Finite Group Databasebeta

← #13088661

Subgroups — #13088661

Order 1440.

456 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C4×D5×C3⋊D3C211yesmaximal, characteristic
2C4⋊(C3⋊ΓL(1,𝔽16))C211yesmaximal, characteristic, complemented
3C32×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4(C2×AGL(1,𝔽5))⋊(C3⋊D3)C212yesmaximal
5AGL(1,𝔽5)×C3⋊Dic3C212yesmaximal, complemented
6#91907—34nomaximal
7C2×D5×C3⋊D3C2211yescharacteristic
8C2×C3⋊ΓL(1,𝔽16)C2212yes—
9C2×C32×AGL(1,𝔽5)C2212yes—
10C4×C3⋊D15C411yescharacteristic, complemented
11C4×C5×C3⋊D3C411yescharacteristic, complemented
12C32×C4×D5C2211yescharacteristic
13#67137C411yescharacteristic, complemented
14C3⋊D3×Dic5C411yescharacteristic, complemented
15D5×C3⋊Dic3C2211yescharacteristic
16#64101—51nomaximal, Hall
17C4×D3×D5—34no—
18C4⋊ΓL(1,𝔽16)—34no—
19C3×C4⋊AGL(1,𝔽5)D314yescomplemented
20(C2×D3)⋊AGL(1,𝔽5)—38no—
21Dic3×AGL(1,𝔽5)—38no—
22C2×C3⋊D15C2×C411yescharacteristic
23C2×C5×C3⋊D3C2×C411yescharacteristic
24C2×C32×D5C2311yescharacteristic
25D5×C3⋊D3—21no—
26D5×C3⋊D3—21no—
27C3⋊ΓL(1,𝔽16)—22no—
28C32×AGL(1,𝔽5)—22no—
29C32×C4×C5C2×C411yescharacteristic
30C3⋊Dic15C2×C411yescharacteristic
31C5×C3⋊Dic3C2×C411yescharacteristic
32C32×Dic5C2×C411yescharacteristic
33#4130—91noHall
34C2×C4×C3⋊D3—51no—
35C32×C4⋊C4—51no—
36#3752—52no—
37C4⋊(C3⋊Dic3)—51no—
38C4×C3⋊Dic3—52no—
39C2×D3×D5—34no—
40C2×ΓL(1,𝔽16)—38no—
41C2×C3×AGL(1,𝔽5)C2×D318yes—
42C4×D15—34no—
43C4×C5×D3—34no—
44C3×C4×D5C2×D314yes—
45#1200—34no—
46D3×Dic5—34no—
47D5×Dic3—34no—
48#892—154no—
49C2×C32×C5C22×C411yescharacteristic
50C3⋊D15—21no—
51C5×C3⋊D3—21no—
52C32×D5#4111yescharacteristic
53C32×D5#4111yescharacteristic
54C2×C4×D5—91no—
55C22⋊AGL(1,𝔽5)—92no—
56C4⋊AGL(1,𝔽5)—91no—
57C4⋊AGL(1,𝔽5)C3⋊D311yescharacteristic, complemented
58C4×AGL(1,𝔽5)—92no—
59C22×C3⋊D3—51no—
60C2×C32×C4—51no—
61C2×C32×C4—52no—
62C2×C3⋊Dic3—51no—
63C2×C3⋊Dic3—52no—
64C4×C3⋊D3—51no—
65C4×C3⋊D3—51no—
66C4×C3⋊D3—51no—
67C4×C3⋊D3AGL(1,𝔽5)11yescharacteristic, complemented
68C2×D15—34no—
69C2×C5×D3—34no—
70C2×C3×D5C22×D314yes—
71D3×D5—64no—
72D3×D5—64no—
73ΓL(1,𝔽16)—68no—
74C3×AGL(1,𝔽5)—28no—
75C3×C4×C5C4×D314yes—
76Dic15—34no—
77C3×Dic5C4×D314yes—
78C5×Dic3—34no—
79C2×C4×D3—154no—
80C3×C4⋊C4—54no—
81(C2×D3)⋊C4—158no—
82C4⋊Dic3—154no—
83C4×Dic3—158no—
84C32×C5#11711yescharacteristic, complemented, Hall
85C22×D5—91no—
86C2×AGL(1,𝔽5)—92no—
87C2×AGL(1,𝔽5)C2×C3⋊D312yes—
88C2×C4×C5—91no—
89C2×Dic5—91no—
90C4×D5—91no—
91C4×D5—91no—
92C4×D5—91no—
93C4×D5C2×C3⋊D311yescharacteristic
94C22×C32—51no—
95C2×C3⋊D3—51no—
96C2×C3⋊D3—101no—
97C2×C3⋊D3—101no—
98C2×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic
99C32×C4—51no—
100C32×C4—102no—
