Finite Group Databasebeta

← D5×#3752 #13086954

Subgroups — D5×#3752 #13086954

Order 1440.

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

normal subgroups only

This group has 792 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
1C22×D5×C3⋊D3C211yesmaximal, characteristic
2C2×C32×C4×D5C211yesmaximal, characteristic, complemented
3C2×D5×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10623908C211yesmaximal, characteristic, complemented
5C5×#3752C211yesmaximal, characteristic, complemented
6#10623772C211yesmaximal, characteristic, complemented
7#10623771C211yesmaximal, characteristic, complemented
8D5×(C2×D3)⋊C4—34nomaximal
9C22×C3⋊D15C2211yescharacteristic
10C22×C5×C3⋊D3C2211yescharacteristic
11C22×C32×D5C2211yescharacteristic
12C2×D5×C3⋊D3—22no—
13C2×D5×C3⋊D3—22no—
14C2×D5×C3⋊D3C414yescomplemented
15C2×C32×C4×C5C2211yescharacteristic, complemented
16C2×C3⋊Dic15C2211yescharacteristic, complemented
17C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
18C2×C32×Dic5C2211yescharacteristic, complemented
19C32×C4×D5—22no—
20D5×C3⋊Dic3—22no—
21C2×#3752—51nomaximal, Hall
22C22×D3×D5—34no—
23C2×C3×C4×D5D314yescomplemented
24C2×D5×Dic3—34no—
25#6714—34no—
26C5×(C2×D3)⋊C4—34no—
27#6667—34no—
28(C2×D3)⋊Dic5—34no—
29C22×C32×C5C2311yescharacteristic
30C2×C3⋊D15—21no—
31C2×C3⋊D15—21no—
32C2×C3⋊D15C2×C412yescomplemented
33C2×C5×C3⋊D3—21no—
34C2×C5×C3⋊D3—21no—
35C2×C5×C3⋊D3C2×C412yescomplemented
36C2×C32×D5C2×C412yes—
37C2×C32×D5D412yes—
38C2×C32×D5D412yes—
39D5×C3⋊D3—28no—
40C32×C4×C5—21no—
41C3⋊Dic15—21no—
42C5×C3⋊Dic3—21no—
43C32×Dic5—21no—
44D5×C22⋊C4—91noHall
45C23×C3⋊D3—51no—
46C22×C32×C4—51no—
47C22×C3⋊Dic3—51no—
48#3752—51no—
49#3752—51no—
50#3752—51no—
51#3752D511yescharacteristic, direct product factor
52C22×D15—34no—
53C22×C5×D3—34no—
54C22×C3×D5C2×D314yes—
55C2×D3×D5—316no—
56C2×D3×D5—68no—
57C2×D3×D5—68no—
58C2×C3×C4×C5C2×D314yescomplemented
59C2×Dic15—34no—
60C2×C5×Dic3—34no—
61C2×C3×Dic5C2×D314yescomplemented
62C3×C4×D5—28no—
63D5×Dic3—68no—
64C2×(C2×D3)⋊C4—154no—
65C2×C32×C5C22×C411yescharacteristic
66C2×C32×C5C2×D411yescharacteristic
67C2×C32×C5C2×D411yescharacteristic
68C3⋊D15—22no—
69C5×C3⋊D3—22no—
70C32×D5C22⋊C414yescomplemented
71C23×D5—91no—
72C2×C4×D5—91no—
73C2×C4×D5C3⋊D311yescharacteristic, complemented
74C5×C22⋊C4—91no—
75C22⋊Dic5—91no—
76#685—91no—
77#685—91no—
78C23×C32—51no—
79C22×C3⋊D3—51no—
80C22×C3⋊D3—54no—
81C22×C3⋊D3—102no—
