Finite Group Databasebeta

← #13088208

Subgroups — #13088208

Order 1440.

780 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C23×C3⋊D15C211yesmaximal, characteristic, complemented
2C2×C32×C5×D4C211yesmaximal, characteristic, complemented
3C2×#67184C212yesmaximal, complemented
4#10623913C211yesmaximal, characteristic, complemented
5#10623908C212yesmaximal, complemented
6#91826—34nomaximal
7C23×C32×C5C2211yescharacteristic
8C23×C32×C5C2211yescharacteristic, complemented
9C22×C3⋊D15—22no—
10C22×C3⋊D15—24no—
11C22×C3⋊D15C2212yes—
12C32×C5×D4—22no—
13C2×C32×C4×C5C2211yescharacteristic, complemented
14#67184—24no—
15C2×C3⋊Dic15C2212yescomplemented
16#64146—51nomaximal, Hall
17C23×D15—34no—
18C2×C3×C5×D4D314yescomplemented
19C2×#1220—38no—
20#6719—34no—
21#6714—38no—
22C22×C32×C5—22no—
23C22×C32×C5—22no—
24C22×C32×C5—21no—
25C22×C32×C5D412yescomplemented
26C22×C32×C5C2311yescharacteristic
27C2×C3⋊D15—28no—
28C2×C3⋊D15—22no—
29C2×C3⋊D15—22no—
30C2×C3⋊D15D414yescomplemented
31C32×C4×C5—21no—
32C3⋊Dic15—22no—
33#4083—91noHall
34C23×C3⋊D3—51no—
35C2×C32×D4D511yescharacteristic, complemented
36C2×#638—52no—
37#3757—51no—
38#3752—52no—
39C23×C3×C5C2×D314yes—
40C23×C3×C5C2×D314yescomplemented
41C22×D15—38no—
42C22×D15—616no—
43C22×D15—68no—
44C3×C5×D4—28no—
45C2×C3×C4×C5C2×D314yescomplemented
46#1220—616no—
47C2×Dic15—38no—
48#937—154no—
49C2×C32×C5—22no—
50C2×C32×C5—41no—
51C2×C32×C5C2×D412yes—
52C2×C32×C5C2×D411yescharacteristic
53C3⋊D15—24no—
54C23×D5—91no—
55C2×C5×D4C3⋊D311yescharacteristic, complemented
56C2×C22⋊D5—92no—
57C22⋊Dic5—91no—
58#685—92no—
59C23×C32C2×D511yescharacteristic
60C23×C32C2×D511yescharacteristic, complemented
61C22×C3⋊D3—52no—
62C22×C3⋊D3—104no—
63C22×C3⋊D3—102no—
64C32×D4—22no—
65C2×C32×C4C2×D511yescharacteristic, complemented
66#638—104no—
67C2×C3⋊Dic3—52no—
68C22×C3×C5—28no—
69C22×C3×C5—28no—
70C22×C3×C5—24no—
71C22×C3×C5#6718yescomplemented
72C22×C3×C5C22×D314yes—
73C2×D15—316no—
74C2×D15—632no—
75C2×D15—68no—
76C2×D15—68no—
77C3×C4×C5—24no—
78Dic15—68no—
79C23×D3—154no—
80C2×C3×D4D1514yescomplemented
81C2×#67—158no—
82#217—154no—
83(C2×D3)⋊C4—158no—
84C32×C5#12011yescharacteristic, complemented, Hall
85C23×C5C2×C3⋊D311yescharacteristic
86C23×C5C2×C3⋊D311yescharacteristic, complemented
87C22×D5—92no—
88C22×D5—184no—
89C22×D5—182no—
90C5×D4—22no—
91C2×C4×C5C2×C3⋊D311yescharacteristic, complemented
92C22⋊D5—184no—
93C2×Dic5—92no—
94C22×C32—22no—
95C22×C32—22no—
96C22×C32—21no—
97C22×C32C22⋊D512yescomplemented
98C22×C32C22×D511yescharacteristic
99C2×C3⋊D3—54no—
100C2×C3⋊D3—108no—
101C2×C3⋊D3—102no—
102C2×C3⋊D3—102no—
103C32×C4—21no—
104C3⋊Dic3—102no—
105#120—451noSylow
106C2×C3×C5—28no—
107C2×C3×C5—44no—
108C2×C3×C5D3×D418yes—
109C2×C3×C5C2×#6714yes—
110D15—616no—
111C23×C3C2×D1514yes—
112C23×C3C2×D1514yescomplemented
