Finite Group Databasebeta

← C2×C5×#67140 #72231485

Subgroups — C2×C5×#67140 #72231485

Order 3600.

840 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 184 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×C52×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C5×C3⋊D15C211yesmaximal, characteristic, complemented
3C2×C32×C5×Dic5C211yesmaximal, characteristic, complemented
4C5×#67140C214yesmaximal, direct product factor
5C2×C5×#1203—34nomaximal
6C22×C32×C52C2211yescharacteristic
7C2×C5×C3⋊D15—22no—
8C2×C5×C3⋊D15C2212yescomplemented
9C2×C52×C3⋊D3—22no—
10C2×C52×C3⋊D3C2212yescomplemented
11C32×C5×Dic5C2212yescomplemented
12C2×C5×C4⋊(C3⋊D3)—51nomaximal
13C2×#67140C511yesmaximal, characteristic, direct product factor
14C22×C5×D15—34no—
15C22×C52×D3—34no—
16C2×C3×C5×Dic5D314yescomplemented
17C5×#1203—316no—
18C2×C32×C52D412yescomplemented
19C2×C32×C52C2311yescharacteristic
20C5×C3⋊D15—22no—
21C52×C3⋊D3—22no—
22C2×C5×C22⋊D5—91noHall
23C22×C3⋊D15C2×C511yescharacteristic, complemented
24C22×C5×C3⋊D3—22no—
25C22×C5×C3⋊D3—51no—
26C22×C5×C3⋊D3D511yescharacteristic, complemented
27C22×C5×C3⋊D3C2×C511yescharacteristic, complemented
28C2×C32×C4×C5—51no—
29C5×C4⋊(C3⋊D3)—54no—
30C2×C32×Dic5C2×C511yescharacteristic, complemented
31#67140C2×C514yesdirect product factor
32C22×C3×C52C2×D314yes—
33C2×C5×D15—38no—
34C2×C5×D15—68no—
35C2×C52×D3—38no—
36C2×C52×D3—68no—
37C3×C5×Dic5C2×D318yescomplemented
38C2×C5×D12—154no—
39C2×#1203—34no—
40C32×C52C2×D411yescharacteristic, complemented, Hall
41C23×C52—91no—
42C22×C5×D5—91no—
43C5×C22⋊D5—94no—
44C2×C5×Dic5C3⋊D311yescharacteristic, complemented
45C22×C32×C5—22no—
46C22×C32×C5C2×D511yescharacteristic
47C22×C32×C5C22×C511yescharacteristic
48C2×C3⋊D15—22no—
49C2×C3⋊D15C22×C512yescomplemented
50C2×C5×C3⋊D3—24no—
51C2×C5×C3⋊D3—24no—
52C2×C5×C3⋊D3—22no—
53C2×C5×C3⋊D3—22no—
54C2×C5×C3⋊D3—24no—
55C2×C5×C3⋊D3—52no—
56C2×C5×C3⋊D3—102no—
57C2×C5×C3⋊D3C2×D512yescomplemented
58C2×C5×C3⋊D3C22×C512yescomplemented
59C32×C4×C5—52no—
60C32×Dic5C22×C512yescomplemented
61C2×C3×C52D1218yescomplemented
62C2×C3×C52C22×D314yes—
63C5×D15—68no—
64C52×D3—68no—
65C2×C4⋊(C3⋊D3)—51noHall
66C22×D15—34no—
67C22×C5×D3—34no—
68C22×C5×D3—34no—
69C22×C5×D3—68no—
70C22×C5×D3—154no—
71C2×C3×C4×C5—54no—
72C5×D12—1516no—
73C2×C3×Dic5C5×D314yescomplemented
74#1203—316no—
75C22×C52—92no—
76C22×C52—182no—
77C22×C52C2×C3⋊D311yescharacteristic
78C2×C5×D5—92no—
79C2×C5×D5—182no—
80C5×Dic5C2×C3⋊D312yescomplemented
81C2×C32×C5—24no—
82C2×C32×C5—22no—
