Finite Group Databasebeta

← #10611001

Subgroups — #10611001

Order 640.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#65811C211yesmaximal, characteristic
2#65804C211yesmaximal, characteristic, complemented
3C5×#431C211yesmaximal, characteristic, complemented
4#65136C214yesmaximal, complemented
5#65026C214yesmaximal, complemented
6#64983C211yesmaximal, characteristic, complemented
7#64983C211yesmaximal, characteristic, complemented
8#64971C211yesmaximal, characteristic, complemented
9#64969C211yesmaximal, characteristic, complemented
10C2×#709—21no—
11C2×C42×C5C2211yescharacteristic
12#4076—21no—
13#4076C2212yes—
14#4072C2212yes—
15#4070—21no—
16#4070—21no—
17#4021—21no—
18#4020C2211yescharacteristic
19C4×D20—21no—
20C4×D20C2212yes—
21#4016C2212yes—
22#4015C2211yescharacteristic
23#4015C2211yescharacteristic
24#4015C2211yescharacteristic
25C4×Dic10—21no—
26C4×Dic10—21no—
27C5×C4⋊C8C2212yes—
28C5×C22⋊C8C2212yes—
29C4×C5×C8C2211yescharacteristic
30C4×C5×C8C2211yescharacteristic
31#3953C2214yes—
32#3953C2214yes—
33#3950C2214yes—
34#3949C2214yes—
35#3228—51nomaximal, Sylow
36C22×C4×C5D411yescharacteristic
37C22×C4×C5D411yescharacteristic
38C22×C4×C5C2311yescharacteristic
39C2×C22⋊D5—22no—
40#709—44no—
41C2×D20—21no—
42C2×D20—21no—
43C2×D20C2312yes—
44C2×C4×D5—22no—
45C2×Dic10—21no—
46C2×Dic10—21no—
47C2×Dic10—21no—
48C2×C5×C8C2312yes—
49C2×C5×C8C2312yes—
50C42×C5D411yescharacteristic
51C42×C5D411yescharacteristic
52C42×C5C2311yescharacteristic
53C42×C5C2311yescharacteristic
54C22⋊Dic5—22no—
55#685—22no—
56#685—22no—
57C4⋊Dic5—21no—
58C4⋊Dic5—21no—
59C4⋊Dic5—21no—
60C4⋊Dic5C2312yes—
61C4⋊Dic5C2312yes—
62C4⋊Dic5C2312yes—
63#683—22no—
64#683—22no—
65C4×Dic5—22no—
66#533—51no—
67#532—51no—
68#500—51no—
69#498—51no—
70#486—51no—
71#486—51no—
72#482—54no—
73#478—54no—
74#431D511yescharacteristic, complemented
75C23×C5C2×D411yescharacteristic
76C22×D5—22no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5C2×D411yescharacteristic
81C2×C4×C5C2×D411yescharacteristic
82C2×C4×C5C2×D411yescharacteristic
83C2×C4×C5C2×D411yescharacteristic
84C2×C4×C5C2×D411yescharacteristic
85C2×C4×C5C2411yescharacteristic
86C22⋊D5—44no—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90D20—24no—
91D20—42no—
92D20—42no—
93C4×D5—44no—
94Dic10—42no—
95Dic10—42no—
96Dic10—42no—
97C5×C8—22no—
98C5×C8—22no—
99C2×#41—101no—
100C4⋊Q8—51no—
101C4⋊Q8—51no—
102C4⋊Q8—51no—
103#127—51no—
104#125—52no—
105#124—101no—
106C22⋊Q8—101no—
107C22⋊Q8—101no—
108#121—52no—
109#121—101no—
110C4×Q8—101no—
111C4×Q8—101no—
112C4×D4—52no—
113C4×D4—101no—
114#117—52no—
115C2×C42C2×D511yescharacteristic
116#107—54no—
117#106—54no—
118C4⋊C8C2×D512yes—
119#102—54no—
120#102—54no—
121C22⋊C8C2×D512yes—
122C4×C8C2×D511yescharacteristic
123C4×C8C2×D511yescharacteristic
124C22×C5—21no—
125C22×C5—21no—
126C22×C5—21no—
127C22×C5C22×D411yescharacteristic
128C2×D5—42no—
129C2×D5—42no—
130C2×D5—42no—
131C4×C5—21no—
132C4×C5—21no—
133C4×C5—21no—
134C4×C5#13512yes—
135C4×C5AGL(1,ℤ/8)12yes—
136C4×C52−1+412yes—
137Dic5—42no—
138Dic5—42no—
139Dic5—42no—
140#41—204no—
141C2×Q8—101no—
142C2×Q8—101no—
143C2×Q8—101no—
144C2×D4—52no—
145C2×D4—102no—
146C2×D4—101no—
147C2×D4—101no—
148C22×C4—102no—
149C22×C4D2011yescharacteristic
150C22×C4D2011yescharacteristic
151C22×C4C22×D511yescharacteristic
152C2×C8C22×D512yes—
153C2×C8C22×D512yes—
154C4⋊C4—52no—
155C4⋊C4—52no—
156C4⋊C4—52no—
157C4⋊C4—101no—
158C4⋊C4—102no—
159C4⋊C4—102no—
160C4⋊C4—101no—
161C4⋊C4—101no—
162C22⋊C4—102no—
163C22⋊C4—102no—
164C22⋊C4—102no—
165C42—102no—
166C42D2011yescharacteristic
167C42D2011yescharacteristic
168C42C22×D511yescharacteristic
169C42C22×D511yescharacteristic
170C2×C5—41no—
171C2×C5#53611yescharacteristic
172C2×C5C2×#13511yescharacteristic
173C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
174D5—82no—
175C23—102no—
176C23C2×D2011yescharacteristic
177Q8—202no—
178Q8—202no—
179Q8—202no—
180D4—104no—
181D4—204no—
182D4—202no—
183D4—202no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—204no—
191C2×C4C2×D2011yescharacteristic
192C2×C4C2×D2011yescharacteristic
193C2×C4C2×D2011yescharacteristic
194C2×C4C2×D2011yescharacteristic
195C2×C4C2×D2011yescharacteristic
196C2×C4C23×D511yescharacteristic
197C8—22no—
198C8—22no—
199C5#322811yescharacteristic, complemented, Sylow
200C22—21no—
201C22—21no—
202C22—21no—
203C22—202no—
204C22—202no—
205C22—202no—
206C22C22×D2011yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—202no—
211C4—202no—
212C4—202no—
213C4#405012yes—
214C4#405412yes—
215C4#415012yes—
216C2—41no—
217C2—402no—
218C2#6586011yescharacteristic
219C2C2×#405011yescharacteristic
220C2C2×#405411yescharacteristic
subgroupG/H
centerC22C22×D20
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#431C2
socleC22×C5C22×D4
prime radical#65811C2
hypercenter#431D5
hypocenterC5#3228
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #65804C2semidirect
3. C5×#431C2semidirect
4. #65136C2semidirect
5. #65026C2semidirect
6. #64983C2semidirect
7. #64983C2semidirect
8. #64971C2semidirect
9. #64969C2semidirect
74. #431D5semidirect
199. C5#3228semidirect

Maximal subgroups

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

subgroupindex
#658112
#658042
C5×#4312
#651362
#650262
#649832
#649712
#649692
#32285