Finite Group Databasebeta

← #10614565

Subgroups — #10614565

Order 640.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D5C211yesmaximal, characteristic
2C2×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#4092C212yesmaximal, complemented
4#65501C211yesmaximal, characteristic, complemented
5#65278C214yesmaximal, complemented
6#65276C212yesmaximal, complemented
7#65244C212yesmaximal
8#65235C212yesmaximal, complemented
9C24×D5C2211yescharacteristic
10C22×C5×Q8C2211yescharacteristic, complemented
11C23×C4×C5C2211yescharacteristic
12C23×Dic5C2211yescharacteristic
13C22×C4×D5—24no—
14C22×C4×D5—28no—
15C22×C4×D5C2212yes—
16C22×C4×D5C2212yes—
17C5×C22⋊Q8—24no—
18C2×C5×C4⋊C4C2211yescharacteristic, complemented
19C2×C5×C4⋊C4C2212yescomplemented
20C2×C5×C22⋊C4C2212yes—
21C2×C22⋊Dic5C2212yes—
22#4092—28no—
23C2×#685C2214yes—
24C2×#685C2214yes—
25C2×C4⋊Dic5C2212yescomplemented
26C2×#683C2214yescomplemented
27#3963C2214yescomplemented
28#3963C2212yescomplemented
29#2848—51nomaximal, Sylow
30C24×C5C2311yescharacteristic
31C23×D5—22no—
32C23×D5—22no—
33C23×D5—24no—
34C23×D5—24no—
35C23×D5Q814yescomplemented
36C23×D5C2312yes—
37C2×C5×Q8—22no—
38C2×C5×Q8—24no—
39C22×C4×C5—24no—
40C22×C4×C5D414yes—
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5—22no—
46C22×Dic5—24no—
47C22×Dic5C2314yes—
48C22×Dic5C2312yes—
49C2×C4×D5—232no—
50C2×C4×D5—24no—
51C2×C4×D5—24no—
52C2×C4×D5—24no—
53C2×C4×D5—24no—
54C2×C4×D5—24no—
55C2×C4×D5D418yes—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—24no—
58C5×C22⋊C4—24no—
59C22⋊Dic5—24no—
60#685—28no—
61#685—28no—
62C4⋊Dic5—24no—
63#683—28no—
64C24×C4—51no—
65C2×C22⋊Q8—52no—
66C2×C22⋊Q8D511yescharacteristic, complemented
67#385—54no—
68#385—52no—
69#380—51no—
70#380—52no—
71#379—52no—
72C23×C5—22no—
73C23×C5—22no—
74C23×C5C2×D412yes—
75C23×C5C2×Q812yes—
76C23×C5#4112yes—
77C23×C5C2411yescharacteristic
78C22×D5—28no—
79C22×D5—216no—
80C22×D5—28no—
81C22×D5—28no—
82C22×D5—22no—
83C22×D5—22no—
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—216no—
87C22×D5C2×D414yes—
88C22×D5C2×Q814yes—
89C22×D5#4114yes—
90C5×Q8—44no—
91C2×C4×C5—22no—
92C2×C4×C5—28no—
93C2×C4×C5—24no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—24no—
97C2×C4×C5C2×D414yes—
98C2×C4×C5C2×D412yes—
99C2×C4×C5C2×D412yes—
100C2×Dic5—28no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—24no—
104C2×Dic5—24no—
105C2×Dic5—24no—
106C2×Dic5C2×D414yes—
107C4×D5—216no—
108C4×D5—216no—
109C25—51no—
110C22×Q8C2×D511yescharacteristic, complemented
111C23×C4—52no—
112C23×C4—51no—
113C23×C4—52no—
114C23×C4—108no—
115C23×C4—104no—
116C23×C4C2×D511yescharacteristic
117C22⋊Q8—24no—
118C22⋊Q8—108no—
119C2×C4⋊C4—52no—
120C2×C4⋊C4—54no—
121C2×C4⋊C4C2×D511yescharacteristic, complemented
122C2×C4⋊C4C2×D512yescomplemented
123C2×C22⋊C4—52no—
124C2×C22⋊C4—54no—
125C2×C22⋊C4—54no—
126C2×C22⋊C4C2×D512yes—
127#95—54no—
128#95—52no—
129C22×C5—22no—
130C22×C5—24no—
131C22×C5—22no—
132C22×C5—24no—
133C22×C5C22⋊Q814yescomplemented
134C22×C5C22×D412yes—
135C22×C5C22×D411yescharacteristic
136C22×C5C22×Q811yescharacteristic
