Finite Group Databasebeta

← #10612644

Subgroups — #10612644

Order 640.

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

normal subgroups only

This group has 402 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#65963C212yesmaximal, complemented
2C5×#470C211yesmaximal, characteristic, complemented
3#65476C212yesmaximal, complemented
4#65424C211yesmaximal, characteristic, complemented
5#65418C211yesmaximal, characteristic, complemented
6#65337C212yesmaximal, complemented
7#65152C212yesmaximal, complemented
8#65100C212yesmaximal, complemented
9#65094C212yesmaximal, complemented
10C2×#713C2212yes—
11C2×#709C2211yescharacteristic
12C2×C5×Q16C2211yescharacteristic
13C2×C5×M16C2211yescharacteristic
14C5×C22⋊Q8C2212yes—
15#4092C2212yes—
16C2×#688C2212yes—
17#4076C2212yes—
18C22×#168C2211yescharacteristic
19C2×D40C2211yescharacteristic
20#4042—22no—
21#4041—22no—
22#4041C2212yes—
23D5×C4⋊C4C2212yes—
24#4030—22no—
25#4030—22no—
26C5×#106C2211yescharacteristic
27C5×#103C2212yes—
28#3967C2212yes—
29#3953C2212yes—
30#3952C2212yes—
31#3952C2212yes—
32#3949C2211yescharacteristic
33#3941C2212yes—
34#3940C2212yes—
35#3204—51nomaximal, Sylow
36C2×C5×Q8C2312yes—
37C22×C4×C5C2311yescharacteristic
38C2×C22⋊D5—22no—
39C2×C22⋊D5D412yes—
40#713—42no—
41#713—42no—
42#709—41no—
43#709—41no—
44C2×D20—22no—
45C2×D20—22no—
46C2×D20D411yescharacteristic
47C2×D20C2312yes—
48C2×C4×D5—22no—
49C2×C4×D5—22no—
50C2×C4×D5C2312yes—
51C2×Dic10D411yescharacteristic
52C5×Q16—41no—
53C5×M16—41no—
54C2×C5×C8C2311yescharacteristic
55C2×C5×C8C2311yescharacteristic
56C5×C4⋊C4—22no—
57C5×C4⋊C4C2312yes—
58C5×C22⋊C4—22no—
59#688—42no—
60#685—22no—
61#685—22no—
62#685—22no—
63C4⋊Dic5C2312yes—
64#683—22no—
65#683—22no—
66C2×#168—41no—
67C2×#168C2312yes—
68D40—41no—
69#537—52no—
70#483—52no—
71#481—52no—
72#470D511yescharacteristic, complemented
73#469—51no—
74#465—52no—
75#452—52no—
76#449—52no—
77#408—51no—
78C23×C5C2×D411yescharacteristic
79C22×D5—22no—
80C22×D5C2×D412yes—
81C5×Q8—22no—
82C5×Q8—22no—
83C2×C4×C5—21no—
84C2×C4×C5—21no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5C2×D411yescharacteristic
88C2×C4×C5C2411yescharacteristic
89C22⋊D5—42no—
90C22⋊D5—82no—
91C2×Dic5—22no—
92C2×Dic5C2×D412yes—
93D20—22no—
94D20—22no—
95D20—41no—
96D20—44no—
97D20—82no—
98C4×D5—22no—
99C4×D5—22no—
100C4×D5—44no—
101C4×D5—82no—
102Dic10—41no—
103C5×C8—21no—
104C5×C8—21no—
105#168—22no—
106C2×#41—51no—
107C2×#41—52no—
108C2×Q16C2×D511yescharacteristic
109C2×SD16—52no—
110C2×D8—51no—
111C2×M16C2×D511yescharacteristic
112C22×C8—51no—
113C22⋊Q8—52no—
114C22⋊Q8—102no—
115C22⋊Q8C2×D512yes—
116#121—52no—
117#121—52no—
118#121—102no—
119#121—102no—
120#121—102no—
121C2×C4⋊C4—52no—
122#106—51no—
123#106—52no—
124#106C2×D511yescharacteristic
125#103—52no—
126#103C2×D512yes—
127#102—52no—
128#102—52no—
129C22⋊C8—52no—
130C22⋊C8—52no—
131C22×C5—21no—
132C22×C5—22no—
133C22×C5C22×D411yescharacteristic
134C2×D5—22no—
135C2×D5—22no—
136C2×D5—22no—
137C2×D5—42no—
138C2×D5—42no—
139C2×D5—42no—
140C4×C5—21no—
141C4×C5—42no—
142C4×C5—42no—
143C4×C5H(5,𝔽2)11yescharacteristic
144C4×C5H(5,𝔽2)11yescharacteristic
145Dic5—22no—
146Dic5—42no—
147#41—201no—
148#41—201no—
149#41—202no—
150#41—202no—
151C2×Q8—51no—
152C2×Q8C22×D512yes—
153C2×D4—52no—
154C2×D4—51no—
155C2×D4—52no—
156C2×D4—102no—
157C2×D4—102no—
158C2×D4—102no—
159C22×C4—52no—
160C22×C4—102no—
161C22×C4—102no—
162C22×C4C22×D511yescharacteristic
163Q16—41no—
164SD16—202no—
165D8—201no—
166M16—41no—
167C2×C8—52no—
168C2×C8—201no—
169C2×C8C22×D511yescharacteristic
170C2×C8C22×D511yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—52no—
173C4⋊C4—102no—
174C4⋊C4—102no—
175C4⋊C4C22×D512yes—
176C22⋊C4—22no—
177C22⋊C4—102no—
178C22⋊C4—102no—
179C22⋊C4—102no—
180C2×C5—41no—
181C2×C5#53411yescharacteristic
182C2×C5#57712yes—
183D5—42no—
184D5—82no—
185C23—52no—
186C23—102no—
187C23D4×D511yescharacteristic
188Q8—22no—
189Q8—22no—
190Q8—201no—
191D4—102no—
192D4—102no—
193D4—201no—
194D4—204no—
195D4—202no—
196D4—402no—
197D4—402no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—52no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—204no—
207C2×C4—402no—
208C2×C4D4×D511yescharacteristic
209C2×C4C23×D511yescharacteristic
210C8—21no—
211C8—21no—
212C8—102no—
213C5#320411yescharacteristic, complemented, Sylow
214C22—22no—
215C22—21no—
216C22—102no—
217C22—102no—
218C22—102no—
219C22—202no—
220C22—202no—
221C22—202no—
222C22C2×D4×D511yescharacteristic
223C4—21no—
224C4—42no—
225C4—42no—
226C4—102no—
227C4—202no—
228C4#414411yescharacteristic
229C4#414911yescharacteristic
230C2—41no—
231C2—202no—
232C2—402no—
233C2#6593911yescharacteristic
234C2#6611212yes—
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#470C2
socleC22×C5C22×D4
hypercenter#470D5
hypocenterC5#3204
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #65963C2semidirect
2. C5×#470C2semidirect
3. #65476C2semidirect
4. #65424C2semidirect
5. #65418C2semidirect
6. #65337C2semidirect
7. #65152C2semidirect
8. #65100C2semidirect
9. #65094C2semidirect
72. #470D5semidirect
213. C5#3204semidirect

Maximal subgroups

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

subgroupindex
#659632
C5×#4702
#654762
#654242
#654182
#653372
#651522
#651002
#650942
#32045