Group #23744852 — subgroups

Subgroups — C₃×C₁₃×#6872 #23744852

Order 9477.

320 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 152 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	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic
2	C₃×C₁₃×C₂₇⋊C₃	C₃	1	2	yes	maximal, complemented
3	C₃²×C₁₃×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6872	C₃	1	9	yes	maximal, direct product factor
5	C₁₃×#737	—	3	2	no	—
6	C₁₃×#737	—	3	4	no	—
7	C₁₃×#737	C₉	1	6	yes	—
8	C₁₃×#737	C₃²	1	3	yes	—
9	C₃×C₁₃×3₋¹⁺²	—	3	2	no	—
10	C₃×C₁₃×3₋¹⁺²	C₉	1	2	yes	—
11	C₃×C₁₃×H(3,𝔽₃)	C₉	1	1	yes	characteristic
12	C₃²×C₉×C₁₃	—	3	1	no	—
13	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
14	C₁₃×C₂₇⋊C₃	—	3	4	no	—
15	C₁₃×C₂₇⋊C₃	C₃²	1	6	yes	complemented
16	C₃×C₁₃×C₂₇	—	3	2	no	—
17	C₃×C₁₃×C₂₇	—	3	1	no	—
18	C₃×C₁₃×C₂₇	—	3	2	no	—
19	C₃×C₁₃×C₂₇	C₃²	1	3	yes	complemented
20	C₃×#6872	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃³×C₁₃	—	3	1	no	—
22	C₃³×C₁₃	C₃×C₉	1	1	yes	characteristic
23	C₁₃×3₋¹⁺²	—	3	4	no	—
24	C₁₃×3₋¹⁺²	—	3	12	no	—
25	C₁₃×3₋¹⁺²	—	3	6	no	—
26	C₁₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
27	C₁₃×H(3,𝔽₃)	—	3	2	no	—
28	C₁₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
29	C₃×C₉×C₁₃	—	3	2	no	—
30	C₃×C₉×C₁₃	—	3	6	no	—
31	C₃×C₉×C₁₃	—	3	3	no	—
32	C₃×C₉×C₁₃	C₃×C₉	1	2	yes	—
33	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
34	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₉×C₁₃	3₋¹⁺²	1	2	yes	—
36	C₃×C₉×C₁₃	3₋¹⁺²	1	4	yes	—
37	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
38	C₁₃×C₂₇	—	3	3	no	—
39	C₁₃×C₂₇	—	3	6	no	—
40	C₃×#737	C₃×C₁₃	1	1	yes	characteristic
41	C₃×C₂₇⋊C₃	C₃×C₁₃	1	2	yes	complemented
42	C₃²×C₂₇	C₃×C₁₃	1	1	yes	characteristic, complemented
43	#6872	C₃×C₁₃	1	9	yes	direct product factor
44	C₃²×C₁₃	—	3	3	no	—
45	C₃²×C₁₃	—	3	3	no	—
46	C₃²×C₁₃	—	9	3	no	—
47	C₃²×C₁₃	C₃²⋊C₉	1	2	yes	—
48	C₃²×C₁₃	C₃²⋊C₉	1	1	yes	characteristic
49	C₃²×C₁₃	C₃²×C₉	1	1	yes	characteristic
50	C₉×C₁₃	—	3	6	no	—
51	C₉×C₁₃	C₃²⋊C₉	1	2	yes	—
52	C₉×C₁₃	C₃²⋊C₉	1	4	yes	—
53	C₉×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₉×C₁₃	C₃×3₋¹⁺²	1	2	yes	—
55	#737	—	3	2	no	—
56	#737	—	3	4	no	—
57	#737	C₉×C₁₃	1	6	yes	—
58	#737	C₃²×C₁₃	1	3	yes	—
59	C₃×3₋¹⁺²	—	3	2	no	—
60	C₃×3₋¹⁺²	C₉×C₁₃	1	2	yes	—
61	C₃×H(3,𝔽₃)	C₉×C₁₃	1	1	yes	characteristic
62	C₃²×C₉	—	3	1	no	—
63	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
64	C₂₇⋊C₃	—	3	4	no	—
65	C₂₇⋊C₃	C₃²×C₁₃	1	6	yes	complemented
66	C₃×C₂₇	—	3	2	no	—
67	C₃×C₂₇	—	3	1	no	—
68	C₃×C₂₇	—	3	2	no	—
69	C₃×C₂₇	C₃²×C₁₃	1	3	yes	complemented
70	C₃×C₁₃	—	3	2	no	—
71	C₃×C₁₃	—	3	1	no	—
72	C₃×C₁₃	—	9	3	no	—
73	C₃×C₁₃	#6872	1	3	yes	direct product factor
74	C₃×C₁₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
75	C₃³	—	3	1	no	—
76	C₃³	C₃×C₉×C₁₃	1	1	yes	characteristic
77	3₋¹⁺²	—	3	4	no	—
78	3₋¹⁺²	—	3	12	no	—
79	3₋¹⁺²	—	3	6	no	—
80	3₋¹⁺²	C₃×C₉×C₁₃	1	6	yes	—
81	H(3,𝔽₃)	—	3	2	no	—
82	H(3,𝔽₃)	C₃×C₉×C₁₃	1	3	yes	—
83	C₃×C₉	—	3	2	no	—
84	C₃×C₉	—	3	6	no	—
85	C₃×C₉	—	3	3	no	—
86	C₃×C₉	C₃×C₉×C₁₃	1	2	yes	—
87	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
88	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
89	C₃×C₉	C₁₃×3₋¹⁺²	1	2	yes	—
90	C₃×C₉	C₁₃×3₋¹⁺²	1	4	yes	—
91	C₃×C₉	C₃³×C₁₃	1	1	yes	characteristic
92	C₂₇	—	3	3	no	—
93	C₂₇	—	3	6	no	—
94	C₁₃	C₃×#6872	1	1	yes	characteristic, direct product factor, Sylow
95	C₃²	—	3	3	no	—
96	C₃²	—	3	3	no	—
97	C₃²	—	9	3	no	—
98	C₃²	C₁₃×C₃²⋊C₉	1	2	yes	—
99	C₃²	C₁₃×C₃²⋊C₉	1	1	yes	characteristic
100	C₃²	C₃²×C₉×C₁₃	1	1	yes	characteristic
101	C₉	—	3	6	no	—
102	C₉	C₁₃×C₃²⋊C₉	1	2	yes	—
103	C₉	C₁₃×C₃²⋊C₉	1	4	yes	—
104	C₉	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
105	C₉	C₃×C₁₃×3₋¹⁺²	1	2	yes	—
106	C₃	—	3	2	no	—
107	C₃	—	3	1	no	—
108	C₃	—	9	3	no	—
109	C₃	C₁₃×#6872	1	3	yes	direct product factor
110	C₃	C₃×C₁₃×C₃²⋊C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₁₃×#737

