Finite Group Databasebeta

← #10613291

Subgroups — #10613291

Order 640.

446 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 121 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#66050C211yesmaximal, characteristic, complemented
3C5×#486C211yesmaximal, characteristic, complemented
4#65368C211yesmaximal, characteristic, complemented
5#65362C211yesmaximal, characteristic, complemented
6#65074C214yesmaximal, complemented
7#65071C214yesmaximal, complemented
8#64983C211yesmaximal, characteristic, complemented
9C4×#676C211yesmaximal, characteristic, complemented
10C2×#713—21no—
11C2×#711—21no—
12C5×C4⋊Q8C2211yescharacteristic
13C5×#127C2211yescharacteristic
14#4093—21no—
15#4084—21no—
16#4084C2212yes—
17#4082—21no—
18D4×Dic5—21no—
19D4×Dic5C2212yes—
20C2×#686C2214yes—
21C2×#676C2212yes—
22#4041—21no—
23#4041C2212yes—
24#4039—21no—
25#4039C2212yes—
26#4020C2211yescharacteristic
27C42×D5C2211yescharacteristic
28#4015C2211yescharacteristic
29C5×#102C2214yes—
30C4×C5×C8C2211yescharacteristic
31#3953C2214yes—
32#3947C2212yes—
33#3940C2214yes—
34C4×#168C2211yescharacteristic
35#3184—51nomaximal, Sylow
36C2×C5×Q8—21no—
37C2×C5×D4—21no—
38C2×C5×D4—21no—
39C2×C5×D4C2312yes—
40C2×C22⋊D5—22no—
41C22×Dic5—22no—
42#713—44no—
43#711—44no—
44C2×D20—22no—
45C2×D20—21no—
46C2×D20—21no—
47C2×D20C2312yes—
48C2×C4×D5—22no—
49C2×C4×D5D411yescharacteristic
50C2×C4×D5D411yescharacteristic
51C2×C4×D5C2311yescharacteristic
52C2×Dic10—21no—
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4C2312yes—
56C42×C5C2311yescharacteristic
57C22⋊Dic5—22no—
58C22⋊Dic5—22no—
59#686—28no—
60#685—22no—
61#685—22no—
62C4⋊Dic5—21no—
63C4⋊Dic5C2312yes—
64C4×Dic5D411yescharacteristic
65C4×Dic5D411yescharacteristic
66C4×Dic5C2311yescharacteristic
67C2×#168C2312yes—
68#676—24no—
69#532—51no—
70#532—51no—
71#498—51no—
72#493—51no—
73#486—51no—
74#486D511yescharacteristic, complemented
75#469—54no—
76#442—54no—
77C4×M16—51no—
78C23×C5—22no—
79C22×D5—22no—
80C22×D5C2×D411yescharacteristic
81C5×Q8—42no—
82C5×D4—24no—
83C5×D4—42no—
84C5×D4—42no—
85C2×C4×C5—22no—
86C2×C4×C5C2×D411yescharacteristic
87C2×C4×C5C2×D411yescharacteristic
88C2×C4×C5C2411yescharacteristic
89C22⋊D5—44no—
90C2×Dic5—22no—
91C2×Dic5—44no—
92C2×Dic5C2×D411yescharacteristic
93C2×Dic5C2×D411yescharacteristic
94C2×Dic5C2×D411yescharacteristic
95D20—24no—
96D20—44no—
97D20—42no—
98D20—42no—
99C4×D5—22no—
100C4×D5—22no—
101C4×D5—22no—
102C4×D5—44no—
103Dic10—42no—
104C5×C8—22no—
105#168#4114yescomplemented
106C2×#41—101no—
107C2×#41—101no—
108C2×D8—54no—
109C2×M16—52no—
110C4⋊Q8—51no—
111C4⋊Q8C2×D511yescharacteristic
112#127—51no—
113#127C2×D511yescharacteristic
114#124—101no—
115#124—101no—
116#121—52no—
117#121—52no—
118#121—101no—
119#121—101no—
120C4×D4—52no—
121C4×D4—52no—
122C4×D4—101no—
123C4×D4—101no—
124C2×C42—51no—
125#106—54no—
126#102—54no—
127#102C2×D514yes—
128#97—52no—
129C4×C8—51no—
130C4×C8C2×D511yescharacteristic
131C22×C5—42no—
132C22×C5—42no—
133C22×C5—42no—
134C22×C5C22×D411yescharacteristic
135C2×D5—21no—
136C2×D5—21no—
137C2×D5—21no—
138C2×D5—42no—
139C2×D5—42no—
140C2×D5—42no—
141C4×C5—42no—
142C4×C5AGL(1,ℤ/8)12yes—
143C4×C5AGL(1,ℤ/8)12yes—
144C4×C5C2×#4112yes—
145Dic5—21no—
146Dic5—21no—
147Dic5—21no—
148Dic5—42no—
149#41—204no—
150#41—204no—
151C2×Q8—21no—
152C2×Q8—101no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—52no—
156C2×D4—102no—
157C2×D4—102no—
158C2×D4—101no—
159C2×D4—101no—
160C2×D4C22×D512yes—
161C22×C4—51no—
162C22×C4—51no—
163C22×C4—51no—
164C22×C4—102no—
165C22×C4—102no—
166D8—108no—
167M16—104no—
168C2×C8—52no—
169C2×C8C22×D512yes—
170C4⋊C4—21no—
171C4⋊C4—52no—
172C4⋊C4—101no—
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—82no—
183C2×C5#53211yescharacteristic
184C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
186D5—41no—
187D5—82no—
188C23—22no—
189C23—51no—
190C23—102no—
191Q8—42no—
192Q8—202no—
193D4—24no—
194D4—42no—
195D4—42no—
196D4—104no—
197D4—204no—
198D4—204no—
199D4—202no—
200D4—202no—
201C2×C4—22no—
202C2×C4—51no—
203C2×C4—51no—
204C2×C4—51no—
205C2×C4—102no—
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—54no—
216C5#318411yescharacteristic, complemented, Sylow
217C22—42no—
218C22—42no—
219C22—42no—
220C22—101no—
221C22—101no—
222C22—101no—
223C22—202no—
224C22—202no—
225C22—202no—
226C22C2×D4×D511yescharacteristic
227C4—42no—
228C4—101no—
229C4—101no—
230C4—101no—
231C4—202no—
232C4#405712yes—
233C4#406012yes—
234C4D5×#4112yes—
235C2—82no—
236C2—201no—
237C2—402no—
238C2#6600211yescharacteristic
239C2C2×#405711yescharacteristic
240C2C2×#406011yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#486C2
socleC22×C5C22×D4
hypercenter#486D5
hypocenterC5#3184
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #66059C2semidirect
2. #66050C2semidirect
3. C5×#486C2semidirect
4. #65368C2semidirect
5. #65362C2semidirect
6. #65074C2semidirect
7. #65071C2semidirect
8. #64983C2semidirect
9. C4×#676C2semidirect
74. #486D5semidirect
105. #168#41semidirect
216. C5#3184semidirect

Maximal subgroups

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

subgroupindex
#660592
#660502
C5×#4862
#653682
#653622
#650742
#650712
#649832
C4×#6762
#31845