113C22×D3—158no—
114C22×D3—3016no—
115C22×D3—308no—
116C3×D4—28no—
117C2×C3×C4C2×D1514yescomplemented
118#67—3016no—
119C2×Dic3—158no—
120C22×C5—22no—
121C22×C5—22no—
122C22×C5—21no—
123C22×C5#63812yescomplemented
124C22×C5C22×C3⋊D311yescharacteristic
125C2×D5—94no—
126C2×D5—188no—
127C2×D5—182no—
128C2×D5—182no—
129C4×C5—21no—
130Dic5—182no—
131C2×C32—22no—
132C2×C32—41no—
133C2×C32D4×D512yes—
134C2×C32C2×C22⋊D511yescharacteristic
135C3⋊D3—104no—
136C24—451no—
137C2×D4—452no—
138C2×D4C3⋊D1511yescharacteristic, complemented
139C22⋊C4—451no—
140C22⋊C4—452no—
141C3×C5#93714yescomplemented
142C22×C3—28no—
143C22×C3—28no—
144C22×C3—24no—
145C22×C3#122018yescomplemented
146C22×C3C22×D1514yes—
147C2×D3—1516no—
148C2×D3—3032no—
149C2×D3—308no—
150C2×D3—308no—
151C3×C4—24no—
152Dic3—308no—
153C2×C5—22no—
154C2×C5—41no—
155C2×C5D4×C3⋊D312yes—
156C2×C5C2×#63811yescharacteristic
157D5—184no—
158C32#408311yescharacteristic, complemented, Sylow
159C23—452no—
160C23—904no—
161C23—902no—
162C23C2×C3⋊D1511yescharacteristic
163C23C2×C3⋊D1511yescharacteristic, complemented
164D4—22no—
165D4—904no—
166C2×C4—452no—
167C2×C4C2×C3⋊D1511yescharacteristic, complemented
168C2×C3—28no—
169C2×C3—44no—
170C2×C3D4×D1518yes—
171C2×C3C2×#122014yes—
172D3—3016no—
173C5#6414611yescharacteristic, complemented, Sylow
174C22—22no—
175C22—22no—
176C22—21no—
177C22—454no—
178C22—908no—
179C22—902no—
180C22—902no—
181C22#6718412yescomplemented
182C22C22×C3⋊D1511yescharacteristic
183C4—21no—
184C4—902no—
185C3#9182614yescomplemented
186C2—22no—
187C2—41no—
188C2—904no—
189C2D4×C3⋊D1512yes—
190C2C2×#6718411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C5×D4C2
hypercenterC2×D4C3⋊D15
hypocenterC32×C5#120
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊D15C2semidirect
2. C2×C32×C5×D4C2semidirect
3. C2×#67184C2semidirect
4. #10623913C2semidirect
5. #10623908C2semidirect
8. C23×C32×C5C22semidirect
13. C2×C32×C4×C5C22semidirect
15. C2×C3⋊Dic15C22semidirect
18. C2×C3×C5×D4D3semidirect
25. C22×C32×C5D4semidirect
30. C2×C3⋊D15D4semidirect
35. C2×C32×D4D5semidirect
40. C23×C3×C5C2×D3semidirect
45. C2×C3×C4×C5C2×D3semidirect
55. C2×C5×D4C3⋊D3semidirect
60. C23×C32C2×D5semidirect
65. C2×C32×C4C2×D5semidirect
71. C22×C3×C5#67semidirect
80. C2×C3×D4D15semidirect
84. C32×C5#120semidirect
86. C23×C5C2×C3⋊D3semidirect
91. C2×C4×C5C2×C3⋊D3semidirect
97. C22×C32C22⋊D5semidirect
112. C23×C3C2×D15semidirect
117. C2×C3×C4C2×D15semidirect
123. C22×C5#638semidirect
138. C2×D4C3⋊D15semidirect
141. C3×C5#937semidirect
145. C22×C3#1220semidirect
158. C32#4083semidirect
163. C23C2×C3⋊D15semidirect
167. C2×C4C2×C3⋊D15semidirect
173. C5#64146semidirect
181. C22#67184semidirect
185. C3#91826semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊D152
C2×C32×C5×D42
C2×#671842
#106239132
#106239082
#918263
#641465