Group #23748619 — subgroups

Subgroups — C₃×#15691697 #23748619

Order 9477.

5588 conjugacy classes of subgroups; only the 2884 normal subgroups are shown.

normal subgroups only

This group has 5588 conjugacy classes of subgroups — too many to draw. Only its 2884 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	#15691697	C₃	1	81	yes	maximal, direct product factor
2	C₃×#11760454	C₃	1	18	yes	maximal, complemented
3	C₃×#11760453	C₃	1	9	yes	maximal, complemented
4	C₃²×#66851	C₃	1	2	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	1	9	yes	maximal, complemented
6	C₃²×C₃²⋊(C₁₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
7	C₃²×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₃⋊C₃	C₃²	1	3	yes	—
9	#11760454	C₃²	1	486	yes	complemented
10	#11760453	C₃²	1	243	yes	complemented
11	C₃×#66851	C₃²	1	6	yes	—
12	C₃×#66851	C₃²	1	6	yes	complemented
13	C₃×#66851	C₃²	1	6	yes	complemented
14	C₃×#66851	C₃²	1	54	yes	complemented
15	C₃×#66848	C₃²	1	27	yes	complemented
16	C₃×#66849	C₃²	1	27	yes	complemented
17	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	243	yes	complemented
18	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
19	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	6	yes	—
20	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	27	yes	—
21	C₃²×C₁₃⋊C₉	C₃²	1	6	yes	complemented
22	C₃×C₉×C₁₃⋊C₃	C₃²	1	27	yes	complemented
23	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
24	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
25	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
26	C₃×C₁₃×3₋¹⁺²	C₃²	1	27	yes	complemented
27	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
28	C₃×3₋¹⁺⁴	—	13	1	no	maximal, Sylow
29	C₃³×C₁₃	—	9	3	no	—
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₃³×C₁₃	C₃³	1	1	yes	characteristic
32	C₃³×C₁₃	C₃³	1	2	yes	—
33	C₃³×C₁₃	C₃³	1	9	yes	—
34	C₃²×C₁₃⋊C₃	—	9	9	no	—
35	C₃²×C₁₃⋊C₃	C₃³	1	9	yes	—
36	C₃²×C₁₃⋊C₃	C₃³	1	27	yes	—
37	C₁₃×3₋¹⁺²	C₃³	1	27	yes	—
38	C₁₃×3₋¹⁺²	C₃³	1	54	yes	complemented
39	C₃×C₉×C₁₃	C₃³	1	9	yes	—
40	C₃×C₉×C₁₃	C₃³	1	27	yes	complemented
41	C₃²⋊(C₁₃⋊C₃)	C₃³	1	27	yes	—
42	C₃²⋊(C₁₃⋊C₃)	C₃³	1	54	yes	—
43	#66851	C₃³	1	54	yes	—
44	#66851	C₃³	1	54	yes	complemented
45	#66851	C₃³	1	54	yes	complemented
46	C₃×C₁₃⋊C₉	C₃³	1	18	yes	—
47	C₃×C₁₃⋊C₉	C₃³	1	54	yes	complemented
48	#66849	C₃³	1	243	yes	complemented
49	#66848	C₃³	1	243	yes	complemented
50	C₉×C₁₃⋊C₃	C₃³	1	243	yes	complemented
51	3₋¹⁺⁴	—	13	81	no	—
52	C₃×#737	—	13	18	no	—
53	C₃×#737	—	13	9	no	—
54	C₃²×3₋¹⁺²	—	13	2	no	—
55	C₃²×3₋¹⁺²	—	13	9	no	—
56	C₃²×3₋¹⁺²	C₁₃⋊C₃	1	1	yes	characteristic, complemented
