Finite Group Databasebeta

← #10614515

Subgroups — #10614515

Order 640.

584 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 161 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
5C2×#3963C211yesmaximal, characteristic, complemented
6#65281C211yesmaximal, characteristic, complemented
7#65278C212yesmaximal, complemented
8#65271C211yesmaximal, characteristic, complemented
9#65267C212yesmaximal, complemented
10#65247C212yesmaximal
11#65231C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C22×C22⋊D5C2211yescharacteristic
15C23×Dic5C2212yes—
16C22×C4×D5C2211yescharacteristic
17C5×C22⋊Q8—24no—
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2212yes—
20C2×C5×C22⋊C4C2212yes—
21C2×C22⋊Dic5C2211yescharacteristic
22C4×C22⋊D5—24no—
23C2×#685C2211yescharacteristic
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#683C2212yes—
27C2×#683C2211yescharacteristic
28C2×#683C2212yes—
29C2×C4×Dic5C2211yescharacteristic
30#3963C2211yescharacteristic
31#3963C2212yes—
32#3963C2212yes—
33#3963C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2564—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⋊D5Q814yescomplemented
51C22×Dic5—24no—
52C22×Dic5—24no—
53C22×Dic5—42no—
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2312yes—
57C22×Dic5C2311yescharacteristic
58C22×Dic5C2312yes—
59C2×C4×D5—22no—
60C2×C4×D5—42no—
61C5×C4⋊C4—22no—
62C5×C4⋊C4—24no—
63C5×C22⋊C4—24no—
64C22⋊Dic5—22no—
65#685—22no—
66#685—24no—
67#685—42no—
68#683—22no—
69#683—24no—
70#683—42no—
71C4×Dic5—22no—
72C2×C22⋊Q8D511yescharacteristic, complemented
73C2×C4×D4—51no—
74#400—51no—
75#399—51no—
76#395—51no—
77#393—51no—
78#387—52no—
79#386—52no—
80#385—52no—
81#380—52no—
82C2×#95—51no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5C2×D412yes—
86C23×C5C2×Q812yes—
87C23×C5#4112yes—
88C23×C5C2411yescharacteristic
89C22×D5—21no—
90C22×D5—22no—
91C22×D5—21no—
92C22×D5—22no—
93C22×D5C2×Q812yes—
94C5×Q8—44no—
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—22no—
102C2×C4×C5—22no—
103C2×C4×C5—44no—
104C2×C4×C5C2×D412yes—
105C2×C4×C5C2×D412yes—
106C22⋊D5—28no—
107C2×Dic5—21no—
108C2×Dic5—22no—
109C2×Dic5—21no—
110C2×Dic5—22no—
111C2×Dic5—21no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—48no—
119C2×Dic5C2×Q812yes—
120C2×Dic5#4114yes—
121C4×D5—44no—
122C22×Q8C2×D511yescharacteristic
123C22×D4—51no—
124C23×C4—52no—
125C23×C4—51no—
126C23×C4C2×D511yescharacteristic
127C22⋊Q8—24no—
128C4×D4—104no—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—52no—
132C2×C4⋊C4C2×D512yes—
133C2×C4⋊C4C2×D511yescharacteristic
134C2×C22⋊C4—51no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4C2×D512yes—
139C2×C42—51no—
140#95—52no—
141#95—51no—
142#95—51no—
143#95—52no—
144#95—52no—
145#95—52no—
146#95—52no—
147C22×C5—22no—
148C22×C5—22no—
149C22×C5—24no—
150C22×C5—24no—
151C22×C5C22⋊Q814yes—
152C22×C5C22×D411yescharacteristic
153C22×C5C22×Q811yescharacteristic
154C22×C5C2×#4111yescharacteristic
155C22×C5C2×#4112yes—
156C22×C5H(5,𝔽2)12yes—
157C2×D5—22no—
158C2×D5—24no—
159C2×D5—41no—
160C2×D5—41no—
161C2×D5—42no—
162C4×C5—22no—
163C4×C5—22no—
164C4×C5—42no—
165C4×C5—42no—
166Dic5—22no—
167Dic5—24no—
168Dic5—41no—
169Dic5—41no—
170Dic5—42no—
171C24—51no—
172C24C22×D511yescharacteristic
173C2×Q8—22no—
174C2×Q8—42no—
175C2×D4—54no—
176C2×D4—102no—
177C2×D4—102no—
178C22×C4—42no—
179C22×C4—51no—
180C22×C4—52no—
181C22×C4—52no—
182C22×C4—51no—
183C22×C4—51no—
184C22×C4—104no—
185C22×C4—104no—
186C22×C4—102no—
187C22×C4—202no—
188C22×C4—202no—
189C22×C4C22⋊D514yes—
190C22×C4C22×D512yes—
191C22×C4C22×D512yes—
192C22×C4C22×D511yescharacteristic
193C22×C4C22×D511yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—104no—
197C4⋊C4—102no—
198C4⋊C4—202no—
199C22⋊C4—24no—
200C22⋊C4—104no—
201C22⋊C4—102no—
202C22⋊C4—102no—
203C22⋊C4—202no—
204C42—102no—
205C2×C5—24no—
206C2×C5C2×C22⋊Q811yescharacteristic
207C2×C5#53211yescharacteristic
208C2×C5#53411yescharacteristic
209C2×C5#55112yes—
210C2×C5#55412yes—
211D5—42no—
212C23—22no—
213C23—22no—
214C23—52no—
215C23—102no—
216C23—101no—
217C23—102no—
218C23—101no—
219C23Q8×D512yes—
220C23#71312yes—
221C23C2×C22⋊D512yes—
222C23C23×D511yescharacteristic
223Q8—44no—
224D4—108no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—44no—
234C2×C4—52no—
235C2×C4—54no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—101no—
239C2×C4—101no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—101no—
246C2×C4—101no—
247C2×C4—204no—
248C2×C4—208no—
249C2×C4C2×C22⋊D512yes—
250C2×C4C2×C22⋊D512yes—
251C5#256411yescharacteristic, complemented, Sylow
252C22—22no—
253C22—22no—
254C22—24no—
255C22—24no—
256C22—102no—
257C22—104no—
258C22—201no—
259C22—201no—
260C22—202no—
261C22#409214yes—
262C22#414412yes—
263C22C2×Q8×D511yescharacteristic
264C22C2×#71311yescharacteristic
265C22D5×#4112yes—
266C22C22×C22⋊D511yescharacteristic
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—102no—
272C4—104no—
273C4—201no—
274C4—201no—
275C4—202no—
276C2—24no—
277C2—202no—
278C2#6597012yes—
279C2#6597112yes—
280C2#6613811yescharacteristic
281C2C2×#409211yescharacteristic
282C2#6616611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC22×C22⋊D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2564
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. C2×#3963C2semidirect
6. #65281C2semidirect
7. #65278C2semidirect
8. #65271C2semidirect
9. #65267C2semidirect
11. #65231C2semidirect
50. C2×C22⋊D5Q8semidirect
72. C2×C22⋊Q8D5semidirect
251. C5#2564semidirect

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
C2×#39632
#652812
#652782
#652712
#652672
#652472
#652312
#25645