Finite Group Databasebeta

← C5×#64090 #13088065

Subgroups — C5×#64090 #13088065

Order 1440.

564 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 174 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×C5×#638C211yesmaximal, characteristic
2C22×C5×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C4×C5×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×C5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#3752C211yesmaximal, characteristic, complemented
6C5×#3750C211yesmaximal, characteristic, complemented
7C4×C5×C3⋊Dic3C211yesmaximal, characteristic, complemented
8C5×#67⋊C4—34nomaximal
9C23×C32×C5C2211yescharacteristic
10C22×C5×C3⋊D3C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C5×#638C414yescomplemented
14C2×C5×C3⋊Dic3—22no—
15C2×C5×C3⋊Dic3C2211yescharacteristic
16C2×C5×C3⋊Dic3C2211yescharacteristic
17C2×C5×C3⋊Dic3C2211yescharacteristic
18C4×C5×C3⋊D3—22no—
19#64090C511yesmaximal, characteristic, direct product factor, Hall
20C2×C5×#67—34no—
21C22×C5×Dic3—34no—
22C2×C4×C5×D3—34no—
23C3×C5×C22⋊C4D314yescomplemented
24C5×(C2×D3)⋊C4—34no—
25C5×Dic3⋊C4—34no—
26C4×C5×Dic3—34no—
27C22×C32×C5—21no—
28C22×C32×C5—21no—
29C22×C32×C5C2×C412yescomplemented
30C22×C32×C5C2311yescharacteristic
31C2×C5×C3⋊D3—21no—
32C2×C5×C3⋊D3—21no—
33C2×C5×C3⋊D3C2×C412yescomplemented
34C32×C4×C5—21no—
35C32×C4×C5—21no—
36C5×C3⋊Dic3—21no—
37C5×C3⋊Dic3C2×C412yescomplemented
38C5×C3⋊Dic3D412yescomplemented
39C4×C5×D4—91noHall
40C2×#638C2×C511yescharacteristic
41C22×C3⋊Dic3C2×C511yescharacteristic, complemented
42C2×C4×C3⋊D3C2×C511yescharacteristic, complemented
43C32×C22⋊C4C2×C511yescharacteristic, complemented
44#3752C2×C511yescharacteristic, complemented
45#3750C2×C511yescharacteristic, complemented
46C4×C3⋊Dic3C2×C511yescharacteristic, complemented
47C23×C3×C5C2×D314yes—
48C22×C5×D3—34no—
49C2×C3×C4×C5C2×D314yes—
50C2×C3×C4×C5C2×D314yes—
51C5×#67—316no—
52C2×C5×Dic3—34no—
53C2×C5×Dic3—34no—
54C2×C5×Dic3—34no—
55C2×C5×Dic3—68no—
56C4×C5×D3—68no—
57#67⋊C4—34no—
58C2×C32×C5—22no—
59C2×C32×C5C22×C411yescharacteristic
60C2×C32×C5C2×D411yescharacteristic
61C2×C32×C5#4111yescharacteristic
62C5×C3⋊D3—22no—
63C2×C5×D4—91no—
64C22×C4×C5—91no—
65C22×C4×C5—91no—
66C5×C4⋊C4—91no—
67C5×C22⋊C4—91no—
68C5×C22⋊C4C3⋊D311yescharacteristic, complemented
69C42×C5—91no—
70C23×C32C22×C511yescharacteristic
71C22×C3⋊D3C22×C511yescharacteristic
72C2×C32×C4C22×C511yescharacteristic
73C2×C32×C4C22×C511yescharacteristic
74#638C4×C514yescomplemented
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3C22×C511yescharacteristic
77C2×C3⋊Dic3C22×C511yescharacteristic
78C2×C3⋊Dic3C22×C511yescharacteristic
79C4×C3⋊D3—22no—
80C22×C3×C5—24no—
81C22×C3×C5—24no—
82C22×C3×C5C4×D318yescomplemented
83C22×C3×C5C22×D314yes—
84C2×C5×D3—38no—
85C2×C5×D3—64no—
86C2×C5×D3—64no—
87C3×C4×C5—24no—
88C3×C4×C5—24no—
89C5×Dic3—38no—
90C5×Dic3—38no—
91C5×Dic3—64no—
92C2×#67—34no—
93C22×Dic3—34no—
94C2×C4×D3—34no—
95C3×C22⋊C4C5×D314yescomplemented
96(C2×D3)⋊C4—34no—
97Dic3⋊C4—34no—
98C4×Dic3—34no—
99C32×C5C4×D411yescharacteristic, complemented, Hall
100C23×C5—91no—
101C23×C5C2×C3⋊D311yescharacteristic
102C5×D4—94no—
103C2×C4×C5—91no—
104C2×C4×C5—91no—
105C2×C4×C5—91no—
106C2×C4×C5—182no—
107C2×C4×C5—182no—
108C2×C4×C5C2×C3⋊D311yescharacteristic
109C2×C4×C5C2×C3⋊D311yescharacteristic
110C22×C32—21no—
111C22×C32—21no—
112C22×C32C2×C4×C512yescomplemented
113C22×C32C23×C511yescharacteristic
114C2×C3⋊D3—21no—
115C2×C3⋊D3—21no—
116C2×C3⋊D3C2×C4×C512yescomplemented
117C32×C4—21no—
118C32×C4—21no—
119C3⋊Dic3—21no—