57	C₃²×H(3,𝔽₃)	—	13	1	no	—
58	C₃²×C₁₃	—	3	3	no	—
59	C₃²×C₁₃	—	3	3	no	—
60	C₃²×C₁₃	—	3	6	no	—
61	C₃²×C₁₃	—	9	9	no	—
62	C₃²×C₁₃	C₃⁴	1	6	yes	—
63	C₃²×C₁₃	C₃⁴	1	3	yes	—
64	C₃²×C₁₃	C₃⁴	1	3	yes	—
65	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
66	C₃×C₁₃⋊C₃	—	3	27	no	—
67	C₃×C₁₃⋊C₃	—	9	27	no	—
68	C₃×C₁₃⋊C₃	C₃⁴	1	27	yes	—
69	C₉×C₁₃	C₃⁴	1	27	yes	—
70	C₁₃⋊C₉	C₃⁴	1	54	yes	—
71	C₃⁴	—	13	3	no	—
72	C₃⁴	C₃×C₁₃⋊C₃	1	1	yes	characteristic
73	#737	—	13	486	no	—
74	#737	—	13	243	no	—
75	C₃×3₋¹⁺²	—	13	6	no	—
76	C₃×3₋¹⁺²	—	13	6	no	—
77	C₃×3₋¹⁺²	—	13	6	no	—
78	C₃×3₋¹⁺²	—	13	54	no	—
79	C₃×3₋¹⁺²	—	13	243	no	—
80	C₃×3₋¹⁺²	—	13	27	no	—
81	C₃×3₋¹⁺²	—	13	27	no	—
82	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	3	yes	—
83	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	6	yes	complemented
84	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	27	yes	complemented
85	C₃×H(3,𝔽₃)	—	13	6	no	—
86	C₃×H(3,𝔽₃)	—	13	3	no	—
87	C₃×H(3,𝔽₃)	—	13	27	no	—
88	C₃²×C₉	—	13	6	no	—
89	C₃²×C₉	—	13	27	no	—
90	C₃²×C₉	C₃×C₁₃⋊C₃	1	3	yes	complemented
91	C₃×C₁₃	—	3	3	no	—
92	C₃×C₁₃	—	3	6	no	—
93	C₃×C₁₃	—	3	3	no	—
94	C₃×C₁₃	3₋¹⁺⁴	1	3	yes	complemented
95	C₃×C₁₃	C₃⁵	1	1	yes	characteristic
96	C₁₃⋊C₃	—	3	27	no	—
97	C₃³	—	9	3	no	—
98	C₃³	—	13	9	no	—
99	C₃³	—	13	27	no	—
100	C₃³	—	117	9	no	—
101	C₃³	C₃²×C₁₃⋊C₃	1	2	yes	—
102	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
103	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
104	C₃³	C₃²×C₁₃⋊C₃	1	9	yes	—
105	3₋¹⁺²	—	13	54	no	—
106	3₋¹⁺²	—	13	54	no	—
107	3₋¹⁺²	—	13	54	no	—
108	3₋¹⁺²	—	13	243	no	—
109	3₋¹⁺²	—	13	243	no	—
110	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	27	yes	—
111	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	54	yes	complemented
112	H(3,𝔽₃)	—	13	54	no	—
113	H(3,𝔽₃)	—	13	27	no	—
114	C₃×C₉	—	13	54	no	—
115	C₃×C₉	—	13	18	no	—
116	C₃×C₉	—	13	243	no	—
117	C₃×C₉	C₃²×C₁₃⋊C₃	1	9	yes	—
118	C₃×C₉	C₃²×C₁₃⋊C₃	1	27	yes	complemented
119	C₁₃	C₃×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
120	C₃²	—	3	3	no	—
121	C₃²	—	3	6	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	9	9	no	—
124	C₃²	—	13	27	no	—
125	C₃²	—	39	27	no	—
126	C₃²	—	117	27	no	—
127	C₃²	C₃³×C₁₃⋊C₃	1	6	yes	—
128	C₃²	C₃³×C₁₃⋊C₃	1	3	yes	—
129	C₃²	C₃³×C₁₃⋊C₃	1	3	yes	—
130	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
131	C₉	—	13	54	no	—
132	C₉	C₃³×C₁₃⋊C₃	1	27	yes	—
133	C₃	—	3	3	no	—
134	C₃	—	3	6	no	—
135	C₃	—	3	3	no	—
136	C₃	—	39	27	no	—
137	C₃	#15691697	1	3	yes	direct product factor
138	C₃	C₃⁴×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691697

