Group #23744847 — subgroups

Subgroups — C₃×C₁₃×3₋¹⁺²⋊C₉ #23744847

Order 9477.

572 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 188 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	1	yes	maximal, characteristic
2	C₃³×C₉×C₁₃	C₃	1	1	yes	maximal, characteristic
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₁₃×3₋¹⁺²⋊C₉	C₃	1	9	yes	maximal, direct product factor
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	6	no	—
9	C₃×C₁₃×3₋¹⁺²	C₉	1	9	yes	complemented
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃²×C₉×C₁₃	—	3	2	no	—
12	C₃²×C₉×C₁₃	—	3	1	no	—
13	C₃²×C₉×C₁₃	—	3	1	no	—
14	C₃²×C₉×C₁₃	—	3	9	no	—
15	C₃²×C₉×C₁₃	—	3	1	no	—
16	C₃²×C₉×C₁₃	—	3	1	no	—
17	C₃²×C₉×C₁₃	C₃²	1	3	yes	—
18	C₁₃×C₃²⋊C₉	—	3	2	no	—
19	C₁₃×C₃²⋊C₉	—	3	2	no	—
20	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
21	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
22	C₃×3₋¹⁺²⋊C₉	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃³×C₁₃	—	3	3	no	—
24	C₃³×C₁₃	—	3	3	no	—
25	C₃³×C₁₃	—	3	3	no	—
26	C₃³×C₁₃	C₃×C₉	1	3	yes	—
27	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
28	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₁₃	3₋¹⁺²	1	6	yes	—
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₁₃×3₋¹⁺²	—	3	18	no	—
32	C₁₃×3₋¹⁺²	—	3	6	no	—
33	C₁₃×3₋¹⁺²	C₃×C₉	1	9	yes	complemented
34	C₃×C₉×C₁₃	—	3	18	no	—
35	C₃×C₉×C₁₃	—	3	9	no	—
36	C₃×C₉×C₁₃	—	3	9	no	—
37	C₃×C₉×C₁₃	—	3	3	no	—
38	C₃×C₉×C₁₃	—	3	3	no	—
39	C₃×C₉×C₁₃	—	3	9	no	—
40	C₃×C₉×C₁₃	—	3	3	no	—
41	C₃×C₉×C₁₃	—	3	3	no	—
42	C₃×C₉×C₁₃	—	9	3	no	—
43	C₃×C₉×C₁₃	—	9	3	no	—
44	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic
45	C₃³×C₉	C₃×C₁₃	1	1	yes	characteristic
46	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic
47	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic
48	3₋¹⁺²⋊C₉	C₃×C₁₃	1	9	yes	direct product factor
49	C₃²×C₁₃	—	3	9	no	—
50	C₃²×C₁₃	—	3	18	no	—
51	C₃²×C₁₃	—	3	2	no	—
52	C₃²×C₁₃	—	3	1	no	—
53	C₃²×C₁₃	—	3	2	no	—
54	C₃²×C₁₃	—	3	1	no	—
55	C₃²×C₁₃	—	3	2	no	—
56	C₃²×C₁₃	—	3	1	no	—
57	C₃²×C₁₃	C₃²⋊C₉	1	6	yes	—
58	C₃²×C₁₃	C₃²⋊C₉	1	3	yes	—
59	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
60	C₃²×C₁₃	#731	1	3	yes	—
61	C₃²×C₁₃	#733	1	3	yes	—
62	C₃²×C₁₃	C₃²×C₉	1	1	yes	characteristic
63	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
64	C₃²×C₁₃	C₃×3₋¹⁺²	1	2	yes	—
65	C₉×C₁₃	—	3	9	no	—
66	C₉×C₁₃	—	3	9	no	—
67	C₉×C₁₃	—	9	3	no	—
68	C₉×C₁₃	—	9	3	no	—
69	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
70	C₃×3₋¹⁺²	—	3	2	no	—
71	C₃×3₋¹⁺²	—	3	6	no	—
72	C₃×3₋¹⁺²	C₉×C₁₃	1	9	yes	complemented
73	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
74	C₃²×C₉	—	3	2	no	—
75	C₃²×C₉	—	3	1	no	—
76	C₃²×C₉	—	3	1	no	—
77	C₃²×C₉	—	3	9	no	—
78	C₃²×C₉	—	3	1	no	—
79	C₃²×C₉	—	3	1	no	—
80	C₃²×C₉	C₃²×C₁₃	1	3	yes	—
81	C₃²⋊C₉	—	3	2	no	—
82	C₃²⋊C₉	—	3	2	no	—
83	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
84	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
85	C₃×C₁₃	—	3	6	no	—
86	C₃×C₁₃	—	3	1	no	—
87	C₃×C₁₃	—	3	2	no	—
88	C₃×C₁₃	3₋¹⁺²⋊C₉	1	9	yes	direct product factor
89	C₃×C₁₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
90	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
91	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
92	C₃×C₁₃	C₃×#733	1	1	yes	characteristic
93	C₃³	—	3	3	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	3	3	no	—
96	C₃³	C₃×C₉×C₁₃	1	3	yes	—
97	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
98	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
99	C₃³	C₁₃×3₋¹⁺²	1	6	yes	—
100	C₃³	C₃³×C₁₃	1	1	yes	characteristic
101	3₋¹⁺²	—	3	18	no	—
102	3₋¹⁺²	—	3	6	no	—
103	3₋¹⁺²	C₃×C₉×C₁₃	1	9	yes	complemented
104	C₃×C₉	—	3	18	no	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	3	9	no	—
107	C₃×C₉	—	3	3	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	3	9	no	—
110	C₃×C₉	—	3	3	no	—
111	C₃×C₉	—	3	3	no	—
112	C₃×C₉	—	9	3	no	—
113	C₃×C₉	—	9	3	no	—
114	C₁₃	C₃×3₋¹⁺²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
115	C₃²	—	3	9	no	—
116	C₃²	—	3	18	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	2	no	—
122	C₃²	—	3	1	no	—
123	C₃²	C₁₃×C₃²⋊C₉	1	6	yes	—
124	C₃²	C₁₃×C₃²⋊C₉	1	3	yes	—
125	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
126	C₃²	C₁₃×#731	1	3	yes	—
127	C₃²	C₁₃×#733	1	3	yes	—
128	C₃²	C₃²×C₉×C₁₃	1	1	yes	characteristic
129	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
130	C₃²	C₃×C₁₃×3₋¹⁺²	1	2	yes	—
131	C₉	—	3	9	no	—
132	C₉	—	3	9	no	—
133	C₉	—	9	3	no	—
134	C₉	—	9	3	no	—
135	C₃	—	3	6	no	—
136	C₃	—	3	1	no	—
137	C₃	—	3	2	no	—
138	C₃	C₁₃×3₋¹⁺²⋊C₉	1	9	yes	direct product factor
139	C₃	C₃×C₁₃×C₃²⋊C₉	1	1	yes	characteristic
140	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
141	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
142	C₃	C₃×C₁₃×#733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃³×C₉×C₁₃

