Finite Group Databasebeta

← #10612601

Subgroups — #10612601

Order 640.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#65940C212yesmaximal, complemented
2C5×#469C211yesmaximal, characteristic, complemented
3#65447C212yesmaximal, complemented
4#65426C211yesmaximal, characteristic, complemented
5#65415C211yesmaximal, characteristic, complemented
6#65331C212yesmaximal, complemented
7#65156C212yesmaximal, complemented
8#65072C212yesmaximal, complemented
9#65068C212yesmaximal, complemented
10C2×#711C2212yes—
11C22×C4×D5C2211yescharacteristic
12C2×C5×D8C2211yescharacteristic, complemented
13C2×C5×M16C2211yescharacteristic
14C5×#121C2212yes—
15#4084C2212yes—
16#4083—22no—
17D4×Dic5—22no—
18C2×#687C2212yescomplemented
19#4070C2212yes—
20C2×#681C2211yescharacteristic
21C2×Dic20C2211yescharacteristic, complemented
22#4043C2212yes—
23#4038C2212yes—
24#4029—22no—
25#4027—22no—
26C5×#106C2211yescharacteristic, complemented
27C5×#102C2212yescomplemented
28#3964C2212yescomplemented
29#3952C2212yes—
30#3952C2212yes—
31#3949C2211yescharacteristic, complemented
32#3948C2212yescomplemented
33#3942C2212yescomplemented
34#3939C2212yescomplemented
35#3197—51nomaximal, Sylow
36C23×D5D411yescharacteristic
37C2×C5×D4—22no—
38C2×C5×D4C2312yes—
39C22×C4×C5C2311yescharacteristic
40C2×C22⋊D5—22no—
41C2×C22⋊D5—22no—
42C22×Dic5—22no—
43C22×Dic5D411yescharacteristic
44#711—42no—
45#711—42no—
46C2×C4×D5—41no—
47C2×C4×D5—41no—
48C2×C4×D5D412yes—
49C2×C4×D5C2312yes—
50C2×Dic10C2312yes—
51C5×D8—41no—
52C5×M16—41no—
53C2×C5×C8C2311yescharacteristic
54C2×C5×C8C2311yescharacteristic
55C5×C4⋊C4C2312yes—
56C5×C22⋊C4—22no—
57C22⋊Dic5—22no—
58C22⋊Dic5—22no—
59#687—42no—
60#685—22no—
61#685—22no—
62C4⋊Dic5—22no—
63C4⋊Dic5C2312yes—
64#683—22no—
65C4×Dic5—22no—
66#681—41no—
67C2×#168C2312yes—
68Dic20—41no—
69#525—52no—
70#485—52no—
71#482—52no—
72#470—51no—
73#469D511yescharacteristic, complemented
74#468—52no—
75#452—52no—
76#449—52no—
77#407—51no—
78C23×C5—22no—
79C23×C5C2×D411yescharacteristic
80C22×D5—41no—
81C22×D5—42no—
82C22×D5C2×D412yes—
83C5×D4—22no—
84C5×D4—22no—
85C5×D4—44no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5C2×D411yescharacteristic
90C2×C4×C5C2411yescharacteristic
91C22⋊D5—44no—
92C22⋊D5—44no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—41no—
96C2×Dic5—44no—
97C2×Dic5C2×D412yes—
98C4×D5—22no—
99C4×D5—22no—
100C4×D5—42no—
101Dic10—22no—
102Dic10—22no—
103C5×C8—21no—
104C5×C8—21no—
105#168—22no—
106C2×#41—52no—
107C23×C4—51no—
108C2×Q16—51no—
109C2×SD16—52no—
110C2×D8C2×D511yescharacteristic, complemented
111C2×M16—51no—
112C2×M16C2×D511yescharacteristic
113#123—102no—
114C22⋊Q8—52no—
115C22⋊Q8—52no—
116#121—52no—
117#121C2×D512yes—
118#120—102no—
119C4×D4—102no—
120C4×D4—102no—
121#117—52no—
122#107—52no—
123#106—51no—
124#106C2×D511yescharacteristic, complemented
