Group #23744974 — subgroups

Subgroups — C₁₃×#10624823 #23744974

Order 9477.

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

normal subgroups only

This group has 368 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	1	yes	maximal, characteristic
2	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic
3	C₁₃×#6891	C₃	1	1	yes	maximal, characteristic
4	C₃×C₁₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₁₃×3₋¹⁺²⋊C₉	C₃	1	9	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	1	no	—
9	C₃×C₁₃×3₋¹⁺²	—	3	9	no	—
10	C₃×C₁₃×3₋¹⁺²	—	3	1	no	—
11	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
12	C₃²×C₉×C₁₃	—	3	1	no	—
13	C₃²×C₉×C₁₃	C₃²	1	3	yes	—
14	C₁₃×C₃²⋊C₉	—	3	2	no	—
15	C₁₃×C₃²⋊C₉	—	3	2	no	—
16	C₁₃×C₃²⋊C₉	—	3	1	no	—
17	C₁₃×C₃²⋊C₉	—	3	2	no	—
18	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₁₃×C₃²⋊C₉	C₃²	1	3	yes	—
20	#10624823	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	H(3,𝔽₃)	1	2	yes	—
26	C₃³×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃³×C₁₃	C₃³	1	1	yes	characteristic
28	C₁₃×3₋¹⁺²	—	3	18	no	—
29	C₁₃×3₋¹⁺²	—	3	9	no	—
30	C₁₃×3₋¹⁺²	—	3	9	no	—
31	C₁₃×3₋¹⁺²	—	9	3	no	—
32	C₃×C₉×C₁₃	—	3	3	no	—
33	C₃×C₉×C₁₃	—	3	9	no	—
34	C₃×C₉×C₁₃	—	3	3	no	—
35	C₃×C₉×C₁₃	—	3	3	no	—
36	C₃×C₉×C₁₃	—	3	3	no	—
37	C₃×C₉×C₁₃	—	9	3	no	—
38	C₃²×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic
39	#6891	C₃×C₁₃	1	1	yes	characteristic
40	#6891	C₃×C₁₃	1	1	yes	characteristic
41	C₃×C₃²⋊C₉	C₃×C₁₃	1	1	yes	characteristic
42	3₋¹⁺²⋊C₉	C₃×C₁₃	1	9	yes	complemented
43	C₃²×C₁₃	—	3	1	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	—	3	1	no	—
49	C₃²×C₁₃	—	3	2	no	—
50	C₃²×C₁₃	—	3	2	no	—
51	C₃²×C₁₃	—	3	2	no	—
52	C₃²×C₁₃	—	3	2	no	—
53	C₃²×C₁₃	—	9	9	no	—
54	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	3	yes	—
55	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×C₁₃	#737	1	3	yes	—
57	C₉×C₁₃	—	3	9	no	—
58	C₉×C₁₃	—	9	3	no	—
59	C₉×C₁₃	—	9	3	no	—
60	C₉×C₁₃	—	9	3	no	—
61	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
62	C₃×3₋¹⁺²	—	3	2	no	—
63	C₃×3₋¹⁺²	—	3	1	no	—
64	C₃×3₋¹⁺²	—	3	9	no	—
65	C₃×3₋¹⁺²	—	3	1	no	—
66	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
67	C₃²×C₉	—	3	1	no	—
68	C₃²×C₉	C₃²×C₁₃	1	3	yes	—
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	1	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
74	C₃²⋊C₉	C₃²×C₁₃	1	3	yes	—
75	C₃×C₁₃	—	3	3	no	—
76	C₃×C₁₃	—	3	3	no	—
77	C₃×C₁₃	—	9	2	no	—
78	C₃×C₁₃	#6893	1	1	yes	characteristic
79	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
80	C₃×C₁₃	#6910	1	1	yes	characteristic
81	C₃×C₁₃	#6912	1	1	yes	characteristic
82	C₃³	—	3	3	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	9	1	no	—
85	C₃³	—	9	1	no	—
86	C₃³	C₁₃×H(3,𝔽₃)	1	2	yes	—
87	C₃³	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
88	C₃³	C₃³×C₁₃	1	1	yes	characteristic
89	3₋¹⁺²	—	3	18	no	—
90	3₋¹⁺²	—	3	9	no	—
91	3₋¹⁺²	—	3	9	no	—
92	3₋¹⁺²	—	9	3	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	3	9	no	—
95	C₃×C₉	—	3	3	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	9	3	no	—
99	C₁₃	#10624823	1	1	yes	characteristic, direct product factor, Sylow
100	C₃²	—	3	1	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	1	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	3	2	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	9	9	no	—
111	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—
112	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
113	C₃²	C₁₃×#737	1	3	yes	—
114	C₉	—	3	9	no	—
115	C₉	—	9	3	no	—
116	C₉	—	9	3	no	—
117	C₉	—	9	3	no	—
118	C₃	—	3	3	no	—
119	C₃	—	3	3	no	—
120	C₃	—	9	2	no	—
121	C₃	C₁₃×#6893	1	1	yes	characteristic
122	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
123	C₃	C₁₃×#6910	1	1	yes	characteristic
124	C₃	C₁₃×#6912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₁₃×3₋¹⁺²

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, characteristic

C₁₃×#6891

C₃

yes

maximal, characteristic

C₃×C₁₃×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₁₃×3₋¹⁺²⋊C₉

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

—

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

C₁₃×C₃²⋊C₉

C₃²

yes

—

#10624823

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

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

—

C₁₃×3₋¹⁺²

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃²×3₋¹⁺²

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

characteristic

#6891

C₃×C₁₃

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₃

yes

characteristic

3₋¹⁺²⋊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

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₃

#737

yes

—

C₉×C₁₃

—

C₉×C₁₃

—

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

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃²⋊C₉

C₃²×C₁₃

yes

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6893

yes

characteristic

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₁₃

yes

C₃×C₁₃

yes

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

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₃

#10624823

yes

characteristic, direct product factor, Sylow

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

—

112

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

113

C₃²

C₁₃×#737

yes

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₃

—

119

C₃

—

120

C₃

—

121

C₃

C₁₃×#6893

yes

characteristic

122

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

characteristic

123

C₃

C₁₃×#6910

yes

characteristic

124

C₃

C₁₃×#6912

yes

characteristic

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

C₃⁴×C₁₃

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
5. C₁₃×3₋¹⁺²⋊C₉	C₃	semidirect
20. #10624823	C₁₃	direct (N × H)
42. 3₋¹⁺²⋊C₉	C₃×C₁₃	semidirect
99. C₁₃	#10624823	direct (N × H)

H ≅ G/N

type

5. C₁₃×3₋¹⁺²⋊C₉

C₃

semidirect

20. #10624823

C₁₃

direct (N × H)

42. 3₋¹⁺²⋊C₉

C₃×C₁₃

semidirect

99. C₁₃

#10624823

direct (N × H)

Subgroups — C₁₃×#10624823 #23744974

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624823 #23744974

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624823 #23744974