Group #23744759 — subgroups

Subgroups — C₁₃×#10624608 #23744759

Order 9477.

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

normal subgroups only

This group has 494 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×C₁₃×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
2	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₁₃×#6857	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
6	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
7	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	18	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	—
11	C₃×C₁₃×H(3,𝔽₃)	—	3	1	no	—
12	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
13	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
14	C₃²×C₉×C₁₃	—	3	2	no	—
15	C₁₃×C₃²⋊C₉	—	3	2	no	—
16	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
17	#10624608	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₃³×C₁₃	—	3	6	no	—
19	C₃³×C₁₃	—	3	3	no	—
20	C₃³×C₁₃	—	9	1	no	—
21	C₃³×C₁₃	—	9	1	no	—
22	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
24	C₃³×C₁₃	C₃³	1	1	yes	characteristic
25	C₁₃×3₋¹⁺²	—	3	18	no	—
26	C₁₃×3₋¹⁺²	—	3	18	no	—
27	C₁₃×3₋¹⁺²	—	3	18	no	—
28	C₁₃×3₋¹⁺²	—	9	3	no	—
29	C₁₃×H(3,𝔽₃)	—	9	3	no	—
30	C₁₃×H(3,𝔽₃)	—	9	3	no	—
31	C₁₃×H(3,𝔽₃)	—	9	6	no	—
32	C₃×C₉×C₁₃	—	3	18	no	—
33	C₃×C₉×C₁₃	—	3	6	no	—
34	C₃×C₉×C₁₃	—	3	3	no	—
35	C₃²×3₋¹⁺²	C₃×C₁₃	1	2	yes	complemented
36	#6891	C₃×C₁₃	1	1	yes	characteristic, complemented
37	C₃²⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
38	#6857	C₃×C₁₃	1	9	yes	complemented
39	C₃²×C₁₃	—	3	2	no	—
40	C₃²×C₁₃	—	3	2	no	—
41	C₃²×C₁₃	—	3	2	no	—
42	C₃²×C₁₃	—	3	2	no	—
43	C₃²×C₁₃	—	3	2	no	—
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	—	9	9	no	—
49	C₃²×C₁₃	—	9	6	no	—
50	C₃²×C₁₃	—	9	3	no	—
51	C₃²×C₁₃	—	9	3	no	—
52	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	6	yes	—
53	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₉×C₁₃	—	3	18	no	—
55	C₉×C₁₃	—	9	3	no	—
56	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
57	C₃×3₋¹⁺²	—	3	2	no	—
58	C₃×3₋¹⁺²	—	3	2	no	—
59	C₃×3₋¹⁺²	—	3	18	no	—
60	C₃×3₋¹⁺²	—	3	1	no	—
61	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
62	C₃×H(3,𝔽₃)	—	3	1	no	—
63	C₃×H(3,𝔽₃)	—	3	2	no	—
64	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
65	C₃²×C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
68	C₃×C₁₃	—	3	6	no	—
69	C₃×C₁₃	—	9	1	no	—
70	C₃×C₁₃	—	9	1	no	—
71	C₃×C₁₃	—	27	3	no	—
72	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
73	C₃×C₁₃	#6909	1	1	yes	characteristic
74	C₃×C₁₃	#6910	1	1	yes	characteristic
75	C₃³	—	3	6	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	9	1	no	—
78	C₃³	—	9	1	no	—
79	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
80	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
81	C₃³	C₃³×C₁₃	1	1	yes	characteristic
82	3₋¹⁺²	—	3	18	no	—
83	3₋¹⁺²	—	3	18	no	—
84	3₋¹⁺²	—	3	18	no	—
85	3₋¹⁺²	—	9	3	no	—
86	H(3,𝔽₃)	—	9	3	no	—
87	H(3,𝔽₃)	—	9	3	no	—
88	H(3,𝔽₃)	—	9	6	no	—
89	C₃×C₉	—	3	18	no	—
90	C₃×C₉	—	3	6	no	—
91	C₃×C₉	—	3	3	no	—
92	C₁₃	#10624608	1	1	yes	characteristic, direct product factor, Sylow
93	C₃²	—	3	2	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	3	2	no	—
96	C₃²	—	3	2	no	—
97	C₃²	—	3	2	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	9	9	no	—
103	C₃²	—	9	6	no	—
104	C₃²	—	9	3	no	—
105	C₃²	—	9	3	no	—
106	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	6	yes	—
107	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
108	C₉	—	3	18	no	—
109	C₉	—	9	3	no	—
110	C₃	—	3	6	no	—
111	C₃	—	9	1	no	—
112	C₃	—	9	1	no	—
113	C₃	—	27	3	no	—
114	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	2	yes	—
115	C₃	C₁₃×#6909	1	1	yes	characteristic
116	C₃	C₁₃×#6910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, complemented

C₁₃×#6891

C₃

yes

maximal, characteristic, complemented

C₁₃×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₁₃×#6857

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)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

#10624608

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

characteristic

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

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

complemented

#6891

C₃×C₁₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic

#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

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

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

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₃

#10624608

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

107

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

108

C₉

—

109

C₉

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

—

114

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

—

115

C₃

C₁₃×#6909

yes

characteristic

116

C₃

C₁₃×#6910

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
prime radical	C₁₃×C₃²⋊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

C₁₃×C₃²⋊H(3,𝔽₃)

C₃

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
2. C₁₃×#6891	C₃	semidirect
4. C₁₃×#6857	C₃	semidirect
17. #10624608	C₁₃	direct (N × H)
35. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
36. #6891	C₃×C₁₃	semidirect
38. #6857	C₃×C₁₃	semidirect
92. C₁₃	#10624608	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

2. C₁₃×#6891

C₃

semidirect

4. C₁₃×#6857

C₃

semidirect

17. #10624608

C₁₃

direct (N × H)

35. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

36. #6891

C₃×C₁₃

semidirect

38. #6857

C₃×C₁₃

semidirect

92. C₁₃

#10624608

direct (N × H)

Subgroups — C₁₃×#10624608 #23744759

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624608 #23744759

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624608 #23744759