Group #23748616 — subgroups

Subgroups — C₁₃⋊C₃×3₋¹⁺⁴ #23748616

Order 9477.

5588 conjugacy classes of subgroups; only the 2881 normal subgroups are shown.

normal subgroups only

This group has 5588 conjugacy classes of subgroups — too many to draw. Only its 2881 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	#15691698	C₃	1	2	yes	maximal, complemented
2	#15691697	C₃	1	24	yes	maximal, complemented
3	#15691696	C₃	1	27	yes	maximal, complemented
4	#15691695	C₃	1	27	yes	maximal, complemented
5	C₁₃⋊C₃×#737	C₃	1	27	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	1	12	yes	maximal, complemented
7	C₃×H(3,𝔽₃)×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic
8	C₁₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃³×C₁₃⋊C₃	C₃²	1	4	yes	—
10	#11760454	C₃²	1	432	yes	complemented
11	#11760453	C₃²	1	216	yes	complemented
12	#11760452	C₃²	1	54	yes	complemented
13	C₃×#66851	C₃²	1	24	yes	complemented
14	C₃×#66848	C₃²	1	36	yes	complemented
15	C₃×#66849	C₃²	1	36	yes	complemented
16	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	9	yes	—
17	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	72	yes	complemented
18	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	72	yes	complemented
19	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	72	yes	complemented
20	3₋¹⁺²×C₁₃⋊C₃	C₃²	1	72	yes	complemented
21	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	8	yes	—
22	H(3,𝔽₃)×C₁₃⋊C₃	C₃²	1	9	yes	—
23	H(3,𝔽₃)×C₁₃⋊C₃	C₃²	1	9	yes	—
24	H(3,𝔽₃)×C₁₃⋊C₃	C₃²	1	9	yes	—
25	C₃×C₉×C₁₃⋊C₃	C₃²	1	36	yes	complemented
26	C₁₃×#737	C₃²	1	27	yes	complemented
27	C₃×C₁₃×3₋¹⁺²	C₃²	1	12	yes	complemented
28	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
29	C₃×3₋¹⁺⁴	—	13	1	no	maximal, Sylow
30	C₃³×C₁₃	C₃³	1	4	yes	—
31	C₃²×C₁₃⋊C₃	—	9	12	no	—
32	C₃²×C₁₃⋊C₃	C₃³	1	12	yes	—
33	C₃²×C₁₃⋊C₃	C₃³	1	1	yes	characteristic
34	C₃²×C₁₃⋊C₃	C₃³	1	12	yes	—
35	C₃²×C₁₃⋊C₃	C₃³	1	12	yes	—
36	C₃²×C₁₃⋊C₃	C₃³	1	8	yes	—
37	C₁₃×3₋¹⁺²	C₃³	1	9	yes	—
38	C₁₃×3₋¹⁺²	C₃³	1	72	yes	complemented
39	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
40	C₃×C₉×C₁₃	C₃³	1	36	yes	complemented
41	C₃²⋊(C₁₃⋊C₃)	C₃³	1	72	yes	—
42	#66851	C₃³	1	18	yes	—
43	#66851	C₃³	1	144	yes	complemented
44	C₃×C₁₃⋊C₉	C₃³	1	72	yes	complemented
45	#66849	C₃³	1	27	yes	—
46	#66849	C₃³	1	216	yes	complemented
47	#66848	C₃³	1	27	yes	—
48	#66848	C₃³	1	216	yes	complemented
49	C₉×C₁₃⋊C₃	C₃³	1	27	yes	—
50	C₉×C₁₃⋊C₃	C₃³	1	216	yes	complemented
51	3₋¹⁺⁴	—	13	27	no	—
52	3₋¹⁺⁴	—	13	27	no	—
53	3₋¹⁺⁴	—	13	24	no	—
54	3₋¹⁺⁴	—	13	2	no	—