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₁₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, direct product factor

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₉

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₃×3₋¹⁺²⋊C₉

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

C₃×C₉

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

3₋¹⁺²

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

C₃×C₉

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic

C₃³×C₉

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic

3₋¹⁺²⋊C₉

C₃×C₁₃

yes

direct product factor

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃²⋊C₉

yes

—

C₃²×C₁₃

C₃²⋊C₉

yes

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

#731

yes

—

C₃²×C₁₃

#733

yes

—

C₃²×C₁₃

C₃²×C₉

yes

characteristic

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₉×C₁₃

yes

complemented

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

3₋¹⁺²⋊C₉

yes

direct product factor

C₃×C₁₃

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₁₃

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₁₃×3₋¹⁺²

yes

—

100

C₃³

C₃³×C₁₃

yes

characteristic

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

C₃×C₉×C₁₃

yes

complemented

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₁₃

C₃×3₋¹⁺²⋊C₉

yes

characteristic, direct product factor, Sylow

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

C₁₃×C₃²⋊C₉

yes

—

124

C₃²

C₁₃×C₃²⋊C₉

yes

—

125

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

126

C₃²

C₁₃×#731

yes

—

127

C₃²

C₁₃×#733

yes

—

128

C₃²

C₃²×C₉×C₁₃

yes

characteristic

129

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

130

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₃

—

136

C₃

—

137

C₃

—

138

C₃

C₁₃×3₋¹⁺²⋊C₉

yes

direct product factor

139

C₃

C₃×C₁₃×C₃²⋊C₉

yes

characteristic

140

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

141

C₃

C₃×C₁₃×#731

yes

characteristic

142

C₃

C₃×C₁₃×#733

yes

characteristic

	subgroup	G/H
center, socle	C₃³×C₁₃	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₁₃
Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃⁴×C₁₃	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃³×C₁₃

H(3,𝔽₃)

commutator

C₃²

C₃²×C₉×C₁₃

Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃⁴×C₁₃

C₃²

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
5. C₁₃×3₋¹⁺²⋊C₉	C₃	direct (N × H)
9. C₃×C₁₃×3₋¹⁺²	C₉	semidirect
22. C₃×3₋¹⁺²⋊C₉	C₁₃	direct (N × H)
33. C₁₃×3₋¹⁺²	C₃×C₉	semidirect
48. 3₋¹⁺²⋊C₉	C₃×C₁₃	direct (N × H)
72. C₃×3₋¹⁺²	C₉×C₁₃	semidirect
88. C₃×C₁₃	3₋¹⁺²⋊C₉	direct (N × H)
103. 3₋¹⁺²	C₃×C₉×C₁₃	semidirect
114. C₁₃	C₃×3₋¹⁺²⋊C₉	direct (N × H)
138. C₃	C₁₃×3₋¹⁺²⋊C₉	direct (N × H)

H ≅ G/N

type

5. C₁₃×3₋¹⁺²⋊C₉

C₃

direct (N × H)

9. C₃×C₁₃×3₋¹⁺²

C₉

semidirect

22. C₃×3₋¹⁺²⋊C₉

C₁₃

direct (N × H)

33. C₁₃×3₋¹⁺²

C₃×C₉

semidirect

48. 3₋¹⁺²⋊C₉

C₃×C₁₃

direct (N × H)

72. C₃×3₋¹⁺²

C₉×C₁₃

semidirect

88. C₃×C₁₃

3₋¹⁺²⋊C₉

direct (N × H)

103. 3₋¹⁺²

C₃×C₉×C₁₃

semidirect

114. C₁₃

C₃×3₋¹⁺²⋊C₉

direct (N × H)

138. C₃

C₁₃×3₋¹⁺²⋊C₉

direct (N × H)

Subgroups — C₃×C₁₃×3₋¹⁺²⋊C₉ #23744847

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×3−1+2⋊C9 #23744847

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×3₋¹⁺²⋊C₉ #23744847