83C2×C32×C5C22⋊D512yescomplemented
84C2×C32×C5C5×D412yescomplemented
85C2×C32×C5C22×D511yescharacteristic
86C2×C32×C5C23×C511yescharacteristic
87C3⋊D15—22no—
88C5×C3⋊D3—24no—
89C5×C3⋊D3—24no—
90C5×C3⋊D3—22no—
91C5×C3⋊D3—22no—
92C5×C3⋊D3—102no—
93C2×C5×D4—451no—
94C2×C22⋊D5—91no—
95C3×C52C2×D1214yescomplemented
96C22×C3⋊D3—51no—
97C22×C3⋊D3C5×D511yescharacteristic, complemented
98C2×C32×C4—51no—
99C4⋊(C3⋊D3)—54no—
100C22×C3×C5—28no—
101C22×C3×C5D3×D514yes—
102C22×C3×C5C2×C5×D314yes—
103C2×D15—38no—
104C2×D15—68no—
105C2×C5×D3—38no—
106C2×C5×D3—38no—
107C2×C5×D3—616no—
108C2×C5×D3—616no—
109C2×C5×D3—616no—
110C2×C5×D3—68no—
111C2×C5×D3—68no—
112C2×C5×D3—158no—
113C2×C5×D3—308no—
114C3×C4×C5—58no—
115C3×Dic5C2×C5×D318yescomplemented
116C2×C52—182no—
117C2×C52C4⋊(C3⋊D3)12yescomplemented
118C2×C52C22×C3⋊D311yescharacteristic
119C5×D5—182no—
120C2×D12—154no—
121C32×C5—22no—
122C32×C5C2×C22⋊D511yescharacteristic, complemented
123C32×C5C2×C5×D411yescharacteristic, complemented
124C23×C5—91no—
125C23×C5—91no—
126C23×C5—182no—
127C23×C5—451no—
128C22×D5—91no—
129C5×D4—454no—
130C2×C4×C5—51no—
131C22⋊D5—94no—
132C2×Dic5C5×C3⋊D311yescharacteristic, complemented
133C22×C32C2×C5×D511yescharacteristic
134C2×C3⋊D3—22no—
135C2×C3⋊D3—52no—
136C2×C3⋊D3—102no—
137C2×C3⋊D3C2×C5×D512yescomplemented
138C32×C4—52no—
139C2×C3×C5—28no—
140C2×C3×C5—216no—
141C2×C3×C5#120318yescomplemented
142C2×C3×C5C5×D1218yescomplemented
143C2×C3×C5C2×D3×D514yes—
144C2×C3×C5C22×C5×D314yes—
145D15—68no—
146C5×D3—616no—
147C5×D3—616no—
148C5×D3—68no—
149C5×D3—68no—
150C5×D3—308no—
151C52C2×C4⋊(C3⋊D3)11yescharacteristic, complemented, Sylow
152C22×D3—34no—
153C22×D3—154no—
154C2×C3×C4—54no—
155D12—1516no—
156C22×C5—22no—
157C22×C5—92no—
158C22×C5—92no—
159C22×C5—184no—
160C22×C5—184no—
161C22×C5—184no—
162C22×C5—182no—
163C22×C5—182no—
164C22×C5—452no—
165C22×C5—902no—
166C22×C5D5×C3⋊D311yescharacteristic
167C22×C5C2×C5×C3⋊D311yescharacteristic
168C2×D5—92no—
169C2×D5—182no—
170C4×C5—52no—
171Dic5C2×C5×C3⋊D312yescomplemented
172C2×C32C5×C22⋊D512yescomplemented
173C2×C32C22×C5×D511yescharacteristic
174C3⋊D3—22no—
175C3⋊D3—102no—
176C2×D4—451noSylow
177C3×C5—28no—
178C3×C5C2×#120314yescomplemented
179C3×C5C2×C5×D1214yescomplemented
180C22×C3C5×D3×D514yes—
181C2×D3—38no—
182C2×D3—68no—
183C2×D3—158no—
184C2×D3—308no—
185C3×C4—58no—
186C2×C5—24no—
187C2×C5—22no—
188C2×C5—184no—
189C2×C5—184no—
190C2×C5—182no—
191C2×C5—182no—
192C2×C5—902no—
193C2×C5#6714012yesdirect product factor
194C2×C5C5×C4⋊(C3⋊D3)12yescomplemented
