Finite Group Databasebeta

← C5×#63983 #13087255

Subgroups — C5×#63983 #13087255

Order 1440.

574 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×C5×D32C211yesmaximal, characteristic, complemented
2C2×C5×#638C211yesmaximal, characteristic, complemented
3C2×C3×C5×#67C212yesmaximal, complemented
4C5×#3722C211yesmaximal, characteristic, complemented
5C5×(C2×D3)⋊Dic3C212yesmaximal, complemented
6C5×#937—32nomaximal
7C23×C32×C5C2211yescharacteristic, complemented
8C22×C5×C3⋊D3C2211yescharacteristic
9C22×C3×C5×D3C2212yes—
10C2×C5×D32—22no—
11C2×C5×D32—24no—
12C5×#638—22no—
13C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
14C3×C5×#67—24no—
15C2×C3×C5×Dic3C2212yescomplemented
16#63983C511yesmaximal, characteristic, direct product factor, Hall
17C23×C5×D3—32no—
18C2×C3×C5×D4—32no—
19C2×C5×#67—32no—
20C2×C5×#67—61no—
21C2×C5×#67D312yescomplemented
22C5×#217—32no—
23C5×(C2×D3)⋊C4—32no—
24C5×(C2×D3)⋊C4—32no—
25C22×C32×C5—21no—
26C22×C32×C5—22no—
27C22×C32×C5C2311yescharacteristic
28C2×C5×C3⋊D3—22no—
29C2×C5×C3⋊D3D412yescomplemented
30C2×C3×C5×D3—22no—
31C2×C3×C5×D3—22no—
32C2×C3×C5×D3D414yescomplemented
33C5×D32—28no—
34C5×C3⋊Dic3—21no—
35C3×C5×Dic3—22no—
36C5×#120—91noHall
37C22×D32C2×C511yescharacteristic, complemented
38C2×#638C2×C511yescharacteristic, complemented
39C2×C3×#67C2×C512yescomplemented
40#3722C2×C511yescharacteristic, complemented
41(C2×D3)⋊Dic3C2×C512yescomplemented
42C23×C3×C5—21no—
43C23×C3×C5—32no—
44C23×C3×C5C2×D312yescomplemented
45C22×C5×D3—32no—
46C22×C5×D3—64no—
47C22×C5×D3—64no—
48C22×C5×D3—64no—
49C22×C5×D3—61no—
50C22×C5×D3C2×D312yes—
51C3×C5×D4—64no—
52C2×C3×C4×C5—32no—
53C5×#67—24no—
54C5×#67—64no—
55C5×#67—64no—
56C2×C5×Dic3—32no—
57C2×C5×Dic3—61no—
58C2×C5×Dic3C2×D312yescomplemented
59#937—32no—
60C2×C32×C5—41no—
61C2×C32×C5C2×D411yescharacteristic
62C2×C32×C5C2×D412yes—
63C5×C3⋊D3—22no—
64C3×C5×D3—24no—
65C24×C5—91no—
66C2×C5×D4—32no—
67C2×C5×D4—91no—
68C5×C22⋊C4—91no—
69C5×C22⋊C4—92no—
70C23×C32C22×C511yescharacteristic, complemented
71C22×C3⋊D3C22×C511yescharacteristic
72C22×C3×D3C22×C512yes—
73C2×D32—22no—
74C2×D32—24no—
75#638—22no—
76C2×C3⋊Dic3C22×C511yescharacteristic, complemented
77C3×#67—24no—
78C2×C3×Dic3C22×C512yescomplemented
79C22×C3×C5—22no—
80C22×C3×C5—22no—
81C22×C3×C5—22no—
82C22×C3×C5—22no—
83C22×C3×C5—21no—
84C22×C3×C5—34no—
85C22×C3×C5—42no—
86C22×C3×C5—62no—
87C22×C3×C5—62no—
88C22×C3×C5C22×D312yes—
89C2×C5×D3—22no—
90C2×C5×D3—22no—
91C2×C5×D3—34no—
92C2×C5×D3—62no—
93C2×C5×D3—616no—
94C2×C5×D3—62no—
95C2×C5×D3—62no—
96C2×C5×D3—122no—
97C2×C5×D3#6714yescomplemented
98C3×C4×C5—62no—
99C5×Dic3—22no—
100C5×Dic3—62no—
101C5×Dic3—62no—
102C23×D3—32no—
103C2×C3×D4—32no—
104C2×#67—32no—
105C2×#67—61no—
106C2×#67C5×D312yescomplemented
107#217—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110C32×C5#12011yescharacteristic, complemented, Hall
111C23×C5—32no—
112C23×C5—91no—
113C23×C5—184no—
114C23×C5—182no—
115C23×C5D3211yescharacteristic, complemented
116C5×D4—64no—
117C5×D4—182no—
118C2×C4×C5—32no—
119C2×C4×C5—91no—
120C22×C32—21no—
121C22×C32—22no—
122C22×C32C23×C511yescharacteristic
123C2×C3⋊D3—22no—
124C2×C3⋊D3C5×D412yescomplemented
125C2×C3×D3—22no—
126C2×C3×D3—22no—
127C2×C3×D3C5×D414yescomplemented
128D32—28no—
129C3⋊Dic3—21no—
130C3×Dic3—22no—
131#120—91noSylow
132C2×C3×C5—22no—
133C2×C3×C5—21no—
134C2×C3×C5—42no—
135C2×C3×C5—42no—
136C2×C3×C5—64no—
137C2×C3×C5D3×D412yes—
138C2×C3×C5D3×D412yes—
139C2×C3×C5C2×#6712yes—
