Group #23744846 — subgroups

Subgroups — C₃×C₁₃×H(3,𝔽₃)⋊C₉ #23744846

Order 9477.

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

normal subgroups only

This group has 680 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×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃³×C₉×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₁₃×H(3,𝔽₃)⋊C₉	C₃	1	9	yes	maximal, direct product factor
5	C₃⁴×C₁₃	—	3	1	no	—
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
8	C₃×C₁₃×H(3,𝔽₃)	—	3	6	no	—
9	C₃×C₁₃×H(3,𝔽₃)	C₉	1	9	yes	complemented
10	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
11	C₃²×C₉×C₁₃	—	3	2	no	—
12	C₃²×C₉×C₁₃	—	3	1	no	—
13	C₃²×C₉×C₁₃	—	3	9	no	—
14	C₃²×C₉×C₁₃	—	3	2	no	—
15	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
16	C₁₃×C₃²⋊C₉	—	3	4	no	—
17	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	complemented
18	C₃×H(3,𝔽₃)⋊C₉	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃³×C₁₃	—	3	3	no	—
20	C₃³×C₁₃	—	3	3	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	3	9	no	—
23	C₃³×C₁₃	—	3	3	no	—
24	C₃³×C₁₃	—	9	3	no	—
25	C₃³×C₁₃	—	9	3	no	—
26	C₃³×C₁₃	—	9	3	no	—
27	C₃³×C₁₃	C₃×C₉	1	3	yes	—
28	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
29	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₁₃	3₋¹⁺²	1	6	yes	—
31	C₃³×C₁₃	C₃³	1	1	yes	characteristic
32	C₁₃×H(3,𝔽₃)	—	3	18	no	—
33	C₁₃×H(3,𝔽₃)	—	3	6	no	—
34	C₁₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	complemented
35	C₃×C₉×C₁₃	—	3	18	no	—
36	C₃×C₉×C₁₃	—	3	9	no	—
37	C₃×C₉×C₁₃	—	3	3	no	—
38	C₃×C₉×C₁₃	—	3	9	no	—
39	C₃×C₉×C₁₃	—	3	6	no	—
40	C₃×C₉×C₁₃	—	9	6	no	—
41	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
42	C₃³×C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
43	C₃×C₃²⋊C₉	C₃×C₁₃	1	2	yes	complemented
44	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	9	yes	direct product factor
45	C₃²×C₁₃	—	3	9	no	—
46	C₃²×C₁₃	—	3	18	no	—
47	C₃²×C₁₃	—	3	2	no	—
48	C₃²×C₁₃	—	3	1	no	—
49	C₃²×C₁₃	—	3	2	no	—
50	C₃²×C₁₃	—	3	1	no	—
51	C₃²×C₁₃	—	3	2	no	—
52	C₃²×C₁₃	—	3	1	no	—
53	C₃²×C₁₃	—	3	9	no	—
54	C₃²×C₁₃	—	9	27	no	—
55	C₃²×C₁₃	—	9	3	no	—
56	C₃²×C₁₃	—	9	3	no	—
57	C₃²×C₁₃	—	9	3	no	—
58	C₃²×C₁₃	C₃²⋊C₉	1	6	yes	—
59	C₃²×C₁₃	C₃²⋊C₉	1	3	yes	—
60	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
61	C₃²×C₁₃	#731	1	3	yes	—
62	C₃²×C₁₃	#732	1	3	yes	—
63	C₃²×C₁₃	C₃²×C₉	1	1	yes	characteristic
64	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₁₃	C₃×3₋¹⁺²	1	2	yes	—