82C22×C3⋊D3—102no—
83C22×C3⋊D3C2×D511yescharacteristic
84C2×C32×C4—51no—
85C2×C32×C4—102no—
86C2×C32×C4C2×D511yescharacteristic, complemented
87C2×C3⋊Dic3—51no—
88C2×C3⋊Dic3—102no—
89C2×C3⋊Dic3C2×D511yescharacteristic, complemented
90C22×C3×C5C22×D314yes—
91C2×D15—38no—
92C2×D15—64no—
93C2×D15—64no—
94C2×C5×D3—38no—
95C2×C5×D3—64no—
96C2×C5×D3—64no—
97C2×C3×D5C4×D318yes—
98C2×C3×D5D1218yes—
99C2×C3×D5#6718yes—
100D3×D5—632no—
101C3×C4×C5—24no—
102Dic15—64no—
103C3×Dic5—24no—
104C5×Dic3—64no—
105C23×D3—154no—
106C22×C3×C4—54no—
107C22×Dic3—154no—
108(C2×D3)⋊C4—34no—
109(C2×D3)⋊C4—154no—
110(C2×D3)⋊C4—154no—
111(C2×D3)⋊C4—154no—
112C32×C5C2×C22⋊C411yescharacteristic, complemented, Hall
113C23×C5—91no—
114C22×D5—91no—
115C22×D5—94no—
116C22×D5—182no—
117C22×D5—182no—
118C22×D5C2×C3⋊D311yescharacteristic
119C2×C4×C5—91no—
120C2×C4×C5C2×C3⋊D311yescharacteristic, complemented
121C2×Dic5—91no—
122C2×Dic5C2×C3⋊D311yescharacteristic, complemented
123C4×D5—22no—
124C4×D5—182no—
125C22×C32—52no—
126C22×C32—52no—
127C22×C32—52no—
128C22×C32C22×D511yescharacteristic
129C2×C3⋊D3—21no—
130C2×C3⋊D3—21no—
131C2×C3⋊D3—52no—
132C2×C3⋊D3—108no—
133C2×C3⋊D3—101no—
134C2×C3⋊D3—101no—
135C2×C3⋊D3C4×D512yescomplemented
136C32×C4—21no—
137C32×C4—101no—
138C3⋊Dic3—21no—
139C3⋊Dic3—101no—
140C2×C22⋊C4—451noSylow
141C2×C3×C5C2×C4×D314yes—
142C2×C3×C5C2×D1214yes—
143C2×C3×C5C2×#6714yes—
144D15—68no—
145C3×D5(C2×D3)⋊C4116yescomplemented
146C5×D3—68no—
147C23×C3—54no—
148C22×D3—34no—
149C22×D3—1516no—
150C22×D3—154no—
151C22×D3—308no—
152C22×D3—308no—
153C2×C3×C4—54no—
154C2×C3×C4—108no—
155C2×C3×C4D3×D514yescomplemented
156C2×Dic3—34no—
157C2×Dic3—154no—
158C2×Dic3—308no—
159C22×C5—92no—
160C22×C5—181no—
161C22×C5—181no—
162C22×C5C22×C3⋊D311yescharacteristic
163C2×D5—92no—
164C2×D5—181no—
165C2×D5—181no—
166C2×D5—188no—
167C2×D5C4×C3⋊D312yes—
168C2×D5C4⋊(C3⋊D3)12yes—
169C2×D5#63812yes—
170C4×C5—21no—
171C4×C5—181no—
172Dic5—21no—
173Dic5—181no—
174C2×C32—54no—
175C2×C32C2×C4×D511yescharacteristic
176C2×C32D4×D511yescharacteristic
177C2×C32D4×D511yescharacteristic
178C3⋊D3—22no—
179C3⋊D3—102no—
180C24—451no—
181C22×C4—51no—
182C22×C4—451no—
183C22⋊C4—91no—
184C22⋊C4—451no—
185C22⋊C4—451no—
186C22⋊C4—451no—
187C3×C5C2×(C2×D3)⋊C414yescomplemented
188C22×C3—58no—
189C22×C3—58no—
190C22×C3—58no—
191C22×C3C2×D3×D514yes—
192C2×D3—38no—
193C2×D3—64no—
194C2×D3—64no—
195C2×D3—158no—
196C2×D3—3032no—
197C2×D3—304no—