140C5×D3—24no—
141C5×D3—64no—
142C5×D3—122no—
143C23×C3—21no—
144C23×C3—32no—
145C23×C3C2×C5×D312yescomplemented
146C22×D3—32no—
147C22×D3—61no—
148C22×D3—64no—
149C22×D3—64no—
150C22×D3—64no—
151C22×D3C2×C5×D312yes—
152C3×D4—64no—
153C2×C3×C4—32no—
154#67—24no—
155#67—64no—
156#67—64no—
157C2×Dic3—32no—
158C2×Dic3—61no—
159C2×Dic3C2×C5×D312yescomplemented
160C22×C5—22no—
161C22×C5—21no—
162C22×C5—34no—
163C22×C5—62no—
164C22×C5—62no—
165C22×C5—92no—
166C22×C5—188no—
167C22×C5—182no—
168C22×C5C2×D3211yescharacteristic
169C4×C5—62no—
170C4×C5—181no—
171C2×C32—41no—
172C2×C32C2×C5×D411yescharacteristic
173C2×C32C2×C5×D412yes—
174C3⋊D3—22no—
175C3×D3—24no—
176C24—91no—
177C2×D4—32no—
178C2×D4—91no—
179C22⋊C4—91no—
180C22⋊C4—92no—
181C3×C5—21no—
182C3×C5#93712yescomplemented
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—22no—
186C22×C3—22no—
187C22×C3—21no—
188C22×C3—34no—
189C22×C3—42no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3C22×C5×D312yes—
193C2×D3—22no—
194C2×D3—22no—
195C2×D3—34no—
196C2×D3—62no—
197C2×D3—616no—
198C2×D3—62no—
199C2×D3—62no—
200C2×D3—122no—
201C2×D3C5×#6714yescomplemented
202C3×C4—62no—
203Dic3—22no—
204Dic3—62no—
205Dic3—62no—
206C2×C5—41no—
207C2×C5—64no—
208C2×C5—182no—
209C2×C5D3×#6712yes—
210C2×C5#381111yescharacteristic
211C32C5×#12011yescharacteristic, complemented, Sylow
212C23—32no—
213C23—91no—
214C23—184no—
215C23—182no—
216C23C5×D3211yescharacteristic, complemented
217D4—64no—
218D4—182no—
219C2×C4—32no—
220C2×C4—91no—
221C2×C3—22no—
222C2×C3—21no—
223C2×C3—42no—
224C2×C3—42no—
225C2×C3—64no—
226C2×C3C5×D3×D412yes—
227C2×C3C5×D3×D412yes—
228C2×C3C2×C5×#6712yes—
229D3—24no—
230D3—64no—
231D3—122no—
232C5#6398311yescharacteristic, direct product factor, Sylow
233C22—22no—
234C22—21no—
235C22—34no—
236C22—62no—
237C22—62no—
238C22—92no—
239C22—188no—
240C22—182no—
241C22C2×C5×D3211yescharacteristic
242C4—62no—
243C4—181no—
244C3—21no—
245C3C5×#93712yescomplemented
246C2—41no—
247C2—64no—
248C2—182no—
249C2C5×D3×#6712yes—
250C2C5×#381111yescharacteristic
subgroupG/H
centerC22×C5C2×D32
commutator, cosocleC22×C32C23×C5
FrattiniC22C2×C5×D32
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23×C5D32
hypocenterC32C5×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
1. C22×C5×D32C2semidirect
2. C2×C5×#638C2semidirect
3. C2×C3×C5×#67C2semidirect
4. C5×#3722C2semidirect
5. C5×(C2×D3)⋊Dic3C2semidirect
7. C23×C32×C5C22semidirect
13. C2×C5×C3⋊Dic3C22semidirect
15. C2×C3×C5×Dic3C22semidirect
16. #63983C5direct (N × H)
21. C2×C5×#67D3semidirect
29. C2×C5×C3⋊D3D4semidirect
32. C2×C3×C5×D3D4semidirect
37. C22×D32C2×C5semidirect
38. C2×#638C2×C5semidirect
39. C2×C3×#67C2×C5semidirect
40. #3722C2×C5semidirect
41. (C2×D3)⋊Dic3C2×C5semidirect
44. C23×C3×C5C2×D3semidirect
58. C2×C5×Dic3C2×D3semidirect
70. C23×C32C22×C5semidirect
76. C2×C3⋊Dic3C22×C5semidirect
78. C2×C3×Dic3C22×C5semidirect
97. C2×C5×D3#67semidirect
106. C2×#67C5×D3semidirect
110. C32×C5#120semidirect
115. C23×C5D32semidirect
124. C2×C3⋊D3C5×D4semidirect
127. C2×C3×D3C5×D4semidirect
145. C23×C3C2×C5×D3semidirect
159. C2×Dic3C2×C5×D3semidirect
182. C3×C5#937semidirect
201. C2×D3C5×#67semidirect
211. C32C5×#120semidirect
216. C23C5×D32semidirect
232. C5#63983direct (N × H)
245. C3C5×#937semidirect

Maximal subgroups

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

subgroupindex
C22×C5×D322
C2×C5×#6382
C2×C3×C5×#672
C5×#37222
C5×(C2×D3)⋊Dic32
C5×#9373
#639835