Finite Group Databasebeta

← #12010026

Subgroups — #12010026

Order 1280.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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#10602387C211yesmaximal, characteristic
3#10602365C211yesmaximal, characteristic
4#10598137C214yesmaximal
5C2×C5×C4⋊Q8C411yescharacteristic, complemented
6C2×C5×#117C2211yescharacteristic
7C5×#400—21no—
8C5×#399—21no—
9C5×#395C412yescomplemented
10C5×#393C2212yes—
11C5×#389—21no—
12C5×#382C411yescharacteristic, complemented
13C5×#380—21no—
14#65500—21no—
15#65215—21no—
16#64748—21no—
17#64746—21no—
18#64746C2212yes—
19#64745C2212yes—
20#11812—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—
39C5×#95—22no—
40C5×#95—21no—
41C5×#95—22no—
42C5×#95C2×C412yes—
43#3963—41no—
44#3963—41no—
45C22⋊#168—22no—
46C4⋊#168—41no—
47#3935—41no—
48#2628D511yescharacteristic
49#1820—51no—
50#1464—51no—
51#1351—54no—
52C24×C5C2×D411yescharacteristic
53C2×C5×Q8—42no—
54C2×C5×Q8—82no—
55C22×C4×C5—22no—
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—22no—
59C22×C4×C5—22no—
60C22×C4×C5—41no—
61C22×C4×C5C22⋊C411yescharacteristic
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5C22⋊C411yescharacteristic
65C22×C4×C5C22×C411yescharacteristic
66C22×C4×C5C2×D411yescharacteristic
67C22×Dic5—42no—
68C5×C4⋊C4—41no—
69C5×C4⋊C4—41no—
70C5×C4⋊C4—42no—
71C5×C4⋊C4—42no—
72C5×C4⋊C4—41no—
73C5×C4⋊C4—41no—
74C5×C4⋊C4—82no—
75C5×C22⋊C4—41no—
76C5×C22⋊C4—41no—
77C5×C22⋊C4—82no—
78C42×C5—21no—
79C42×C5—21no—
80C42×C5—21no—
81C42×C5—21no—
82C22⋊Dic5—42no—
83C22⋊Dic5—42no—
84C2×#168—42no—
85C2×C4⋊Q8Dic511yescharacteristic, complemented
86C2×#117C2×D511yescharacteristic
87#432—101no—
88#400—21no—
89#399—21no—
90#395Dic512yescomplemented
91#393C2×D512yes—
92#389—21no—
93#382Dic511yescharacteristic, complemented
94#381—101no—
95#380—21no—
96#342—52no—
97#342—101no—
98#324—101no—
99#323—52no—
100C23×C5—21no—
101C23×C5—21no—
102C23×C5—41no—
103C23×C5—41no—
104C23×C5C2×C22⋊C411yescharacteristic
105C23×C5#13712yes—
106C5×Q8—84no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—21no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5—21no—
116C2×C4×C5—42no—
117C2×C4×C5—42no—
118C2×C4×C5—41no—
119C2×C4×C5—41no—
120C2×C4×C5—41no—
121C2×C4×C5—41no—
122C2×C4×C5—42no—
123C2×C4×C5—42no—
124C2×C4×C5—42no—
125C2×C4×C5—42no—
126C2×C4×C5—42no—
127C2×C4×C5—42no—
128C2×C4×C5#9911yescharacteristic
129C2×C4×C5#9911yescharacteristic
130C2×Dic5—82no—
131C2×Dic5—82no—
132C2×Dic5—82no—
133#168—82no—
134C22×Q8C2×Dic512yes—
135C23×C4C22⋊D511yescharacteristic
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138#117—41no—
139#117—41no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C2×Dic511yescharacteristic
145C2×C4⋊C4C22⋊D511yescharacteristic
146C2×C22⋊C4—22no—
147C2×C22⋊C4—102no—
148C2×C22⋊C4C22⋊D511yescharacteristic
149C2×C22⋊C4C22×D511yescharacteristic
150C2×C42C2×Dic511yescharacteristic
151C2×C42C22⋊D511yescharacteristic
152C4⋊C8—201no—
153C22⋊C8—102no—
154#97—201no—
155#95—22no—
156#95—21no—
157#95—22no—
158#95—201no—
159#95—201no—
160#95C2×Dic512yes—
161C22×C5—21no—
162C22×C5—21no—
163C22×C5—22no—
164C22×C5—42no—
165C22×C5—41no—
166C22×C5—41no—
167C22×C5—81no—
168C22×C5C2×#9911yescharacteristic
169C22×C5#41911yescharacteristic
170C22×C5#42111yescharacteristic
171C4×C5—21no—
172C4×C5—21no—
173C4×C5—41no—
174C4×C5—41no—
175C4×C5—41no—
176C4×C5—41no—
177C4×C5—41no—
178C4×C5—82no—
179C4×C5—82no—
180Dic5—162no—
181C24C2×C22⋊D511yescharacteristic
182C2×Q8—42no—
183C2×Q8—82no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—202no—
191C22×C4C22⋊Dic511yescharacteristic
192C22×C4C22⋊Dic511yescharacteristic
193C22×C4C22⋊Dic511yescharacteristic
194C22×C4C22⋊Dic511yescharacteristic
195C22×C4C22×Dic511yescharacteristic
196C22×C4C2×C22⋊D511yescharacteristic
197C2×C8—202no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—82no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—82no—
208C22⋊C4—202no—
209C22⋊C4—202no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C2×C5—21no—
215C2×C5—21no—
216C2×C5—42no—
217C2×C5#151111yescharacteristic
218C2×C5#212211yescharacteristic
219C2×C5#212911yescharacteristic
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23#408812yes—
225C23C2×C22⋊Dic511yescharacteristic
226Q8—84no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—402no—
249C2×C4—402no—
250C2×C4—402no—
251C2×C4#396611yescharacteristic
252C2×C4#396611yescharacteristic
253C8—402no—
254C5#1181211yescharacteristic, complemented, Sylow
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22C2×#396611yescharacteristic
263C22#6551411yescharacteristic
264C22#6552511yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4—802no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2#1060248211yescharacteristic
279C2#1060265211yescharacteristic
280C2#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#11812
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
9. C5×#395C4semidirect
12. C5×#382C4semidirect
85. C2×C4⋊Q8Dic5semidirect
90. #395Dic5semidirect
93. #382Dic5semidirect
254. C5#11812semidirect

Maximal subgroups

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

subgroupindex
C5×#26282
#106023872
#106023652
#105981372
#118125