101C32×C4C2×AGL(1,𝔽5)11yescharacteristic
102C3⋊Dic3—51no—
103C3⋊Dic3—102no—
104C3⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
105#117—451noSylow
106C2×C3×C5C2×C4×D314yes—
107D15—64no—
108C3×D5#23714yes—
109C3×D5#23914yes—
110C5×D3—64no—
111C22×D3—154no—
112C2×C3×C4—54no—
113C2×C3×C4—58no—
114C2×Dic3—154no—
115C2×Dic3—158no—
116C4×D3—34no—
117C4×D3—154no—
118C4×D3—154no—
119C4×D3—154no—
120C22×C5—91no—
121C2×D5—91no—
122C2×D5—181no—
123C2×D5—181no—
124C2×D5C22×C3⋊D311yescharacteristic
125AGL(1,𝔽5)—22no—
126AGL(1,𝔽5)—182no—
127C4×C5—91no—
128C4×C5C4×C3⋊D311yescharacteristic
129Dic5—91no—
130Dic5C4×C3⋊D311yescharacteristic
131C2×C32—51no—
132C2×C32—51no—
133C2×C32C22×AGL(1,𝔽5)11yescharacteristic
134C3⋊D3—21no—
135C3⋊D3—101no—
136C22×C4—451no—
137C4⋊C4—51no—
138C4⋊C4—451no—
139C22⋊C4—452no—
140C42—452no—
141C3×C5#89214yescomplemented
142C22×C3—54no—
143C2×D3—34no—
144C2×D3—154no—
145C2×D3—304no—
146C2×D3—304no—
147C3×C4—54no—
148C3×C4—108no—
149C3×C4D3×AGL(1,𝔽5)14yes—
150Dic3—34no—
151Dic3—154no—
152Dic3—308no—
153C2×C5—181no—
154C2×C5C2×C4×C3⋊D311yescharacteristic
155D5—181no—
156D5#383011yescharacteristic
157D5#383211yescharacteristic
158C32#413011yescharacteristic, complemented, Sylow
159C23—451no—
160C2×C4—51no—
161C2×C4—52no—
162C2×C4—91no—
163C2×C4—451no—
164C2×C4—451no—
165C2×C4—451no—
166C2×C4—451no—
167C2×C4—452no—
168C2×C3—54no—
169C2×C3—54no—
170C2×C3C2×D3×AGL(1,𝔽5)14yes—
171D3—64no—
172D3—304no—
173C5#6410111yescharacteristic, complemented, Sylow
174C22—51no—
175C22—91no—
176C22—451no—
177C22—901no—
178C22—901no—
179C4—51no—
180C4—91no—
181C4—102no—
182C4—451no—
183C4—902no—
184C4C3⋊D3×AGL(1,𝔽5)11yescharacteristic
185C3#9190714yescomplemented
186C2—51no—
187C2—51no—
188C2—181no—
189C2—901no—
190C2C2×C3⋊D3×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊D3×AGL(1,𝔽5)
commutator, socleC2×C32×C5C22×C4
FittingC32×C4×C5C2×C4
prime radicalC2×D5×C3⋊D3C22
cosocleC2×C32×D5C23
hypercenterC4C3⋊D3×AGL(1,𝔽5)
hypocenterC32×C5#117
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C4⋊(C3⋊ΓL(1,𝔽16))C2semidirect
3. C32×C4⋊AGL(1,𝔽5)C2semidirect
5. AGL(1,𝔽5)×C3⋊Dic3C2semidirect
10. C4×C3⋊D15C4semidirect
11. C4×C5×C3⋊D3C4semidirect
13. #67137C4semidirect
14. C3⋊D3×Dic5C4semidirect
19. C3×C4⋊AGL(1,𝔽5)D3semidirect
57. C4⋊AGL(1,𝔽5)C3⋊D3semidirect
67. C4×C3⋊D3AGL(1,𝔽5)semidirect
84. C32×C5#117semidirect
141. C3×C5#892semidirect
158. C32#4130semidirect
173. C5#64101semidirect
185. C3#91907semidirect

Maximal subgroups

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

subgroupindex
C4×D5×C3⋊D32
C4⋊(C3⋊ΓL(1,𝔽16))2
C32×C4⋊AGL(1,𝔽5)2
(C2×AGL(1,𝔽5))⋊(C3⋊D3)2
AGL(1,𝔽5)×C3⋊Dic32
#919073
#641015