Finite Group Databasebeta

← #10614739

Subgroups — #10614739

Order 640.

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

normal subgroups only

This group has 696 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×C5×#124C211yesmaximal, characteristic, complemented
2C2×#4092C211yesmaximal, characteristic, complemented
3C2×#4084C211yesmaximal, characteristic
4C2×C42×D5C211yesmaximal, characteristic, complemented
5#65519C211yesmaximal, characteristic, complemented
6#65510C211yesmaximal, characteristic, complemented
7#65245C214yesmaximal, complemented
8#65233C214yesmaximal, complemented
9#65225C211yesmaximal, characteristic, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×C5×D4C2211yescharacteristic
12C22×C22⋊D5C2212yes—
13C22×C4×D5C2212yes—
14C22×C4×D5C2211yescharacteristic
15C5×#124—24no—
16C2×C5×C22⋊C4C2214yes—
17C2×C42×C5C2211yescharacteristic
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2214yes—
20#4092—24no—
21#4084—24no—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×C4⋊Dic5C2211yescharacteristic
25C2×#683C2212yes—
26C2×C4×Dic5C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28C42×D5—24no—
29#3963C2214yes—
30#3963C2214yes—
31#2850—51nomaximal, Sylow
32C24×C5C2312yes—
33C23×D5C2311yescharacteristic
34C2×C5×Q8—22no—
35C2×C5×Q8—22no—
36C2×C5×Q8—22no—
37C2×C5×D4—22no—
38C2×C5×D4—22no—
39C2×C5×D4—22no—
40C22×C4×C5C2312yes—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C2×C22⋊D5—24no—
44C2×C22⋊D5—28no—
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2312yes—
49C2×C4×D5—24no—
50C2×C4×D5—24no—
51C2×C4×D5—22no—
52C2×C4×D5—22no—
53C2×C4×D5—24no—
54C2×C4×D5D414yes—
55C5×C22⋊C4—28no—
56C42×C5D414yescomplemented
57C22⋊Dic5—24no—
58C22⋊Dic5—28no—
59#685—24no—
60C4⋊Dic5—22no—
61C4⋊Dic5—22no—
62#683—24no—
63C4×Dic5—24no—
64C4×Dic5—22no—
65C2×#124D511yescharacteristic, complemented
66C2×C22⋊Q8—51no—
67C2×#121—51no—
68C22×C42—51no—
69#391—51no—
70#390—51no—
71#388—54no—
72#386—54no—
73#382—51no—
74C23×C5—22no—
75C23×C5—22no—
76C23×C5—24no—
77C23×C5—24no—
78C23×C5—22no—
79C23×C5C2411yescharacteristic
80C22×D5—21no—
81C22×D5—21no—
82C22×D5—22no—
83C22×D5C2×D412yes—
84C22×D5#4114yes—
85C5×Q8—44no—
86C5×D4—44no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5C2×D412yes—
93C2×C4×C5C2×D414yes—
94C2×C4×C5C2×D412yes—
95C2×C4×C5#4112yes—
96C22⋊D5—48no—
97C2×Dic5—21no—
98C2×Dic5—22no—
99C2×Dic5—21no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—24no—
104C2×Dic5—24no—
105C2×Dic5C2×D412yes—
106C2×Dic5#4114yes—
107C4×D5—216no—
108C4×D5—28no—
109C22×Q8C2×D511yescharacteristic
110C22×D4—52no—
111C22×D4C2×D511yescharacteristic
112C23×C4—51no—
113C23×C4—52no—
114#124—24no—
115C22⋊Q8—104no—
116#121—104no—
117C2×C4⋊C4—51no—
118C2×C4⋊C4—51no—
119C2×C4⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4—54no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4C2×D514yes—
124C2×C42—52no—
125C2×C42—51no—
126C2×C42—104no—
127C2×C42C2×D511yescharacteristic
128#95—54no—
129#95—54no—
130C22×C5—42no—
131C22×C5—44no—
132C22×C5—42no—
133C22×C5—44no—
134C22×C5—42no—
135C22×C5C22×D411yescharacteristic
136C22×C5C22×D411yescharacteristic
137C22×C5C2×#4111yescharacteristic
138C22×C5C2×#4112yes—
139C22×C5C2×#4112yes—
140C2×D5—22no—
141C2×D5—24no—
142C2×D5—24no—
143C2×D5—22no—
144C2×D5#12414yescomplemented
145C4×C5—24no—
146C4×C5—42no—
147C4×C5#12114yes—
148Dic5—22no—
149Dic5—24no—
150Dic5—42no—
151Dic5—42no—
152C24—51no—
153C24C22×D512yes—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—104no—
161C2×D4—108no—
162C22×C4—52no—
163C22×C4—51no—
164C22×C4—54no—
165C22×C4—52no—
166C22×C4—52no—
167C22×C4—104no—
168C22×C4—104no—
169C22×C4—102no—
170C22×C4—104no—
171C22×C4—102no—
172C22×C4C22×D512yes—
173C22×C4C22×D512yes—
174C22×C4C22×D511yescharacteristic
175C4⋊C4—104no—
176C4⋊C4—102no—
177C4⋊C4—102no—
178C22⋊C4—28no—
179C22⋊C4—104no—
180C22⋊C4—104no—
181C22⋊C4—108no—
182C42—104no—
183C42—102no—
184C42C22⋊D514yescomplemented
185C2×C5—82no—
186C2×C5C2×#12111yescharacteristic
187C2×C5#52512yes—
188C2×C5C2×#12411yescharacteristic
189C2×C5#52912yes—
190C2×C5#53211yescharacteristic
191D5—24no—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—24no—
196C23—22no—
197C23—52no—
198C23—54no—
199C23—101no—
200C23—102no—
201C23—101no—
202C23C23×D511yescharacteristic
203Q8—44no—
204D4—44no—
205D4—208no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—52no—
212C2×C4—54no—
213C2×C4—1016no—
214C2×C4—101no—
215C2×C4—102no—
216C2×C4—101no—
217C2×C4—102no—
218C2×C4—108no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—104no—
222C2×C4—104no—
223C2×C4D4×D512yes—
224C2×C4#71112yes—
225C2×C4C2×C22⋊D514yes—
226C2×C4C2×C22⋊D512yes—
227C5#285011yescharacteristic, complemented, Sylow
228C22—42no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—44no—
233C22—54no—
234C22—102no—
235C22—104no—
236C22—102no—
237C22—104no—
238C22C2×D4×D511yescharacteristic
239C22C2×#71111yescharacteristic
240C22D5×#4112yes—
241C22D5×#4112yes—
242C22C22×C22⋊D511yescharacteristic
243C4—24no—
244C4—42no—
245C4—102no—
246C4—104no—
247C4—202no—
248C4—202no—
249C4#408414yes—
250C2—82no—
251C2—104no—
252C2#6600211yescharacteristic
253C2D5×#12411yescharacteristic
254C2#6601412yes—
255C2C2×#408411yescharacteristic
256C2#6616212yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#124C2
prime radicalC2×#4084C2
hypercenterC2×#124D5
hypocenterC5#2850
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C5×#124C2semidirect
2. C2×#4092C2semidirect
4. C2×C42×D5C2semidirect
5. #65519C2semidirect
6. #65510C2semidirect
7. #65245C2semidirect
8. #65233C2semidirect
9. #65225C2semidirect
56. C42×C5D4semidirect
65. C2×#124D5semidirect
144. C2×D5#124semidirect
184. C42C22⋊D5semidirect
227. C5#2850semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1242
C2×#40922
C2×#40842
C2×C42×D52
#655192
#655102
#652452
#652332
#652252
#28505