5C2×C411yescharacteristic
43C2×C42×C5D411yescharacteristic
44C2×C22⋊Dic5—22no—
45C5×#95—21no—
46C5×#95—21no—
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95C2×C411yescharacteristic
51C5×#95C2×C411yescharacteristic
52#3963—41no—
53#3963—41no—
54C22⋊#168—22no—
55C4⋊#168—41no—
56C4×#168—41no—
57#2611D511yescharacteristic
58#1820—51no—
59#1458—51no—
60#1352—52no—
61#1344—52no—
62C24×C5C2×D411yescharacteristic
63C22×C4×C5—21no—
64C22×C4×C5—22no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—41no—
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22×C411yescharacteristic
76C22×C4×C5C2×D411yescharacteristic
77C22×Dic5—42no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—42no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—41no—
87C5×C22⋊C4—41no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—42no—
90C5×C22⋊C4—42no—
91C42×C5—21no—
92C42×C5—21no—
93C42×C5—21no—
94C42×C5—21no—
95C42×C5—42no—
96C22⋊Dic5—42no—
97C22⋊Dic5—42no—
98C2×#168—42no—
99C2×#125Dic511yescharacteristic, complemented
100C2×#117C2×D511yescharacteristic
101#431—101no—
102#400Dic511yescharacteristic, complemented
103#399C2×D511yescharacteristic
104#398Dic511yescharacteristic, complemented
105#397C2×D511yescharacteristic
106#389—21no—
107#388—21no—
108#387—21no—
109#385—21no—
110#381—101no—
111#376Dic511yescharacteristic, complemented
112#342—52no—
113#342—101no—
114#324—101no—
115#323—52no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—41no—
119C23×C5—41no—
120C23×C5C2×C22⋊C411yescharacteristic
121C23×C5#13512yes—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—22no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—41no—
137C2×C4×C5—41no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5#9911yescharacteristic
150C2×C4×C5#9911yescharacteristic
151C2×Dic5—82no—
152C2×Dic5—82no—
153C2×Dic5—82no—
154#168—82no—
155C23×C4C22⋊D511yescharacteristic
156#125—41no—
157#125—41no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×Dic511yescharacteristic
165C2×C4⋊C4C2×Dic511yescharacteristic
166C2×C4⋊C4C2×Dic511yescharacteristic
167C2×C4⋊C4C22⋊D511yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—102no—
171C2×C22⋊C4C22⋊D511yescharacteristic
172C2×C22⋊C4C22×D511yescharacteristic
173C2×C42—21no—
174C2×C42C2×Dic511yescharacteristic
175C2×C42C22⋊D511yescharacteristic
176C4⋊C8—201no—
177C22⋊C8—102no—
178C4×C8—201no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—201no—
185#95—201no—
186#95C2×Dic511yescharacteristic
187#95C2×Dic511yescharacteristic
188C22×C5—21no—
189C22×C5—21no—
190C22×C5—22no—
191C22×C5—42no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—81no—
195C22×C5C2×#9911yescharacteristic
196C22×C5#41611yescharacteristic
197C22×C5#42111yescharacteristic
198C4×C5—21no—
199C4×C5—21no—
200C4×C5—42no—
201C4×C5—41no—
202C4×C5—41no—
203C4×C5—41no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5—81no—
207C4×C5—81no—
208C4×C5—81no—
209Dic5—162no—
210C24C2×C22⋊D511yescharacteristic
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—202no—
220C22×C4C22⋊Dic511yescharacteristic
221C22×C4C22⋊Dic511yescharacteristic
222C22×C4C22⋊Dic511yescharacteristic
223C22×C4C22⋊Dic511yescharacteristic
224C22×C4C22×Dic511yescharacteristic
225C22×C4C2×C22⋊D511yescharacteristic
226C2×C8—202no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—202no—
241C22⋊C4—202no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—42no—
247C2×C5—21no—
248C2×C5—21no—
249C2×C5—42no—
250C2×C5#151611yescharacteristic
251C2×C5#212311yescharacteristic
252C2×C5#212611yescharacteristic
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23#409612yes—
258C23C2×C22⋊Dic511yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—402no—
287C2×C4—402no—
288C2×C4—402no—
289C2×C4#396611yescharacteristic
290C2×C4#396611yescharacteristic
291C8—402no—
292C5#1184611yescharacteristic, complemented, Sylow
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#396611yescharacteristic
301C22#6552211yescharacteristic
302C22#6552511yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—802no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#1060250311yescharacteristic
319C2#1060261211yescharacteristic
320C2#1060266311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2611C2
socleC22×C5C2×#99
prime radicalC24×C5C2×D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2611D5
hypocenterC5#11846
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
6. C2×C5×#125C4semidirect
8. C5×#400C4semidirect
10. C5×#398C4semidirect
16. C5×#376C4semidirect
99. C2×#125Dic5semidirect
102. #400Dic5semidirect
104. #398Dic5semidirect
111. #376Dic5semidirect
292. C5#11846semidirect

Maximal subgroups

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

subgroupindex
C5×#26112
#106023872
#106023642
#105981422
#105981402
#118465