Group #23748600 — subgroups

Subgroups — #23748600

Order 9477.

948 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 286 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	3	yes	maximal, complemented
2	#15691679	C₃	1	18	yes	maximal, complemented
3	#15691661	C₃	1	3	yes	maximal, complemented
4	#15691656	C₃	1	3	yes	maximal, complemented
5	#15691651	C₃	1	9	yes	maximal, complemented
6	C₃×#11760443	C₃	1	2	yes	maximal, complemented
7	C₃×#11760441	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	#11760454	—	3	18	no	—
10	#11760453	C₃²	1	27	yes	complemented
11	C₃×#66851	C₃²	1	6	yes	—
12	3₋¹⁺²×C₁₃⋊C₃	—	3	9	no	—
13	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
14	#11760443	C₃²	1	6	yes	complemented
15	#11760443	C₃²	1	12	yes	complemented
16	#11760441	C₃²	1	3	yes	—
17	#11760441	C₃²	1	6	yes	—
18	#11760438	C₃²	1	27	yes	complemented
19	#11760435	C₃²	1	27	yes	complemented
20	C₁₃×#737	—	3	3	no	—
21	C₁₃×#737	—	3	6	no	—
22	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
23	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
24	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
25	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
26	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
27	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
28	#10624936	—	13	1	no	maximal, Sylow
29	C₃³×C₁₃	C₃³	1	3	yes	—
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₃²×C₁₃⋊C₃	—	3	3	no	—
32	C₁₃×3₋¹⁺²	—	3	2	no	—
33	C₁₃×3₋¹⁺²	—	3	1	no	—
34	C₁₃×3₋¹⁺²	—	3	18	no	—
35	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
36	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
37	C₁₃×H(3,𝔽₃)	—	3	2	no	—
38	C₁₃×H(3,𝔽₃)	—	3	1	no	—
39	C₃×C₉×C₁₃	—	3	2	no	—
40	C₃×C₉×C₁₃	—	3	1	no	—
41	C₃×C₉×C₁₃	—	3	9	no	—
42	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
43	C₃×C₉×C₁₃	C₃³	1	2	yes	—
44	C₃×C₉×C₁₃	C₃³	1	3	yes	—
45	C₃×C₉×C₁₃	C₃³	1	3	yes	—
46	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
47	C₃²⋊(C₁₃⋊C₃)	C₃³	1	9	yes	—
48	#66851	—	3	6	no	—
49	#66851	—	3	6	no	—
50	#66851	C₃³	1	18	yes	—
51	C₃×C₁₃⋊C₉	—	3	6	no	—
52	#66849	—	3	27	no	—
53	#66848	—	3	27	no	—
54	C₉×C₁₃⋊C₃	—	3	27	no	—
55	3₋¹⁺⁴	—	13	3	no	—
56	C₃×#737	C₁₃⋊C₃	1	1	yes	characteristic, complemented
57	#6913	—	13	18	no	—
58	#6910	—	13	3	no	—
59	#6909	—	13	3	no	—
60	#6908	—	13	9	no	—
61	C₃×#733	—	13	2	no	—
62	C₃×#732	—	13	1	no	—
63	C₃²×C₁₃	—	3	3	no	—
64	C₃²×C₁₃	—	9	1	no	—
65	C₃²×C₁₃	—	9	3	no	—
66	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
