Group #23744742 — subgroups

Subgroups — C₁₃×#10624591 #23744742

Order 9477.

464 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 86 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×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₁₃×#6891	C₃	1	2	yes	maximal, complemented
3	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6857	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	C₃²	1	6	yes	complemented
8	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
9	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
10	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
11	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
12	C₃²×C₉×C₁₃	—	3	1	no	—
13	C₁₃×C₃²⋊C₉	—	3	1	no	—
14	C₁₃×C₃²⋊C₉	—	3	1	no	—
15	C₁₃×C₃²⋊C₉	—	3	2	no	—
16	C₁₃×C₃²⋊C₉	—	3	2	no	—
17	C₁₃×C₃²⋊C₉	—	3	2	no	—
18	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	complemented
19	#10624591	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃³×C₁₃	—	3	9	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	3	3	no	—
23	C₃³×C₁₃	—	3	3	no	—
24	C₃³×C₁₃	—	9	2	no	—
25	C₃³×C₁₃	—	9	6	no	—
26	C₃³×C₁₃	—	9	3	no	—
27	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₁₃	C₃³	1	1	yes	characteristic
31	C₁₃×3₋¹⁺²	—	9	6	no	—
32	C₁₃×H(3,𝔽₃)	—	3	9	no	—
33	C₁₃×H(3,𝔽₃)	—	3	9	no	—
34	C₁₃×H(3,𝔽₃)	—	3	9	no	—
35	C₃×C₉×C₁₃	—	3	6	no	—
36	C₃×C₉×C₁₃	—	3	3	no	—
37	C₃×C₉×C₁₃	—	9	3	no	—
38	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
39	#6891	C₃×C₁₃	1	2	yes	complemented
40	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
41	#6857	C₃×C₁₃	1	9	yes	complemented
42	C₃²×C₁₃	—	3	1	no	—
43	C₃²×C₁₃	—	3	1	no	—
44	C₃²×C₁₃	—	3	2	no	—
45	C₃²×C₁₃	—	3	2	no	—
46	C₃²×C₁₃	—	3	1	no	—
47	C₃²×C₁₃	—	3	1	no	—
48	C₃²×C₁₃	—	3	9	no	—
49	C₃²×C₁₃	—	3	1	no	—
50	C₃²×C₁₃	—	3	1	no	—
51	C₃²×C₁₃	—	3	2	no	—
52	C₃²×C₁₃	—	3	2	no	—
53	C₃²×C₁₃	—	9	9	no	—
54	C₃²×C₁₃	—	9	27	no	—
55	C₃²×C₁₃	—	9	3	no	—
56	C₃²×C₁₃	—	9	6	no	—
57	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
58	C₃²×C₁₃	#731	1	3	yes	—
59	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₉×C₁₃	—	9	3	no	—
61	C₉×C₁₃	—	9	6	no	—
62	C₃⁴	—	3	1	no	—
63	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
64	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	complemented
65	C₃×H(3,𝔽₃)	—	3	1	no	—
66	C₃×H(3,𝔽₃)	—	3	1	no	—
67	C₃×H(3,𝔽₃)	—	3	9	no	—
68	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
69	C₃²×C₉	—	3	1	no	—
70	C₃²⋊C₉	—	3	1	no	—
71	C₃²⋊C₉	—	3	1	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃²⋊C₉	—	3	2	no	—
74	C₃²⋊C₉	—	3	2	no	—
75	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	complemented
76	C₃×C₁₃	—	3	3	no	—
77	C₃×C₁₃	—	3	3	no	—
78	C₃×C₁₃	—	9	1	no	—
79	C₃×C₁₃	—	9	1	no	—
80	C₃×C₁₃	—	9	9	no	—
81	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
82	C₃×C₁₃	C₃×#731	1	1	yes	characteristic
83	C₃×C₁₃	#6909	1	2	yes	—
84	C₃³	—	3	9	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	3	3	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	9	6	no	—
90	C₃³	—	9	3	no	—
91	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
92	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃³	C₃³×C₁₃	1	1	yes	characteristic
95	3₋¹⁺²	—	9	6	no	—
96	H(3,𝔽₃)	—	3	9	no	—
97	H(3,𝔽₃)	—	3	9	no	—
98	H(3,𝔽₃)	—	3	9	no	—
99	C₃×C₉	—	3	6	no	—
100	C₃×C₉	—	3	3	no	—
101	C₃×C₉	—	9	3	no	—
102	C₁₃	#10624591	1	1	yes	characteristic, direct product factor, Sylow
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	9	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	2	no	—
114	C₃²	—	9	9	no	—
115	C₃²	—	9	27	no	—
116	C₃²	—	9	3	no	—
117	C₃²	—	9	6	no	—
118	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
119	C₃²	C₁₃×#731	1	3	yes	—
120	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
121	C₉	—	9	3	no	—
122	C₉	—	9	6	no	—
123	C₃	—	3	3	no	—
124	C₃	—	3	3	no	—
125	C₃	—	9	1	no	—
126	C₃	—	9	1	no	—
127	C₃	—	9	9	no	—
128	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
129	C₃	C₃×C₁₃×#731	1	1	yes	characteristic
130	C₃	C₁₃×#6909	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, complemented

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₁₃×#6857

C₃

yes

maximal, complemented

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

complemented

#10624591

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

yes

C₃³×C₁₃

yes

C₃³×C₁₃

yes

C₃³×C₁₃

C₃³

yes

characteristic

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

—

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

complemented

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic, complemented

#6857

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

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

#731

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₁₃

#10624591

yes

characteristic, direct product factor, Sylow

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

119

C₃²

C₁₃×#731

yes

—

120

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

121

C₉

—

122

C₉

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

129

C₃

C₃×C₁₃×#731

yes

characteristic

130

C₃

C₁₃×#6909

yes

—

	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 ▷ C₃² ▷ 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 ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#6891	C₃	semidirect
3. C₃×C₁₃×C₃²⋊C₉	C₃	semidirect
4. C₁₃×#6857	C₃	semidirect
7. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
18. C₁₃×C₃²⋊C₉	C₃²	semidirect
19. #10624591	C₁₃	direct (N × H)
39. #6891	C₃×C₁₃	semidirect
40. C₃×C₃²⋊C₉	C₃×C₁₃	semidirect
41. #6857	C₃×C₁₃	semidirect
64. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
75. C₃²⋊C₉	C₃²×C₁₃	semidirect
102. C₁₃	#10624591	direct (N × H)

H ≅ G/N

type

2. C₁₃×#6891

C₃

semidirect

3. C₃×C₁₃×C₃²⋊C₉

C₃

semidirect

4. C₁₃×#6857

C₃

semidirect

7. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

18. C₁₃×C₃²⋊C₉

C₃²

semidirect

19. #10624591

C₁₃

direct (N × H)

39. #6891

C₃×C₁₃

semidirect

40. C₃×C₃²⋊C₉

C₃×C₁₃

semidirect

41. #6857

C₃×C₁₃

semidirect

64. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

75. C₃²⋊C₉

C₃²×C₁₃

semidirect

102. C₁₃

#10624591

direct (N × H)

Subgroups — C₁₃×#10624591 #23744742

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624591 #23744742

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624591 #23744742