Group #23744930 — subgroups

Subgroups — C₁₃×#10624779 #23744930

Order 9477.

572 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₃²×C₁₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₁₃×#6891	C₃	1	2	yes	maximal, complemented
4	C₁₃×H(3,𝔽₃)⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₁₃	—	3	1	no	—
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
10	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
11	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
12	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
13	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
14	C₃²×C₉×C₁₃	C₃²	1	3	yes	complemented
15	C₁₃×C₃²⋊C₉	—	3	2	no	—
16	C₁₃×C₃²⋊C₉	—	3	2	no	—
17	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	complemented
18	#10624779	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃³×C₁₃	—	3	9	no	—
20	C₃³×C₁₃	—	3	3	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	3	3	no	—
23	C₃³×C₁₃	—	9	1	no	—
24	C₃³×C₁₃	—	9	1	no	—
25	C₃³×C₁₃	—	9	3	no	—
26	C₃³×C₁₃	—	9	3	no	—
27	C₃³×C₁₃	—	9	3	no	—
28	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
29	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₁₃×3₋¹⁺²	—	3	18	no	—
32	C₁₃×3₋¹⁺²	—	3	9	no	—
33	C₁₃×3₋¹⁺²	—	9	6	no	—
34	C₁₃×H(3,𝔽₃)	—	3	18	no	—
35	C₁₃×H(3,𝔽₃)	—	3	6	no	—
36	C₁₃×H(3,𝔽₃)	3₋¹⁺²	1	9	yes	complemented
37	C₃×C₉×C₁₃	—	3	3	no	—
38	C₃×C₉×C₁₃	—	3	9	no	—
39	C₃×C₉×C₁₃	—	3	6	no	—
40	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
41	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
42	#6891	C₃×C₁₃	1	2	yes	complemented
43	H(3,𝔽₃)⋊C₉	C₃×C₁₃	1	9	yes	complemented
44	C₃²×C₁₃	—	3	1	no	—
45	C₃²×C₁₃	—	3	1	no	—
46	C₃²×C₁₃	—	3	1	no	—
47	C₃²×C₁₃	—	3	1	no	—
48	C₃²×C₁₃	—	3	1	no	—
49	C₃²×C₁₃	—	3	1	no	—
50	C₃²×C₁₃	—	3	2	no	—
51	C₃²×C₁₃	—	3	9	no	—
52	C₃²×C₁₃	—	3	2	no	—
53	C₃²×C₁₃	—	3	2	no	—
54	C₃²×C₁₃	—	3	2	no	—
55	C₃²×C₁₃	—	9	9	no	—
56	C₃²×C₁₃	—	9	27	no	—
57	C₃²×C₁₃	—	9	3	no	—
58	C₃²×C₁₃	—	9	3	no	—
59	C₃²×C₁₃	—	9	3	no	—
60	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
61	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	C₃²×C₁₃	C₃×3₋¹⁺²	1	3	yes	—
63	C₉×C₁₃	—	3	9	no	—
64	C₉×C₁₃	—	9	6	no	—
65	C₃⁴	—	3	1	no	—
66	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
67	C₃×3₋¹⁺²	—	3	2	no	—
68	C₃×3₋¹⁺²	—	3	1	no	—
69	C₃×3₋¹⁺²	—	3	9	no	—
70	C₃×3₋¹⁺²	—	3	2	no	—
71	C₃×H(3,𝔽₃)	—	3	2	no	—
72	C₃×H(3,𝔽₃)	—	3	9	no	—
73	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
74	C₃²×C₉	C₃²×C₁₃	1	3	yes	complemented
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	complemented
78	C₃×C₁₃	—	3	3	no	—
79	C₃×C₁₃	—	3	3	no	—
80	C₃×C₁₃	—	9	2	no	—
81	C₃×C₁₃	—	9	9	no	—
82	C₃×C₁₃	#6891	1	1	yes	characteristic
83	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
84	C₃×C₁₃	#6910	1	1	yes	characteristic
85	C₃×C₁₃	#6911	1	1	yes	characteristic
86	C₃³	—	3	9	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	3	3	no	—
89	C₃³	—	3	3	no	—
90	C₃³	—	9	1	no	—
91	C₃³	—	9	1	no	—
92	C₃³	—	9	3	no	—
93	C₃³	—	9	3	no	—
94	C₃³	—	9	3	no	—
95	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
96	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
97	C₃³	C₃³×C₁₃	1	1	yes	characteristic
98	3₋¹⁺²	—	3	18	no	—
99	3₋¹⁺²	—	3	9	no	—
100	3₋¹⁺²	—	9	6	no	—
101	H(3,𝔽₃)	—	3	18	no	—
102	H(3,𝔽₃)	—	3	6	no	—
103	H(3,𝔽₃)	C₁₃×3₋¹⁺²	1	9	yes	complemented
104	C₃×C₉	—	3	3	no	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	3	6	no	—
107	C₁₃	#10624779	1	1	yes	characteristic, direct product factor, Sylow
108	C₃²	—	3	1	no	—
109	C₃²	—	3	1	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	2	no	—
115	C₃²	—	3	9	no	—
116	C₃²	—	3	2	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	9	9	no	—
120	C₃²	—	9	27	no	—
121	C₃²	—	9	3	no	—
122	C₃²	—	9	3	no	—
123	C₃²	—	9	3	no	—
124	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
125	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
126	C₃²	C₃×C₁₃×3₋¹⁺²	1	3	yes	—
127	C₉	—	3	9	no	—
128	C₉	—	9	6	no	—
129	C₃	—	3	3	no	—
130	C₃	—	3	3	no	—
131	C₃	—	9	2	no	—
132	C₃	—	9	9	no	—
133	C₃	C₁₃×#6891	1	1	yes	characteristic
134	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
135	C₃	C₁₃×#6910	1	1	yes	characteristic
136	C₃	C₁₃×#6911	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, complemented

