Finite Group Databasebeta

← C3×#72213641 #72247227

Subgroups — C3×#72213641 #72247227

Order 14520.

228 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C3×#11757244C511yesmaximal, characteristic, complemented, Hall
2#72213641C311yesmaximal, characteristic, direct product factor, Hall
3C2×C3×#11925892C211yesmaximal, characteristic
4C3×#14559054C211yesmaximal, characteristic, complemented
5C3×#14559054C211yesmaximal, characteristic, complemented
6C2×C3×#10597243C2211yescharacteristic
7C3×#11925892C412yescomplemented
8C2×#11925892C2×C311yescharacteristic
9#14559054C2×C311yescharacteristic, complemented
10#14559054C2×C311yescharacteristic, complemented
11C3×#10597243C2×C411yescharacteristic, complemented, Hall
12C2×C3×C11⋊D11C2×C511yescharacteristic
13C3×C11×Dic11C2×C511yescharacteristic, complemented
14C3×C11×Dic11C2×C511yescharacteristic, complemented
15C3×C4×AGL(1,𝔽11)—111nomaximal
16C3×C4×AGL(1,𝔽11)—111nomaximal
17C2×#10597243C22×C311yescharacteristic
18#11925892C3×C412yescomplemented
19#11757244C3×C511yescharacteristic, complemented, Hall
20C2×C3×C112C22×C511yescharacteristic
21C3×C11⋊D11C4×C512yescomplemented
22C2×C3×AGL(1,𝔽11)—111no—
23C2×C3×AGL(1,𝔽11)—111no—
24C2×C3×AGL(1,𝔽11)—225no—
25C3×C4×C11⋊C5—111no—
26C3×C4×C11⋊C5—111no—
27C3×Dic11⋊C5—111no—
28C3×Dic11⋊C5—221no—
29#10597243C2×C3×C411yescharacteristic, complemented, Hall
30C2×C11⋊D11C2×C3×C511yescharacteristic
31C11×Dic11C2×C3×C511yescharacteristic, complemented
32C11×Dic11C2×C3×C511yescharacteristic, complemented
33C4×AGL(1,𝔽11)—111no—
34C4×AGL(1,𝔽11)—111no—
35C3×C112C2×C4×C511yescharacteristic, complemented, Hall
36C2×C3×C11⋊C5—111no—
37C2×C3×C11⋊C5—221no—
38C2×C3×C11⋊C5—445no—
39C3×AGL(1,𝔽11)—112no—
40C3×AGL(1,𝔽11)—112no—
41C3×AGL(1,𝔽11)—2210no—
42C3×C4×D11—111no—
43C3×C4×D11—111no—
44C2×C112C22×C3×C511yescharacteristic
45C11⋊D11C3×C4×C512yescomplemented
46C2×AGL(1,𝔽11)—111no—
47C2×AGL(1,𝔽11)—111no—
48C2×AGL(1,𝔽11)—225no—
49C4×C11⋊C5—111no—
50C4×C11⋊C5—111no—
51Dic11⋊C5—111no—
52Dic11⋊C5—221no—
53C3×C11⋊C5—111no—
54C3×C11⋊C5—221no—
55C3×C11⋊C5—445no—
56C2×C3×D11—111no—
57C2×C3×D11—111no—
58C2×C3×D11—225no—
59C3×C4×C11—111no—
60C3×C4×C11—111no—
61C3×Dic11AGL(1,𝔽11)11yescharacteristic, complemented
62C3×Dic11AGL(1,𝔽11)11yescharacteristic, complemented
63C112C2×C3×C4×C511yescharacteristic, complemented, Sylow
64C2×C3×C4×C5—1211noHall
65C2×C11⋊C5—111no—
66C2×C11⋊C5—221no—
67C2×C11⋊C5—445no—
68AGL(1,𝔽11)—112no—
69AGL(1,𝔽11)—112no—
70AGL(1,𝔽11)—2210no—
71C4×D11—111no—
72C4×D11—111no—
73C2×C3×C11—25no—
74C2×C3×C11C2×AGL(1,𝔽11)11yescharacteristic
