Group #23744890 — subgroups

Subgroups — C₁₃×#10624739 #23744890

Order 9477.

458 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 98 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×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₁₃×#733	C₃	1	2	yes	maximal, complemented
3	C₃×C₁₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃×#731	C₃	1	3	yes	maximal, complemented
5	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
6	C₁₃×#6893	C₃	1	3	yes	maximal, complemented
7	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
8	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
10	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
11	C₃×C₁₃×3₋¹⁺²	—	3	3	no	—
12	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	—
13	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
14	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
15	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
17	C₁₃×#733	—	3	6	no	—
18	C₁₃×#732	—	3	3	no	—
19	C₁₃×#731	—	3	9	no	—
20	C₁₃×AΣL(1,𝔽₂₇)	—	3	9	no	—
21	C₁₃×C₃²⋊C₉	—	3	9	no	—
22	#10624739	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃³×C₁₃	—	3	1	no	—
24	C₃³×C₁₃	—	3	1	no	—
25	C₃³×C₁₃	—	3	1	no	—
26	C₃³×C₁₃	—	3	3	no	—
27	C₃³×C₁₃	—	9	3	no	—
28	C₃³×C₁₃	H(3,𝔽₃)	1	3	yes	—
29	C₃³×C₁₃	C₃³	1	1	yes	characteristic
30	C₁₃×3₋¹⁺²	—	3	27	no	—
31	C₁₃×3₋¹⁺²	—	3	6	no	—
32	C₁₃×3₋¹⁺²	—	3	6	no	—
33	C₁₃×3₋¹⁺²	—	3	6	no	—
34	C₁₃×H(3,𝔽₃)	—	3	3	no	—
35	C₁₃×H(3,𝔽₃)	—	3	3	no	—
36	C₁₃×H(3,𝔽₃)	—	3	3	no	—
37	C₃×C₉×C₁₃	—	3	1	no	—
38	C₃×C₉×C₁₃	—	3	1	no	—
39	C₃×C₉×C₁₃	—	3	1	no	—
40	C₃×C₉×C₁₃	—	3	1	no	—
41	C₃×C₉×C₁₃	—	3	1	no	—
42	C₃×C₉×C₁₃	—	3	1	no	—
43	C₃×C₉×C₁₃	—	3	1	no	—
44	C₃×C₉×C₁₃	—	3	1	no	—
45	C₃×C₉×C₁₃	—	3	1	no	—
46	C₃×C₉×C₁₃	—	3	6	no	—
47	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
48	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
49	C₃×C₉×C₁₃	H(3,𝔽₃)	1	3	yes	—
50	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
51	C₃×#733	C₃×C₁₃	1	2	yes	complemented
52	C₃×#732	C₃×C₁₃	1	1	yes	characteristic, complemented
53	C₃×#731	C₃×C₁₃	1	3	yes	complemented
54	C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	1	3	yes	complemented
55	#6893	C₃×C₁₃	1	3	yes	complemented
56	C₃²×C₁₃	—	3	3	no	—
57	C₃²×C₁₃	—	3	3	no	—
58	C₃²×C₁₃	—	9	3	no	—
59	C₃²×C₁₃	—	9	3	no	—
60	C₃²×C₁₃	—	9	3	no	—
61	C₃²×C₁₃	—	9	3	no	—
62	C₃²×C₁₃	—	9	9	no	—
63	C₃²×C₁₃	—	9	3	no	—
64	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₉×C₁₃	—	3	3	no	—
69	C₉×C₁₃	—	3	3	no	—
70	C₉×C₁₃	—	3	3	no	—
71	C₉×C₁₃	—	9	6	no	—
72	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
73	C₃×3₋¹⁺²	—	3	3	no	—
74	C₃×3₋¹⁺²	—	3	3	no	—
75	C₃×3₋¹⁺²	—	3	3	no	—
76	C₃×3₋¹⁺²	—	3	3	no	—
77	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
78	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
79	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
80	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
81	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
82	#733	—	3	6	no	—
83	#732	—	3	3	no	—
84	#731	—	3	9	no	—
85	AΣL(1,𝔽₂₇)	—	3	9	no	—
86	C₃²⋊C₉	—	3	9	no	—
87	C₃×C₁₃	—	3	1	no	—
88	C₃×C₁₃	—	3	1	no	—
89	C₃×C₁₃	—	3	1	no	—
90	C₃×C₁₃	—	9	3	no	—
91	C₃×C₁₃	—	27	3	no	—
92	C₃×C₁₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
93	C₃×C₁₃	#6913	1	3	yes	—
94	C₃³	—	3	1	no	—
95	C₃³	—	3	1	no	—
96	C₃³	—	3	1	no	—
97	C₃³	—	3	3	no	—
98	C₃³	—	9	3	no	—
99	C₃³	C₁₃×H(3,𝔽₃)	1	3	yes	—
100	C₃³	C₃³×C₁₃	1	1	yes	characteristic
101	3₋¹⁺²	—	3	27	no	—
102	3₋¹⁺²	—	3	6	no	—
103	3₋¹⁺²	—	3	6	no	—
104	3₋¹⁺²	—	3	6	no	—
105	H(3,𝔽₃)	—	3	3	no	—
106	H(3,𝔽₃)	—	3	3	no	—
107	H(3,𝔽₃)	—	3	3	no	—
108	C₃×C₉	—	3	1	no	—
109	C₃×C₉	—	3	1	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	1	no	—
116	C₃×C₉	—	3	1	no	—
117	C₃×C₉	—	3	6	no	—
118	C₃×C₉	C₁₃×H(3,𝔽₃)	1	3	yes	—
119	C₃×C₉	C₁₃×H(3,𝔽₃)	1	3	yes	—
120	C₃×C₉	C₁₃×H(3,𝔽₃)	1	3	yes	—
121	C₁₃	#10624739	1	1	yes	characteristic, direct product factor, Sylow
122	C₃²	—	3	3	no	—
123	C₃²	—	3	3	no	—
124	C₃²	—	9	3	no	—
125	C₃²	—	9	3	no	—
126	C₃²	—	9	3	no	—
127	C₃²	—	9	3	no	—
128	C₃²	—	9	3	no	—
129	C₃²	—	9	9	no	—
130	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
131	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
132	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
133	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
134	C₉	—	3	3	no	—
135	C₉	—	3	3	no	—
136	C₉	—	3	3	no	—
137	C₉	—	9	6	no	—
138	C₃	—	3	1	no	—
139	C₃	—	3	1	no	—
140	C₃	—	3	1	no	—
141	C₃	—	9	3	no	—
142	C₃	—	27	3	no	—
143	C₃	C₁₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
144	C₃	C₁₃×#6913	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×#733

