Finite Group Databasebeta

← #13088663

Subgroups — #13088663

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#10623965C212yesmaximal, complemented
5(C2×AGL(1,𝔽5))⋊(C3⋊D3)C212yesmaximal
6#91909—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#64081—51nomaximal, Hall
17C4×D3×D5—34no—
18C4×ΓL(1,𝔽16)—34no—
19C3×C4×AGL(1,𝔽5)D314yescomplemented
20Dic3⋊AGL(1,𝔽5)—38no—
21(C2×D3)⋊AGL(1,𝔽5)—38no—
22C2×C3⋊D15C2×C411yescharacteristic
23C2×C5×C3⋊D3C2×C411yescharacteristic
24C2×C32×D5C2311yescharacteristic
25D5×C3⋊D3—22no—
26C3⋊ΓL(1,𝔽16)—22no—
27C32×AGL(1,𝔽5)—22no—
28C32×C4×C5C2×C411yescharacteristic
29C3⋊Dic15C2×C411yescharacteristic
30C5×C3⋊Dic3C2×C411yescharacteristic
31C32×Dic5C2×C411yescharacteristic
32#4130—91noHall
33C2×C4×C3⋊D3—51no—
34C32×C42—51no—
35#3752—52no—
36#3750—52no—
37C4×C3⋊Dic3—51no—
38C2×D3×D5—34no—
39C2×ΓL(1,𝔽16)—38no—
40C2×C3×AGL(1,𝔽5)C2×D318yes—
41C4×D15—34no—
42C4×C5×D3—34no—
43C3×C4×D5C2×D314yes—
44#1200—34no—
45D3×Dic5—34no—
46D5×Dic3—34no—
47#872—154no—
48C2×C32×C5C22×C411yescharacteristic
49C3⋊D15—21no—
50C5×C3⋊D3—21no—
51C32×D5#4112yes—
52C2×C4×D5—91no—
53C22⋊AGL(1,𝔽5)—92no—
54C4⋊AGL(1,𝔽5)—92no—
55C4×AGL(1,𝔽5)—91no—
56C4×AGL(1,𝔽5)C3⋊D311yescharacteristic, complemented
57C22×C3⋊D3—51no—
58C2×C32×C4—51no—
59C2×C32×C4—52no—
60C2×C3⋊Dic3—51no—
61C2×C3⋊Dic3—52no—
62C4×C3⋊D3—51no—
63C4×C3⋊D3—51no—
64C4×C3⋊D3—51no—
65C4×C3⋊D3AGL(1,𝔽5)11yescharacteristic, complemented
66C2×D15—34no—
67C2×C5×D3—34no—
68C2×C3×D5C22×D314yes—
69D3×D5—68no—
70ΓL(1,𝔽16)—68no—
71C3×AGL(1,𝔽5)—28no—
72C3×C4×C5C4×D314yes—
73Dic15—34no—
74C3×Dic5C4×D314yes—
75C5×Dic3—34no—
76C2×C4×D3—154no—
77C3×C42—54no—
78(C2×D3)⋊C4—158no—
79Dic3⋊C4—158no—
80C4×Dic3—154no—
81C32×C5#11711yescharacteristic, complemented, Hall
82C22×D5—91no—
83C2×AGL(1,𝔽5)—92no—
84C2×AGL(1,𝔽5)C2×C3⋊D312yes—
85C2×C4×C5—91no—
86C2×Dic5—91no—
87C4×D5—91no—
88C4×D5—91no—
89C4×D5—91no—
90C4×D5C2×C3⋊D311yescharacteristic
91C22×C32—51no—
92C2×C3⋊D3—51no—
93C2×C3⋊D3—102no—
94C2×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic
95C32×C4—51no—
96C32×C4—102no—
97C32×C4C2×AGL(1,𝔽5)11yescharacteristic
98C3⋊Dic3—51no—
99C3⋊Dic3—102no—
100C3⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
101#117—451noSylow
102C2×C3×C5C2×C4×D314yes—
103D15—64no—
104C3×D5#23518yes—
105C5×D3—64no—
106C22×D3—154no—
107C2×C3×C4—54no—
108C2×C3×C4—58no—
109C2×Dic3—154no—
110C2×Dic3—158no—
111C4×D3—34no—
112C4×D3—154no—
113C4×D3—154no—
114C4×D3—154no—
115C22×C5—91no—
116C2×D5—91no—
117C2×D5—182no—
118C2×D5C22×C3⋊D311yescharacteristic
119AGL(1,𝔽5)—22no—
120AGL(1,𝔽5)—182no—
121C4×C5—91no—
122C4×C5C4×C3⋊D311yescharacteristic
123Dic5—91no—
124Dic5C4×C3⋊D311yescharacteristic
125C2×C32—52no—
126C2×C32C22×AGL(1,𝔽5)11yescharacteristic
127C3⋊D3—21no—
128C3⋊D3—101no—
129C22×C4—451no—
130C4⋊C4—452no—
131C22⋊C4—452no—
132C42—51no—
133C42—451no—
134C3×C5#87214yescomplemented
135C22×C3—54no—
136C2×D3—34no—
137C2×D3—154no—
138C2×D3—308no—
139C3×C4—54no—
140C3×C4—108no—
141C3×C4D3×AGL(1,𝔽5)14yes—
142Dic3—34no—
143Dic3—154no—
144Dic3—308no—
145C2×C5—181no—
146C2×C5C2×C4×C3⋊D311yescharacteristic
147D5—181no—
148D5#382812yes—
149C32#413011yescharacteristic, complemented, Sylow
150C23—451no—
151C2×C4—51no—
152C2×C4—52no—
153C2×C4—91no—
154C2×C4—451no—
155C2×C4—451no—
156C2×C4—451no—
157C2×C4—451no—
158C2×C4—452no—
159C2×C3—58no—
160C2×C3C2×D3×AGL(1,𝔽5)14yes—
161D3—64no—
162D3—304no—
163C5#6408111yescharacteristic, complemented, Sylow
164C22—51no—
165C22—91no—
166C22—451no—
167C22—902no—
168C4—51no—
169C4—91no—
170C4—102no—
171C4—451no—
172C4—902no—
173C4C3⋊D3×AGL(1,𝔽5)11yescharacteristic
174C3#9190914yescomplemented
175C2—52no—
176C2—181no—
177C2—901no—
178C2C2×C3⋊D3×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, hypercenterC4C3⋊D3×AGL(1,𝔽5)
commutator, socleC2×C32×C5C22×C4
FrattiniC2C2×C3⋊D3×AGL(1,𝔽5)
FittingC32×C4×C5C2×C4
prime radicalC2×D5×C3⋊D3C22
cosocleC2×C32×D5C23
hypocenterC32×C5#117
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ 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
4. #10623965C2semidirect
10. C4×C3⋊D15C4semidirect
11. C4×C5×C3⋊D3C4semidirect
13. #67137C4semidirect
14. C3⋊D3×Dic5C4semidirect
19. C3×C4×AGL(1,𝔽5)D3semidirect
56. C4×AGL(1,𝔽5)C3⋊D3semidirect
65. C4×C3⋊D3AGL(1,𝔽5)semidirect
81. C32×C5#117semidirect
134. C3×C5#872semidirect
149. C32#4130semidirect
163. C5#64081semidirect
174. C3#91909semidirect

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
#106239652
(C2×AGL(1,𝔽5))⋊(C3⋊D3)2
#919093
#640815