Finite Group Databasebeta

← C3×C31×C3⋊#1042 #42124905

Subgroups — C3×C31×C3⋊#1042 #42124905

Order 27900.

124 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C32×C5×D5×C31C211yesmaximal, characteristic
2C31×C3⋊#1042C311yesmaximal, characteristic, direct product factor
3C3×C31×#1042—31nomaximal
4C32×C52×C31C411yescharacteristic, complemented, Hall
5C3×C31×ΓL(1,𝔽16)—51nomaximal
6C3×C31×Dic15—51nomaximal
7C3×C5×D5×C31—21no—
8C3×C5×D5×C31D311yescharacteristic
9C3×C5×D5×C31C2×C311yescharacteristic
10C31×#1042—31noHall
11C32×D5×C31D511yescharacteristic
12C2×C32×C5×C31—51no—
13C3×C52×C31—21no—
14C3×C52×C31Dic311yescharacteristic, complemented
15C3×C52×C31C3×C411yescharacteristic, complemented
16C31×ΓL(1,𝔽16)—51no—
17C3×AGL(1,𝔽5)×C31—151no—
18C31×Dic15—51no—
19C3×Dic5×C31—151no—
20C5×D5×C31C3×D311yescharacteristic
21C32×C5×C31—41no—
22C32×C5×C31Dic511yescharacteristic, complemented
23C32×C5×C31AGL(1,𝔽5)11yescharacteristic, complemented
24C3×Dic3×C31—251noHall
25C2×C3×C5×C31—51no—
26C2×C3×C5×C31—51no—
27C2×C3×C5×C31—101no—
28C3×D5×C31—21no—
29C3×D5×C31C3×D511yescharacteristic
30C3×D5×C31D1511yescharacteristic
31C3×C3⋊#1042C3111yesmaximal, characteristic, direct product factor, Hall
32C52×C31C3×Dic311yescharacteristic, complemented, Hall
33AGL(1,𝔽5)×C31—151no—
34Dic5×C31—151no—
35C2×C32×C31—51no—
36C3×C5×C31—21no—
37C3×C5×C31—21no—
38C3×C5×C31—42no—
39C3×C5×C31—41no—
40C3×C5×C31—41no—
41C3×C5×C31C3×Dic511yescharacteristic, complemented
42C3×C5×C31Dic1511yescharacteristic, complemented
43C3×C5×C31C3×AGL(1,𝔽5)11yescharacteristic, complemented
44C3×C5×C31ΓL(1,𝔽16)11yescharacteristic, complemented
45C32×C5×D5C2×C3111yescharacteristic
46C3×C4×C31—751no—
47Dic3×C31—251no—
48C2×C5×C31—51no—
49D5×C31C3×D1511yescharacteristic
50C3⋊#1042C3×C3111yescharacteristic, direct product factor
51C3×#1042—31no—
52C32×C31#104211yescharacteristic, complemented, Hall
53C32×C52C4×C3111yescharacteristic, complemented, Hall
54C2×C3×C31—51no—
55C2×C3×C31—51no—
56C2×C3×C31—101no—
57C3×ΓL(1,𝔽16)—51no—
58C3×Dic15—51no—
59C5×C31—41no—
60C5×C31C3×Dic1511yescharacteristic, complemented
61C5×C31C3×ΓL(1,𝔽16)11yescharacteristic, complemented
62C3×C5×D5—21no—
63C3×C5×D5D3×C3111yescharacteristic
64C3×C5×D5C2×C3×C3111yescharacteristic
65C4×C31—751noHall
66#1042—31noHall
67C3×C31—21no—
68C3×C31C3×#104211yescharacteristic, complemented
69C3×C31C3⋊#104211yescharacteristic, direct product factor
70C2×C32×C5—51no—
71C32×D5D5×C3111yescharacteristic
72C3×C52—21no—
73C3×C52Dic3×C3111yescharacteristic, complemented
74C3×C52C3×C4×C3111yescharacteristic, complemented
75C2×C31—51no—
76ΓL(1,𝔽16)—51no—
77C3×AGL(1,𝔽5)—151no—
78Dic15—51no—
79C3×Dic5—151no—
80C5×D5C3×D3×C3111yescharacteristic
81C32×C5—41no—
82C32×C5Dic5×C3111yescharacteristic, complemented
83C32×C5AGL(1,𝔽5)×C3111yescharacteristic, complemented
84C3×Dic3—251noHall
85C31C3×C3⋊#104211yescharacteristic, direct product factor, Sylow
86C2×C3×C5—51no—
87C2×C3×C5—51no—
88C2×C3×C5—101no—
89C3×D5—21no—
90C3×D5C3×D5×C3111yescharacteristic
91C3×D5D15×C3111yescharacteristic
92C52C3×Dic3×C3111yescharacteristic, complemented, Sylow
93AGL(1,𝔽5)—151no—
94Dic5—151no—
95C2×C32—51no—
96C3×C5—21no—
97C3×C5—21no—
98C3×C5—42no—
99C3×C5—41no—
100C3×C5—41no—
101C3×C5C3×Dic5×C3111yescharacteristic, complemented
102C3×C5C31×Dic1511yescharacteristic, complemented
103C3×C5C3×AGL(1,𝔽5)×C3111yescharacteristic, complemented
104C3×C5C31×ΓL(1,𝔽16)11yescharacteristic, complemented
105C3×C4—751no—
106Dic3—251no—
107C2×C5—51no—
108D5C3×D15×C3111yescharacteristic
109C32C31×#104211yescharacteristic, complemented, Sylow
110C2×C3—51no—
111C2×C3—51no—
112C2×C3—101no—
113C5—41no—
114C5C3×C31×Dic1511yescharacteristic, complemented
115C5C3×C31×ΓL(1,𝔽16)11yescharacteristic, complemented
116C4—751noSylow
117C3—21no—
118C3C3×C31×#104211yescharacteristic, complemented
119C3C31×C3⋊#104211yescharacteristic, direct product factor
120C2—51no—
subgroupG/H
center, hypercenterC3×C31C3⋊#1042
commutator, hypocenterC3×C52C3×C4×C31
FrattiniC1G
Fitting, socleC32×C52×C31C4
prime radicalC32×C5×D5×C31C2
cosocleC3×C5×D5C2×C3×C31
LCSG ▷ C3×C52
UCSC1 ◁ C3×C31