195C2×C5C2×D5×C3⋊D311yescharacteristic
196C2×C5C22×C5×C3⋊D311yescharacteristic
197D5—182no—
198C32C2×C5×C22⋊D511yescharacteristic, complemented, Sylow
199C23—91no—
200C23—451no—
201D4—454no—
202C2×C4—51no—
203C2×C3C5×#120318yescomplemented
204C2×C3C2×C5×D3×D514yes—
205D3—68no—
206D3—308no—
207C5—22no—
208C5C2×#6714011yescharacteristic, direct product factor
209C5C2×C5×C4⋊(C3⋊D3)11yescharacteristic, complemented
210C22—92no—
211C22—182no—
212C22—452no—
213C22—902no—
214C22C5×D5×C3⋊D311yescharacteristic
215C4—52no—
216C3C2×C5×#120314yescomplemented
217C2—182no—
218C2—902no—
219C2C5×#6714012yesdirect product factor
220C2C2×C5×D5×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22×C5D5×C3⋊D3
commutator, cosocleC2×C32×C5C23×C5
FrattiniC2C2×C5×D5×C3⋊D3
Fitting, socleC22×C32×C52C22
hypocenterC32×C5C2×C5×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
1. C22×C52×C3⋊D3C2semidirect
2. C22×C5×C3⋊D15C2semidirect
3. C2×C32×C5×Dic5C2semidirect
4. C5×#67140C2direct (N × H)
8. C2×C5×C3⋊D15C22semidirect
10. C2×C52×C3⋊D3C22semidirect
11. C32×C5×Dic5C22semidirect
13. C2×#67140C5direct (N × H)
16. C2×C3×C5×Dic5D3semidirect
18. C2×C32×C52D4semidirect
23. C22×C3⋊D15C2×C5semidirect
26. C22×C5×C3⋊D3D5semidirect
27. C22×C5×C3⋊D3C2×C5semidirect
30. C2×C32×Dic5C2×C5semidirect
31. #67140C2×C5direct (N × H)
37. C3×C5×Dic5C2×D3semidirect
40. C32×C52C2×D4semidirect
44. C2×C5×Dic5C3⋊D3semidirect
49. C2×C3⋊D15C22×C5semidirect
57. C2×C5×C3⋊D3C2×D5semidirect
58. C2×C5×C3⋊D3C22×C5semidirect
60. C32×Dic5C22×C5semidirect
61. C2×C3×C52D12semidirect
73. C2×C3×Dic5C5×D3semidirect
80. C5×Dic5C2×C3⋊D3semidirect
83. C2×C32×C5C22⋊D5semidirect
84. C2×C32×C5C5×D4semidirect
95. C3×C52C2×D12semidirect
97. C22×C3⋊D3C5×D5semidirect
115. C3×Dic5C2×C5×D3semidirect
117. C2×C52C4⋊(C3⋊D3)semidirect
122. C32×C5C2×C22⋊D5semidirect
123. C32×C5C2×C5×D4semidirect
132. C2×Dic5C5×C3⋊D3semidirect
137. C2×C3⋊D3C2×C5×D5semidirect
141. C2×C3×C5#1203semidirect
142. C2×C3×C5C5×D12semidirect
151. C52C2×C4⋊(C3⋊D3)semidirect
171. Dic5C2×C5×C3⋊D3semidirect
172. C2×C32C5×C22⋊D5semidirect
178. C3×C5C2×#1203semidirect
179. C3×C5C2×C5×D12semidirect
193. C2×C5#67140direct (N × H)
194. C2×C5C5×C4⋊(C3⋊D3)semidirect
198. C32C2×C5×C22⋊D5semidirect
203. C2×C3C5×#1203semidirect
208. C5C2×#67140direct (N × H)
209. C5C2×C5×C4⋊(C3⋊D3)semidirect
216. C3C2×C5×#1203semidirect
219. C2C5×#67140direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C52×C3⋊D32
C22×C5×C3⋊D152
C2×C32×C5×Dic52
C5×#671402
C2×C5×#12033
C2×C5×C4⋊(C3⋊D3)5
C2×#671405