Group #23744906 — subgroups

Subgroups — C₁₃×#10624755 #23744906

Order 9477.

446 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₁₃×#6912	C₃	1	3	yes	maximal, complemented
3	C₁₃×#6910	C₃	1	3	yes	maximal, complemented
4	C₁₃×#6909	C₃	1	3	yes	maximal, complemented
5	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
6	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
7	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
8	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
9	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
12	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃²×C₉×C₁₃	—	3	1	no	—
14	C₁₃×#733	—	3	9	no	—
15	C₁₃×#731	—	3	9	no	—
16	C₁₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
17	C₁₃×AΣL(1,𝔽₂₇)	—	3	6	no	—
18	C₁₃×AΣL(1,𝔽₂₇)	—	3	9	no	—
19	#10624755	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃³×C₁₃	—	3	3	no	—
21	C₃³×C₁₃	—	3	3	no	—
22	C₃³×C₁₃	—	9	3	no	—
23	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
25	C₃³×C₁₃	C₃³	1	1	yes	characteristic
26	C₁₃×3₋¹⁺²	—	3	18	no	—
27	C₁₃×3₋¹⁺²	—	3	6	no	—
28	C₁₃×3₋¹⁺²	—	3	3	no	—
29	C₁₃×3₋¹⁺²	—	3	3	no	—
30	C₁₃×3₋¹⁺²	—	3	3	no	—
31	C₁₃×3₋¹⁺²	—	3	3	no	—
32	C₁₃×3₋¹⁺²	—	3	3	no	—
33	C₁₃×3₋¹⁺²	—	3	3	no	—
34	C₁₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
35	C₁₃×H(3,𝔽₃)	—	3	3	no	—
36	C₁₃×H(3,𝔽₃)	—	3	3	no	—
37	C₁₃×H(3,𝔽₃)	—	3	3	no	—
38	C₃×C₉×C₁₃	—	3	9	no	—
39	C₃×C₉×C₁₃	—	3	3	no	—
40	C₃×C₉×C₁₃	—	3	3	no	—
41	C₃×C₉×C₁₃	—	3	3	no	—
42	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
43	#6912	C₃×C₁₃	1	3	yes	complemented
44	#6910	C₃×C₁₃	1	3	yes	complemented
45	#6909	C₃×C₁₃	1	3	yes	complemented
46	C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	1	3	yes	complemented
47	C₃²×C₁₃	—	3	3	no	—
48	C₃²×C₁₃	—	3	1	no	—
49	C₃²×C₁₃	—	3	2	no	—
50	C₃²×C₁₃	—	9	1	no	—
51	C₃²×C₁₃	—	9	9	no	—
52	C₃²×C₁₃	—	9	2	no	—
53	C₃²×C₁₃	—	9	3	no	—
54	C₃²×C₁₃	—	27	3	no	—
55	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
57	C₉×C₁₃	—	3	9	no	—
58	C₉×C₁₃	—	9	3	no	—
59	C₉×C₁₃	—	9	3	no	—
60	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
61	C₃×3₋¹⁺²	—	3	2	no	—
62	C₃×3₋¹⁺²	—	3	9	no	—
63	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
64	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
65	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
66	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
67	C₃²×C₉	—	3	1	no	—
68	#733	—	3	9	no	—
69	#731	—	3	9	no	—
70	AΣL(1,𝔽₂₇)	—	3	3	no	—
71	AΣL(1,𝔽₂₇)	—	3	6	no	—
72	AΣL(1,𝔽₂₇)	—	3	9	no	—
73	C₃×C₁₃	—	3	3	no	—
74	C₃×C₁₃	—	3	1	no	—
75	C₃×C₁₃	—	3	3	no	—
76	C₃×C₁₃	—	9	2	no	—
77	C₃×C₁₃	—	27	3	no	—
78	C₃×C₁₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
79	C₃³	—	3	3	no	—
80	C₃³	—	3	3	no	—
81	C₃³	—	9	3	no	—
82	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
83	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
84	C₃³	C₃³×C₁₃	1	1	yes	characteristic
85	3₋¹⁺²	—	3	18	no	—
86	3₋¹⁺²	—	3	6	no	—
87	3₋¹⁺²	—	3	3	no	—
88	3₋¹⁺²	—	3	3	no	—
89	3₋¹⁺²	—	3	3	no	—
90	3₋¹⁺²	—	3	3	no	—
91	3₋¹⁺²	—	3	3	no	—
92	3₋¹⁺²	—	3	3	no	—
93	3₋¹⁺²	C₁₃×H(3,𝔽₃)	1	9	yes	—
94	H(3,𝔽₃)	—	3	3	no	—
95	H(3,𝔽₃)	—	3	3	no	—
96	H(3,𝔽₃)	—	3	3	no	—
97	C₃×C₉	—	3	9	no	—
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	3	3	no	—
100	C₃×C₉	—	3	3	no	—
101	C₁₃	#10624755	1	1	yes	characteristic, direct product factor, Sylow
102	C₃²	—	3	3	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	2	no	—
105	C₃²	—	9	1	no	—
106	C₃²	—	9	9	no	—
107	C₃²	—	9	2	no	—
108	C₃²	—	9	3	no	—
109	C₃²	—	27	3	no	—
110	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
111	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	3	yes	—
112	C₉	—	3	9	no	—
113	C₉	—	9	3	no	—
114	C₉	—	9	3	no	—
115	C₃	—	3	1	no	—
116	C₃	—	3	3	no	—
117	C₃	—	3	3	no	—
118	C₃	—	9	2	no	—
119	C₃	—	27	3	no	—
120	C₃	C₁₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₁₃×#6912