Complements

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

NH ≅ G/Ntype
2. C31×C3⋊#1042C3direct (N × H)
4. C32×C52×C31C4semidirect
14. C3×C52×C31Dic3semidirect
15. C3×C52×C31C3×C4semidirect
22. C32×C5×C31Dic5semidirect
23. C32×C5×C31AGL(1,𝔽5)semidirect
31. C3×C3⋊#1042C31direct (N × H)
32. C52×C31C3×Dic3semidirect
41. C3×C5×C31C3×Dic5semidirect
42. C3×C5×C31Dic15semidirect
43. C3×C5×C31C3×AGL(1,𝔽5)semidirect
44. C3×C5×C31ΓL(1,𝔽16)semidirect
50. C3⋊#1042C3×C31direct (N × H)
52. C32×C31#1042semidirect
53. C32×C52C4×C31semidirect
60. C5×C31C3×Dic15semidirect
61. C5×C31C3×ΓL(1,𝔽16)semidirect
68. C3×C31C3×#1042semidirect
69. C3×C31C3⋊#1042direct (N × H)
73. C3×C52Dic3×C31semidirect
74. C3×C52C3×C4×C31semidirect
82. C32×C5Dic5×C31semidirect
83. C32×C5AGL(1,𝔽5)×C31semidirect
85. C31C3×C3⋊#1042direct (N × H)
92. C52C3×Dic3×C31semidirect
101. C3×C5C3×Dic5×C31semidirect
102. C3×C5C31×Dic15semidirect
103. C3×C5C3×AGL(1,𝔽5)×C31semidirect
104. C3×C5C31×ΓL(1,𝔽16)semidirect
109. C32C31×#1042semidirect
114. C5C3×C31×Dic15semidirect
115. C5C3×C31×ΓL(1,𝔽16)semidirect
118. C3C3×C31×#1042semidirect
119. C3C31×C3⋊#1042direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C5×D5×C312
C31×C3⋊#10423
C3×C31×#10423
C3×C31×ΓL(1,𝔽16)5
C3×C31×Dic155
C3×C3⋊#104231