67	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
69	C₃×C₁₃⋊C₃	—	3	9	no	—
70	C₃×C₁₃⋊C₃	—	27	3	no	—
71	C₉×C₁₃	—	3	1	no	—
72	C₉×C₁₃	—	3	2	no	—
73	C₉×C₁₃	—	3	3	no	—
74	C₉×C₁₃	C₃×H(3,𝔽₃)	1	9	yes	—
75	C₁₃⋊C₉	—	3	18	no	—
76	#737	—	3	3	no	—
77	#737	—	3	6	no	—
78	#737	—	13	27	no	—
79	#737	—	39	18	no	—
80	C₃×3₋¹⁺²	—	13	6	no	—
81	C₃×3₋¹⁺²	—	39	9	no	—
82	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
83	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	1	yes	characteristic
84	C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	1	6	yes	complemented
85	C₃×H(3,𝔽₃)	—	13	3	no	—
86	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
87	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
88	C₃²×C₉	C₃×C₁₃⋊C₃	1	3	yes	complemented
89	#733	—	13	12	no	—
90	#733	—	13	6	no	—
91	#732	—	13	6	no	—
92	#732	—	13	3	no	—
93	#731	—	13	27	no	—
94	AΣL(1,𝔽₂₇)	—	13	27	no	—
95	C₃×C₁₃	—	3	1	no	—
96	C₃×C₁₃	—	3	1	no	—
97	C₃×C₁₃	—	3	2	no	—
98	C₃×C₁₃	—	9	3	no	—
99	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
100	C₁₃⋊C₃	—	9	9	no	—
101	C₃³	—	39	3	no	—
102	C₃³	C₃²×C₁₃⋊C₃	1	3	yes	—
103	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
104	3₋¹⁺²	—	3	2	no	—
105	3₋¹⁺²	—	3	1	no	—
106	3₋¹⁺²	—	3	18	no	—
107	3₋¹⁺²	—	13	18	no	—
108	3₋¹⁺²	—	39	6	no	—
109	3₋¹⁺²	—	39	6	no	—
110	3₋¹⁺²	—	39	27	no	—
111	3₋¹⁺²	—	39	27	no	—
112	3₋¹⁺²	C₃²⋊(C₁₃⋊C₃)	1	3	yes	—
113	3₋¹⁺²	C₃²⋊(C₁₃⋊C₃)	1	6	yes	—
114	H(3,𝔽₃)	—	3	2	no	—
115	H(3,𝔽₃)	—	3	1	no	—
116	H(3,𝔽₃)	—	13	9	no	—
117	H(3,𝔽₃)	—	39	6	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	1	no	—
120	C₃×C₉	—	3	9	no	—
121	C₃×C₉	—	39	6	no	—
122	C₃×C₉	—	39	27	no	—
123	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
124	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
125	C₃×C₉	C₃²×C₁₃⋊C₃	1	2	yes	—
126	C₃×C₉	C₃²×C₁₃⋊C₃	1	3	yes	—
127	C₁₃	#10624936	1	1	yes	characteristic, complemented, Sylow
128	C₃²	—	3	3	no	—
129	C₃²	—	9	1	no	—
130	C₃²	—	9	3	no	—
131	C₃²	—	39	9	no	—
132	C₃²	—	351	3	no	—
133	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	2	yes	—
134	C₃²	C₃×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
135	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
136	C₉	—	3	1	no	—
137	C₉	—	3	2	no	—
138	C₉	—	3	3	no	—
139	C₉	—	39	18	no	—
140	C₉	C₃×C₃²⋊(C₁₃⋊C₃)	1	9	yes	—
141	C₃	—	3	1	no	—
142	C₃	—	3	1	no	—
143	C₃	—	3	2	no	—
144	C₃	—	9	3	no	—
145	C₃	—	117	9	no	—
146	C₃	C₃²×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691697

