Finite Group Databasebeta

← #12009826

Subgroups — #12009826

Order 1280.

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

normal subgroups only

This group has 420 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#10602380C211yesmaximal, characteristic
3#10602365C211yesmaximal, characteristic
4#10598143C212yesmaximal, complemented
5#10598141C212yesmaximal, complemented
6C2×C5×#125C411yescharacteristic
7C2×C5×#117C2211yescharacteristic
8C22×#681—21no—
9C5×#400C2211yescharacteristic
10C5×#399C411yescharacteristic
11C5×#398C2211yescharacteristic
12C5×#397C411yescharacteristic
13C5×#389—21no—
14C5×#388—21no—
15C5×#387—21no—
16C5×#385—21no—
17C5×#376C411yescharacteristic
18#65215—21no—
19#64750—21no—
20#64750C2212yes—
21#64748C2212yes—
22#64745—21no—
23#11767—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⋊C4D411yescharacteristic
36C2×C5×C4⋊C4C2311yescharacteristic
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4C2×C411yescharacteristic
40C2×C5×C22⋊C4D411yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5C2×C411yescharacteristic
43C2×C42×C5D411yescharacteristic
44C2×#681—24no—
45C2×#681—22no—
46C2×#681—41no—
47C2×#681—42no—
48C22×#168—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#1742—51no—
61#1464—51no—
62#1353—52no—
63#1346—52no—
64C24×C5C22⋊C411yescharacteristic
65C22×C4×C5—22no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
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×C5C2×D411yescharacteristic
79C22×C4×C5C2×D411yescharacteristic
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—42no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—41no—
90C5×C22⋊C4—41no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—42no—
94C42×C5—21no—
95C42×C5—21no—
96C42×C5—21no—
97C42×C5—21no—
98C42×C5—42no—
99#681—24no—
100#681—44no—
101#681—42no—
102C2×#168—44no—
103C2×#168—42no—
104C2×#168—42no—
105C22×M16—101no—
106C2×#125Dic511yescharacteristic
107C2×#117C2×D511yescharacteristic
108#432—101no—
109#400C2×D511yescharacteristic
110#399Dic511yescharacteristic
111#398C2×D511yescharacteristic
112#397Dic511yescharacteristic
113#389—21no—
114#388—21no—
115#387—21no—
116#385—21no—
117#376Dic511yescharacteristic
118#343—52no—
119#343—101no—
120#324—52no—
121#323—101no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5#10412yes—
127C23×C5C2×C22⋊C411yescharacteristic
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—22no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
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—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5#9911yescharacteristic
157C2×C4×C5#9911yescharacteristic
158C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C5#13711yescharacteristic
160#168—44no—
161#168—82no—
162C23×C4C22⋊D511yescharacteristic
163C2×M16—104no—
164C2×M16—102no—
165C2×M16—201no—
166C2×M16—202no—
167C22×C8—201no—
168#125—41no—
169#125—41no—
170#117—41no—
171#117—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×Dic511yescharacteristic
177C2×C4⋊C4C2×Dic511yescharacteristic
178C2×C4⋊C4C22⋊D511yescharacteristic
179C2×C4⋊C4C22×D511yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4C2×Dic511yescharacteristic
183C2×C22⋊C4C22⋊D511yescharacteristic
184C2×C42—21no—
185C2×C42C2×Dic511yescharacteristic
186C2×C42C22⋊D511yescharacteristic
187C4⋊C8—201no—
188C22⋊C8—102no—
189#97—201no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95C2×Dic511yescharacteristic
196#95C2×Dic511yescharacteristic
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—22no—
200C22×C5—42no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—81no—
204C22×C5C2×#9911yescharacteristic
205C22×C5#41711yescharacteristic
206C22×C5C2×#10411yescharacteristic
207C4×C5—21no—
208C4×C5—21no—
209C4×C5—21no—
210C4×C5—21no—
211C4×C5—42no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—41no—
215C4×C5—41no—
216C4×C5—81no—
217C4×C5—81no—
218C4×C5—81no—
219C24C22⋊Dic511yescharacteristic
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4C22⋊Dic511yescharacteristic
230C22×C4C22⋊Dic511yescharacteristic
231C22×C4C22⋊Dic511yescharacteristic
232C22×C4C22×Dic511yescharacteristic
233C22×C4C2×C22⋊D511yescharacteristic
234C22×C4C2×C22⋊D511yescharacteristic
235M16—104no—
236M16—204no—
237M16—202no—
238C2×C8—204no—
239C2×C8—202no—
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—41no—
250C4⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—42no—
260C2×C5—21no—
261C2×C5—21no—
262C2×C5—42no—
263C2×C5#150411yescharacteristic
264C2×C5#212311yescharacteristic
265C2×C5#213211yescharacteristic
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23#396912yes—
271C23C2×C22⋊Dic511yescharacteristic
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
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#396611yescharacteristic
301C2×C4#396611yescharacteristic
302C2×C4#409511yescharacteristic
303C2×C4#409711yescharacteristic
304C8—204no—
305C8—402no—
306C5#1176711yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#396611yescharacteristic
315C22#6552111yescharacteristic
316C22C2×#396911yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#1060250111yescharacteristic
333C2#1060261311yescharacteristic
334C2#1060266311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2611C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2611D5
hypocenterC5#11767
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
4. #10598143C2semidirect
5. #10598141C2semidirect
306. C5#11767semidirect

Maximal subgroups

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

subgroupindex
C5×#26112
#106023802
#106023652
#105981432
#105981412
#117675