198C2×D3—304no—
199C3×C4—24no—
200C3×C4—104no—
201Dic3—64no—
202Dic3—304no—
203C2×C5—182no—
204C2×C5C2×C4×C3⋊D311yescharacteristic
205C2×C5C2×C4⋊(C3⋊D3)11yescharacteristic
206C2×C5C2×#63811yescharacteristic
207D5—182no—
208D5#375214yesdirect product factor
209C32D5×C22⋊C411yescharacteristic, complemented, Sylow
210C23—51no—
211C23—91no—
212C23—454no—
213C23—451no—
214C23—902no—
215C23—902no—
216C2×C4—51no—
217C2×C4—91no—
218C2×C4—102no—
219C2×C4—451no—
220C2×C4—902no—
221C2×C4D5×C3⋊D311yescharacteristic, complemented
222C2×C3—516no—
223C2×C3C4×D3×D514yes—
224C2×C3D5×D1214yes—
225C2×C3D5×#6714yes—
226D3—68no—
227D3—308no—
228C5C2×#375211yescharacteristic, complemented, Sylow
229C22—52no—
230C22—52no—
231C22—52no—
232C22—92no—
233C22—181no—
234C22—181no—
235C22—452no—
236C22—908no—
237C22—901no—
238C22—901no—
239C22C2×D5×C3⋊D311yescharacteristic
240C4—21no—
241C4—101no—
242C4—181no—
243C4—901no—
244C3D5×(C2×D3)⋊C414yescomplemented
245C2—54no—
246C2—182no—
247C2—902no—
248C2C4×D5×C3⋊D311yescharacteristic
249C2D5×C4⋊(C3⋊D3)11yescharacteristic
250C2D5×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×C3⋊D3
commutatorC2×C32×C5C22×C4
FittingC2×C32×C4×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC22×D5×C3⋊D3C2
hypercenterC2×C4D5×C3⋊D3
hypocenterC32×C5C2×C22⋊C4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×D5C2semidirect
3. C2×D5×C3⋊Dic3C2semidirect
4. #10623908C2semidirect
5. C5×#3752C2semidirect
6. #10623772C2semidirect
7. #10623771C2semidirect
14. C2×D5×C3⋊D3C4semidirect
15. C2×C32×C4×C5C22semidirect
16. C2×C3⋊Dic15C22semidirect
17. C2×C5×C3⋊Dic3C22semidirect
18. C2×C32×Dic5C22semidirect
23. C2×C3×C4×D5D3semidirect
32. C2×C3⋊D15C2×C4semidirect
35. C2×C5×C3⋊D3C2×C4semidirect
51. #3752D5direct (N × H)
58. C2×C3×C4×C5C2×D3semidirect
61. C2×C3×Dic5C2×D3semidirect
70. C32×D5C22⋊C4semidirect
73. C2×C4×D5C3⋊D3semidirect
86. C2×C32×C4C2×D5semidirect
89. C2×C3⋊Dic3C2×D5semidirect
112. C32×C5C2×C22⋊C4semidirect
120. C2×C4×C5C2×C3⋊D3semidirect
122. C2×Dic5C2×C3⋊D3semidirect
135. C2×C3⋊D3C4×D5semidirect
145. C3×D5(C2×D3)⋊C4semidirect
155. C2×C3×C4D3×D5semidirect
187. C3×C5C2×(C2×D3)⋊C4semidirect
208. D5#3752direct (N × H)
209. C32D5×C22⋊C4semidirect
221. C2×C4D5×C3⋊D3semidirect
228. C5C2×#3752semidirect
244. C3D5×(C2×D3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×D5×C3⋊D32
C2×C32×C4×D52
C2×D5×C3⋊Dic32
#106239082
C5×#37522
#106237722
#106237712
D5×(C2×D3)⋊C43
C2×#37525