Finite Group Databasebeta

← #10604912

Subgroups — #10604912

Order 640.

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

normal subgroups only

This group has 326 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×#4084C211yesmaximal, characteristic
2#65764C211yesmaximal, characteristic, complemented
3#65755C211yesmaximal, characteristic, complemented
4#64920C214yesmaximal, complemented
5C22×C5×D4C411yescharacteristic, complemented
6C22×C22⋊D5—21no—
7C22×C4×D5C2211yescharacteristic
8C2×C22⋊AGL(1,𝔽5)—21no—
9C2×#704—21no—
10C2×C22⋊Dic5—21no—
11#4084—21no—
12#4084—21no—
13#4084C412yescomplemented
14#4084C2212yes—
15C2×C4⋊Dic5C411yescharacteristic, complemented
16(C2×AGL(1,𝔽5))⋊C4—21no—
17(C2×AGL(1,𝔽5))⋊C4C2212yescomplemented
18#4003—21no—
19#4003C2212yescomplemented
20#1510—51nomaximal, Sylow
21C24×C5—21no—
22C23×D5D411yescharacteristic
23C22×AGL(1,𝔽5)—22no—
24C2×C5×D4—21no—
25C2×C5×D4—42no—
26C2×C5×D4C2×C412yes—
27C22×C4×C5C2×C411yescharacteristic
28C2×C22⋊D5—22no—
29C2×C22⋊D5—22no—
30C2×C22⋊D5—44no—
31C22×Dic5—21no—
32C22×Dic5D411yescharacteristic
33C2×C4×D5—42no—
34C2×C4×D5C2×C411yescharacteristic
35C2×C4×D5D411yescharacteristic
36C2×C4×D5D411yescharacteristic
37C2×C4×D5C2311yescharacteristic
38C22⋊AGL(1,𝔽5)—42no—
39#704—42no—
40C2×#170—22no—
41C22⋊Dic5—22no—
42C22⋊Dic5—22no—
43C4⋊Dic5—21no—
44C4⋊Dic5C2×C412yes—
45C2×#121—51no—
46#407—51no—
47#381—51no—
48#327—54no—
49C23×C5—22no—
50C23×C5—42no—
51C23×C5—42no—
52C23×C5—42no—
53C23×C5C22⋊C411yescharacteristic
54C22×D5—22no—
55C22×D5—22no—
56C22×D5—41no—
57C22×D5C22⋊C411yescharacteristic
58C22×D5C2×D411yescharacteristic
59C2×AGL(1,𝔽5)—42no—
60C2×AGL(1,𝔽5)—42no—
61C2×AGL(1,𝔽5)—42no—
62C5×D4—42no—
63C5×D4—42no—
64C2×C4×C5—22no—
65C2×C4×C5C22⋊C411yescharacteristic
66C2×C4×C5C22×C411yescharacteristic
67C22⋊D5—44no—
68C22⋊D5—44no—
69C2×Dic5—22no—
70C2×Dic5—22no—
71C2×Dic5—42no—
72C2×Dic5C22⋊C411yescharacteristic
73C2×Dic5C2×D411yescharacteristic
74C4×D5—41no—
75C4×D5—41no—
76C4×D5—41no—
77C4×D5—41no—
78#170—42no—
79C22×D4—101no—
80C22×D4AGL(1,𝔽5)11yescharacteristic, complemented
81C23×C4—51no—
82C2×M16—101no—
83#121—52no—
84#121—52no—
85#121—101no—
86#121—101no—
87C2×C4⋊C4—51no—
88C2×C22⋊C4—101no—
89C2×C22⋊C4—101no—
90C22⋊C8—52no—
91C22⋊C8—101no—
92#95—52no—
93#95—101no—
94C22×C5—21no—
95C22×C5—21no—
96C22×C5—42no—
97C22×C5—42no—
98C22×C5—42no—
99C22×C5—84no—
100C22×C5#9912yes—
101C22×C5C2×C22⋊C411yescharacteristic
102C2×D5—21no—
103C2×D5—21no—
104C2×D5—21no—
105C2×D5—21no—
106C2×D5—21no—
107C2×D5—44no—
108C2×D5AGL(1,ℤ/8)12yes—
109AGL(1,𝔽5)—82no—
110C4×C5—21no—
111C4×C5—21no—
112Dic5—21no—
113Dic5—21no—
114Dic5—42no—
115C24—21no—
116C24—51no—
117C2×D4—21no—
118C2×D4—42no—
119C2×D4—102no—
120C2×D4—102no—
121C2×D4—204no—
122C2×D4C2×AGL(1,𝔽5)12yes—
123C22×C4—51no—
124C22×C4—51no—
125C22×C4—51no—
126C22×C4—51no—
127C22×C4—51no—
128C22×C4—101no—
129C22×C4—102no—
130C22×C4—202no—
131C22×C4C2×AGL(1,𝔽5)11yescharacteristic
132M16—202no—
133C2×C8—102no—
134C4⋊C4—52no—
135C4⋊C4—101no—
136C22⋊C4—102no—
137C22⋊C4—102no—
138C22⋊C4—202no—
139C2×C5—22no—
140C2×C5—82no—
141C2×C5C2×#9911yescharacteristic
142C2×C5#41811yescharacteristic
143C2×C5#42111yescharacteristic
144D5—22no—
145D5—41no—
146C23—22no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—51no—
151C23—51no—
152C23—102no—
153C23—102no—
154C23—201no—
155C23C22⋊AGL(1,𝔽5)11yescharacteristic
156D4—42no—
157D4—42no—
158D4—204no—
159D4—204no—
160C2×C4—22no—
161C2×C4—51no—
162C2×C4—51no—
163C2×C4—102no—
164C2×C4—102no—
165C2×C4—201no—
166C2×C4—201no—
167C2×C4—201no—
168C2×C4—201no—
169C2×C4—202no—
170C2×C4—202no—
171C2×C4—202no—
172C2×C4—202no—
173C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
174C2×C4C22×AGL(1,𝔽5)11yescharacteristic
175C8—202no—
176C5#151011yescharacteristic, complemented, Sylow
177C22—21no—
178C22—21no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—52no—
183C22—84no—
184C22—101no—
185C22—101no—
186C22—101no—
187C22—101no—
188C22—101no—
189C22—204no—
190C22#401112yes—
191C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
192C4—21no—
193C4—21no—
194C4—101no—
195C4—101no—
196C4—202no—
197C4—402no—
198C2—22no—
199C2—82no—
200C2—102no—
201C2—201no—
202C2#6576711yescharacteristic
203C2#6576911yescharacteristic
204C2C2×#401111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC22×C5×D4C4
socleC22×C5C2×C22⋊C4
prime radicalC2×#4084C2
cosocleC2×C4×D5C23
hypercenterC22×D4AGL(1,𝔽5)
hypocenterC5#1510
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #65764C2semidirect
3. #65755C2semidirect
4. #64920C2semidirect
5. C22×C5×D4C4semidirect
13. #4084C4semidirect
15. C2×C4⋊Dic5C4semidirect
17. (C2×AGL(1,𝔽5))⋊C4C22semidirect
19. #4003C22semidirect
80. C22×D4AGL(1,𝔽5)semidirect
176. C5#1510semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
#657642
#657552
#649202
#15105