Finite Group Databasebeta

← #13086864

Subgroups — #13086864

Order 1440.

456 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 126 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×C3⋊D3×Dic5C211yesmaximal, characteristic
2C4×C3⋊Dic15C211yesmaximal, characteristic, complemented
3C5×#3752C211yesmaximal, characteristic
4C32×#683C211yesmaximal, characteristic, complemented
5#10623773C211yesmaximal, characteristic, complemented
6#10623771C211yesmaximal, characteristic
7Dic5×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#91400—34nomaximal
9C22×C5×C3⋊D3C2211yescharacteristic
10C2×C32×C4×C5C2211yescharacteristic
11C2×C3⋊Dic15C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C32×Dic5C2211yescharacteristic
15C2×C32×Dic5C2211yescharacteristic
16C3⋊D3×Dic5C414yescomplemented
17#64101—51nomaximal, Hall
18C2×D3×Dic5—34no—
19C4×Dic15—34no—
20C5×(C2×D3)⋊C4—34no—
21C3×#683D314yescomplemented
22#6668—34no—
23(C2×D3)⋊Dic5—34no—
24Dic3×Dic5—34no—
25C22×C32×C5C2311yescharacteristic
26C2×C5×C3⋊D3—21no—
27C2×C5×C3⋊D3—21no—
28C2×C5×C3⋊D3C2×C412yes—
29C32×C4×C5—21no—
30C3⋊Dic15—21no—
31C3⋊Dic15C2×C412yes—
32C5×C3⋊Dic3—21no—
33C32×Dic5—21no—
34C32×Dic5C2×C412yes—
35#4023—91noHall
36C2×C4×C3⋊D3—51no—
37C32×C4⋊C4—51no—
38#3752—51no—
39#3752D511yescharacteristic
40C4⋊(C3⋊Dic3)—51no—
41C4×C3⋊Dic3—51no—
42C4×C3⋊Dic3—51no—
43C22×C5×D3—34no—
44C2×C3×C4×C5C2×D314yes—
45C2×Dic15—34no—
46C2×Dic15—34no—
47C2×C5×Dic3—34no—
48C2×C3×Dic5C2×D314yes—
49C2×C3×Dic5C2×D314yes—
50D3×Dic5—316no—
51#892—154no—
52C2×C32×C5C22×C411yescharacteristic
53C2×C32×C5#4111yescharacteristic
54C2×C32×C5#4111yescharacteristic
55C5×C3⋊D3—22no—
56C22×Dic5—91no—
57C5×C22⋊C4—91no—
58C22⋊Dic5—91no—
59#683—91no—
60#683C3⋊D311yescharacteristic, complemented
61C4×Dic5—91no—
62C4×Dic5—91no—
63C22×C3⋊D3C2×D511yescharacteristic
64C2×C32×C4—51no—
65C2×C32×C4—51no—
66C2×C32×C4C2×D511yescharacteristic
67C2×C3⋊Dic3—51no—
68C2×C3⋊Dic3—51no—
69C2×C3⋊Dic3C2×D511yescharacteristic
70C4×C3⋊D3—54no—
71C22×C3×C5C22×D314yes—
72C2×C5×D3—38no—
73C2×C5×D3—64no—
74C2×C5×D3—64no—
75C3×C4×C5—24no—
76Dic15—38no—
77Dic15—64no—
78C3×Dic5—24no—
79C3×Dic5C4×D318yes—
80C5×Dic3—64no—
81C2×C4×D3—154no—
82C3×C4⋊C4—54no—
83(C2×D3)⋊C4—34no—
84(C2×D3)⋊C4—154no—
85C4⋊Dic3—154no—
86C4×Dic3—154no—
87C4×Dic3—154no—
88C32×C5#11711yescharacteristic, complemented, Hall
89C23×C5—91no—
90C2×C4×C5—91no—
91C2×C4×C5C2×C3⋊D311yescharacteristic
92C2×Dic5—91no—
93C2×Dic5—94no—
94C2×Dic5—91no—
95C2×Dic5C2×C3⋊D311yescharacteristic