C₃

yes

maximal, complemented

C₁₃×#6910

C₃

yes

maximal, complemented

C₁₃×#6909

C₃

yes

maximal, complemented

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

yes

maximal, complemented

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₉×C₁₃

—

C₁₃×#733

—

C₁₃×#731

—

C₁₃×AΣL(1,𝔽₂₇)

—

C₁₃×AΣL(1,𝔽₂₇)

—

C₁₃×AΣL(1,𝔽₂₇)

—

#10624755

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

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

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic, complemented

#6912

C₃×C₁₃

yes

complemented

#6910

C₃×C₁₃

yes

complemented

#6909

C₃×C₁₃

yes

complemented

C₃×AΣL(1,𝔽₂₇)

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

—

C₉×C₁₃

—

C₉×C₁₃

—

C₉×C₁₃

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃²×C₉

—

#733

—

#731

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²⋊H(3,𝔽₃)

yes

characteristic

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

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₁₃×H(3,𝔽₃)

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₁₃

#10624755

yes

characteristic, direct product factor, Sylow

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

111

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

C₁₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

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

subgroup

G/H

center, socle

C₃×C₁₃

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₁₃ ◁ C₃³×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃²×C₁₃×3₋¹⁺²	C₃	semidirect
2. C₁₃×#6912	C₃	semidirect
3. C₁₃×#6910	C₃	semidirect
4. C₁₃×#6909	C₃	semidirect
5. C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	semidirect
19. #10624755	C₁₃	direct (N × H)
42. C₃²×3₋¹⁺²	C₃×C₁₃	semidirect
43. #6912	C₃×C₁₃	semidirect
44. #6910	C₃×C₁₃	semidirect
45. #6909	C₃×C₁₃	semidirect
46. C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	semidirect
101. C₁₃	#10624755	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×3₋¹⁺²

C₃

semidirect

2. C₁₃×#6912

C₃

semidirect

3. C₁₃×#6910

C₃

semidirect

4. C₁₃×#6909

C₃

semidirect

5. C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

semidirect

19. #10624755

C₁₃

direct (N × H)

42. C₃²×3₋¹⁺²

C₃×C₁₃

semidirect

43. #6912

C₃×C₁₃

semidirect

44. #6910

C₃×C₁₃

semidirect

45. #6909

C₃×C₁₃

semidirect

46. C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

semidirect

101. C₁₃

#10624755

direct (N × H)

Subgroups — C₁₃×#10624755 #23744906

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624755 #23744906

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624755 #23744906