Finite Group Databasebeta

← #12008977

Subgroups — #12008977

Order 1280.

426 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2487C211yesmaximal, characteristic
2#10602396C211yesmaximal, characteristic
3#10602393C211yesmaximal, characteristic
4#10598127C214yesmaximal, complemented
5C2×C4×C5×D4C2211yescharacteristic
6C22×C5×C4⋊C4C411yescharacteristic
7C5×#400—21no—
8C5×#399—21no—
9C5×#389—21no—
10C5×#386C412yes—
11C5×#385C2212yes—
12C5×#380—21no—
13C5×#380C411yescharacteristic
14C2×C22⋊#168—21no—
15#65215—21no—
16#64748—21no—
17#64745—21no—
18#64745C2212yes—
19#64745C2212yes—
20#11576—51nomaximal, Sylow
21C22×C5×D4C2×C411yescharacteristic
22C23×C4×C5C2×C411yescharacteristic
23C23×C4×C5C2×C412yes—
24C23×C4×C5D411yescharacteristic
25C4×C5×D4—41no—
26C4×C5×D4—41no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4—41no—
32C2×C5×C4⋊C4—41no—
33C2×C5×C4⋊C4D411yescharacteristic
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4C2×C412yes—
36C2×C5×C22⋊C4D411yescharacteristic
37C2×C5×C22⋊C4C2311yescharacteristic
38C2×C42×C5D411yescharacteristic
39C22×#168—41no—
40C5×#95—21no—
41C5×#95—22no—
42C5×#95—22no—
43C22⋊#168—22no—
44C22⋊#168—22no—
45C22⋊#168—41no—
46C4⋊#168—41no—
47#3935—41no—
48#2487D511yescharacteristic
49#1466—51no—
50#1459—51no—
51#1326—54no—
52C24×C5C22⋊C411yescharacteristic
53C24×C5C22×C411yescharacteristic
54C2×C5×D4—22no—
55C2×C5×D4—22no—
56C2×C5×D4—41no—
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—42no—
64C22×C4×C5—42no—
65C22×C4×C5—42no—
66C22×C4×C5—82no—
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C2×D411yescharacteristic
71C22×C4×C5C2×D411yescharacteristic
72C5×C4⋊C4—41no—
73C5×C4⋊C4—42no—
74C5×C4⋊C4—42no—
75C5×C4⋊C4—41no—
76C5×C4⋊C4—41no—
77C5×C4⋊C4—82no—
78C5×C22⋊C4—24no—
79C5×C22⋊C4—41no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—82no—
82C42×C5—21no—
83C42×C5—21no—
84C2×#168—42no—
85C2×#168—42no—
86C2×#168—82no—
87C2×C4×D4C2×D511yescharacteristic
88C22×C4⋊C4Dic511yescharacteristic
89#432—101no—
90C2×C22⋊C8—101no—
91#400—21no—
92#399—21no—
93#389—21no—
94#386Dic512yes—
95#385C2×D512yes—
96#380—21no—
97#380Dic511yescharacteristic
98#324—101no—
99#323—52no—
100#323—52no—
101#323—101no—
102C23×C5—21no—
103C23×C5—21no—
104C23×C5—21no—
105C23×C5—21no—
106C23×C5—41no—
107C23×C5—41no—
108C23×C5—41no—
109C23×C5—41no—
110C23×C5#10112yes—
111C23×C5C2×C22⋊C411yescharacteristic
112C23×C5#13112yes—
113C5×D4—44no—
114C2×C4×C5—22no—
115C2×C4×C5—21no—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—42no—
122C2×C4×C5—42no—
123C2×C4×C5—42no—
124C2×C4×C5—41no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—81no—
134C2×C4×C5—84no—
135C2×C4×C5#9911yescharacteristic
136C2×C4×C5#9911yescharacteristic
137#168—82no—
138#168—82no—
139C22×D4C2×Dic511yescharacteristic
140C23×C4C2×Dic511yescharacteristic
141C23×C4C2×Dic512yes—
142C23×C4C22⋊D511yescharacteristic
143C22×C8—201no—
144C4×D4—41no—
145C4×D4—41no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4C22⋊D511yescharacteristic
153C2×C22⋊C4—22no—
154C2×C22⋊C4C2×Dic512yes—
155C2×C22⋊C4C22⋊D511yescharacteristic
156C2×C22⋊C4C22×D511yescharacteristic
157C2×C42C22⋊D511yescharacteristic
158C4⋊C8—201no—
159C22⋊C8—102no—
160C22⋊C8—102no—
161C22⋊C8—201no—
162#97—201no—
163#95—21no—
164#95—22no—
165#95—22no—
166C22×C5—21no—
167C22×C5—21no—
168C22×C5—22no—
169C22×C5—22no—
170C22×C5—42no—
171C22×C5—42no—
172C22×C5—42no—
173C22×C5—41no—
174C22×C5—41no—
175C22×C5—41no—
176C22×C5—41no—
177C22×C5—81no—
178C22×C5#40811yescharacteristic
179C22×C5C2×#9911yescharacteristic
180C22×C5C2×#10111yescharacteristic
181C4×C5—21no—
182C4×C5—41no—
183C4×C5—41no—
184C4×C5—41no—
185C4×C5—41no—
186C4×C5—82no—
187C4×C5—82no—
188C4×C5#35212yes—
189C24C22⋊Dic511yescharacteristic
190C24C22×Dic511yescharacteristic
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—41no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4C22⋊Dic511yescharacteristic
205C22×C4C22⋊Dic511yescharacteristic
206C22×C4C22⋊Dic511yescharacteristic
207C22×C4C2×C22⋊D511yescharacteristic
208C22×C4C2×C22⋊D511yescharacteristic
209C2×C8—202no—
210C2×C8—202no—
211C2×C8—402no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—82no—
218C22⋊C4—24no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—82no—
222C42—21no—
223C42—21no—
224C2×C5—21no—
225C2×C5—21no—
226C2×C5—42no—
227C2×C5—42no—
228C2×C5#147211yescharacteristic
229C2×C5C2×#35211yescharacteristic
230C2×C5#212211yescharacteristic
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23#396812yes—
240C23#409412yes—
241C23C2×C22⋊Dic511yescharacteristic
242D4—44no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—84no—
264C2×C4#396611yescharacteristic
265C2×C4#396611yescharacteristic
266C8—402no—
267C8—402no—
268C5#1157611yescharacteristic, complemented, Sylow
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22C2×#396611yescharacteristic
282C22C2×#396811yescharacteristic
283C22#6552311yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—82no—
290C4—82no—
291C4#6475912yes—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—42no—
296C2#1060248411yescharacteristic
297C2C2×#6475911yescharacteristic
298C2#1060265211yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC24×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2487C2
socleC22×C5C2×#99
prime radicalC22×C5×D4C2×C4
cosocleC2×C5×C22⋊C4C23
hypercenter#2487D5
hypocenterC5#11576
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #10598127C2semidirect
268. C5#11576semidirect

Maximal subgroups

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

subgroupindex
C5×#24872
#106023962
#106023932
#105981272
#115765