C₃

yes

maximal, direct product factor

C₃×#11760454

C₃

yes

maximal, complemented

C₃×#11760453

C₃

yes

maximal, complemented

C₃²×#66851

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₁₃⋊C₃

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₁₃⋊C₃)

C₃

yes

maximal, characteristic

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₃⋊C₃

C₃²

yes

—

#11760454

C₃²

486

yes

complemented

#11760453

C₃²

243

yes

complemented

C₃×#66851

C₃²

yes

—

C₃×#66851

C₃²

yes

complemented

C₃×#66851

C₃²

yes

complemented

C₃×#66851

C₃²

yes

complemented

C₃×#66848

C₃²

yes

complemented

C₃×#66849

C₃²

yes

complemented

3₋¹⁺²×C₁₃⋊C₃

C₃²

243

yes

complemented

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

C₃²×C₁₃⋊C₉

C₃²

yes

complemented

C₃×C₉×C₁₃⋊C₃

C₃²

yes

complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₃×3₋¹⁺⁴

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₃³×C₁₃

C₃³

yes

—

C₃³×C₁₃

C₃³

yes

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×3₋¹⁺²

C₃³

yes

—

C₁₃×3₋¹⁺²

C₃³

yes

complemented

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

complemented

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

C₃³

yes

—

#66851

C₃³

yes

complemented

#66851

C₃³

yes

complemented

C₃×C₁₃⋊C₉

C₃³

yes

—

C₃×C₁₃⋊C₉

C₃³

yes

complemented

#66849

C₃³

243

yes

complemented

#66848

C₃³

243

yes

complemented

C₉×C₁₃⋊C₃

C₃³

243

yes

complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₁₃⋊C₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃⁴

yes

—

C₃²×C₁₃

C₃⁴

yes

—

C₃²×C₁₃

C₃⁴

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

C₃⁴

yes

—

C₉×C₁₃

C₃⁴

yes

—

C₁₃⋊C₉

C₃⁴

yes

—

C₃⁴

—

C₃⁴

C₃×C₁₃⋊C₃

yes

characteristic

#737

—

486

—

#737

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

complemented

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₃⋊C₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

3₋¹⁺⁴

yes

complemented

C₃×C₁₃

C₃⁵

yes

characteristic

C₁₃⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

100

C₃³

—

117

—

101

C₃³

C₃²×C₁₃⋊C₃

yes

—

102

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

103

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

104

C₃³

C₃²×C₁₃⋊C₃

yes

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

243

—

109

3₋¹⁺²

—

243

—

110

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

—

111

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

complemented

112

H(3,𝔽₃)

—

113

H(3,𝔽₃)

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

243

—

117

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

118

C₃×C₉

C₃²×C₁₃⋊C₃

yes

complemented

119

C₁₃

C₃×3₋¹⁺⁴

yes

characteristic, complemented, Sylow

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

117

—

127

C₃²

C₃³×C₁₃⋊C₃

yes

—

128

C₃²

C₃³×C₁₃⋊C₃

yes

—

129

C₃²

C₃³×C₁₃⋊C₃

yes

—

130

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

131

C₉

—

132

C₉

C₃³×C₁₃⋊C₃

yes

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

#15691697

yes

direct product factor

138

C₃

C₃⁴×C₁₃⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	C₃³×C₁₃⋊C₃
commutator, cosocle	C₃×C₁₃	C₃⁵
Frattini	C₃	C₃⁴×C₁₃⋊C₃
Fitting	C₃²×C₁₃×3₋¹⁺²	C₃
socle	C₃²×C₁₃	C₃⁴
prime radical	C₃²×C₃²⋊(C₁₃⋊C₃)	C₃
hypercenter	C₃²×3₋¹⁺²	C₁₃⋊C₃
hypocenter	C₁₃	C₃×3₋¹⁺⁴
LCS	G ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃³×C₁₃⋊C₃

commutator, cosocle

C₃×C₁₃

C₃⁵

Frattini

C₃

C₃⁴×C₁₃⋊C₃

Fitting

C₃²×C₁₃×3₋¹⁺²

C₃

socle

C₃²×C₁₃

C₃⁴

prime radical

C₃²×C₃²⋊(C₁₃⋊C₃)