C₃

yes

maximal, complemented

C₃×C₁₃×#732

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×#731

C₃

yes

maximal, complemented

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

yes

maximal, complemented

C₁₃×#6893

C₃

yes

maximal, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×#733

—

C₁₃×#732

—

C₁₃×#731

—

C₁₃×AΣL(1,𝔽₂₇)

—

C₁₃×C₃²⋊C₉

—

#10624739

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

C₃×#733

C₃×C₁₃

yes

complemented

C₃×#732

C₃×C₁₃

yes

characteristic, complemented

C₃×#731

C₃×C₁₃

yes

complemented

C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

yes

complemented

#6893

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₃²×C₉

C₃²×C₁₃

yes

—

—

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₁₃

#6913

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

100

C₃³

C₃³×C₁₃

yes

characteristic

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

119

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

120

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

121

C₁₃

#10624739

yes

characteristic, direct product factor, Sylow

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

131

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

132

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

133

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

134

C₉

—

135

C₉

—

136

C₉

—

137

C₉

—

138

C₃

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

C₁₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

144

C₃

C₁₃×#6913

yes

—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
2. C₃×C₁₃×#733	C₃	semidirect
3. C₃×C₁₃×#732	C₃	semidirect
4. C₃×C₁₃×#731	C₃	semidirect
5. C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	semidirect
6. C₁₃×#6893	C₃	semidirect
22. #10624739	C₁₃	direct (N × H)
50. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
51. C₃×#733	C₃×C₁₃	semidirect
52. C₃×#732	C₃×C₁₃	semidirect
53. C₃×#731	C₃×C₁₃	semidirect
54. C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	semidirect
55. #6893	C₃×C₁₃	semidirect
121. C₁₃	#10624739	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

2. C₃×C₁₃×#733

C₃

semidirect

3. C₃×C₁₃×#732

C₃

semidirect

4. C₃×C₁₃×#731

C₃

semidirect

5. C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

semidirect

6. C₁₃×#6893

C₃

semidirect

22. #10624739

C₁₃

direct (N × H)

50. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

51. C₃×#733

C₃×C₁₃

semidirect

52. C₃×#732

C₃×C₁₃

semidirect

53. C₃×#731

C₃×C₁₃

semidirect

54. C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

semidirect

55. #6893

C₃×C₁₃

semidirect

121. C₁₃

#10624739

direct (N × H)

Subgroups — C₁₃×#10624739 #23744890

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624739 #23744890

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624739 #23744890