Finite Group Databasebeta

← #10604957

Subgroups — #10604957

Order 640.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#4092C211yesmaximal, characteristic
2#65764C211yesmaximal, characteristic
3#65755C211yesmaximal, characteristic
4#64926C214yesmaximal, complemented
5C22×C5×Q8C411yescharacteristic, complemented
6C22×C4×D5C2211yescharacteristic
7C2×C22⋊AGL(1,𝔽5)—21no—
8C2×#704—21no—
9#4092—21no—
10#4092—21no—
11#4092C412yescomplemented
12#4092C2212yes—
13C2×#685—21no—
14C2×C4⋊Dic5C411yescharacteristic, complemented
15C2×#683—21no—
16(C2×AGL(1,𝔽5))⋊C4—21no—
17(C2×AGL(1,𝔽5))⋊C4C2212yes—
18#4003—21no—
19#4003C2212yes—
20#1511—51nomaximal, Sylow
21C23×D5D411yescharacteristic
22C22×AGL(1,𝔽5)—22no—
23C2×C5×Q8—21no—
24C2×C5×Q8—42no—
25C2×C5×Q8C2×C412yes—
26C22×C4×C5—21no—
27C22×C4×C5C2×C411yescharacteristic
28C22×Dic5—21no—
29C22×Dic5D411yescharacteristic
30C2×C4×D5—42no—
31C2×C4×D5C2×C411yescharacteristic
32C2×C4×D5D411yescharacteristic
33C2×C4×D5D411yescharacteristic
34C2×C4×D5C2311yescharacteristic
35C22⋊AGL(1,𝔽5)—42no—
36#704—42no—
37C2×#170—22no—
38#685—22no—
39#685—22no—
40C4⋊Dic5—21no—
41C4⋊Dic5C2×C412yes—
42#683—22no—
43#683—22no—
44C2×C22⋊Q8—51no—
45#407—51no—
46#381—51no—
47#328—54no—
48C23×C5C22⋊C411yescharacteristic
49C22×D5—22no—
50C22×D5—22no—
51C22×D5—41no—
52C22×D5C22⋊C411yescharacteristic
53C22×D5C2×D411yescharacteristic
54C2×AGL(1,𝔽5)—42no—
55C2×AGL(1,𝔽5)—42no—
56C2×AGL(1,𝔽5)—42no—
57C5×Q8—42no—
58C5×Q8—42no—
59C2×C4×C5—22no—
60C2×C4×C5—22no—
61C2×C4×C5—42no—
62C2×C4×C5C22⋊C411yescharacteristic
63C2×C4×C5C22×C411yescharacteristic
64C2×Dic5—22no—
65C2×Dic5—22no—
66C2×Dic5—42no—
67C2×Dic5C22⋊C411yescharacteristic
68C2×Dic5C2×D411yescharacteristic
69C4×D5—41no—
70C4×D5—41no—
71C4×D5—41no—
72C4×D5—41no—
73#170—42no—
74C22×Q8AGL(1,𝔽5)11yescharacteristic, complemented
75C23×C4—51no—
76C2×M16—101no—
77C22⋊Q8—52no—
78C22⋊Q8—52no—
79C22⋊Q8—101no—
80C22⋊Q8—101no—
81C2×C4⋊C4—51no—
82C2×C4⋊C4—101no—
83C2×C22⋊C4—101no—
84C2×C22⋊C4—101no—
85C22⋊C8—52no—
86C22⋊C8—101no—
87#95—52no—
88#95—101no—
89C22×C5—21no—
90C22×C5—21no—
91C22×C5#9912yes—
92C22×C5C2×C22⋊C411yescharacteristic
93C2×D5—21no—
94C2×D5—21no—
95C2×D5—21no—
96C2×D5—21no—
97C2×D5—21no—
98C2×D5—44no—
99C2×D5#13712yes—
100AGL(1,𝔽5)—82no—
101C4×C5—21no—
102C4×C5—21no—
103C4×C5—42no—
104Dic5—21no—
105Dic5—21no—
106Dic5—42no—
107C24—51no—
108C2×Q8—21no—
109C2×Q8—42no—
110C2×Q8C2×AGL(1,𝔽5)12yes—
111C22×C4—21no—
112C22×C4—51no—
113C22×C4—51no—
114C22×C4—51no—
115C22×C4—51no—
116C22×C4—51no—
117C22×C4—101no—
118C22×C4—102no—
119C22×C4—202no—
120C22×C4C2×AGL(1,𝔽5)11yescharacteristic
121M16—202no—
122C2×C8—102no—
123C4⋊C4—52no—
124C4⋊C4—102no—
125C4⋊C4—102no—
126C4⋊C4—101no—
127C22⋊C4—102no—
128C22⋊C4—102no—
129C22⋊C4—202no—
130C2×C5—22no—
131C2×C5C2×#9911yescharacteristic
132C2×C5#41911yescharacteristic
133C2×C5#42111yescharacteristic
134D5—22no—
135D5—41no—
136C23—51no—
137C23—51no—
138C23—102no—
139C23—102no—
140C23—201no—
141C23C22⋊AGL(1,𝔽5)11yescharacteristic
142Q8—42no—
143Q8—42no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—51no—
148C2×C4—51no—
149C2×C4—102no—
150C2×C4—102no—
151C2×C4—201no—
152C2×C4—201no—
153C2×C4—201no—
154C2×C4—201no—
155C2×C4—202no—
156C2×C4—202no—
157C2×C4—202no—
158C2×C4—202no—
159C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
160C2×C4C22×AGL(1,𝔽5)11yescharacteristic
161C8—202no—
162C5#151111yescharacteristic, complemented, Sylow
163C22—21no—
164C22—21no—
165C22—52no—
166C22—101no—
167C22—101no—
168C22—101no—
169C22—101no—
170C22—101no—
171C22—204no—
172C22#401112yes—
173C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—101no—
178C4—101no—
179C4—202no—
180C4—402no—
181C2—22no—
182C2—102no—
183C2—201no—
184C2#6578211yescharacteristic
185C2#6578411yescharacteristic
186C2C2×#401111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC22×C5×Q8C4
socleC22×C5C2×C22⋊C4
prime radicalC23×D5D4
cosocleC2×C4×D5C23
hypercenterC22×Q8AGL(1,𝔽5)
hypocenterC5#1511
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
4. #64926C2semidirect
5. C22×C5×Q8C4semidirect
11. #4092C4semidirect
14. C2×C4⋊Dic5C4semidirect
74. C22×Q8AGL(1,𝔽5)semidirect
162. C5#1511semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
#657642
#657552
#649262
#15115