66	C₉×C₁₃	—	3	9	no	—
67	C₉×C₁₃	—	9	6	no	—
68	C₃⁴	—	3	1	no	—
69	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
70	C₃×H(3,𝔽₃)	—	3	2	no	—
71	C₃×H(3,𝔽₃)	—	3	6	no	—
72	C₃×H(3,𝔽₃)	C₉×C₁₃	1	9	yes	complemented
73	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
74	C₃²×C₉	—	3	2	no	—
75	C₃²×C₉	—	3	1	no	—
76	C₃²×C₉	—	3	9	no	—
77	C₃²×C₉	—	3	2	no	—
78	C₃²×C₉	C₃²×C₁₃	1	3	yes	complemented
79	C₃²⋊C₉	—	3	4	no	—
80	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	complemented
81	C₃×C₁₃	—	3	6	no	—
82	C₃×C₁₃	—	3	1	no	—
83	C₃×C₁₃	—	3	2	no	—
84	C₃×C₁₃	—	9	9	no	—
85	C₃×C₁₃	H(3,𝔽₃)⋊C₉	1	9	yes	direct product factor
86	C₃×C₁₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
87	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
88	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
89	C₃×C₁₃	C₃×#732	1	1	yes	characteristic
90	C₃³	—	3	3	no	—
91	C₃³	—	3	3	no	—
92	C₃³	—	3	3	no	—
93	C₃³	—	3	9	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	9	3	no	—
96	C₃³	—	9	3	no	—
97	C₃³	—	9	3	no	—
98	C₃³	C₃×C₉×C₁₃	1	3	yes	—
99	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
100	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
101	C₃³	C₁₃×3₋¹⁺²	1	6	yes	—
102	C₃³	C₃³×C₁₃	1	1	yes	characteristic
103	H(3,𝔽₃)	—	3	18	no	—
104	H(3,𝔽₃)	—	3	6	no	—
105	H(3,𝔽₃)	C₃×C₉×C₁₃	1	9	yes	complemented
106	C₃×C₉	—	3	18	no	—
107	C₃×C₉	—	3	9	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	3	9	no	—
110	C₃×C₉	—	3	6	no	—
111	C₃×C₉	—	9	6	no	—
112	C₁₃	C₃×H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
113	C₃²	—	3	9	no	—
114	C₃²	—	3	18	no	—
115	C₃²	—	3	2	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	9	no	—
122	C₃²	—	9	27	no	—
123	C₃²	—	9	3	no	—
124	C₃²	—	9	3	no	—
125	C₃²	—	9	3	no	—
126	C₃²	C₁₃×C₃²⋊C₉	1	6	yes	—
127	C₃²	C₁₃×C₃²⋊C₉	1	3	yes	—
128	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
129	C₃²	C₁₃×#731	1	3	yes	—
130	C₃²	C₁₃×#732	1	3	yes	—
131	C₃²	C₃²×C₉×C₁₃	1	1	yes	characteristic
132	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
133	C₃²	C₃×C₁₃×3₋¹⁺²	1	2	yes	—
134	C₉	—	3	9	no	—
135	C₉	—	9	6	no	—
136	C₃	—	3	6	no	—
137	C₃	—	3	1	no	—
138	C₃	—	3	2	no	—
139	C₃	—	9	9	no	—
140	C₃	C₁₃×H(3,𝔽₃)⋊C₉	1	9	yes	direct product factor
141	C₃	C₃×C₁₃×C₃²⋊C₉	1	1	yes	characteristic
142	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
143	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
144	C₃	C₃×C₁₃×#732	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃³×C₉×C₁₃

