Group #23748594 — subgroups

Subgroups — #23748594

Order 9477.

948 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 268 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	3	yes	maximal, complemented
2	C₃×#11760453	C₃	1	1	yes	maximal, characteristic, complemented
3	#15691676	C₃	1	9	yes	maximal, complemented
4	#15691675	C₃	1	9	yes	maximal, complemented
5	#15691660	C₃	1	3	yes	maximal, complemented
6	#15691655	C₃	1	3	yes	maximal, complemented
7	#15691649	C₃	1	9	yes	maximal, complemented
8	C₃×#11760443	C₃	1	2	yes	maximal, complemented
9	C₃×C₁₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
10	#11760453	—	3	3	no	—
11	#11760453	—	3	6	no	—
12	#11760452	—	3	18	no	—
13	#11760452	C₃²	1	27	yes	complemented
14	C₃×#66851	—	3	9	no	—
15	C₃×#66851	C₃²	1	6	yes	—
16	C₃×#66851	C₃²	1	1	yes	characteristic
17	C₃×#66851	C₃²	1	1	yes	characteristic
18	C₃×#66848	C₃²	1	3	yes	complemented
19	C₃×#66849	C₃²	1	3	yes	complemented
20	C₃×C₃²⋊(C₁₃⋊C₃)	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₁₃⋊C₃	C₃²	1	3	yes	complemented
22	#11760443	C₃²	1	6	yes	complemented
23	#11760443	C₃²	1	6	yes	complemented
24	#11760443	C₃²	1	6	yes	complemented
25	#11760437	C₃²	1	27	yes	complemented
26	#11760434	C₃²	1	27	yes	complemented
27	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
28	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
29	C₁₃×#732	C₃²	1	3	yes	—
30	C₁₃×#732	C₃²	1	3	yes	—
31	C₁₃×#732	C₃²	1	3	yes	—
32	#10624936	—	13	1	no	maximal, Sylow
33	C₃³×C₁₃	—	3	3	no	—
34	C₃³×C₁₃	C₃³	1	1	yes	characteristic
35	C₃²×C₁₃⋊C₃	C₃³	1	3	yes	—
36	C₁₃×H(3,𝔽₃)	—	3	6	no	—
37	C₁₃×H(3,𝔽₃)	C₃³	1	9	yes	—
38	C₃×C₉×C₁₃	—	3	2	no	—
39	C₃×C₉×C₁₃	—	3	1	no	—
40	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
41	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
42	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
43	C₃²⋊(C₁₃⋊C₃)	—	3	2	no	—
44	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
45	#66851	—	3	6	no	—
46	#66851	—	3	6	no	—
47	#66851	—	3	2	no	—
48	#66851	—	3	1	no	—
49	#66851	—	3	54	no	—
50	#66851	H(3,𝔽₃)	1	3	yes	—
51	#66851	H(3,𝔽₃)	1	6	yes	—
52	#66851	C₃³	1	18	yes	—
53	C₃×C₁₃⋊C₉	—	3	6	no	—
54	C₃×C₁₃⋊C₉	—	3	27	no	—
55	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
56	C₃×C₁₃⋊C₉	C₃³	1	3	yes	—
57	#66849	—	3	9	no	—
58	#66848	—	3	9	no	—
59	C₉×C₁₃⋊C₃	—	3	9	no	—
60	3₋¹⁺⁴	—	13	3	no	—
61	C₃×#737	—	13	1	no	—
62	#6913	—	13	9	no	—
63	#6913	—	13	9	no	—
64	#6910	—	13	3	no	—
65	#6909	—	13	3	no	—
66	#6908	—	13	9	no	—
67	C₃×#733	—	13	2	no	—
68	C₃×#732	C₁₃⋊C₃	1	1	yes	characteristic, complemented
69	C₃²×C₁₃	—	3	9	no	—
70	C₃²×C₁₃	—	9	1	no	—
71	C₃²×C₁₃	—	27	3	no	—
72	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	2	yes	—
73	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
75	C₃×C₁₃⋊C₃	—	3	3	no	—
76	C₃×C₁₃⋊C₃	—	9	3	no	—
77	C₉×C₁₃	—	3	1	no	—
78	C₉×C₁₃	—	3	1	no	—
79	C₉×C₁₃	—	3	1	no	—
80	C₁₃⋊C₉	—	3	3	no	—
81	C₁₃⋊C₉	—	3	18	no	—
82	C₁₃⋊C₉	C₃×H(3,𝔽₃)	1	9	yes	—
83	#737	—	13	27	no	—
84	#737	—	39	3	no	—
85	#737	—	39	6	no	—
86	#737	—	39	18	no	—
87	C₃×3₋¹⁺²	—	13	1	no	—
88	C₃×3₋¹⁺²	—	13	1	no	—
89	C₃×3₋¹⁺²	—	13	3	no	—
90	C₃×3₋¹⁺²	—	13	3	no	—
91	C₃×3₋¹⁺²	—	13	6	no	—
92	C₃×3₋¹⁺²	—	39	9	no	—
93	C₃×H(3,𝔽₃)	—	13	1	no	—
94	C₃×H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	—
95	C₃²×C₉	—	13	3	no	—
96	C₃²×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
97	#733	—	13	6	no	—
98	#733	—	13	6	no	—
99	#733	—	13	6	no	—
100	#732	C₃×C₁₃⋊C₃	1	3	yes	—
101	#732	C₃×C₁₃⋊C₃	1	3	yes	—
102	#732	C₃×C₁₃⋊C₃	1	3	yes	—
103	#731	—	13	27	no	—
104	AΣL(1,𝔽₂₇)	—	13	27	no	—
105	C₃×C₁₃	—	3	1	no	—
106	C₃×C₁₃	—	3	1	no	—
107	C₃×C₁₃	—	3	2	no	—
108	C₃×C₁₃	—	9	9	no	—
109	C₃×C₁₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
110	C₁₃⋊C₃	—	9	3	no	—
111	C₃³	—	3	3	no	—
112	C₃³	—	13	3	no	—
113	C₃³	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
114	3₋¹⁺²	—	13	3	no	—
115	3₋¹⁺²	—	13	6	no	—
116	3₋¹⁺²	—	13	18	no	—
117	3₋¹⁺²	—	39	9	no	—
118	3₋¹⁺²	—	39	9	no	—
119	3₋¹⁺²	—	39	54	no	—
120	3₋¹⁺²	—	39	6	no	—
121	3₋¹⁺²	—	39	2	no	—
122	3₋¹⁺²	—	39	1	no	—
123	3₋¹⁺²	—	39	6	no	—
124	H(3,𝔽₃)	—	3	6	no	—
125	H(3,𝔽₃)	—	39	2	no	—
126	H(3,𝔽₃)	—	39	1	no	—
127	H(3,𝔽₃)	C₃²×C₁₃⋊C₃	1	9	yes	—
128	C₃×C₉	—	3	2	no	—
129	C₃×C₉	—	3	1	no	—
130	C₃×C₉	—	13	3	no	—
131	C₃×C₉	—	13	3	no	—
132	C₃×C₉	—	39	9	no	—
133	C₃×C₉	—	39	27	no	—
134	C₃×C₉	—	39	6	no	—
135	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
136	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
137	C₃×C₉	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
138	C₁₃	#10624936	1	1	yes	characteristic, complemented, Sylow
139	C₃²	—	3	9	no	—
140	C₃²	—	9	1	no	—
141	C₃²	—	27	3	no	—
142	C₃²	—	39	3	no	—
143	C₃²	—	117	3	no	—
144	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	2	yes	—
145	C₃²	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic
146	C₃²	C₃³×C₁₃⋊C₃	1	1	yes	characteristic
147	C₉	—	3	1	no	—
148	C₉	—	3	1	no	—
149	C₉	—	3	1	no	—
150	C₉	—	13	9	no	—
151	C₉	—	39	3	no	—
152	C₉	—	39	18	no	—
153	C₃	—	3	1	no	—
154	C₃	—	3	1	no	—
155	C₃	—	3	2	no	—
156	C₃	—	9	9	no	—
157	C₃	—	117	3	no	—
158	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15691698

