Group #23744905 — subgroups

Subgroups — C₁₃×#10624754 #23744905

Order 9477.

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

normal subgroups only

This group has 554 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×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#6911	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	—	3	1	no	—
7	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
8	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	complemented
9	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	complemented
10	C₃×C₁₃×H(3,𝔽₃)	—	3	2	no	—
11	C₃×C₁₃×H(3,𝔽₃)	—	3	9	no	—
12	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	complemented
14	C₁₃×#732	—	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	#10624754	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	3	no	—
25	C₃³×C₁₃	—	9	9	no	—
26	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃³×C₁₃	H(3,𝔽₃)	1	2	yes	—
28	C₃³×C₁₃	C₃³	1	1	yes	characteristic
29	C₁₃×3₋¹⁺²	—	3	6	no	—
30	C₁₃×3₋¹⁺²	—	3	3	no	—
31	C₁₃×3₋¹⁺²	—	3	3	no	—
32	C₁₃×3₋¹⁺²	—	3	6	no	—
33	C₁₃×H(3,𝔽₃)	—	3	18	no	—
34	C₁₃×H(3,𝔽₃)	—	3	6	no	—
35	C₁₃×H(3,𝔽₃)	—	3	3	no	—
36	C₁₃×H(3,𝔽₃)	—	3	6	no	—
37	C₁₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
38	C₃×C₉×C₁₃	—	3	3	no	—
39	C₃×C₉×C₁₃	—	3	3	no	—
40	C₃²×H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
41	#6911	C₃×C₁₃	1	3	yes	complemented
42	#6910	C₃×C₁₃	1	3	yes	complemented
43	#6909	C₃×C₁₃	1	3	yes	complemented
44	C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	1	3	yes	complemented
45	C₃²×C₁₃	—	3	3	no	—
46	C₃²×C₁₃	—	3	1	no	—
47	C₃²×C₁₃	—	3	2	no	—
48	C₃²×C₁₃	—	3	9	no	—
49	C₃²×C₁₃	—	9	1	no	—
50	C₃²×C₁₃	—	9	9	no	—
51	C₃²×C₁₃	—	9	2	no	—
52	C₃²×C₁₃	—	9	27	no	—
53	C₃²×C₁₃	—	9	9	no	—
54	C₃²×C₁₃	—	9	3	no	—
55	C₃²×C₁₃	—	27	3	no	—
56	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
57	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₉×C₁₃	—	9	3	no	—
59	C₉×C₁₃	—	9	3	no	—
60	C₃⁴	—	3	1	no	—
61	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
62	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	complemented
63	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	complemented
64	C₃×H(3,𝔽₃)	—	3	2	no	—
65	C₃×H(3,𝔽₃)	—	3	9	no	—
66	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	—
67	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	complemented
68	#732	—	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	—	9	9	no	—
78	C₃×C₁₃	—	27	3	no	—
79	C₃×C₁₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
80	C₃³	—	3	9	no	—
81	C₃³	—	3	3	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	9	3	no	—
85	C₃³	—	9	9	no	—
86	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
87	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
88	C₃³	C₃³×C₁₃	1	1	yes	characteristic
89	3₋¹⁺²	—	3	6	no	—
90	3₋¹⁺²	—	3	3	no	—
91	3₋¹⁺²	—	3	3	no	—
92	3₋¹⁺²	—	3	6	no	—
93	H(3,𝔽₃)	—	3	18	no	—
94	H(3,𝔽₃)	—	3	6	no	—
95	H(3,𝔽₃)	—	3	3	no	—
96	H(3,𝔽₃)	—	3	6	no	—
97	H(3,𝔽₃)	C₁₃×H(3,𝔽₃)	1	9	yes	—
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	3	3	no	—
100	C₁₃	#10624754	1	1	yes	characteristic, direct product factor, Sylow
101	C₃²	—	3	3	no	—
102	C₃²	—	3	1	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	3	9	no	—
105	C₃²	—	9	1	no	—
106	C₃²	—	9	9	no	—
107	C₃²	—	9	2	no	—
108	C₃²	—	9	9	no	—
109	C₃²	—	9	27	no	—
110	C₃²	—	9	3	no	—
111	C₃²	—	27	3	no	—
112	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	3	yes	—
113	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
114	C₉	—	9	3	no	—
115	C₉	—	9	3	no	—
116	C₃	—	3	1	no	—
117	C₃	—	3	3	no	—
118	C₃	—	3	3	no	—
119	C₃	—	9	9	no	—
120	C₃	—	9	2	no	—
121	C₃	—	27	3	no	—
122	C₃	C₁₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₁₃×#6911

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

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

complemented

C₁₃×#732

—

C₁₃×#731

—

C₁₃×AΣL(1,𝔽₂₇)

—

C₁₃×AΣL(1,𝔽₂₇)

—

C₁₃×AΣL(1,𝔽₂₇)

—

#10624754

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

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)

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

#6911

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

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

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

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

complemented

#732

—

#731

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃³

—

C₃³

—

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

—

—

—

—

—

C₁₃×H(3,𝔽₃)

yes

—

C₃×C₉

—

C₃×C₉

—

100

C₁₃

#10624754

yes

characteristic, direct product factor, Sylow

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

113

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

114

C₉

—

115

C₉

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

—

121

C₃

—

122

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×H(3,𝔽₃)	C₃	semidirect
2. C₁₃×#6911	C₃	semidirect
3. C₁₃×#6910	C₃	semidirect
4. C₁₃×#6909	C₃	semidirect
5. C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	semidirect
8. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
9. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
13. C₃×C₁₃×H(3,𝔽₃)	C₃²	semidirect
19. #10624754	C₁₃	direct (N × H)
40. C₃²×H(3,𝔽₃)	C₃×C₁₃	semidirect
41. #6911	C₃×C₁₃	semidirect
42. #6910	C₃×C₁₃	semidirect
43. #6909	C₃×C₁₃	semidirect
44. C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	semidirect
62. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
63. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
67. C₃×H(3,𝔽₃)	C₃²×C₁₃	semidirect
100. C₁₃	#10624754	direct (N × H)

H ≅ G/N

type

1. C₃²×C₁₃×H(3,𝔽₃)

C₃

semidirect

2. C₁₃×#6911

C₃

semidirect

3. C₁₃×#6910

C₃

semidirect

4. C₁₃×#6909

C₃

semidirect

5. C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

semidirect

8. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

9. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

13. C₃×C₁₃×H(3,𝔽₃)

C₃²

semidirect

19. #10624754

C₁₃

direct (N × H)

40. C₃²×H(3,𝔽₃)

C₃×C₁₃

semidirect

41. #6911

C₃×C₁₃

semidirect

42. #6910

C₃×C₁₃

semidirect

43. #6909

C₃×C₁₃

semidirect

44. C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

semidirect

62. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

63. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

67. C₃×H(3,𝔽₃)

C₃²×C₁₃

semidirect

100. C₁₃

#10624754

direct (N × H)

Subgroups — C₁₃×#10624754 #23744905

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624754 #23744905

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624754 #23744905