Finite Group Databasebeta

← #11997050

Subgroups — #11997050

Order 1280.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2504C211yesmaximal, characteristic
2#10613886C211yesmaximal, characteristic, complemented
3#10602467C212yesmaximal
4#10602355C211yesmaximal, characteristic, complemented
5#10598107C212yesmaximal, complemented
6C2×C5×#121C2211yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#397—22no—
9C5×#390C412yescomplemented
10C5×#386—21no—
11C5×#386—21no—
12C5×#382C412yescomplemented
13C4×C5×C22⋊C4C2211yescharacteristic
14#65501—21no—
15#65500C2212yes—
16C4×C22⋊Dic5—21no—
17#65496C2212yes—
18#65270—24no—
19#64757—24no—
20#64750—22no—
21#9465—51nomaximal, Sylow
22C22×C5×D4—21no—
23C22×C5×D4C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5C2311yescharacteristic
27C5×#121—22no—
28C5×#121—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C4⋊C4D412yes—
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—21no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—22no—
38C2×C42×C5C2×C412yes—
39C2×C22⋊Dic5—22no—
40C2×C4⋊Dic5—22no—
41C2×C4×Dic5—22no—
42C2×#681—22no—
43C5×#95—22no—
44C5×#95—22no—
45C5×#95—21no—
46#3963—22no—
47#3963—22no—
48#3963—22no—
49#3963D414yes—
50C22⋊#168—22no—
51C22⋊#168—22no—
52#2504D511yescharacteristic
53#2313—51no—
54#1821—51no—
55#1510—52no—
56#1284—52no—
57C24×C5—21no—
58C24×C5C2×D411yescharacteristic
59C2×C5×D4—22no—
60C2×C5×D4—42no—
61C2×C5×D4—44no—
62C2×C5×D4—41no—
63C2×C5×D4—41no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—22no—
69C22×C4×C5—21no—
70C22×C4×C5—42no—
71C22×C4×C5—81no—
72C22×C4×C5—81no—
73C22×C4×C5C22⋊C412yes—
74C22×C4×C5C22⋊C412yes—
75C22×C4×C5C22×C411yescharacteristic
76C22×C4×C5C2×D412yes—
77C22×C4×C5C2×D411yescharacteristic
78C22×C4×C5#4112yes—
79C22×Dic5—24no—
80C5×C4⋊C4—24no—
81C5×C4⋊C4—24no—
82C5×C4⋊C4—22no—
83C5×C22⋊C4—42no—
84C5×C22⋊C4—42no—
85C5×C22⋊C4—82no—
86C5×C22⋊C4—82no—
87C42×C5—42no—
88C22⋊Dic5—44no—
89C4⋊Dic5—82no—
90C4×Dic5—82no—
91#681—44no—
92C2×#168—42no—
93C2×#121C2×D511yescharacteristic
94C22×C4⋊C4C2×D511yescharacteristic
95#407—52no—
96#397—22no—
97#390Dic512yescomplemented
98#387—104no—
99#386—21no—
100#386—21no—
101#382Dic512yescomplemented
102#381—52no—
103#380—101no—
104C4×C22⋊C4—101no—
105C4×C22⋊C4C2×D511yescharacteristic
106#343—102no—
107#327—104no—
108C23×C5—22no—
109C23×C5—22no—
110C23×C5—41no—
111C23×C5—42no—
112C23×C5—42no—
113C23×C5—41no—
114C23×C5—42no—
115C23×C5C2×C22⋊C411yescharacteristic
116C23×C5#12011yescharacteristic
117C23×C5#12311yescharacteristic
118C5×D4—82no—
119C5×D4—82no—
120C5×D4—82no—
121C2×C4×C5—22no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—22no—
125C2×C4×C5—42no—
126C2×C4×C5—42no—
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—42no—
134C2×C4×C5—41no—
135C2×C4×C5—42no—
136C2×C4×C5—81no—
137C2×C4×C5—81no—
138C2×C4×C5C4×D412yes—
139C2×C4×C5#12112yes—
140C2×Dic5—44no—
141C2×Dic5—44no—
142C2×Dic5—44no—
143#168—82no—
144C22×D4—21no—
145C22×D4C2×Dic511yescharacteristic
146C23×C4C22⋊D511yescharacteristic
147C23×C4C22⋊D511yescharacteristic
148C23×C4C22×D511yescharacteristic
149C2×M16—102no—
150#121—22no—
151#121—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—102no—
155C2×C4⋊C4C2×Dic511yescharacteristic
156C2×C4⋊C4C2×Dic512yes—
157C2×C4⋊C4C22⋊D512yes—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—102no—
163C2×C42—102no—
164C2×C42C2×Dic512yes—
165C22⋊C8—102no—
166C22⋊C8—102no—
167#95—22no—
168#95—22no—
169#95—21no—
170#95—54no—
171#95—102no—
172#95—102no—
173#95—102no—
174C22×C5—21no—
175C22×C5—21no—
176C22×C5—21no—
177C22×C5—21no—
178C22×C5—21no—
179C22×C5—21no—
180C22×C5—21no—
181C22×C5—44no—
182C22×C5—84no—
183C22×C5—81no—
184C22×C5—81no—
185C22×C5—81no—
186C22×C5#38611yescharacteristic
187C22×C5#41012yes—
188C22×C5#45412yes—
189C4×C5—22no—
190C4×C5—22no—
191C4×C5—42no—
192C4×C5—42no—
193C4×C5—82no—
194C4×C5—81no—
195Dic5—84no—
196C24—21no—
197C24C2×C22⋊D511yescharacteristic
198C2×D4—22no—
199C2×D4—42no—
200C2×D4—44no—
201C2×D4—41no—
202C2×D4—41no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—104no—
213C22×C4C22⋊Dic512yes—
214C22×C4C22⋊Dic512yes—
215C22×C4D4×D512yes—
216C22×C4#71112yes—
217C22×C4C22×Dic511yescharacteristic
218C22×C4C2×C22⋊D511yescharacteristic
219M16—204no—
220C2×C8—202no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—22no—
224C4⋊C4—402no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—204no—
230C42—42no—
231C42—402no—
232C2×C5—22no—
233C2×C5—22no—
234C2×C5—41no—
235C2×C5—161no—
236C2×C5#188811yescharacteristic
237C2×C5#189911yescharacteristic
238C2×C5#191111yescharacteristic
239C23—22no—
240C23—22no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—42no—
246C23#408111yescharacteristic
247C23#408311yescharacteristic
248C23C2×C22⋊Dic511yescharacteristic
249D4—82no—
250D4—82no—
251D4—82no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—204no—
270C2×C4—204no—
271C2×C4—204no—
272C2×C4D4×Dic512yes—
273C2×C4#408512yes—
274C8—402no—
275C5#946511yescharacteristic, complemented, Sylow
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#6535112yes—
289C22#6550911yescharacteristic
290C22#6552912yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—81no—
297C4—404no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#1060249811yescharacteristic
303C2#1060251711yescharacteristic
304C2#1060264611yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2504C2
socleC22×C5#386
prime radicalC2×C5×#121C22
cosocleC23×C4×C5C23
hypercenter#2504D5
hypocenterC5#9465
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
2. #10613886C2semidirect
4. #10602355C2semidirect
5. #10598107C2semidirect
9. C5×#390C4semidirect
12. C5×#382C4semidirect
97. #390Dic5semidirect
101. #382Dic5semidirect
275. C5#9465semidirect

Maximal subgroups

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

subgroupindex
C5×#25042
#106138862
#106024672
#106023552
#105981072
#94655