120C3⋊Dic3C2×C4×C512yescomplemented
121C3⋊Dic3C5×D412yescomplemented
122C4×D4—91noSylow
123C2×C3×C5—28no—
124C2×C3×C5C2×C4×D314yes—
125C2×C3×C5D3×D414yes—
126C2×C3×C5#23714yes—
127C5×D3—68no—
128C23×C3C2×C5×D314yes—
129C22×D3—34no—
130C2×C3×C4C2×C5×D314yes—
131C2×C3×C4C2×C5×D314yes—
132#67—316no—
133C2×Dic3—34no—
134C2×Dic3—34no—
135C2×Dic3—34no—
136C2×Dic3—68no—
137C4×D3—68no—
138C22×C5—21no—
139C22×C5—21no—
140C22×C5—92no—
141C22×C5—181no—
142C22×C5—181no—
143C22×C5C4×C3⋊D312yescomplemented
144C22×C5C22×C3⋊D311yescharacteristic
145C4×C5—21no—
146C4×C5—21no—
147C4×C5—92no—
148C4×C5—92no—
149C4×C5—181no—
150C2×C32—22no—
151C2×C32C22×C4×C511yescharacteristic
152C2×C32C2×C5×D411yescharacteristic
153C2×C32C5×#4111yescharacteristic
154C3⋊D3—22no—
155C2×D4—91no—
156C22×C4—91no—
157C22×C4—91no—
158C4⋊C4—91no—
159C22⋊C4—91no—
160C22⋊C4C5×C3⋊D311yescharacteristic, complemented
161C42—91no—
162C3×C5#67⋊C414yescomplemented
163C22×C3—24no—
164C22×C3—24no—
165C22×C3C4×C5×D318yescomplemented
166C22×C3C22×C5×D314yes—
167C2×D3—38no—
168C2×D3—64no—
169C2×D3—64no—
170C3×C4—24no—
171C3×C4—24no—
172Dic3—38no—
173Dic3—38no—
174Dic3—64no—
175C2×C5—22no—
176C2×C5—182no—
177C2×C5C2×C4×C3⋊D311yescharacteristic
178C2×C5D4×C3⋊D311yescharacteristic
179C2×C5#383011yescharacteristic
180C32C4×C5×D411yescharacteristic, complemented, Sylow
181C23—91no—
182C23C2×C5×C3⋊D311yescharacteristic
183D4—94no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—182no—
188C2×C4—182no—
189C2×C4C2×C5×C3⋊D311yescharacteristic
190C2×C4C2×C5×C3⋊D311yescharacteristic
191C2×C3—28no—
192C2×C3C2×C4×C5×D314yes—
193C2×C3C5×D3×D414yes—
194C2×C3C5×#23714yes—
195D3—68no—
196C5#6409011yescharacteristic, direct product factor, Sylow
197C22—21no—
198C22—21no—
199C22—92no—
200C22—181no—
201C22—181no—
202C22C4×C5×C3⋊D312yescomplemented
203C22C22×C5×C3⋊D311yescharacteristic
204C4—21no—
205C4—21no—
206C4—92no—
207C4—92no—
208C4—181no—
209C3C5×#67⋊C414yescomplemented
210C2—22no—
211C2—182no—
212C2C2×C4×C5×C3⋊D311yescharacteristic
213C2C5×D4×C3⋊D311yescharacteristic
214C2C5×#383011yescharacteristic
subgroupG/H
centerC22×C5C22×C3⋊D3
commutatorC2×C32C22×C4×C5
FrattiniC22C22×C5×C3⋊D3
FittingC32×C5×C22⋊C4C2
socleC22×C32×C5C23
prime radicalC2×C5×#638C2
cosocleC22×C32C23×C5
hypercenterC5×C22⋊C4C3⋊D3
hypocenterC32C4×C5×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C5×C3⋊Dic3C2semidirect
3. C2×C4×C5×C3⋊D3C2semidirect
4. C32×C5×C22⋊C4C2semidirect
5. C5×#3752C2semidirect
6. C5×#3750C2semidirect
7. C4×C5×C3⋊Dic3C2semidirect
13. C5×#638C4semidirect
19. #64090C5direct (N × H)
23. C3×C5×C22⋊C4D3semidirect
29. C22×C32×C5C2×C4semidirect
33. C2×C5×C3⋊D3C2×C4semidirect
37. C5×C3⋊Dic3C2×C4semidirect
38. C5×C3⋊Dic3D4semidirect
41. C22×C3⋊Dic3C2×C5semidirect
42. C2×C4×C3⋊D3C2×C5semidirect
43. C32×C22⋊C4C2×C5semidirect
44. #3752C2×C5semidirect
45. #3750C2×C5semidirect
46. C4×C3⋊Dic3C2×C5semidirect
68. C5×C22⋊C4C3⋊D3semidirect
74. #638C4×C5semidirect
82. C22×C3×C5C4×D3semidirect
95. C3×C22⋊C4C5×D3semidirect
99. C32×C5C4×D4semidirect
112. C22×C32C2×C4×C5semidirect
116. C2×C3⋊D3C2×C4×C5semidirect
120. C3⋊Dic3C2×C4×C5semidirect
121. C3⋊Dic3C5×D4semidirect
143. C22×C5C4×C3⋊D3semidirect
160. C22⋊C4C5×C3⋊D3semidirect
162. C3×C5#67⋊C4semidirect
165. C22×C3C4×C5×D3semidirect
180. C32C4×C5×D4semidirect
196. C5#64090direct (N × H)
202. C22C4×C5×C3⋊D3semidirect
209. C3C5×#67⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#6382
C22×C5×C3⋊Dic32
C2×C4×C5×C3⋊D32
C32×C5×C22⋊C42
C5×#37522
C5×#37502
C4×C5×C3⋊Dic32
C5×#67⋊C43
#640905