Finite Group Databasebeta

← #10604429

Subgroups — #10604429

Order 640.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#65912C211yesmaximal, characteristic
2#65689C211yesmaximal, characteristic
3#65680C211yesmaximal, characteristic
4#64926C212yesmaximal
5#64869C212yesmaximal
6C2×C4⋊AGL(1,𝔽5)—21no—
7C4×C5×Q8C411yescharacteristic, complemented
8#4092C2211yescharacteristic
9#4090C411yescharacteristic, complemented
10#4043C2211yescharacteristic
11#4042—21no—
12#4035—21no—
13#4034C411yescharacteristic, complemented
14#4018—21no—
15C42×D5C2211yescharacteristic
16C4×Dic10—21no—
17C4×Dic10C411yescharacteristic, complemented
18(C2×AGL(1,𝔽5))⋊C4—21no—
19(C2×AGL(1,𝔽5))⋊C4C2212yes—
20#4003C2212yes—
21C4⋊#170—21no—
22C4×#170—21no—
23#1518—51nomaximal, Sylow
24C22×AGL(1,𝔽5)—22no—
25C2×C5×Q8C2×C411yescharacteristic
26C2×C4×D5D411yescharacteristic
27C2×C4×D5D411yescharacteristic
28C2×C4×D5C2311yescharacteristic
29C2×Dic10—21no—
30C2×Dic10C2×C411yescharacteristic
31C2×#170—22no—
32C4⋊AGL(1,𝔽5)—42no—
33C5×C4⋊C4—21no—
34C5×C4⋊C4C2×C411yescharacteristic
35C42×C5—21no—
36C42×C5C2×C411yescharacteristic
37#685—21no—
38#685—21no—
39C4⋊Dic5—21no—
40C4⋊Dic5C2×C411yescharacteristic
41#683—21no—
42#683—21no—
43#683—21no—
44#683—21no—
45#683—21no—
46C4×Dic5—21no—
47C4×Dic5C2×C411yescharacteristic
48C4×Dic5D411yescharacteristic
49C4×Dic5D411yescharacteristic
50#533—51no—
51#431—51no—
52#382—51no—
53#328—52no—
54#328—52no—
55C22×D5C2×D411yescharacteristic
56C2×AGL(1,𝔽5)—42no—
57C2×AGL(1,𝔽5)—42no—
58C2×AGL(1,𝔽5)—42no—
59C5×Q8—41no—
60C2×C4×C5—21no—
61C2×C4×C5—21no—
62C2×C4×C5C22⋊C411yescharacteristic
63C2×C4×C5C22⋊C411yescharacteristic
64C2×C4×C5C22×C411yescharacteristic
65C2×Dic5—21no—
66C2×Dic5—21no—
67C2×Dic5C22⋊C411yescharacteristic
68C2×Dic5C22⋊C411yescharacteristic
69C2×Dic5C2×D411yescharacteristic
70C4×D5—41no—
71C4×D5—41no—
72C4×D5SD1612yes—
73C4×D5Q1612yes—
74Dic10—42no—
75Dic10—41no—
76#170—42no—
77C4⋊Q8—51no—
78C4⋊Q8—51no—
79#125—101no—
80C22⋊Q8—51no—
81C22⋊Q8—51no—
82C22⋊Q8—101no—
83C4×Q8—51no—
84C4×Q8—101no—
85C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
86#117—101no—
87C2×C4⋊C4—101no—
88C2×C42—51no—
89C4⋊C8—101no—
90C22⋊C8—52no—
91C4×C8—101no—
92#95—52no—
93#95—101no—
94C22×C5C2×C22⋊C411yescharacteristic
95C2×D5—21no—
96C2×D5—21no—
97C2×D5C2×SD1611yescharacteristic
98C2×D5C2×Q1611yescharacteristic
99AGL(1,𝔽5)—82no—
100C4×C5—21no—
101C4×C5—21no—
102C4×C5—41no—
103C4×C5—41no—
104C4×C5#10312yes—
105Dic5—21no—
106Dic5—21no—
107Dic5—41no—
108Dic5—41no—
109Dic5#10312yes—
110C2×Q8—51no—
111C2×Q8—101no—
112C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
113C22×C4—51no—
114C22×C4—51no—
115C22×C4—51no—
116C22×C4—102no—
117C2×C8—102no—
118C4⋊C4—21no—
119C4⋊C4—51no—
120C4⋊C4—101no—
121C4⋊C4—101no—
122C4⋊C4—101no—
123C4⋊C4—101no—
124C4⋊C4—101no—
125C4⋊C4—101no—
126C4⋊C4—202no—
127C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
128C22⋊C4—101no—
129C22⋊C4—101no—
130C42—21no—
131C42—51no—
132C42—51no—
133C42—51no—
134C42—101no—
135C42C2×AGL(1,𝔽5)11yescharacteristic
136C2×C5#41011yescharacteristic
137C2×C5C2×#10311yescharacteristic
138C2×C5#42111yescharacteristic
139D5—21no—
140D5—21no—
141C23—51no—
142Q8—41no—
143Q8—201no—
144Q8—202no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—51no—
148C2×C4—52no—
149C2×C4—52no—
150C2×C4—51no—
151C2×C4—51no—
152C2×C4—101no—
153C2×C4—101no—
154C2×C4—201no—
155C2×C4—201no—
156C2×C4—202no—
157C2×C4—202no—
158C2×C4—202no—
159C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
160C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
161C2×C4C22×AGL(1,𝔽5)11yescharacteristic
162C8—202no—
163C5#151811yescharacteristic, complemented, Sylow
164C22—51no—
165C22—51no—
166C22—101no—
167C22—101no—
168C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—52no—
174C4—101no—
175C4—101no—
176C4—201no—
177C4—201no—
178C4—402no—
179C4#400912yes—
180C2—101no—
181C2—101no—
182C2#6574511yescharacteristic
183C2C2×#400911yescharacteristic
184C2#6579511yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC4×C5×Q8C4
socleC22×C5C2×C22⋊C4
prime radicalC22×D5C2×D4
cosocleC2×C4×D5C23
hypercenterC4×Q8AGL(1,𝔽5)
hypocenterC5#1518
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
7. C4×C5×Q8C4semidirect
9. #4090C4semidirect
13. #4034C4semidirect
17. C4×Dic10C4semidirect
85. C4×Q8AGL(1,𝔽5)semidirect
163. C5#1518semidirect

Maximal subgroups

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

subgroupindex
#659122
#656892
#656802
#649262
#648692
#15185