Finite Group Databasebeta

← #10613334

Subgroups — #10613334

Order 640.

438 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#66059C211yesmaximal, characteristic, complemented
2#66059C211yesmaximal, characteristic, complemented
3C5×#487C211yesmaximal, characteristic, complemented
4#65377C211yesmaximal, characteristic, complemented
5#65374C211yesmaximal, characteristic, complemented
6#65108C214yesmaximal, complemented
7#65100C214yesmaximal, complemented
8#64984C211yesmaximal, characteristic, complemented
9#64974C211yesmaximal, characteristic, complemented
10C2×#713—21no—
11C2×#713C2212yes—
12C5×C4⋊Q8C2211yescharacteristic
13C5×C4⋊Q8C2211yescharacteristic
14#4093—21no—
15#4093C2212yes—
16C2×#688C2214yes—
17C2×D40C2214yes—
18#4041—21no—
19#4041—21no—
20#4041—21no—
21#4041C2212yes—
22#4039—21no—
23#4039—21no—
24#4039—21no—
25#4039C2212yes—
26#4020C2211yescharacteristic
27#4020C2211yescharacteristic
28C42×D5C2211yescharacteristic
29C5×#103C2214yes—
30C4×C5×C8C2211yescharacteristic
31#3952C2212yes—
32#3946C2212yes—
33#3941C2214yes—
34C4×#168C2211yescharacteristic
35#3207—51nomaximal, Sylow
36C2×C5×Q8—21no—
37C2×C5×Q8C2312yes—
38#713—44no—
39#713—44no—
40C2×D20—22no—
41C2×D20—22no—
42C2×D20—21no—
43C2×D20—21no—
44C2×D20—21no—
45C2×D20—21no—
46C2×D20C2312yes—
47C2×D20C2312yes—
48C2×C4×D5—22no—
49C2×C4×D5—22no—
50C2×C4×D5D411yescharacteristic
51C2×C4×D5D411yescharacteristic
52C2×C4×D5C2311yescharacteristic
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4—21no—
56C5×C4⋊C4—21no—
57C5×C4⋊C4C2312yes—
58C42×C5C2311yescharacteristic
59#688—44no—
60#685—22no—
61#685—22no—
62#685—22no—
63#685—22no—
64C4×Dic5D411yescharacteristic
65C4×Dic5D411yescharacteristic
66C4×Dic5C2311yescharacteristic
67C2×#168C2312yes—
68D40—44no—
69#532—51no—
70#532—51no—
71#493—51no—
72#492—51no—
73#487D511yescharacteristic, complemented
74#486—51no—
75#463—54no—
76#449—54no—
77#431—51no—
78C22×D5—22no—
79C22×D5—22no—
80C22×D5C2×D411yescharacteristic
81C5×Q8—24no—
82C5×Q8—42no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5C2×D411yescharacteristic
86C2×C4×C5C2×D411yescharacteristic
87C2×C4×C5C2411yescharacteristic
88C2×Dic5C2×D411yescharacteristic
89C2×Dic5C2×D411yescharacteristic
90C2×Dic5C2×D411yescharacteristic
91D20—24no—
92D20—24no—
93D20—44no—
94D20—44no—
95D20—42no—
96D20—42no—
97D20—42no—
98D20—42no—
99C4×D5—22no—
100C4×D5—22no—
101C4×D5—22no—
102C4×D5—44no—
103C4×D5—44no—
104C5×C8—22no—
105#168—22no—
106C2×#41—52no—
107C2×#41—101no—
108C2×SD16—54no—
109C2×D8—54no—
110C4⋊Q8C2×D511yescharacteristic
111C4⋊Q8C2×D511yescharacteristic
112#127—51no—
113#127—51no—
114#124—52no—
115#124—101no—
116#121—52no—
117#121—101no—
118#121—101no—
119#121—101no—
120C4×D4—52no—
121C4×D4—101no—
122C4×D4—101no—
123C4×D4—101no—
124C2×C42—51no—
125C4⋊C8—52no—
126#103C2×D514yes—
127#102—54no—
128C22⋊C8—52no—
129C4×C8—51no—
130C4×C8C2×D511yescharacteristic
131C22×C5C22×D411yescharacteristic
132C2×D5—21no—
133C2×D5—21no—
134C2×D5—21no—
135C2×D5—42no—
136C2×D5—42no—
137C2×D5—42no—
138C2×D5—42no—
139C2×D5—42no—
140C2×D5—42no—
141C4×C5—42no—
142C4×C5—42no—
143C4×C5#13512yes—
144C4×C5AGL(1,ℤ/8)12yes—
145C4×C5H(5,𝔽2)12yes—
146Dic5—21no—
147Dic5—21no—
148Dic5—21no—
149#41—204no—
150#41—204no—
151C2×Q8—21no—
152C2×Q8C22×D512yes—
153C2×D4—52no—
154C2×D4—52no—
155C2×D4—102no—
156C2×D4—102no—
157C2×D4—101no—
158C2×D4—101no—
159C2×D4—101no—
160C2×D4—101no—
161C22×C4—51no—
162C22×C4—51no—
163C22×C4—51no—
164C22×C4—102no—
165C22×C4—102no—
166SD16—204no—
167D8—204no—
168C2×C8—52no—
169C2×C8C22×D512yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22×D512yes—
174C22⋊C4—102no—
175C22⋊C4—102no—
176C22⋊C4—102no—
177C22⋊C4—102no—
178C42—51no—
179C42—51no—
180C42—51no—
181C42C22×D511yescharacteristic
182C2×C5#53511yescharacteristic
183C2×C5C2×#13511yescharacteristic
184C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
185D5—41no—
186D5—82no—
187D5—82no—
188C23—51no—
189C23—102no—
190C23—102no—
191Q8—24no—
192Q8—42no—
193D4—104no—
194D4—104no—
195D4—204no—
196D4—204no—
197D4—202no—
198D4—202no—
199D4—202no—
200D4—202no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—51no—
204C2×C4—51no—
205C2×C4—51no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—204no—
210C2×C4—204no—
211C2×C4D4×D511yescharacteristic
212C2×C4D4×D511yescharacteristic
213C2×C4C23×D511yescharacteristic
214C8—22no—
215C8—102no—
216C5#320711yescharacteristic, complemented, Sylow
217C22—101no—
218C22—101no—
219C22—101no—
220C22—202no—
221C22—202no—
222C22—202no—
223C22—202no—
224C22—202no—
225C22—202no—
226C22C2×D4×D511yescharacteristic
227C4—42no—
228C4—42no—
229C4—101no—
230C4—101no—
231C4—101no—
232C4#406012yes—
233C4#406512yes—
234C4#414912yes—
235C2—201no—
236C2—402no—
237C2—402no—
238C2#6600811yescharacteristic
239C2C2×#406011yescharacteristic
240C2C2×#406511yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#487C2
socleC22×C5C22×D4
hypercenter#487D5
hypocenterC5#3207
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #66059C2semidirect
2. #66059C2semidirect
3. C5×#487C2semidirect
4. #65377C2semidirect
5. #65374C2semidirect
6. #65108C2semidirect
7. #65100C2semidirect
8. #64984C2semidirect
9. #64974C2semidirect
73. #487D5semidirect
216. C5#3207semidirect

Maximal subgroups

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

subgroupindex
#660592
C5×#4872
#653772
#653742
#651082
#651002
#649842
#649742
#32075