Finite Group Databasebeta

← C22×C5×#611 #13085343

Subgroups — C22×C5×#611 #13085343

Order 1440.

708 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 264 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
1C24×C5×C9C211yesmaximal, characteristic, complemented
2C23×C5×D9C211yesmaximal, characteristic, complemented
3C2×C5×#611C2112yesmaximal, direct product factor
4C22×C5×Dic9C211yesmaximal, characteristic, complemented
5C22×C5×#67—31nomaximal
6C23×C5×C9—24no—
7C23×C5×C9C2211yescharacteristic
8C23×C5×C9C2216yescomplemented
9C22×C5×D9—24no—
10C22×C5×D9C2216yescomplemented
11C5×#611C22116yesdirect product factor
12C2×C5×Dic9C2216yescomplemented
13C22×#611C511yesmaximal, characteristic, direct product factor, Hall
14C24×C3×C5D311yescharacteristic
15C23×C5×D3—31no—
16C2×C5×#67—312no—
17C22×C5×Dic3—31no—
18C22×C5×C9—212no—
19C22×C5×C9D414yescomplemented
20C22×C5×C9C2313yes—
21C22×C5×C9C2314yescomplemented
22C2×C5×D9—212no—
23C2×C5×D9C2314yescomplemented
24C5×Dic9C2314yescomplemented
25C22×C5×D4—91noHall
26C24×C9C2×C511yescharacteristic, complemented
27C23×D9C2×C511yescharacteristic, complemented
28C2×#611C2×C5112yesdirect product factor
29C22×Dic9C2×C511yescharacteristic, complemented
30C23×C3×C5—24no—
31C23×C3×C5C2×D316yes—
32C23×C3×C5C2×D311yescharacteristic
33C22×C5×D3—36no—
34C22×C5×D3—64no—
35C5×#67—316no—
36C2×C5×Dic3—36no—
37C22×#67—31no—
38C2×C5×C9—24no—
39C2×C5×C9C2×D416yescomplemented
40C2×C5×C9C2411yescharacteristic
41C5×D9—24no—
42C24×C5—91no—
43C24×C5D911yescharacteristic, complemented
44C2×C5×D4—912no—
45C22×C4×C5—91no—
46C23×C9—24no—
47C23×C9C22×C511yescharacteristic
48C23×C9C22×C516yescomplemented
49C22×D9—24no—
50C22×D9C22×C516yescomplemented
51#611C22×C5116yesdirect product factor
52C2×Dic9C22×C516yescomplemented
53C22×C3×C5—212no—
54C22×C3×C5#6714yes—
55C22×C3×C5C22×D313yes—
56C22×C3×C5C22×D314yes—
57C2×C5×D3—34no—
58C2×C5×D3—612no—
59C5×Dic3—34no—
60C24×C3C5×D311yescharacteristic
61C23×D3—31no—
62C2×#67—312no—
63C22×Dic3—31no—
64C5×C9C22×D411yescharacteristic, complemented, Hall
65C23×C5—24no—
66C23×C5—96no—
67C23×C5—184no—
68C23×C5C2×D911yescharacteristic
69C23×C5C2×D916yescomplemented
70C5×D4—916no—
71C2×C4×C5—96no—
72C22×C9—212no—
73C22×C9C5×D414yescomplemented
74C22×C9C23×C513yes—
75C22×C9C23×C514yescomplemented
76C2×D9—212no—
77C2×D9C23×C514yescomplemented
78Dic9C23×C514yescomplemented
79C22×D4—91noSylow
80C2×C3×C5—24no—
81C2×C3×C5C2×#6716yes—
82C2×C3×C5C23×D311yescharacteristic
83C5×D3—64no—
84C23×C3—24no—
85C23×C3C2×C5×D311yescharacteristic
86C23×C3C2×C5×D316yes—
87C22×D3—36no—
88C22×D3—64no—
89#67—316no—
90C2×Dic3—36no—
91C22×C5—212no—
92C22×C5—94no—
93C22×C5—1812no—
94C22×C5#61114yesdirect product factor
95C22×C5C22×D913yes—
96C22×C5C22×D914yescomplemented
97C4×C5—94no—
98C2×C9—24no—
99C2×C9C2×C5×D416yescomplemented
100C2×C9C24×C511yescharacteristic
101D9—24no—
102C24—91no—
103C24C5×D911yescharacteristic, complemented
104C2×D4—912no—
105C22×C4—91no—
106C3×C5C22×#6711yescharacteristic
107C22×C3—212no—
108C22×C3C5×#6714yes—
109C22×C3C22×C5×D313yes—
110C22×C3C22×C5×D314yes—
111C2×D3—34no—
112C2×D3—612no—
113Dic3—34no—
114C2×C5—24no—
115C2×C5—184no—
116C2×C5C2×#61116yesdirect product factor
117C2×C5C23×D911yescharacteristic
118C9C22×C5×D411yescharacteristic, complemented, Sylow
119C23—24no—
120C23—96no—
121C23—184no—
122C23C2×C5×D911yescharacteristic
123C23C2×C5×D916yescomplemented
124D4—916no—
125C2×C4—96no—
126C2×C3—24no—
127C2×C3C2×C5×#6716yes—
128C2×C3C23×C5×D311yescharacteristic
129D3—64no—
130C5C22×#61111yescharacteristic, direct product factor, Sylow
131C22—212no—
132C22—94no—
133C22—1812no—
134C22C5×#61114yesdirect product factor
135C22C22×C5×D913yes—
136C22C22×C5×D914yescomplemented
137C4—94no—
138C3C22×C5×#6711yescharacteristic
139C2—24no—
140C2—184no—
141C2C2×C5×#61116yesdirect product factor
142C2C23×C5×D911yescharacteristic
subgroupG/H
centerC23×C5C2×D9
commutator, cosocleC2×C9C24×C5
FrattiniC2×C3C23×C5×D3
FittingC24×C5×C9C2
socleC23×C3×C5C2×D3
hypercenterC24×C5D9
hypocenterC9C22×C5×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23×C5 ◁ C24×C5

