Finite Group Databasebeta

← #12009791

Subgroups — #12009791

Order 1280.

436 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 91 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#10602385C211yesmaximal, characteristic
3#10602365C211yesmaximal, characteristic
4#10598142C212yesmaximal
5#10598140C212yesmaximal
6C2×C5×#125C411yescharacteristic, complemented
7C2×C5×#117C2211yescharacteristic
8C2×#4072—21no—
9C5×#400C411yescharacteristic, complemented
10C5×#399C2211yescharacteristic
11C5×#398C411yescharacteristic, complemented
12C5×#397C2211yescharacteristic
13C5×#389—21no—
14C5×#388—21no—
15C5×#387—21no—
16C5×#385—21no—
17C5×#376C411yescharacteristic, complemented
18#65215—21no—
19#64748—21no—
20#64746—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11760—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—
45#4072—24no—
46#4072—42no—
47C2×C4⋊Dic5—41no—
48C2×C4×Dic5—41no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95—21no—
52C5×#95—21no—
53C5×#95—21no—
54C5×#95C2×C411yescharacteristic
55C5×#95C2×C411yescharacteristic
56C22⋊#168—22no—
57C4⋊#168—41no—
58#3935—41no—
59#2611D511yescharacteristic
60#1740—51no—
61#1464—51no—
62#1352—52no—
63#1344—52no—
64C24×C5C2×D411yescharacteristic
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22×C411yescharacteristic
79C22×C4×C5C2×D411yescharacteristic
80C22×Dic5—42no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—42no—
86C5×C4⋊C4—42no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—42no—
90C5×C22⋊C4—41no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—41no—
94C5×C22⋊C4—41no—
95C42×C5—21no—
96C42×C5—21no—
97C42×C5—21no—
98C42×C5—21no—
99C42×C5—42no—
100C22⋊Dic5—24no—
101C22⋊Dic5—42no—
102C4⋊Dic5—44no—
103C4×Dic5—44no—
104C2×#168—42no—
105C2×#125Dic511yescharacteristic, complemented
106C2×#117—101no—
107C2×#117C2×D511yescharacteristic
108#432—101no—
109#400Dic511yescharacteristic, complemented
110#399C2×D511yescharacteristic
111#398Dic511yescharacteristic, complemented
112#397C2×D511yescharacteristic
113#389—21no—
114#388—21no—
115#387—21no—
116#385—21no—
117#376Dic511yescharacteristic, complemented
118#342—52no—
119#342—101no—
120#324—101no—
121#323—52no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5C2×C22⋊C411yescharacteristic
127C23×C5AGL(1,ℤ/8)11yescharacteristic
128C23×C5#13711yescharacteristic
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—22no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
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—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5#9911yescharacteristic
158C2×C4×C5#9911yescharacteristic
159C2×C4×C5#10412yes—
160C2×Dic5—44no—
161C2×Dic5—82no—
162C2×Dic5—82no—
163#168—82no—
164C23×C4C22⋊D511yescharacteristic
165#125—41no—
166#125—41no—
167#117—41no—
168#117—41no—
169#117—104no—
170#117—202no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—201no—
176C2×C4⋊C4C2×Dic511yescharacteristic
177C2×C4⋊C4C2×Dic511yescharacteristic
178C2×C4⋊C4C2×Dic511yescharacteristic
179C2×C4⋊C4C22⋊D511yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—102no—
183C2×C22⋊C4C22⋊D511yescharacteristic
184C2×C22⋊C4C22×D511yescharacteristic
185C2×C42—21no—
186C2×C42—201no—
187C2×C42C2×Dic511yescharacteristic
188C2×C42C22⋊D511yescharacteristic
189C4⋊C8—201no—
190C22⋊C8—102no—
191#97—201no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95C2×Dic511yescharacteristic
198#95C2×Dic511yescharacteristic
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—81no—
208C22×C5C2×#9911yescharacteristic
209C22×C5#41711yescharacteristic
210C22×C5C2×#10411yescharacteristic
211C4×C5—21no—
212C4×C5—21no—
213C4×C5—22no—
214C4×C5—42no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5—41no—
218C4×C5—41no—
219C4×C5—81no—
220C4×C5—81no—
221C4×C5—81no—
222Dic5—84no—
223C24C2×C22⋊D511yescharacteristic
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—202no—
234C22×C4C22⋊Dic511yescharacteristic
235C22×C4C22⋊Dic511yescharacteristic
236C22×C4C22⋊Dic511yescharacteristic
237C22×C4C22⋊Dic511yescharacteristic
238C22×C4C22×Dic511yescharacteristic
239C22×C4C2×C22⋊D511yescharacteristic
240C2×C8—202no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—204no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—42no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—104no—
257C22⋊C4—202no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—42no—
263C42—204no—
264C2×C5—21no—
265C2×C5—21no—
266C2×C5—41no—
267C2×C5—41no—
268C2×C5#150411yescharacteristic
269C2×C5#212311yescharacteristic
270C2×C5#212611yescharacteristic
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#409511yescharacteristic
276C23#409711yescharacteristic
277C23C2×C22⋊Dic511yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—204no—
307C2×C4—402no—
308C2×C4—402no—
309C2×C4#396611yescharacteristic
310C2×C4#396611yescharacteristic
311C2×C4#396912yes—
312C8—402no—
313C5#1176011yescharacteristic, complemented, Sylow
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#396611yescharacteristic
324C22#6552111yescharacteristic
325C22C2×#396911yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—404no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#1060250111yescharacteristic
343C2#1060261211yescharacteristic
344C2#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#11760
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
9. C5×#400C4semidirect
11. C5×#398C4semidirect
17. C5×#376C4semidirect
105. C2×#125Dic5semidirect
109. #400Dic5semidirect
111. #398Dic5semidirect
117. #376Dic5semidirect
313. C5#11760semidirect

Maximal subgroups

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

subgroupindex
C5×#26112
#106023852
#106023652
#105981422
#105981402
#117605