Group #23745076 — subgroups

Subgroups — C₃×C₁₃×#6910 #23745076

Order 9477.

1088 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₁₃×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
2	C₃²×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₁₃×#6910	C₃	1	27	yes	maximal, direct product factor
5	C₃×C₁₃×#731	C₃	1	9	yes	maximal, complemented
6	C₃⁴×C₁₃	—	3	1	no	—
7	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
8	C₃×C₁₃×3₋¹⁺²	—	3	12	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	6	no	—
10	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
11	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
12	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
13	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
14	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
15	C₃×C₁₃×3₋¹⁺²	C₃²	1	18	yes	complemented
16	C₃×C₁₃×3₋¹⁺²	C₃²	1	9	yes	complemented
17	C₃×C₁₃×H(3,𝔽₃)	—	3	6	no	—
18	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
19	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
20	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	9	yes	—
21	C₃²×C₉×C₁₃	—	3	2	no	—
22	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
23	C₁₃×#731	C₃²	1	81	yes	complemented
24	C₃×#6910	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃³×C₁₃	—	3	9	no	—
26	C₃³×C₁₃	—	3	3	no	—
27	C₃³×C₁₃	—	9	3	no	—
28	C₃³×C₁₃	—	9	9	no	—
29	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
30	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₃	H(3,𝔽₃)	1	6	yes	—
32	C₃³×C₁₃	C₃³	1	1	yes	characteristic
33	C₃³×C₁₃	C₃³	1	3	yes	—
34	C₁₃×3₋¹⁺²	—	3	12	no	—
35	C₁₃×3₋¹⁺²	—	3	18	no	—
36	C₁₃×3₋¹⁺²	—	3	18	no	—
37	C₁₃×3₋¹⁺²	—	3	18	no	—
38	C₁₃×3₋¹⁺²	—	3	9	no	—
39	C₁₃×3₋¹⁺²	C₃³	1	18	yes	complemented
40	C₁₃×H(3,𝔽₃)	—	3	6	no	—
41	C₁₃×H(3,𝔽₃)	—	3	18	no	—
42	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
43	C₃×C₉×C₁₃	—	3	18	no	—
44	C₃×C₉×C₁₃	—	3	6	no	—
45	C₃×C₉×C₁₃	—	3	6	no	—
46	C₃×C₉×C₁₃	—	3	3	no	—
47	C₃×C₉×C₁₃	C₃³	1	9	yes	complemented
48	C₃²×3₋¹⁺²	C₃×C₁₃	1	2	yes	complemented
49	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
50	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
51	#6910	C₃×C₁₃	1	27	yes	direct product factor
52	C₃×#731	C₃×C₁₃	1	9	yes	complemented
53	C₃²×C₁₃	—	3	3	no	—
54	C₃²×C₁₃	—	3	6	no	—
55	C₃²×C₁₃	—	3	3	no	—
56	C₃²×C₁₃	—	3	9	no	—
57	C₃²×C₁₃	—	9	6	no	—
58	C₃²×C₁₃	—	9	3	no	—
59	C₃²×C₁₃	—	9	27	no	—
60	C₃²×C₁₃	—	9	9	no	—
61	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	6	yes	—
62	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
63	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
64	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
66	C₉×C₁₃	—	3	18	no	—
67	C₉×C₁₃	—	3	9	no	—
68	C₃⁴	—	3	1	no	—
69	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
70	C₃×3₋¹⁺²	—	3	2	no	—
71	C₃×3₋¹⁺²	—	3	2	no	—
72	C₃×3₋¹⁺²	—	3	12	no	—
73	C₃×3₋¹⁺²	—	3	2	no	—
74	C₃×3₋¹⁺²	—	3	1	no	—
75	C₃×3₋¹⁺²	—	3	6	no	—
76	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	complemented
77	C₃×3₋¹⁺²	C₃²×C₁₃	1	18	yes	complemented
78	C₃×3₋¹⁺²	C₃²×C₁₃	1	9	yes	complemented
79	C₃×H(3,𝔽₃)	—	3	2	no	—
80	C₃×H(3,𝔽₃)	—	3	6	no	—
81	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
82	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	9	yes	—
83	C₃²×C₉	—	3	2	no	—
84	C₃²×C₉	C₃²×C₁₃	1	3	yes	complemented
85	#731	C₃²×C₁₃	1	81	yes	complemented
86	C₃×C₁₃	—	3	3	no	—
87	C₃×C₁₃	—	3	6	no	—
88	C₃×C₁₃	—	3	2	no	—
89	C₃×C₁₃	—	3	1	no	—
90	C₃×C₁₃	—	9	9	no	—
91	C₃×C₁₃	#6910	1	3	yes	direct product factor
92	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃³	—	3	9	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	9	3	no	—
96	C₃³	—	9	9	no	—
97	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
98	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
99	C₃³	C₁₃×H(3,𝔽₃)	1	6	yes	—
100	C₃³	C₃³×C₁₃	1	1	yes	characteristic
101	C₃³	C₃³×C₁₃	1	3	yes	—
102	3₋¹⁺²	—	3	12	no	—
103	3₋¹⁺²	—	3	18	no	—
104	3₋¹⁺²	—	3	18	no	—
105	3₋¹⁺²	—	3	18	no	—
106	3₋¹⁺²	—	3	9	no	—
107	3₋¹⁺²	C₃³×C₁₃	1	18	yes	complemented
108	H(3,𝔽₃)	—	3	6	no	—
109	H(3,𝔽₃)	—	3	18	no	—
110	H(3,𝔽₃)	C₃³×C₁₃	1	9	yes	—
111	C₃×C₉	—	3	18	no	—
112	C₃×C₉	—	3	6	no	—
113	C₃×C₉	—	3	6	no	—
114	C₃×C₉	—	3	3	no	—
115	C₃×C₉	C₃³×C₁₃	1	9	yes	complemented
116	C₁₃	C₃×#6910	1	1	yes	characteristic, direct product factor, Sylow
117	C₃²	—	3	3	no	—
118	C₃²	—	3	6	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	9	no	—
121	C₃²	—	9	6	no	—
122	C₃²	—	9	3	no	—
123	C₃²	—	9	27	no	—
124	C₃²	—	9	9	no	—
125	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	6	yes	—
126	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	2	yes	—
127	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	3	yes	—
128	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
129	C₃²	C₃⁴×C₁₃	1	1	yes	characteristic
130	C₉	—	3	18	no	—
131	C₉	—	3	9	no	—
132	C₃	—	3	3	no	—
133	C₃	—	3	6	no	—
134	C₃	—	3	2	no	—
135	C₃	—	3	1	no	—
136	C₃	—	9	9	no	—
137	C₃	C₁₃×#6910	1	3	yes	direct product factor
138	C₃	C₃²×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, complemented