96C2×Dic5C2×C3⋊D311yescharacteristic
97C22×C32C22×D511yescharacteristic
98C2×C3⋊D3—21no—
99C2×C3⋊D3—21no—
100C2×C3⋊D3C4×D512yes—
101C32×C4—21no—
102C32×C4—52no—
103C32×C4—101no—
104C3⋊Dic3—21no—
105C3⋊Dic3—52no—
106C3⋊Dic3—101no—
107#117—451noSylow
108C2×C3×C5C2×C4×D314yes—
109C2×C3×C5#23714yes—
110C2×C3×C5#23914yes—
111C5×D3—68no—
112C22×D3—34no—
113C2×C3×C4—54no—
114C2×C3×C4—54no—
115C2×C3×C4D3×D514yes—
116C2×Dic3—34no—
117C2×Dic3—154no—
118C2×Dic3—154no—
119C4×D3—1516no—
120C22×C5—92no—
121C22×C5—181no—
122C22×C5—181no—
123C22×C5C22×C3⋊D311yescharacteristic
124C4×C5—21no—
125C4×C5—181no—
126Dic5—21no—
127Dic5—92no—
128Dic5—181no—
129Dic5C4×C3⋊D312yes—
130C2×C32C2×C4×D511yescharacteristic
131C2×C32#71111yescharacteristic
132C2×C32#71111yescharacteristic
133C3⋊D3—22no—
134C22×C4—451no—
135C4⋊C4—51no—
136C4⋊C4—451no—
137C22⋊C4—91no—
138C22⋊C4—451no—
139C42—451no—
140C42—451no—
141C3×C5#89214yescomplemented
142C22×C3C2×D3×D514yes—
143C2×D3—38no—
144C2×D3—64no—
145C2×D3—64no—
146C3×C4—24no—
147C3×C4—58no—
148C3×C4—104no—
149Dic3—64no—
150Dic3—158no—
151Dic3—304no—
152C2×C5—182no—
153C2×C5C2×C4×C3⋊D311yescharacteristic
154C2×C5#383011yescharacteristic
155C2×C5#383211yescharacteristic
156C32#402311yescharacteristic, complemented, Sylow
157C23—91no—
158C2×C4—51no—
159C2×C4—51no—
160C2×C4—91no—
161C2×C4—454no—
162C2×C4—451no—
163C2×C4—451no—
164C2×C4D5×C3⋊D311yescharacteristic
165C2×C3#676814yes—
166C2×C3C4×D3×D514yes—
167C2×C3#678014yes—
168D3—68no—
169C5#6410111yescharacteristic, complemented, Sylow
170C22—92no—
171C22—181no—
172C22—181no—
173C22C2×D5×C3⋊D311yescharacteristic
174C4—21no—
175C4—52no—
176C4—101no—
177C4—181no—
178C4—452no—
179C4—901no—
180C3#9140014yescomplemented
181C2—182no—
182C2#1062415111yescharacteristic
183C2C4×D5×C3⋊D311yescharacteristic
184C2#1062416311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×C3⋊D3
commutatorC2×C32×C5C22×C4
FittingC2×C32×C4×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC22×C5×C3⋊D3C22
hypercenterC2×C4D5×C3⋊D3
hypocenterC32×C5#117
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. C4×C3⋊Dic15C2semidirect
4. C32×#683C2semidirect
5. #10623773C2semidirect
7. Dic5×C3⋊Dic3C2semidirect
16. C3⋊D3×Dic5C4semidirect
21. C3×#683D3semidirect
60. #683C3⋊D3semidirect
88. C32×C5#117semidirect
141. C3×C5#892semidirect
156. C32#4023semidirect
169. C5#64101semidirect
180. C3#91400semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic52
C4×C3⋊Dic152
C5×#37522
C32×#6832
#106237732
#106237712
Dic5×C3⋊Dic32
#914003
#641015