55	3₋¹⁺⁴	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
56	C₃×#737	—	13	27	no	—
57	C₃²×3₋¹⁺²	—	13	12	no	—
58	C₃²×H(3,𝔽₃)	—	13	1	no	—
59	C₃²×C₁₃	—	9	4	no	—
60	C₃²×C₁₃	C₃⁴	1	12	yes	—
61	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
62	C₃×C₁₃⋊C₃	—	3	3	no	—
63	C₃×C₁₃⋊C₃	—	3	36	no	—
64	C₃×C₁₃⋊C₃	—	9	12	no	—
65	C₃×C₁₃⋊C₃	—	9	12	no	—
66	C₃×C₁₃⋊C₃	—	9	8	no	—
67	C₃×C₁₃⋊C₃	C₃⁴	1	24	yes	—
68	C₃×C₁₃⋊C₃	C₃⁴	1	1	yes	characteristic
69	C₃×C₁₃⋊C₃	C₃⁴	1	1	yes	characteristic
70	C₃×C₁₃⋊C₃	C₃⁴	1	1	yes	characteristic
71	C₉×C₁₃	C₃⁴	1	27	yes	—
72	C₁₃⋊C₉	C₃⁴	1	54	yes	—
73	C₃⁴	—	13	4	no	—
74	#737	—	13	432	no	—
75	#737	—	13	54	no	—
76	#737	—	13	216	no	—
77	#737	C₃×C₁₃⋊C₃	1	27	yes	complemented
78	C₃×3₋¹⁺²	—	13	24	no	—
79	C₃×3₋¹⁺²	—	13	9	no	—
80	C₃×3₋¹⁺²	—	13	72	no	—
81	C₃×3₋¹⁺²	—	13	72	no	—
82	C₃×3₋¹⁺²	—	13	72	no	—
83	C₃×3₋¹⁺²	—	13	72	no	—
84	C₃×3₋¹⁺²	—	13	36	no	—
85	C₃×3₋¹⁺²	—	13	36	no	—
86	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	12	yes	complemented
87	C₃×H(3,𝔽₃)	—	13	8	no	—
88	C₃×H(3,𝔽₃)	—	13	9	no	—
89	C₃×H(3,𝔽₃)	—	13	9	no	—
90	C₃×H(3,𝔽₃)	—	13	9	no	—
91	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
92	C₃²×C₉	—	13	36	no	—
93	C₃×C₁₃	—	3	1	no	—
94	C₃×C₁₃	—	3	12	no	—
95	C₃×C₁₃	C₃⁵	1	1	yes	characteristic
96	C₁₃⋊C₃	—	3	1	no	—
97	C₁₃⋊C₃	—	3	1	no	—
98	C₁₃⋊C₃	—	3	24	no	—
99	C₁₃⋊C₃	3₋¹⁺⁴	1	3	yes	direct product factor
100	C₃³	—	13	12	no	—
101	C₃³	—	13	1	no	—
102	C₃³	—	13	8	no	—
103	C₃³	—	13	12	no	—
104	C₃³	—	13	12	no	—
105	C₃³	—	117	12	no	—
106	C₃³	C₃²×C₁₃⋊C₃	1	4	yes	—
107	3₋¹⁺²	—	13	18	no	—
108	3₋¹⁺²	—	13	144	no	—
109	3₋¹⁺²	—	13	27	no	—
110	3₋¹⁺²	—	13	216	no	—
111	3₋¹⁺²	—	13	27	no	—
112	3₋¹⁺²	—	13	216	no	—
113	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	9	yes	—
114	3₋¹⁺²	C₃²×C₁₃⋊C₃	1	72	yes	complemented
115	H(3,𝔽₃)	—	13	72	no	—
116	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
117	C₃×C₉	—	13	72	no	—
118	C₃×C₉	—	13	27	no	—
119	C₃×C₉	—	13	216	no	—
120	C₃×C₉	C₃²×C₁₃⋊C₃	1	36	yes	complemented
121	C₁₃	C₃×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
122	C₃²	—	9	4	no	—
123	C₃²	—	13	1	no	—
124	C₃²	—	13	24	no	—
125	C₃²	—	13	1	no	—
126	C₃²	—	13	1	no	—
127	C₃²	—	39	3	no	—
128	C₃²	—	39	36	no	—
129	C₃²	—	117	8	no	—
130	C₃²	—	117	12	no	—
131	C₃²	—	117	12	no	—
132	C₃²	C₃³×C₁₃⋊C₃	1	12	yes	—
133	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
134	C₉	—	13	54	no	—
135	C₉	C₃³×C₁₃⋊C₃	1	27	yes	—
136	C₃	—	3	1	no	—
137	C₃	—	3	12	no	—
138	C₃	—	13	3	no	—
139	C₃	—	39	1	no	—
140	C₃	—	39	1	no	—
141	C₃	—	39	24	no	—
142	C₃	C₃⁴×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691698

