Finite Group Databasebeta

← #10612499

Subgroups — #10612499

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×#467C211yesmaximal, characteristic, complemented
3#65476C212yesmaximal, complemented
4#65404C211yesmaximal, characteristic, complemented
5#65399C211yesmaximal, characteristic, complemented
6#65338C212yesmaximal, complemented
7#65171C212yesmaximal, complemented
8#65100C212yesmaximal, complemented
9#65098C212yesmaximal, complemented
10C2×#713C2212yes—
11C2×#709C2211yescharacteristic
12C2×C5×Q16C2211yescharacteristic
13C22×C5×C8C2211yescharacteristic
14C5×C22⋊Q8C2212yes—
15#4092C2212yes—
16C2×#688C2212yes—
17#4076C2212yes—
18C2×#681C2211yescharacteristic
19C2×D40C2211yescharacteristic
20#4042—22no—
21#4041—22no—
22#4041C2212yes—
23D5×C4⋊C4C2212yes—
24#4030—22no—
25#4030—22no—
26C5×#107C2211yescharacteristic
27C5×#103C2212yes—
28#3967C2212yes—
29#3953C2212yes—
30#3952C2212yes—
31#3952C2212yes—
32#3950C2211yescharacteristic
33#3941C2212yes—
34#3939C2212yes—
35#3203—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—
53C2×C5×C8—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—
66#681—41no—
67C2×#168C2312yes—
68D40—41no—
69#537—52no—
70#484—52no—
71#480—52no—
72#468—52no—
73#467D511yescharacteristic, complemented
74#466—51no—
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×M16—51no—
112C22×C8C2×D511yescharacteristic
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#107—51no—
123#107—52no—
124#107C2×D511yescharacteristic
125#103—52no—
126#103C2×D512yes—
127#102—52no—
128#102—52no—
129C22⋊C8—52no—
130C22⋊C8—52no—
131C22×C5—21no—
132C22×C5—21no—
133C22×C5—21no—
134C22×C5C22×D411yescharacteristic
135C2×D5—22no—
136C2×D5—22no—
137C2×D5—22no—
138C2×D5—42no—
139C2×D5—42no—
140C2×D5—42no—
141C4×C5—21no—
142C4×C5—42no—
143C4×C5—42no—
144C4×C5H(5,𝔽2)11yescharacteristic
145C4×C5H(5,𝔽2)11yescharacteristic
146Dic5—22no—
147Dic5—42no—
148#41—201no—
149#41—201no—
150#41—202no—
151#41—202no—
152C2×Q8—51no—
153C2×Q8C22×D512yes—
154C2×D4—52no—
155C2×D4—51no—
156C2×D4—52no—
157C2×D4—102no—
158C2×D4—102no—
159C2×D4—102no—
160C22×C4—52no—
161C22×C4—102no—
162C22×C4—102no—
163C22×C4C22×D511yescharacteristic
164Q16—41no—
165SD16—202no—
166D8—201no—
167M16—201no—
168C2×C8—41no—
169C2×C8—52no—
170C2×C8C22×D511yescharacteristic
171C2×C8C22×D511yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—52no—
174C4⋊C4—102no—
175C4⋊C4—102no—
176C4⋊C4C22×D512yes—
177C22⋊C4—22no—
178C22⋊C4—102no—
179C22⋊C4—102no—
180C22⋊C4—102no—
181C2×C5—41no—
182C2×C5#53411yescharacteristic
183C2×C5#57611yescharacteristic
184C2×C5#57811yescharacteristic
185D5—42no—
186D5—82no—
187C23—52no—
188C23—102no—
189C23D4×D511yescharacteristic
190Q8—22no—
191Q8—22no—
192Q8—201no—
193D4—102no—
194D4—102no—
195D4—201no—
196D4—204no—
197D4—202no—
198D4—402no—
199D4—402no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—52no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—204no—
209C2×C4—402no—
210C2×C4D4×D511yescharacteristic
211C2×C4C23×D511yescharacteristic
212C8—21no—
213C8—21no—
214C8—102no—
215C5#320311yescharacteristic, complemented, Sylow
216C22—21no—
217C22—21no—
218C22—21no—
219C22—102no—
220C22—102no—
221C22—102no—
222C22—202no—
223C22—202no—
224C22—202no—
225C22C2×D4×D511yescharacteristic
226C4—21no—
227C4—42no—
228C4—42no—
229C4—102no—
230C4—202no—
231C4#414411yescharacteristic
232C4#414911yescharacteristic
233C2—41no—
234C2—202no—
235C2—402no—
236C2#6593911yescharacteristic
237C2#6610011yescharacteristic
238C2#6610311yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#467C2
socleC22×C5C22×D4
hypercenter#467D5
hypocenterC5#3203
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #65963C2semidirect
2. C5×#467C2semidirect
3. #65476C2semidirect
4. #65404C2semidirect
5. #65399C2semidirect
6. #65338C2semidirect
7. #65171C2semidirect
8. #65100C2semidirect
9. #65098C2semidirect
73. #467D5semidirect
215. C5#3203semidirect

Maximal subgroups

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

subgroupindex
#659632
C5×#4672
#654762
#654042
#653992
#653382
#651712
#651002
#650982
#32035