Finite Group Databasebeta

← #12010173

Subgroups — #12010173

Order 1280.

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

normal subgroups only

This group has 390 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×#2599C211yesmaximal, characteristic
2#10602387C211yesmaximal, characteristic
3#10602364C211yesmaximal, characteristic
4#10598140C212yesmaximal
5#10598137C212yesmaximal
6C2×C4×C5×Q8C411yescharacteristic, complemented
7C2×C5×#117C2211yescharacteristic
8C5×#398C411yescharacteristic, complemented
9C5×#397C2211yescharacteristic
10C5×#395C411yescharacteristic, complemented
11C5×#393C2211yescharacteristic
12C5×#388—21no—
13C5×#387—21no—
14C5×#387—21no—
15C5×#385—21no—
16C5×#383C411yescharacteristic, complemented
17#65500—21no—
18#65214—21no—
19#64748—21no—
20#64746—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11842—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C4×C5×Q8—41no—
27C4×C5×Q8—41no—
28C5×#117—41no—
29C5×#117—41no—
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⋊C4D411yescharacteristic
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4D411yescharacteristic
39C2×C5×C22⋊C4C2311yescharacteristic
40C2×C42×C5—21no—
41C2×C42×C5C2×C411yescharacteristic
42C2×C42×C5D411yescharacteristic
43C2×C22⋊Dic5—22no—
44C5×#95—21no—
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#2599D511yescharacteristic
58#1820—51no—
59#1458—51no—
60#1351—52no—
61#1344—52no—
62C24×C5C2×D411yescharacteristic
63C2×C5×Q8—42no—
64C2×C5×Q8—41no—
65C22×C4×C5—21no—
66C22×C4×C5—22no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—41no—
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22×C411yescharacteristic
78C22×C4×C5C2×D411yescharacteristic
79C22×Dic5—42no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
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×C4×Q8Dic511yescharacteristic, complemented
100C2×#117C2×D511yescharacteristic
101#431—101no—
102#398Dic511yescharacteristic, complemented
103#397C2×D511yescharacteristic
104#395Dic511yescharacteristic, complemented
105#393C2×D511yescharacteristic
106#388—21no—
107#387—21no—
108#387—21no—
109#385—21no—
110#383Dic511yescharacteristic, complemented
111#381—101no—
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—
122C5×Q8—42no—
123C5×Q8—42no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—22no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
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—41no—
150C2×C4×C5—41no—
151C2×C4×C5#9911yescharacteristic
152C2×C4×C5#9911yescharacteristic
153C2×Dic5—82no—
154C2×Dic5—82no—
155C2×Dic5—82no—
156#168—82no—
157C22×Q8C2×Dic511yescharacteristic
158C23×C4C22⋊D511yescharacteristic
159C4×Q8—41no—
160C4×Q8—41no—
161#117—41no—
162#117—41no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C2×Dic511yescharacteristic
167C2×C4⋊C4C2×Dic511yescharacteristic
168C2×C4⋊C4C22⋊D511yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—102no—
172C2×C22⋊C4C22⋊D511yescharacteristic
173C2×C22⋊C4C22×D511yescharacteristic
174C2×C42—21no—
175C2×C42C2×Dic511yescharacteristic
176C2×C42C22⋊D511yescharacteristic
177C4⋊C8—201no—
178C22⋊C8—102no—
179C4×C8—201no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—201no—
187#95—201no—
188#95C2×Dic511yescharacteristic
189#95C2×Dic511yescharacteristic
190C22×C5—21no—
191C22×C5—21no—
192C22×C5—22no—
193C22×C5—42no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—81no—
197C22×C5C2×#9911yescharacteristic
198C22×C5#41611yescharacteristic
199C22×C5#42111yescharacteristic
200C4×C5—21no—
201C4×C5—21no—
202C4×C5—41no—
203C4×C5—42no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5—81no—
209C4×C5—81no—
210C4×C5—81no—
211Dic5—162no—
212C24C2×C22⋊D511yescharacteristic
213C2×Q8—42no—
214C2×Q8—41no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—202no—
224C22×C4C22⋊Dic511yescharacteristic
225C22×C4C22⋊Dic511yescharacteristic
226C22×C4C22⋊Dic511yescharacteristic
227C22×C4C22⋊Dic511yescharacteristic
228C22×C4C22×Dic511yescharacteristic
229C22×C4C2×C22⋊D511yescharacteristic
230C2×C8—202no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—202no—
243C22⋊C4—202no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—42no—
249C2×C5—21no—
250C2×C5—21no—
251C2×C5—42no—
252C2×C5#151611yescharacteristic
253C2×C5#212011yescharacteristic
254C2×C5#212911yescharacteristic
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23#409612yes—
260C23C2×C22⋊Dic511yescharacteristic
261Q8—42no—
262Q8—42no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
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—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—402no—
291C2×C4—402no—
292C2×C4—402no—
293C2×C4#396611yescharacteristic
294C2×C4#396611yescharacteristic
295C8—402no—
296C5#1184211yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#396611yescharacteristic
305C22#6552211yescharacteristic
306C22#6552511yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—802no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#1060250311yescharacteristic
323C2#1060250911yescharacteristic
324C2#1060266411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2599C2
socleC22×C5C2×#99
prime radicalC24×C5C2×D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2599D5
hypocenterC5#11842
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
6. C2×C4×C5×Q8C4semidirect
8. C5×#398C4semidirect
10. C5×#395C4semidirect
16. C5×#383C4semidirect
99. C2×C4×Q8Dic5semidirect
102. #398Dic5semidirect
104. #395Dic5semidirect
110. #383Dic5semidirect
296. C5#11842semidirect

Maximal subgroups

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

subgroupindex
C5×#25992
#106023872
#106023642
#105981402
#105981372
#118425