C₃

yes

maximal, complemented

C₃×#11760453

C₃

yes

maximal, characteristic, complemented

#15691676

C₃

yes

maximal, complemented

#15691675

C₃

yes

maximal, complemented

#15691660

C₃

yes

maximal, complemented

#15691655

C₃

yes

maximal, complemented

#15691649

C₃

yes

maximal, complemented

C₃×#11760443

C₃

yes

maximal, complemented

C₃×C₁₃×#732

C₃

yes

maximal, characteristic, complemented

—

—

—

C₃²

yes

complemented

C₃×#66851

—

C₃×#66851

C₃²

yes

—

C₃×#66851

C₃²

yes

characteristic

C₃×#66851

C₃²

yes

characteristic

C₃×#66848

C₃²

yes

complemented

C₃×#66849

C₃²

yes

complemented

C₃×C₃²⋊(C₁₃⋊C₃)

C₃²

yes

characteristic

C₃×C₉×C₁₃⋊C₃

C₃²

yes

complemented

#11760443

C₃²

yes

complemented

#11760443

C₃²

yes

complemented

#11760443

C₃²

yes

complemented

#11760437

C₃²

yes

complemented

#11760434

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×#732

C₃²

yes

—

C₁₃×#732

C₃²

yes

—

C₁₃×#732

C₃²

yes

—

#10624936

—

maximal, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃³

yes

characteristic

C₃²×C₁₃⋊C₃

C₃³

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₃²⋊(C₁₃⋊C₃)

—

C₃²⋊(C₁₃⋊C₃)