125#103—52no—
126#103—52no—
127#102—52no—
128#102C2×D512yescomplemented
129C22⋊C8—52no—
130C22⋊C8—52no—
131C22×C5—21no—
132C22×C5—22no—
133C22×C5—42no—
134C22×C5—42no—
135C22×C5—42no—
136C22×C5C22×D411yescharacteristic
137C2×D5—22no—
138C2×D5—22no—
139C2×D5—22no—
140C2×D5—44no—
141C4×C5—21no—
142C4×C5—42no—
143C4×C5H(5,𝔽2)11yescharacteristic
144C4×C5H(5,𝔽2)11yescharacteristic
145Dic5—22no—
146Dic5—42no—
147Dic5—42no—
148C24—51no—
149#41—202no—
150#41—202no—
151C2×Q8—52no—
152C2×D4—22no—
153C2×D4—102no—
154C2×D4—102no—
155C2×D4C22×D512yes—
156C22×C4—51no—
157C22×C4—52no—
158C22×C4—52no—
159C22×C4—102no—
160C22×C4—201no—
161C22×C4—201no—
162C22×C4C22×D511yescharacteristic
163Q16—201no—
164SD16—202no—
165D8—41no—
166M16—41no—
167M16—201no—
168C2×C8—52no—
169C2×C8C22×D511yescharacteristic
170C2×C8C22×D511yescharacteristic
171C4⋊C4—52no—
172C4⋊C4—102no—
173C4⋊C4—102no—
174C4⋊C4C22×D512yes—
175C22⋊C4—22no—
176C22⋊C4—102no—
177C22⋊C4—102no—
178C22⋊C4—102no—
179C22⋊C4—102no—
180C42—102no—
181C2×C5—41no—
182C2×C5—82no—
183C2×C5#53411yescharacteristic
184C2×C5#57512yes—
185D5—42no—
186C23—22no—
187C23—52no—
188C23—201no—
189C23—202no—
190C23D4×D511yescharacteristic
191Q8—102no—
192Q8—102no—
193D4—22no—
194D4—22no—
195D4—44no—
196D4—204no—
197D4—204no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—52no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—202no—
207C2×C4—201no—
208C2×C4—204no—
209C2×C4D4×D511yescharacteristic
210C2×C4C23×D511yescharacteristic
211C8—21no—
212C8—21no—
213C8—102no—
214C5#319711yescharacteristic, complemented, Sylow
215C22—22no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—102no—
221C22—102no—
222C22—102no—
223C22—204no—
224C22C2×D4×D511yescharacteristic
225C4—21no—
226C4—42no—
227C4—102no—
228C4—202no—
229C4—202no—
230C4#414411yescharacteristic
231C4#414911yescharacteristic
232C2—41no—
233C2—82no—
234C2—202no—
235C2#6593911yescharacteristic
236C2#6610712yes—
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#469C2
socleC22×C5C22×D4
hypercenter#469D5
hypocenterC5#3197
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #65940C2semidirect
2. C5×#469C2semidirect
3. #65447C2semidirect
4. #65426C2semidirect
5. #65415C2semidirect
6. #65331C2semidirect
7. #65156C2semidirect
8. #65072C2semidirect
9. #65068C2semidirect
12. C2×C5×D8C22semidirect
18. C2×#687C22semidirect
21. C2×Dic20C22semidirect
26. C5×#106C22semidirect
27. C5×#102C22semidirect
28. #3964C22semidirect
31. #3949C22semidirect
32. #3948C22semidirect
33. #3942C22semidirect
34. #3939C22semidirect
73. #469D5semidirect
110. C2×D8C2×D5semidirect
124. #106C2×D5semidirect
128. #102C2×D5semidirect
214. C5#3197semidirect

Maximal subgroups

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

subgroupindex
#659402
C5×#4692
#654472
#654262
#654152
#653312
#651562
#650722
#650682
#31975