Finite Group Databasebeta

← #10614554

Subgroups — #10614554

Order 640.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D5C211yesmaximal, characteristic
3#65520C211yesmaximal, characteristic, complemented
4#65518C211yesmaximal, characteristic, complemented
5#65500C211yesmaximal, characteristic, complemented
6#65282C211yesmaximal, characteristic, complemented
7#65278C212yesmaximal, complemented
8#65269C211yesmaximal, characteristic, complemented
9#65267C212yesmaximal, complemented
10#65248C212yesmaximal
11#65241C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C22×C22⋊D5C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C5×C22⋊Q8—24no—
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2212yes—
19C2×C5×C22⋊C4C2212yes—
20C2×C22⋊Dic5C2211yescharacteristic
21C2×C22⋊Dic5C2212yes—
22C4×C22⋊D5—24no—
23C2×#685C2211yescharacteristic
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#683C2211yescharacteristic
27C2×#683C2212yes—
28C2×#683C2212yes—
29C2×C4×Dic5C2211yescharacteristic
30#3963C2212yes—
31#3963C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2638—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C2×C5×Q8—22no—
41C2×C5×Q8—42no—
42C22×C4×C5—42no—
43C22×C4×C5D414yes—
44C22×C4×C5C2312yes—
45C22×C4×C5C2312yes—
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2311yescharacteristic
48C2×C22⋊D5—22no—
49C2×C22⋊D5—22no—
50C2×C22⋊D5—22no—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2312yes—
54C22×Dic5C2312yes—
55C22×Dic5C2311yescharacteristic
56C2×C4×D5—22no—
57C2×C4×D5—42no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—24no—
60C5×C22⋊C4—24no—
61C22⋊Dic5—22no—
62C22⋊Dic5—42no—
63#685—24no—
64#685—22no—
65#685—42no—
66#683—22no—
67#683—24no—
68#683—42no—
69C4×Dic5—22no—
70C2×C22⋊Q8D511yescharacteristic, complemented
71C2×C4×D4—51no—
72#398—51no—
73#397—51no—
74#397—52no—
75#395—51no—
76#394—52no—
77#393—51no—
78#387—52no—
79#385—52no—
80#381—51no—
81C23×C5—21no—
82C23×C5—21no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5C2×D412yes—
86C23×C5C2411yescharacteristic
87C22×D5—21no—
88C22×D5—21no—
89C22×D5—22no—
90C22×D5—22no—
91C22×D5#4112yes—
92C5×Q8—44no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—44no—
102C2×C4×C5C2×D412yes—
103C2×C4×C5C2×D412yes—
104C22⋊D5—28no—
105C2×Dic5—21no—
106C2×Dic5—21no—
107C2×Dic5—21no—
108C2×Dic5—21no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5#4112yes—
118C4×D5—44no—
119C22×Q8C2×D511yescharacteristic
120C22×D4—51no—
121C23×C4—51no—
122C23×C4C2×D511yescharacteristic
123C22⋊Q8—24no—
124C4×D4—104no—
125C2×C4⋊C4—51no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—52no—
128C2×C4⋊C4C2×D512yes—
129C2×C4⋊C4C2×D511yescharacteristic
130C2×C22⋊C4—51no—
131C2×C22⋊C4—51no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4—52no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4C2×D512yes—
136C2×C42—51no—
137#95—52no—
138#95—51no—
139#95—51no—
140#95—52no—
141#95—52no—
142#95—52no—
143#95—52no—
144C22×C5—22no—
145C22×C5—22no—
146C22×C5—22no—
147C22×C5—42no—
148C22×C5—42no—
149C22×C5C22×D411yescharacteristic
150C22×C5C2×#4111yescharacteristic
151C22×C5C2×#4111yescharacteristic
152C22×C5H(5,𝔽2)12yes—
153C22×C52−1+412yes—
154C2×D5—22no—
155C2×D5—24no—
156C2×D5—41no—
157C2×D5—41no—
158C2×D5—42no—
159C4×C5—22no—
160C4×C5—22no—
161C4×C5—42no—
162C4×C5—42no—
163Dic5—22no—
164Dic5—41no—
165Dic5—41no—
166Dic5—42no—
167Dic5—42no—
168C24—51no—
169C24C22×D511yescharacteristic
170C2×Q8—22no—
171C2×Q8—42no—
172C2×D4—102no—
173C2×D4—102no—
174C2×D4—102no—
175C22×C4—42no—
176C22×C4—51no—
177C22×C4—51no—
178C22×C4—52no—
179C22×C4—52no—
180C22×C4—51no—
181C22×C4—102no—
182C22×C4—202no—
183C22×C4C22⋊D514yes—
184C22×C4C22×D512yes—
185C22×C4C22×D512yes—
186C22×C4C22×D511yescharacteristic
187C22×C4C22×D511yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—24no—
190C4⋊C4—104no—
191C4⋊C4—102no—
192C4⋊C4—202no—
193C22⋊C4—24no—
194C22⋊C4—104no—
195C22⋊C4—102no—
196C22⋊C4—102no—
197C22⋊C4—202no—
198C22⋊C4—202no—
199C42—102no—
200C2×C5—42no—
201C2×C5#52511yescharacteristic
202C2×C5#53411yescharacteristic
203C2×C5#53611yescharacteristic
204C2×C5#53912yes—
205C2×C5#54212yes—
206D5—42no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—52no—
212C23—102no—
213C23—102no—
214C23—101no—
215C23—101no—
216C23C2×C22⋊D512yes—
217C23C23×D511yescharacteristic
218Q8—44no—
219D4—108no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—44no—
229C2×C4—52no—
230C2×C4—101no—
231C2×C4—101no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—101no—
241C2×C4—101no—
242C2×C4—204no—
243C2×C4C2×C22⋊D512yes—
244C2×C4C2×C22⋊D512yes—
245C5#263811yescharacteristic, complemented, Sylow
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—102no—
252C22—104no—
253C22—201no—
254C22—201no—
255C22—202no—
256C22#414412yes—
257C22#414712yes—
258C22D5×#4111yescharacteristic
259C22D5×#4111yescharacteristic
260C22C22×C22⋊D511yescharacteristic
261C4—22no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—102no—
266C4—201no—
267C4—201no—
268C4—202no—
269C4—202no—
270C2—42no—
271C2—202no—
272C2#6597212yes—
273C2#6597412yes—
274C2#6613811yescharacteristic
275C2#6615011yescharacteristic
276C2#6616211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC22×C22⋊D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2638
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C22⋊Q8C2semidirect
3. #65520C2semidirect
4. #65518C2semidirect
5. #65500C2semidirect
6. #65282C2semidirect
7. #65278C2semidirect
8. #65269C2semidirect
9. #65267C2semidirect
11. #65241C2semidirect
70. C2×C22⋊Q8D5semidirect
245. C5#2638semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C2×C4×C22⋊D52
#655202
#655182
#655002
#652822
#652782
#652692
#652672
#652482
#652412
#26385