—

—

—

—

—

—

yes

—

#66851

H(3,𝔽₃)

yes

—

#66851

C₃³

yes

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

—

C₃×C₁₃⋊C₉

C₃³

yes

—

C₃×C₁₃⋊C₉

C₃³

yes

—

#66849

—

#66848

—

C₉×C₁₃⋊C₃

—

3₋¹⁺⁴

—

—

—

—

—

—

—

—

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

—

C₁₃⋊C₉

—

C₁₃⋊C₉

—

C₁₃⋊C₉

C₃×H(3,𝔽₃)

yes

—

—

—

—

—

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₃⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₃⋊C₃

yes

—

—

—

100

#732

C₃×C₁₃⋊C₃

yes

—

101

#732

C₃×C₁₃⋊C₃

yes

—

102

#732

C₃×C₁₃⋊C₃

yes

—

103

#731

—

104

AΣL(1,𝔽₂₇)

—

105

C₃×C₁₃

—

106

C₃×C₁₃

—

107

C₃×C₁₃

—

108

C₃×C₁₃

—

109

C₃×C₁₃

C₃²×H(3,𝔽₃)

yes

characteristic

110

C₁₃⋊C₃

—

111

C₃³

—

112

C₃³

—

113

C₃³

C₃²×C₁₃⋊C₃

yes

characteristic

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

—

119

3₋¹⁺²

—

120

3₋¹⁺²

—

121

3₋¹⁺²

—

122

3₋¹⁺²

—

123

3₋¹⁺²

—

124

H(3,𝔽₃)

—

125

H(3,𝔽₃)

—

126

H(3,𝔽₃)

—

127

H(3,𝔽₃)

C₃²×C₁₃⋊C₃

yes

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₃×C₉

C₃²×C₁₃⋊C₃

yes

characteristic

136

C₃×C₉

C₃²×C₁₃⋊C₃

yes

characteristic

137

C₃×C₉

C₃²×C₁₃⋊C₃

yes

characteristic

138

C₁₃

#10624936

yes

characteristic, complemented, Sylow

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

117

—

144

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

—

145

C₃²

H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

146

C₃²

C₃³×C₁₃⋊C₃

yes

characteristic

147

C₉

—

148

C₉

—

149

C₉

—

150

C₉

—

151

C₉

—

152

C₉

—

153

C₃

—

154

C₃

—

155

C₃

—

156

C₃

—

157

C₃

—

117

—

158

C₃

C₃×H(3,𝔽₃)×C₁₃⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₁₃⋊C₃
commutator, cosocle	C₃²×C₁₃	C₃⁴
Frattini	C₃²	C₃³×C₁₃⋊C₃
Fitting	C₃×C₁₃×#732	C₃
socle	C₃×C₁₃	C₃²×H(3,𝔽₃)
hypercenter	C₃×#732	C₁₃⋊C₃
hypocenter	C₁₃	#10624936
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#732

subgroup

G/H

center

C₃

C₃×H(3,𝔽₃)×C₁₃⋊C₃

commutator, cosocle

C₃²×C₁₃

C₃⁴

Frattini

C₃²

C₃³×C₁₃⋊C₃

Fitting

C₃×C₁₃×#732

C₃

socle

C₃×C₁₃

C₃²×H(3,𝔽₃)

C₁₃⋊C₃

C₁₃

G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#732

N	H ≅ G/N	type
1. #15691698	C₃	semidirect
2. C₃×#11760453	C₃	semidirect
3. #15691676	C₃	semidirect
4. #15691675	C₃	semidirect
5. #15691660	C₃	semidirect
6. #15691655	C₃	semidirect
7. #15691649	C₃	semidirect
8. C₃×#11760443	C₃	semidirect
9. C₃×C₁₃×#732	C₃	semidirect
13. #11760452	C₃²	semidirect
18. C₃×#66848	C₃²	semidirect
19. C₃×#66849	C₃²	semidirect
21. C₃×C₉×C₁₃⋊C₃	C₃²	semidirect
22. #11760443	C₃²	semidirect
23. #11760443	C₃²	semidirect
24. #11760443	C₃²	semidirect
25. #11760437	C₃²	semidirect
26. #11760434	C₃²	semidirect
68. C₃×#732	C₁₃⋊C₃	semidirect
138. C₁₃	#10624936	semidirect

H ≅ G/N

type

1. #15691698

C₃

semidirect

2. C₃×#11760453

C₃

semidirect

3. #15691676

C₃

semidirect

4. #15691675

C₃

semidirect

5. #15691660

C₃

semidirect

6. #15691655

C₃

semidirect

7. #15691649

C₃

semidirect

8. C₃×#11760443

C₃

semidirect

9. C₃×C₁₃×#732

C₃

semidirect

13. #11760452

C₃²

semidirect

18. C₃×#66848

C₃²

semidirect

19. C₃×#66849

C₃²

semidirect

21. C₃×C₉×C₁₃⋊C₃

C₃²

semidirect

22. #11760443

C₃²

semidirect

23. #11760443

C₃²

semidirect

24. #11760443

C₃²

semidirect

25. #11760437

C₃²

semidirect

26. #11760434

C₃²

semidirect

68. C₃×#732

C₁₃⋊C₃

semidirect

138. C₁₃

#10624936

semidirect

Subgroups — #23748594

Conjugacy classes of subgroups

Complements