75C2×C3×C11C2×AGL(1,𝔽11)11yescharacteristic
76C3×D11—112no—
77C3×D11—112no—
78C3×D11—2210no—
79C22×C3×C5—1211no—
80C3×C4×C5—2421no—
81C3×C4×C5—2421no—
82C11⋊C5—111no—
83C11⋊C5—221no—
84C11⋊C5—445no—
85C2×D11—111no—
86C2×D11—111no—
87C2×D11—225no—
88C4×C11—111no—
89C4×C11—111no—
90Dic11C3×AGL(1,𝔽11)11yescharacteristic, complemented
91Dic11C3×AGL(1,𝔽11)11yescharacteristic, complemented
92C2×C4×C5—1211noHall
93C3×C11—25no—
94C3×C11C4×AGL(1,𝔽11)11yescharacteristic, complemented
95C3×C11C4×AGL(1,𝔽11)11yescharacteristic, complemented
96C2×C3×C5—1212no—
97C2×C3×C5—2421no—
98C2×C3×C4—1211noHall
99C2×C11—25no—
100C2×C11C2×C3×AGL(1,𝔽11)11yescharacteristic
101C2×C11C2×C3×AGL(1,𝔽11)11yescharacteristic
102D11—112no—
103D11—112no—
104D11—2210no—
105C22×C5—1211no—
106C4×C5—2421no—
107C4×C5—2421no—
108C3×C5—2421noHall
109C22×C3—1211no—
110C3×C4—111no—
111C3×C4—111no—
112C11—25no—
113C11C3×C4×AGL(1,𝔽11)11yescharacteristic, complemented
114C11C3×C4×AGL(1,𝔽11)11yescharacteristic, complemented
115C2×C5—1212no—
116C2×C5—2421no—
117C2×C4—1211noSylow
118C2×C3—1212no—
119C2×C3#11925892⋊C211yescharacteristic
120C5—2421noSylow
121C22—1211no—
122C4—111no—
123C4—111no—
124C3#7221364111yescharacteristic, direct product factor, Sylow
125C2—1212no—
126C2C3×#11925892⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2×C3#11925892⋊C2
commutator, hypocenterC112C2×C3×C4×C5
FrattiniC2C3×#11925892⋊C2
Fitting, socleC2×C3×C112C22×C5
prime radicalC2×C3×#11925892C2
cosocleC2×C112C22×C3×C5
LCSG ▷ C112
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×#11757244C5semidirect
2. #72213641C3direct (N × H)
4. C3×#14559054C2semidirect
5. C3×#14559054C2semidirect
7. C3×#11925892C4semidirect
9. #14559054C2×C3semidirect
10. #14559054C2×C3semidirect
11. C3×#10597243C2×C4semidirect
13. C3×C11×Dic11C2×C5semidirect
14. C3×C11×Dic11C2×C5semidirect
18. #11925892C3×C4semidirect
19. #11757244C3×C5semidirect
21. C3×C11⋊D11C4×C5semidirect
29. #10597243C2×C3×C4semidirect
31. C11×Dic11C2×C3×C5semidirect
32. C11×Dic11C2×C3×C5semidirect
35. C3×C112C2×C4×C5semidirect
45. C11⋊D11C3×C4×C5semidirect
61. C3×Dic11AGL(1,𝔽11)semidirect
62. C3×Dic11AGL(1,𝔽11)semidirect
63. C112C2×C3×C4×C5semidirect
90. Dic11C3×AGL(1,𝔽11)semidirect
91. Dic11C3×AGL(1,𝔽11)semidirect
94. C3×C11C4×AGL(1,𝔽11)semidirect
95. C3×C11C4×AGL(1,𝔽11)semidirect
113. C11C3×C4×AGL(1,𝔽11)semidirect
114. C11C3×C4×AGL(1,𝔽11)semidirect
124. C3#72213641direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#117572445
#722136413
C2×C3×#119258922
C3×#145590542
C3×C4×AGL(1,𝔽11)11