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₁₃×#6910

C₃

yes

maximal, direct product factor

C₃×C₁₃×#731

C₃

yes

maximal, complemented

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₁₃×#731

C₃²

yes

complemented

C₃×#6910

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃³×C₁₃

C₃³

yes

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

C₃³

yes

complemented

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

complemented

C₃²×3₋¹⁺²

C₃×C₁₃

yes

complemented

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

#6910

C₃×C₁₃

yes

direct product factor

C₃×#731

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃⁴

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

complemented

#731

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6910

yes

direct product factor

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₁₃×H(3,𝔽₃)

yes

—

100

C₃³

C₃³×C₁₃

yes

characteristic

101

C₃³

C₃³×C₁₃

yes

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

C₃³×C₁₃

yes

complemented

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

—

110

H(3,𝔽₃)

C₃³×C₁₃

yes

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

C₃³×C₁₃

yes

complemented

116

C₁₃

C₃×#6910

yes

characteristic, direct product factor, Sylow

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

126

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

127

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

128

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

129

C₃²

C₃⁴×C₁₃

yes

characteristic

130

C₉

—

131

C₉

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

C₁₃×#6910

yes

direct product factor

138

C₃

C₃²×C₁₃×H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₁₃
prime radical	C₃²×C₁₃×H(3,𝔽₃)	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃²

C₃⁴×C₁₃

prime radical

C₃²×C₁₃×H(3,𝔽₃)

C₃

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
2. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
4. C₁₃×#6910	C₃	direct (N × H)
5. C₃×C₁₃×#731	C₃	semidirect
14. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
15. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
16. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
22. C₃²×C₉×C₁₃	C₃²	semidirect
23. C₁₃×#731	C₃²	semidirect
24. C₃×#6910	C₁₃	direct (N × H)
39. C₁₃×3₋¹⁺²	C₃³	semidirect
47. C₃×C₉×C₁₃	C₃³	semidirect
48. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
49. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
51. #6910	C₃×C₁₃	direct (N × H)
52. C₃×#731	C₃×C₁₃	semidirect
76. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
77. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
78. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
84. C₃²×C₉	C₃²×C₁₃	semidirect
85. #731	C₃²×C₁₃	semidirect
91. C₃×C₁₃	#6910	direct (N × H)
107. 3₋¹⁺²	C₃³×C₁₃	semidirect
115. C₃×C₉	C₃³×C₁₃	semidirect
116. C₁₃	C₃×#6910	direct (N × H)
137. C₃	C₁₃×#6910	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

2. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

4. C₁₃×#6910

C₃

direct (N × H)

5. C₃×C₁₃×#731

C₃

semidirect

14. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

15. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

16. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

22. C₃²×C₉×C₁₃

C₃²

semidirect

23. C₁₃×#731

C₃²

semidirect

24. C₃×#6910

C₁₃

direct (N × H)

39. C₁₃×3₋¹⁺²

C₃³

semidirect

47. C₃×C₉×C₁₃

C₃³

semidirect

48. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

49. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

51. #6910

C₃×C₁₃

direct (N × H)

52. C₃×#731

C₃×C₁₃

semidirect

76. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

77. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

78. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

84. C₃²×C₉

C₃²×C₁₃

semidirect

85. #731

C₃²×C₁₃

semidirect

91. C₃×C₁₃

#6910

direct (N × H)

107. 3₋¹⁺²

C₃³×C₁₃

semidirect

115. C₃×C₉

C₃³×C₁₃

semidirect

116. C₁₃

C₃×#6910

direct (N × H)

137. C₃

C₁₃×#6910

direct (N × H)

Subgroups — C₃×C₁₃×#6910 #23745076

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×#6910 #23745076

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×#6910 #23745076