C₃

yes

maximal, complemented

#15691697

C₃

yes

maximal, complemented

#15691696

C₃

yes

maximal, complemented

#15691695

C₃

yes

maximal, complemented

C₁₃⋊C₃×#737

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₁₃⋊C₃

C₃

yes

maximal, complemented

C₃×H(3,𝔽₃)×C₁₃⋊C₃

C₃

yes

maximal, characteristic

C₁₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₃⋊C₃

C₃²

yes

—

#11760454

C₃²

432

yes

complemented

#11760453

C₃²

216

yes

complemented

#11760452

C₃²

yes

complemented

C₃×#66851

C₃²

yes

complemented

C₃×#66848

C₃²

yes

complemented

C₃×#66849

C₃²

yes

complemented

3₋¹⁺²×C₁₃⋊C₃

C₃²

yes

—

3₋¹⁺²×C₁₃⋊C₃

C₃²

yes

complemented

3₋¹⁺²×C₁₃⋊C₃

C₃²

yes

complemented

3₋¹⁺²×C₁₃⋊C₃

C₃²

yes

complemented

3₋¹⁺²×C₁₃⋊C₃

C₃²

yes

complemented

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

H(3,𝔽₃)×C₁₃⋊C₃

C₃²

yes

—

H(3,𝔽₃)×C₁₃⋊C₃

C₃²

yes

—

H(3,𝔽₃)×C₁₃⋊C₃

C₃²

yes

—

C₃×C₉×C₁₃⋊C₃

C₃²

yes

complemented

C₁₃×#737

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×3₋¹⁺⁴

—

maximal, Sylow

C₃³×C₁₃

C₃³

yes

—

C₃²×C₁₃⋊C₃

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₃²×C₁₃⋊C₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×3₋¹⁺²

C₃³

yes

—

C₁₃×3₋¹⁺²

C₃³

yes

complemented

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

complemented

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

C₃³

yes

—

#66851

C₃³

144

yes

complemented

C₃×C₁₃⋊C₉

C₃³

yes

complemented

#66849

C₃³

yes

—

#66849

C₃³

216

yes

complemented

#66848

C₃³

yes

—

#66848

C₃³

216

yes

complemented

C₉×C₁₃⋊C₃

C₃³

yes

—

C₉×C₁₃⋊C₃

C₃³

216

yes

complemented

3₋¹⁺⁴

—

3₋¹⁺⁴

—

3₋¹⁺⁴

—

3₋¹⁺⁴

—

3₋¹⁺⁴

C₁₃⋊C₃

yes

characteristic, direct product factor

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃⁴

yes

—

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

C₃⁴

yes

—

C₃×C₁₃⋊C₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

C₃⁴

yes

characteristic

C₉×C₁₃

C₃⁴

yes

—

C₁₃⋊C₉

C₃⁴

yes

—

C₃⁴

—

#737

—

432

—

#737

—

#737

—

216

—

#737

C₃×C₁₃⋊C₃

yes

complemented

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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

characteristic

C₃²×C₉

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃⁵

yes

characteristic

C₁₃⋊C₃

—

C₁₃⋊C₃

—

C₁₃⋊C₃

—

C₁₃⋊C₃

3₋¹⁺⁴

yes

direct product factor

100

C₃³

—

101

C₃³

—

102

C₃³

—

103

C₃³

—

104

C₃³

—

105

C₃³

—

117

—

106

C₃³

C₃²×C₁₃⋊C₃

yes

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

144

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

216

—

111

3₋¹⁺²

—

112

3₋¹⁺²

—

216

—

113

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

—

114

3₋¹⁺²

C₃²×C₁₃⋊C₃

yes

complemented

115

H(3,𝔽₃)

—

116

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

216

—

120

C₃×C₉

C₃²×C₁₃⋊C₃

yes

complemented

121

C₁₃

C₃×3₋¹⁺⁴

yes

characteristic, complemented, Sylow

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

117

—

130

C₃²

—

117

—

131

C₃²

—

117

—

132

C₃²

C₃³×C₁₃⋊C₃

yes

—

133

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

134

C₉

—

135

C₉

C₃³×C₁₃⋊C₃

yes

—

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

C₃⁴×C₁₃⋊C₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₃	C₃⁴×C₁₃⋊C₃
commutator, socle, cosocle	C₃×C₁₃	C₃⁵
Fitting	C₁₃×3₋¹⁺⁴	C₃
prime radical	C₃×H(3,𝔽₃)×C₁₃⋊C₃	C₃
hypercenter	3₋¹⁺⁴	C₁₃⋊C₃
hypocenter	C₁₃	C₃×3₋¹⁺⁴
LCS	G ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺⁴

