Finite Group Databasebeta

← #12009589

Subgroups — #12009589

Order 1280.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2628C211yesmaximal, characteristic
2#10602385C211yesmaximal, characteristic
3#10602364C211yesmaximal, characteristic
4#10598137C214yesmaximal
5C2×C5×C4⋊Q8C411yescharacteristic, complemented
6C2×C5×#117C2211yescharacteristic
7C2×#4072—21no—
8C5×#400—21no—
9C5×#399—21no—
10C5×#395C412yescomplemented
11C5×#393C2212yes—
12C5×#389—21no—
13C5×#382C411yescharacteristic, complemented
14C5×#380—21no—
15#65214—21no—
16#64748—21no—
17#64746—21no—
18#64746C2212yes—
19#64745C2212yes—
20#11717—51nomaximal, Sylow
21C22×C5×Q8C2×C412yes—
22C23×C4×C5D411yescharacteristic
23C5×C4⋊Q8—41no—
24C5×C4⋊Q8—41no—
25C5×#117—41no—
26C5×#117—41no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C5×C22⋊C4—22no—
34C2×C5×C22⋊C4D411yescharacteristic
35C2×C5×C22⋊C4C2311yescharacteristic
36C2×C42×C5C2×C411yescharacteristic
37C2×C42×C5D411yescharacteristic
38C2×C22⋊Dic5—22no—
39#4072—24no—
40#4072—42no—
41C2×C4⋊Dic5—41no—
42C2×C4×Dic5—41no—
43C5×#95—22no—
44C5×#95—21no—
45C5×#95—22no—
46C5×#95C2×C412yes—
47C22⋊#168—22no—
48C4⋊#168—41no—
49C4×#168—41no—
50#2628D511yescharacteristic
51#1740—51no—
52#1458—51no—
53#1351—54no—
54C24×C5C2×D411yescharacteristic
55C2×C5×Q8—42no—
56C2×C5×Q8—82no—
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5C22⋊C411yescharacteristic
65C22×C4×C5C22⋊C411yescharacteristic
66C22×C4×C5SD1612yes—
67C22×C4×C5Q1612yes—
68C22×C4×C5C22×C411yescharacteristic
69C22×C4×C5C2×D411yescharacteristic
70C22×Dic5—42no—
71C5×C4⋊C4—41no—
72C5×C4⋊C4—41no—
73C5×C4⋊C4—42no—
74C5×C4⋊C4—41no—
75C5×C4⋊C4—41no—
76C5×C4⋊C4—42no—
77C5×C4⋊C4—82no—
78C5×C22⋊C4—41no—
79C5×C22⋊C4—41no—
80C5×C22⋊C4—82no—
81C42×C5—21no—
82C42×C5—21no—
83C42×C5—21no—
84C42×C5—21no—
85C22⋊Dic5—24no—
86C22⋊Dic5—42no—
87C4⋊Dic5—44no—
88C4×Dic5—44no—
89C2×#168—42no—
90C2×C4⋊Q8Dic511yescharacteristic, complemented
91C2×#117—101no—
92C2×#117C2×D511yescharacteristic
93#431—101no—
94#400—21no—
95#399—21no—
96#395Dic512yescomplemented
97#393C2×D512yes—
98#389—21no—
99#382Dic511yescharacteristic, complemented
100#380—21no—
101#342—52no—
102#342—101no—
103#324—101no—
104#323—52no—
105C23×C5—21no—
106C23×C5—21no—
107C23×C5—41no—
108C23×C5—41no—
109C23×C5C2×C22⋊C411yescharacteristic
110C23×C5C2×SD1611yescharacteristic
111C23×C5C2×Q1611yescharacteristic
112C5×Q8—84no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—22no—
118C2×C4×C5—22no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—41no—
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#9911yescharacteristic
134C2×C4×C5#9911yescharacteristic
135C2×C4×C5#10312yes—
136C2×C4×C5#10312yes—
137C2×C4×C5#10412yes—
138C2×Dic5—44no—
139C2×Dic5—82no—
140C2×Dic5—82no—
141#168—82no—
142C22×Q8C2×Dic512yes—
143C23×C4C22⋊D511yescharacteristic
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146#117—41no—
147#117—41no—
148#117—104no—
149#117—202no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—201no—
155C2×C4⋊C4C2×Dic511yescharacteristic
156C2×C4⋊C4C22⋊D511yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—102no—
159C2×C22⋊C4C22⋊D511yescharacteristic
160C2×C22⋊C4C22×D511yescharacteristic
161C2×C42—201no—
162C2×C42C2×Dic511yescharacteristic
163C2×C42C22⋊D511yescharacteristic
164C4⋊C8—201no—
165C22⋊C8—102no—
166C4×C8—201no—
167#95—22no—
168#95—21no—
169#95—22no—
170#95C2×Dic512yes—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—41no—
176C22×C5—41no—
177C22×C5—41no—
178C22×C5—41no—
179C22×C5—81no—
180C22×C5C2×#9911yescharacteristic
181C22×C5C2×#10311yescharacteristic
182C22×C5C2×#10411yescharacteristic
183C4×C5—22no—
184C4×C5—41no—
185C4×C5—41no—
186C4×C5—41no—
187C4×C5—41no—
188C4×C5—82no—
189C4×C5—82no—
190C4×C5#32812yes—
191C4×C5#32812yes—
192Dic5—84no—
193C24C2×C22⋊D511yescharacteristic
194C2×Q8—42no—
195C2×Q8—82no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—202no—
202C22×C4#68812yes—
203C22×C4#68912yes—
204C22×C4C22⋊Dic511yescharacteristic
205C22×C4C22⋊Dic511yescharacteristic
206C22×C4C22⋊Dic511yescharacteristic
207C22×C4C22⋊Dic511yescharacteristic
208C22×C4C22×Dic511yescharacteristic
209C22×C4C2×C22⋊D511yescharacteristic
210C2×C8—202no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C4⋊C4—204no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—82no—
222C22⋊C4—104no—
223C22⋊C4—202no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—204no—
229C2×C5—21no—
230C2×C5—21no—
231C2×C5—41no—
232C2×C5—41no—
233C2×C5C2×#32811yescharacteristic
234C2×C5#212211yescharacteristic
235C2×C5#212911yescharacteristic
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×#68811yescharacteristic
241C23C2×#68911yescharacteristic
242C23C2×C22⋊Dic511yescharacteristic
243Q8—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—204no—
265C2×C4—402no—
266C2×C4—402no—
267C2×C4#396611yescharacteristic
268C2×C4#396611yescharacteristic
269C2×C4#396712yes—
270C2×C4#396712yes—
271C2×C4#396912yes—
272C8—402no—
273C5#1171711yescharacteristic, complemented, Sylow
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#396611yescharacteristic
284C22C2×#396711yescharacteristic
285C22C2×#396911yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4—404no—
294C4#6475812yes—
295C4#6475812yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#6475811yescharacteristic
301C2#1060265211yescharacteristic
302C2#1060266411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2628C2
socleC22×C5C2×#99
prime radicalC24×C5C2×D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2628D5
hypocenterC5#11717
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
5. C2×C5×C4⋊Q8C4semidirect
10. C5×#395C4semidirect
13. C5×#382C4semidirect
90. C2×C4⋊Q8Dic5semidirect
96. #395Dic5semidirect
99. #382Dic5semidirect
273. C5#11717semidirect

Maximal subgroups

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

subgroupindex
C5×#26282
#106023852
#106023642
#105981372
#117175