Finite Group Databasebeta

← #10608066

Subgroups — #10608066

Order 640.

708 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 169 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×#4084C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#4029C214yesmaximal, complemented
4C5×#390C211yesmaximal, characteristic, complemented
5#65510C211yesmaximal, characteristic, complemented
6#65265C211yesmaximal, characteristic, complemented
7#65238C214yesmaximal, complemented
8C4×#685C211yesmaximal, characteristic, complemented
9#65224C211yesmaximal, characteristic, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C4×D5C2212yes—
13C22×C4×D5C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2214yes—
16C2×C42×C5C2211yescharacteristic
17C2×C22⋊Dic5C2214yes—
18C2×C22⋊Dic5C2212yes—
19#4084—24no—
20C2×#685C2212yes—
21C2×C4⋊Dic5C2214yes—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×C4⋊Dic5C2211yescharacteristic
24C2×#683C2214yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27D5×C4⋊C4—24no—
28#4029—216no—
29#3963C2212yes—
30#2501—51nomaximal, Sylow
31C24×C5C2312yes—
32C23×D5C2311yescharacteristic
33C2×C5×D4—22no—
34C2×C5×D4—22no—
35C2×C5×D4—22no—
36C22×C4×C5C2312yes—
37C22×C4×C5C2312yes—
38C22×C4×C5C2311yescharacteristic
39C2×C22⋊D5—28no—
40C2×C22⋊D5—24no—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2311yescharacteristic
44C22×Dic5C2312yes—
45C2×C4×D5—22no—
46C2×C4×D5—22no—
47C2×C4×D5—28no—
48C2×C4×D5—24no—
49C2×C4×D5D414yes—
50C5×C4⋊C4—22no—
51C5×C22⋊C4—28no—
52C42×C5—22no—
53C22⋊Dic5—28no—
54C22⋊Dic5—24no—
55#685D418yescomplemented
56C4⋊Dic5—28no—
57C4⋊Dic5—22no—
58C4⋊Dic5—22no—
59#683—28no—
60#683—24no—
61C4×Dic5—22no—
62C2×#123—54no—
63C2×#121—51no—
64C22×C4⋊C4—51no—
65#399—54no—
66#390—51no—
67#390D511yescharacteristic, complemented
68#384—51no—
69#384—51no—
70C4×C22⋊C4—51no—
71C23×C5—24no—
72C23×C5—22no—
73C23×C5—22no—
74C23×C5—22no—
75C23×C5—24no—
76C23×C5C2411yescharacteristic
77C22×D5—21no—
78C22×D5—21no—
79C22×D5—22no—
80C22×D5C2×D414yes—
81C22×D5C2×D412yes—
82C5×D4—44no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—24no—
86C2×C4×C5—22no—
87C2×C4×C5C2×D414yes—
88C2×C4×C5C2×D412yes—
89C2×C4×C5#4112yes—
90C2×C4×C5#4112yes—
91C22⋊D5—48no—
92C2×Dic5—22no—
93C2×Dic5—21no—
94C2×Dic5—21no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—24no—
98C2×Dic5—24no—
99C2×Dic5—22no—
100C2×Dic5C2×D414yes—
101C2×Dic5C2×D412yes—
102C4×D5—28no—
103C4×D5—48no—
104C22×D4—52no—
105C22×D4C2×D511yescharacteristic
106C23×C4—51no—
107C23×C4—52no—
108#123—1016no—
109#121—104no—
110C2×C4⋊C4—51no—
111C2×C4⋊C4—52no—
112C2×C4⋊C4—51no—
113C2×C4⋊C4—54no—
114C2×C4⋊C4—54no—
115C2×C4⋊C4—104no—
116C2×C4⋊C4C2×D511yescharacteristic
117C2×C22⋊C4—52no—
118C2×C22⋊C4—52no—
119C2×C22⋊C4—54no—
120C2×C22⋊C4C2×D514yes—
121C2×C42—51no—