137C22×C5C2×#4111yescharacteristic
138C22×C5C2×#4112yes—
139C2×D5—232no—
140C2×D5—24no—
141C2×D5—24no—
142C2×D5—24no—
143C2×D5—24no—
144C2×D5—24no—
145C2×D5—24no—
146C2×D5C22⋊Q818yescomplemented
147C4×C5—22no—
148C4×C5—42no—
149C4×C5—42no—
150C4×C5#12014yes—
151Dic5—24no—
152Dic5—44no—
153C24—52no—
154C24—54no—
155C24—104no—
156C24—104no—
157C24—102no—
158C24—102no—
159C24C22×D511yescharacteristic
160C2×Q8—24no—
161C2×Q8—22no—
162C22×C4—24no—
163C22×C4—52no—
164C22×C4—58no—
165C22×C4—54no—
166C22×C4—104no—
167C22×C4—104no—
168C22×C4—1032no—
169C22×C4—104no—
170C22×C4—102no—
171C22×C4—104no—
172C22×C4—104no—
173C22×C4—104no—
174C22×C4C22⋊D514yes—
175C22×C4C22×D512yes—
176C22×C4C22×D512yes—
177C22×C4C22×D511yescharacteristic
178C22×C4C22×D511yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—108no—
182C4⋊C4—104no—
183C22⋊C4—24no—
184C22⋊C4—108no—
185C22⋊C4—108no—
186C22⋊C4—104no—
187C2×C5—24no—
188C2×C5C2×#12011yescharacteristic
189C2×C5C2×C22⋊Q811yescharacteristic
190C2×C5C2×C22⋊Q812yes—
191C2×C5#52511yescharacteristic
192C2×C5#52512yes—
193D5—28no—
194C23—22no—
195C23—22no—
196C23—54no—
197C23—54no—
198C23—54no—
199C23—1016no—
200C23—108no—
201C23—108no—
202C23—1016no—
203C23—102no—
204C23—102no—
205C23—102no—
206C23—102no—
207C23—108no—
208C23Q8×D512yes—
209C23#71312yes—
210C23C2×C22⋊D512yes—
211C23C23×D511yescharacteristic
212Q8—44no—
213C2×C4—22no—
214C2×C4—28no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—54no—
220C2×C4—1016no—
221C2×C4—108no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—1016no—
225C2×C4—104no—
226C2×C4—104no—
227C2×C4—104no—
228C2×C4D4×D514yes—
229C2×C4C2×C22⋊D512yes—
230C2×C4C2×C22⋊D512yes—
231C5#284811yescharacteristic, complemented, Sylow
232C22—22no—
233C22—22no—
234C22—24no—
235C22—24no—
236C22—58no—
237C22—1032no—
238C22—104no—
239C22—104no—
240C22—104no—
241C22—104no—
242C22—104no—
243C22—104no—
244C22#409214yescomplemented
245C22C2×D4×D512yes—
246C22C2×Q8×D511yescharacteristic
247C22C2×#71311yescharacteristic
248C22D5×#4112yes—
249C22C22×C22⋊D511yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—104no—
254C4—204no—
255C4#408314yes—
256C2—24no—
257C2—108no—
258C2D5×C22⋊Q812yes—
259C2#6596112yes—
260C2C2×#408311yescharacteristic
261C2C2×#409211yescharacteristic
262C2#6616211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2848
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C5×C22⋊Q8C2semidirect
3. C2×#4092C2semidirect
4. #65501C2semidirect
5. #65278C2semidirect
6. #65276C2semidirect
8. #65235C2semidirect
10. C22×C5×Q8C22semidirect
18. C2×C5×C4⋊C4C22semidirect
19. C2×C5×C4⋊C4C22semidirect
25. C2×C4⋊Dic5C22semidirect
26. C2×#683C22semidirect
27. #3963C22semidirect
28. #3963C22semidirect
35. C23×D5Q8semidirect
66. C2×C22⋊Q8D5semidirect
110. C22×Q8C2×D5semidirect
121. C2×C4⋊C4C2×D5semidirect
122. C2×C4⋊C4C2×D5semidirect
133. C22×C5C22⋊Q8semidirect
146. C2×D5C22⋊Q8semidirect
231. C5#2848semidirect
244. C22#4092semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D52
C2×C5×C22⋊Q82
C2×#40922
#655012
#652782
#652762
#652442
#652352
#28485