Complements

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

NH ≅ G/Ntype
1. C24×C5×C9C2semidirect
2. C23×C5×D9C2semidirect
3. C2×C5×#611C2direct (N × H)
4. C22×C5×Dic9C2semidirect
8. C23×C5×C9C22semidirect
10. C22×C5×D9C22semidirect
11. C5×#611C22direct (N × H)
12. C2×C5×Dic9C22semidirect
13. C22×#611C5direct (N × H)
19. C22×C5×C9D4semidirect
21. C22×C5×C9C23semidirect
23. C2×C5×D9C23semidirect
24. C5×Dic9C23semidirect
26. C24×C9C2×C5semidirect
27. C23×D9C2×C5semidirect
28. C2×#611C2×C5direct (N × H)
29. C22×Dic9C2×C5semidirect
39. C2×C5×C9C2×D4semidirect
43. C24×C5D9semidirect
48. C23×C9C22×C5semidirect
50. C22×D9C22×C5semidirect
51. #611C22×C5direct (N × H)
52. C2×Dic9C22×C5semidirect
64. C5×C9C22×D4semidirect
69. C23×C5C2×D9semidirect
73. C22×C9C5×D4semidirect
75. C22×C9C23×C5semidirect
77. C2×D9C23×C5semidirect
78. Dic9C23×C5semidirect
94. C22×C5#611direct (N × H)
96. C22×C5C22×D9semidirect
99. C2×C9C2×C5×D4semidirect
103. C24C5×D9semidirect
116. C2×C5C2×#611direct (N × H)
118. C9C22×C5×D4semidirect
123. C23C2×C5×D9semidirect
130. C5C22×#611direct (N × H)
134. C22C5×#611direct (N × H)
136. C22C22×C5×D9semidirect
141. C2C2×C5×#611direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C5×C92
C23×C5×D92
C2×C5×#6112
C22×C5×Dic92
C22×C5×#673
C22×#6115