C₃

yes

maximal, complemented

#15691679

C₃

yes

maximal, complemented

#15691661

C₃

yes

maximal, complemented

#15691656

C₃

yes

maximal, complemented

#15691651

C₃

yes

maximal, complemented

C₃×#11760443

C₃

yes

maximal, complemented

C₃×#11760441

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×#737

C₃

yes

maximal, characteristic, complemented

#11760454

—

#11760453

C₃²

yes

complemented

C₃×#66851

C₃²

yes

—

3₋¹⁺²×C₁₃⋊C₃

—

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

—

#11760443

C₃²

yes

complemented

#11760443

C₃²

yes

complemented

#11760441

C₃²

yes

—

#11760441

C₃²

yes

—

#11760438

C₃²

yes

complemented

#11760435

C₃²

yes

complemented

C₁₃×#737

—

C₁₃×#737

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

complemented

#10624936

—

maximal, Sylow

C₃³×C₁₃

C₃³

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃×C₉×C₁₃

C₃³

yes

—

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

C₃³

yes

—

#66851

—

#66851

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

—

#66849

—

#66848

—

C₉×C₁₃⋊C₃

—

3₋¹⁺⁴

—

C₃×#737

C₁₃⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃⁴

yes

characteristic

C₃×C₁₃⋊C₃

—

C₃×C₁₃⋊C₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₁₃⋊C₉

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₃⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₃⋊C₃

yes

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

100

C₁₃⋊C₃

—

101

C₃³

—

102

C₃³

C₃²×C₁₃⋊C₃

yes

—

103

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

3₋¹⁺²

—

112

3₋¹⁺²

C₃²⋊(C₁₃⋊C₃)

yes

—

113

3₋¹⁺²

C₃²⋊(C₁₃⋊C₃)

yes

—

114

H(3,𝔽₃)

—

115

H(3,𝔽₃)

—

116

H(3,𝔽₃)

—

117

H(3,𝔽₃)

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

124

C₃×C₉

C₃²×C₁₃⋊C₃

yes

characteristic

125

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

126

C₃×C₉

C₃²×C₁₃⋊C₃

yes

—

127

C₁₃

#10624936

yes

characteristic, complemented, Sylow

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

351

—

133

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

134

C₃²

C₃×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

135

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

C₃×C₃²⋊(C₁₃⋊C₃)

yes

—

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

—

117

—

146

C₃

C₃²×C₃²⋊(C₁₃⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₁₃⋊C₃)
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting	C₃×C₁₃×#737	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₁₃⋊C₃
hypocenter	C₁₃	#10624936
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

subgroup

G/H

center

C₃

C₃²×C₃²⋊(C₁₃⋊C₃)

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting

C₃×C₁₃×#737

C₃

socle

C₃×C₁₃

C₃²×H(3,𝔽₃)

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

N	H ≅ G/N	type
1. #15691697	C₃	semidirect
2. #15691679	C₃	semidirect
3. #15691661	C₃	semidirect
4. #15691656	C₃	semidirect
5. #15691651	C₃	semidirect
6. C₃×#11760443	C₃	semidirect
7. C₃×#11760441	C₃	semidirect
8. C₃×C₁₃×#737	C₃	semidirect
10. #11760453	C₃²	semidirect
14. #11760443	C₃²	semidirect
15. #11760443	C₃²	semidirect
18. #11760438	C₃²	semidirect
19. #11760435	C₃²	semidirect
24. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
27. C₃²×C₉×C₁₃	C₃²	semidirect
56. C₃×#737	C₁₃⋊C₃	semidirect
84. C₃×3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
88. C₃²×C₉	C₃×C₁₃⋊C₃	semidirect
127. C₁₃	#10624936	semidirect

H ≅ G/N

type

1. #15691697

C₃

semidirect

2. #15691679

C₃

semidirect

3. #15691661

C₃

semidirect

4. #15691656

C₃

semidirect

5. #15691651

C₃

semidirect

6. C₃×#11760443

C₃

semidirect

7. C₃×#11760441

C₃

semidirect

8. C₃×C₁₃×#737

C₃

semidirect

10. #11760453

C₃²

semidirect

14. #11760443

C₃²

semidirect

15. #11760443

C₃²

semidirect

18. #11760438

C₃²

semidirect

19. #11760435

C₃²

semidirect

24. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

27. C₃²×C₉×C₁₃

C₃²

semidirect

56. C₃×#737

C₁₃⋊C₃

semidirect

84. C₃×3₋¹⁺²

C₃×C₁₃⋊C₃

semidirect

88. C₃²×C₉

C₃×C₁₃⋊C₃

semidirect

127. C₁₃

#10624936

semidirect

Subgroups — #23748600

Conjugacy classes of subgroups

Complements