C₃

hypercenter

C₃²×3₋¹⁺²

C₁₃⋊C₃

hypocenter

C₁₃

C₃×3₋¹⁺⁴

LCS

G ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. #15691697	C₃	direct (N × H)
2. C₃×#11760454	C₃	semidirect
3. C₃×#11760453	C₃	semidirect
4. C₃²×#66851	C₃	semidirect
5. C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	semidirect
7. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
9. #11760454	C₃²	semidirect
10. #11760453	C₃²	semidirect
12. C₃×#66851	C₃²	semidirect
13. C₃×#66851	C₃²	semidirect
14. C₃×#66851	C₃²	semidirect
15. C₃×#66848	C₃²	semidirect
16. C₃×#66849	C₃²	semidirect
17. 3₋¹⁺²×C₁₃⋊C₃	C₃²	semidirect
21. C₃²×C₁₃⋊C₉	C₃²	semidirect
22. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
25. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
26. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
27. C₃²×C₉×C₁₃	C₃²	semidirect
38. C₁₃×3₋¹⁺²	C₃³	semidirect
40. C₃×C₉×C₁₃	C₃³	semidirect
44. #66851	C₃³	semidirect
45. #66851	C₃³	semidirect
47. C₃×C₁₃⋊C₉	C₃³	semidirect
48. #66849	C₃³	semidirect
49. #66848	C₃³	semidirect
50. C₉×C₁₃⋊C₃	C₃³	semidirect
56. C₃²×3₋¹⁺²	C₁₃⋊C₃	semidirect
83. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
84. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
90. C₃²×C₉	C₃×C₁₃⋊C₃	semidirect
94. C₃×C₁₃	3₋¹⁺⁴	semidirect
111. 3₋¹⁺²	C₃²×C₁₃⋊C₃	semidirect
118. C₃×C₉	C₃²×C₁₃⋊C₃	semidirect
119. C₁₃	C₃×3₋¹⁺⁴	semidirect
137. C₃	#15691697	direct (N × H)

H ≅ G/N

type

1. #15691697

C₃

direct (N × H)

2. C₃×#11760454

C₃

semidirect

3. C₃×#11760453

C₃

semidirect

4. C₃²×#66851

C₃

semidirect

5. C₃×3₋¹⁺²×C₁₃⋊C₃

C₃

semidirect

7. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

9. #11760454

C₃²

semidirect

10. #11760453

C₃²

semidirect

12. C₃×#66851

C₃²

semidirect

13. C₃×#66851

C₃²

semidirect

14. C₃×#66851

C₃²

semidirect

15. C₃×#66848

C₃²

semidirect

16. C₃×#66849

C₃²

semidirect

17. 3₋¹⁺²×C₁₃⋊C₃

C₃²

semidirect

21. C₃²×C₁₃⋊C₉

C₃²

semidirect

22. C₃×C₉×C₁₃⋊C₃

C₃²

semidirect

25. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

26. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

27. C₃²×C₉×C₁₃

C₃²

semidirect

38. C₁₃×3₋¹⁺²

C₃³

semidirect

40. C₃×C₉×C₁₃

C₃³

semidirect

44. #66851

C₃³

semidirect

45. #66851

C₃³

semidirect

47. C₃×C₁₃⋊C₉

C₃³

semidirect

48. #66849

C₃³

semidirect

49. #66848

C₃³

semidirect

50. C₉×C₁₃⋊C₃

C₃³

semidirect

56. C₃²×3₋¹⁺²

C₁₃⋊C₃

semidirect

83. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

84. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

90. C₃²×C₉

C₃×C₁₃⋊C₃

semidirect

94. C₃×C₁₃

3₋¹⁺⁴

semidirect

111. 3₋¹⁺²

C₃²×C₁₃⋊C₃

semidirect

118. C₃×C₉

C₃²×C₁₃⋊C₃

semidirect

119. C₁₃

C₃×3₋¹⁺⁴

semidirect

137. C₃

#15691697

direct (N × H)

Subgroups — C₃×#15691697 #23748619

Conjugacy classes of subgroups

Complements

Subgroups — C3×#15691697 #23748619

Conjugacy classes of subgroups

Complements

Subgroups — C₃×#15691697 #23748619