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, complemented

C₁₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, direct product factor

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₉

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

C₃×H(3,𝔽₃)⋊C₉

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

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×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃×C₉

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

C₃³×C₉

C₃×C₁₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

C₃×C₁₃

yes

complemented

H(3,𝔽₃)⋊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×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

#732

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

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₉×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

H(3,𝔽₃)⋊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

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₁₃

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

100

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

101

C₃³

C₁₃×3₋¹⁺²

yes

—

102

C₃³

C₃³×C₁₃

yes

characteristic

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

C₃×C₉×C₁₃

yes

complemented

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₁₃

C₃×H(3,𝔽₃)⋊C₉

yes

characteristic, direct product factor, Sylow

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

C₁₃×C₃²⋊C₉

yes

—

127

C₃²

C₁₃×C₃²⋊C₉

yes

—

128

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

129

C₃²

C₁₃×#731

yes

—

130

C₃²

C₁₃×#732

yes

—

131

C₃²

C₃²×C₉×C₁₃

yes

characteristic

132

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

133

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

134

C₉

—

135

C₉

—

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

C₁₃×H(3,𝔽₃)⋊C₉

yes

direct product factor

141

C₃

C₃×C₁₃×C₃²⋊C₉

yes

characteristic

142

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

143

C₃

C₃×C₁₃×#731

yes

characteristic

144

C₃

C₃×C₁₃×#732

yes

characteristic

	subgroup	G/H
center, socle	C₃³×C₁₃	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₁₃
Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃²×C₁₃×H(3,𝔽₃)	C₃
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

C₃²×C₁₃×H(3,𝔽₃)

C₃

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₃³×C₉×C₁₃	C₃	semidirect
3. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
4. C₁₃×H(3,𝔽₃)⋊C₉	C₃	direct (N × H)
9. C₃×C₁₃×H(3,𝔽₃)	C₉	semidirect
15. C₃²×C₉×C₁₃	C₃²	semidirect
17. C₁₃×C₃²⋊C₉	C₃²	semidirect
18. C₃×H(3,𝔽₃)⋊C₉	C₁₃	direct (N × H)
34. C₁₃×H(3,𝔽₃)	C₃×C₉	semidirect
42. C₃³×C₉	C₃×C₁₃	semidirect
43. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
44. H(3,𝔽₃)⋊C₉	C₃×C₁₃	direct (N × H)
72. C₃×H(3,𝔽₃)	C₉×C₁₃	semidirect
78. C₃²×C₉	C₃²×C₁₃	semidirect
80. C₃²⋊C₉	C₃²×C₁₃	semidirect
85. C₃×C₁₃	H(3,𝔽₃)⋊C₉	direct (N × H)
105. H(3,𝔽₃)	C₃×C₉×C₁₃	semidirect
112. C₁₃	C₃×H(3,𝔽₃)⋊C₉	direct (N × H)
140. C₃	C₁₃×H(3,𝔽₃)⋊C₉	direct (N × H)

H ≅ G/N

type

2. C₃³×C₉×C₁₃

C₃

semidirect

3. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

4. C₁₃×H(3,𝔽₃)⋊C₉

C₃

direct (N × H)

9. C₃×C₁₃×H(3,𝔽₃)

C₉

semidirect

15. C₃²×C₉×C₁₃

C₃²

semidirect

17. C₁₃×C₃²⋊C₉

C₃²

semidirect

18. C₃×H(3,𝔽₃)⋊C₉

C₁₃

direct (N × H)

34. C₁₃×H(3,𝔽₃)

C₃×C₉

semidirect

42. C₃³×C₉

C₃×C₁₃

semidirect

43. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

44. H(3,𝔽₃)⋊C₉

C₃×C₁₃

direct (N × H)

72. C₃×H(3,𝔽₃)

C₉×C₁₃

semidirect

78. C₃²×C₉

C₃²×C₁₃

semidirect

80. C₃²⋊C₉

C₃²×C₁₃

semidirect

85. C₃×C₁₃

H(3,𝔽₃)⋊C₉

direct (N × H)

105. H(3,𝔽₃)

C₃×C₉×C₁₃

semidirect

112. C₁₃

C₃×H(3,𝔽₃)⋊C₉

direct (N × H)

140. C₃

C₁₃×H(3,𝔽₃)⋊C₉

direct (N × H)

Subgroups — C₃×C₁₃×H(3,𝔽₃)⋊C₉ #23744846

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×H(3,𝔽3)⋊C9 #23744846

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×H(3,𝔽₃)⋊C₉ #23744846