C₁₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal, complemented

C₃⁴×C₁₃

—

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×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

C₃²

yes

complemented

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

#10624779

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

3₋¹⁺²

yes

complemented

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

complemented

H(3,𝔽₃)⋊C₉

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

C₃×3₋¹⁺²

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

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×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

C₃²×C₁₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6891

yes

characteristic

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

C₁₃×3₋¹⁺²

yes

complemented

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₁₃

#10624779

yes

characteristic, direct product factor, Sylow

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

125

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

126

C₃²

C₃×C₁₃×3₋¹⁺²

yes

—

127

C₉

—

128

C₉

—

129

C₃

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

C₁₃×#6891

yes

characteristic

134

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

135

C₃

C₁₃×#6910

yes

characteristic

136

C₃

C₁₃×#6911

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₃²×C₁₃×H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

prime radical

C₃²×C₁₃×H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
3. C₁₃×#6891	C₃	semidirect
4. C₁₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
14. C₃²×C₉×C₁₃	C₃²	semidirect
17. C₁₃×C₃²⋊C₉	C₃²	semidirect
18. #10624779	C₁₃	direct (N × H)
36. C₁₃×H(3,𝔽₃)	3₋¹⁺²	semidirect
40. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
42. #6891	C₃×C₁₃	semidirect
43. H(3,𝔽₃)⋊C₉	C₃×C₁₃	semidirect
74. C₃²×C₉	C₃²×C₁₃	semidirect
77. C₃²⋊C₉	C₃²×C₁₃	semidirect
103. H(3,𝔽₃)	C₁₃×3₋¹⁺²	semidirect
107. C₁₃	#10624779	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

3. C₁₃×#6891

C₃

semidirect

4. C₁₃×H(3,𝔽₃)⋊C₉

C₃

semidirect

14. C₃²×C₉×C₁₃

C₃²

semidirect

17. C₁₃×C₃²⋊C₉

C₃²

semidirect

18. #10624779

C₁₃

direct (N × H)

36. C₁₃×H(3,𝔽₃)

3₋¹⁺²

semidirect

40. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

42. #6891

C₃×C₁₃

semidirect

43. H(3,𝔽₃)⋊C₉

C₃×C₁₃

semidirect

74. C₃²×C₉

C₃²×C₁₃

semidirect

77. C₃²⋊C₉

C₃²×C₁₃

semidirect

103. H(3,𝔽₃)

C₁₃×3₋¹⁺²

semidirect

107. C₁₃

#10624779

direct (N × H)

Subgroups — C₁₃×#10624779 #23744930

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624779 #23744930

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624779 #23744930