subgroup

G/H

center, Frattini

C₃

C₃⁴×C₁₃⋊C₃

commutator, socle, cosocle

C₃×C₁₃

C₃⁵

Fitting

C₁₃×3₋¹⁺⁴

C₃

prime radical

C₃×H(3,𝔽₃)×C₁₃⋊C₃

C₃

hypercenter

3₋¹⁺⁴

C₁₃⋊C₃

hypocenter

C₁₃

C₃×3₋¹⁺⁴

LCS

G ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃ ◁ 3₋¹⁺⁴

N	H ≅ G/N	type
1. #15691698	C₃	semidirect
2. #15691697	C₃	semidirect
3. #15691696	C₃	semidirect
4. #15691695	C₃	semidirect
5. C₁₃⋊C₃×#737	C₃	semidirect
6. C₃×3₋¹⁺²×C₁₃⋊C₃	C₃	semidirect
8. C₁₃×3₋¹⁺⁴	C₃	semidirect
10. #11760454	C₃²	semidirect
11. #11760453	C₃²	semidirect
12. #11760452	C₃²	semidirect
13. C₃×#66851	C₃²	semidirect
14. C₃×#66848	C₃²	semidirect
15. C₃×#66849	C₃²	semidirect
17. 3₋¹⁺²×C₁₃⋊C₃	C₃²	semidirect
18. 3₋¹⁺²×C₁₃⋊C₃	C₃²	semidirect
19. 3₋¹⁺²×C₁₃⋊C₃	C₃²	semidirect
20. 3₋¹⁺²×C₁₃⋊C₃	C₃²	semidirect
25. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
26. C₁₃×#737	C₃²	semidirect
27. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
38. C₁₃×3₋¹⁺²	C₃³	semidirect
40. C₃×C₉×C₁₃	C₃³	semidirect
43. #66851	C₃³	semidirect
44. C₃×C₁₃⋊C₉	C₃³	semidirect
46. #66849	C₃³	semidirect
48. #66848	C₃³	semidirect
50. C₉×C₁₃⋊C₃	C₃³	semidirect
55. 3₋¹⁺⁴	C₁₃⋊C₃	direct (N × H)
77. #737	C₃×C₁₃⋊C₃	semidirect
86. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
99. C₁₃⋊C₃	3₋¹⁺⁴	direct (N × H)
114. 3₋¹⁺²	C₃²×C₁₃⋊C₃	semidirect
120. C₃×C₉	C₃²×C₁₃⋊C₃	semidirect
121. C₁₃	C₃×3₋¹⁺⁴	semidirect

H ≅ G/N

type

1. #15691698

C₃

semidirect

2. #15691697

C₃

semidirect

3. #15691696

C₃

semidirect

4. #15691695

C₃

semidirect

5. C₁₃⋊C₃×#737

C₃

semidirect

6. C₃×3₋¹⁺²×C₁₃⋊C₃

C₃

semidirect

8. C₁₃×3₋¹⁺⁴

C₃

semidirect

10. #11760454

C₃²

semidirect

11. #11760453

C₃²

semidirect

12. #11760452

C₃²

semidirect

13. C₃×#66851

C₃²

semidirect

14. C₃×#66848

C₃²

semidirect

15. C₃×#66849

C₃²

semidirect

17. 3₋¹⁺²×C₁₃⋊C₃

C₃²

semidirect

18. 3₋¹⁺²×C₁₃⋊C₃

C₃²

semidirect

19. 3₋¹⁺²×C₁₃⋊C₃

C₃²

semidirect

20. 3₋¹⁺²×C₁₃⋊C₃

C₃²

semidirect

25. C₃×C₉×C₁₃⋊C₃

C₃²

semidirect

26. C₁₃×#737

C₃²

semidirect

27. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

38. C₁₃×3₋¹⁺²

C₃³

semidirect

40. C₃×C₉×C₁₃

C₃³

semidirect

43. #66851

C₃³

semidirect

44. C₃×C₁₃⋊C₉

C₃³

semidirect

46. #66849

C₃³

semidirect

48. #66848

C₃³

semidirect

50. C₉×C₁₃⋊C₃

C₃³

semidirect

55. 3₋¹⁺⁴

C₁₃⋊C₃

direct (N × H)

77. #737

C₃×C₁₃⋊C₃

semidirect

86. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

99. C₁₃⋊C₃

3₋¹⁺⁴

direct (N × H)

114. 3₋¹⁺²

C₃²×C₁₃⋊C₃

semidirect

120. C₃×C₉

C₃²×C₁₃⋊C₃

semidirect

121. C₁₃

C₃×3₋¹⁺⁴

semidirect

Subgroups — C₁₃⋊C₃×3₋¹⁺⁴ #23748616

Conjugacy classes of subgroups

Complements

Subgroups — C13⋊C3×3−1+4 #23748616

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃⋊C₃×3₋¹⁺⁴ #23748616