122C2×C42C2×D511yescharacteristic
123#95—52no—
124C22×C5—42no—
125C22×C5—42no—
126C22×C5—42no—
127C22×C5—44no—
128C22×C5—44no—
129C22×C5C22×D411yescharacteristic
130C22×C5C22×D412yes—
131C22×C5C2×#4111yescharacteristic
132C22×C5C2×#4111yescharacteristic
133C22×C52−1+412yes—
134C2×D5—22no—
135C2×D5—22no—
136C2×D5—24no—
137C2×D5—24no—
138C2×D5#12714yes—
139C4×C5—24no—
140C4×C5—42no—
141C4×C5#12314yescomplemented
142Dic5—22no—
143Dic5—24no—
144Dic5—42no—
145Dic5—42no—
146C24—51no—
147C24C22×D512yes—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—104no—
152C2×D4—108no—
153C22×C4—51no—
154C22×C4—52no—
155C22×C4—54no—
156C22×C4—52no—
157C22×C4—52no—
158C22×C4—104no—
159C22×C4—108no—
160C22×C4—102no—
161C22×C4—102no—
162C22×C4C22×D512yes—
163C22×C4C22×D512yes—
164C22×C4C22×D511yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—104no—
167C4⋊C4—108no—
168C4⋊C4—102no—
169C4⋊C4—102no—
170C4⋊C4—108no—
171C22⋊C4—28no—
172C22⋊C4—58no—
173C22⋊C4—104no—
174C22⋊C4—108no—
175C42—22no—
176C42—102no—
177C2×C5—82no—
178C2×C5C2×#12311yescharacteristic
179C2×C5C2×#12711yescharacteristic
180C2×C5#53611yescharacteristic
181C2×C5#54712yes—
182C2×C5#54712yes—
183D5—24no—
184C23—22no—
185C23—24no—
186C23—22no—
187C23—22no—
188C23—24no—
189C23—54no—
190C23—52no—
191C23—101no—
192C23—101no—
193C23—102no—
194C23C23×D511yescharacteristic
195D4—44no—
196D4—208no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—54no—
202C2×C4—52no—
203C2×C4—102no—
204C2×C4—101no—
205C2×C4—101no—
206C2×C4—102no—
207C2×C4—108no—
208C2×C4—104no—
209C2×C4—102no—
210C2×C4—102no—
211C2×C4—104no—
212C2×C4—208no—
213C2×C4#70912yes—
214C2×C4D4×D514yes—
215C2×C4D4×D512yes—
216C2×C4#71112yes—
217C5#250111yescharacteristic, complemented, Sylow
218C22—42no—
219C22—42no—
220C22—44no—
221C22—44no—
222C22—42no—
223C22—54no—
224C22—102no—
225C22—104no—
226C22—104no—
227C22—102no—
228C22C2×#70911yescharacteristic
229C22C2×D4×D511yescharacteristic
230C22C2×D4×D512yes—
231C22C2×#71111yescharacteristic
232C22#415012yes—
233C4—24no—
234C4—42no—
235C4—102no—
236C4—104no—
237C4—202no—
238C4—202no—
239C4#402914yescomplemented
240C2—82no—
241C2—104no—
242C2C2×#402911yescharacteristic
243C2#6588512yes—
244C2#6594512yes—
245C2#6600911yescharacteristic
246C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC2×#4084C2
hypercenter#390D5
hypocenterC5#2501
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊C4C2semidirect
3. C2×#4029C2semidirect
4. C5×#390C2semidirect
5. #65510C2semidirect
6. #65265C2semidirect
7. #65238C2semidirect
8. C4×#685C2semidirect
9. #65224C2semidirect
55. #685D4semidirect
67. #390D5semidirect
141. C4×C5#123semidirect
217. C5#2501semidirect
239. C4#4029semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×D5×C4⋊C42
C2×#40292
C5×#3902
#655102
#652652
#652382
C4×#6852
#652242
#25015