C₃

yes

maximal, characteristic

C₃×C₁₃×C₂₇⋊C₃

C₃

yes

maximal, complemented

C₃²×C₁₃×C₂₇

C₃

yes

maximal, characteristic, complemented

C₁₃×#6872

C₃

yes

maximal, direct product factor

C₁₃×#737

—

C₁₃×#737

—

C₁₃×#737

C₉

yes

—

C₁₃×#737

C₃²

yes

—

C₃×C₁₃×3₋¹⁺²

—

C₃×C₁₃×3₋¹⁺²

C₉

yes

—

C₃×C₁₃×H(3,𝔽₃)

C₉

yes

characteristic

C₃²×C₉×C₁₃

—

C₃²×C₉×C₁₃

C₃²

yes

characteristic

C₁₃×C₂₇⋊C₃

—

C₁₃×C₂₇⋊C₃

C₃²

yes

complemented

C₃×C₁₃×C₂₇

—

C₃×C₁₃×C₂₇

—

C₃×C₁₃×C₂₇

—

C₃×C₁₃×C₂₇

C₃²

yes

complemented

C₃×#6872

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

C₃×C₉

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

—

C₁₃×3₋¹⁺²

C₃×C₉

yes

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

C₃×C₉

yes

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

—

C₃×C₉×C₁₃

C₃×C₉

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₃

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₁₃

3₋¹⁺²

yes

—

C₃×C₉×C₁₃

3₋¹⁺²

yes

—

C₃×C₉×C₁₃

C₃³

yes

characteristic

C₁₃×C₂₇

—

C₁₃×C₂₇

—

C₃×#737

C₃×C₁₃

yes

characteristic

C₃×C₂₇⋊C₃

C₃×C₁₃

yes

complemented

C₃²×C₂₇

C₃×C₁₃

yes

characteristic, complemented

#6872

C₃×C₁₃

yes

direct product factor

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃²⋊C₉

yes

—

C₃²×C₁₃

C₃²⋊C₉

yes

characteristic

C₃²×C₁₃

C₃²×C₉

yes

characteristic

C₉×C₁₃

—

C₉×C₁₃

C₃²⋊C₉

yes

—

C₉×C₁₃

C₃²⋊C₉

yes

—

C₉×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

C₃×3₋¹⁺²

yes

—

#737

—

#737

—

#737

C₉×C₁₃

yes

—

#737

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₉×C₁₃

yes

—

C₃×H(3,𝔽₃)

C₉×C₁₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃²×C₁₃

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

C₃²×C₁₃

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

#6872

yes

direct product factor

C₃×C₁₃

C₃×C₃²⋊C₉

yes

characteristic

C₃³

—

C₃³

C₃×C₉×C₁₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×C₉×C₁₃

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×C₉×C₁₃

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₉×C₁₃

yes

—

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₁₃×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₁₃×3₋¹⁺²

yes

—

C₃×C₉

C₁₃×3₋¹⁺²

yes

—

C₃×C₉

C₃³×C₁₃

yes

characteristic

C₂₇

—

C₂₇

—

C₁₃

C₃×#6872

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

C₁₃×C₃²⋊C₉

yes

—

C₃²

C₁₃×C₃²⋊C₉

yes

characteristic

100

C₃²

C₃²×C₉×C₁₃

yes

characteristic

101

C₉

—

102

C₉

C₁₃×C₃²⋊C₉

yes

—

103

C₉

C₁₃×C₃²⋊C₉

yes

—

104

C₉

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

105

C₉

C₃×C₁₃×3₋¹⁺²

yes

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

C₁₃×#6872

yes

direct product factor

110

C₃

C₃×C₁₃×C₃²⋊C₉

yes

characteristic

	subgroup	G/H
center	C₃×C₉×C₁₃	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₁₃
Frattini, cosocle	C₃×C₉	C₃³×C₁₃
socle	C₃²×C₁₃	C₃²⋊C₉
prime radical	C₃×C₁₃×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

subgroup

G/H

center

C₃×C₉×C₁₃

H(3,𝔽₃)

commutator

C₃²

C₃²×C₉×C₁₃

Frattini, cosocle

C₃×C₉

C₃³×C₁₃

socle

C₃²×C₁₃

C₃²⋊C₉

prime radical

C₃×C₁₃×H(3,𝔽₃)

C₉

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₉×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₃×C₁₃×C₂₇⋊C₃	C₃	semidirect
3. C₃²×C₁₃×C₂₇	C₃	semidirect
4. C₁₃×#6872	C₃	direct (N × H)
15. C₁₃×C₂₇⋊C₃	C₃²	semidirect
19. C₃×C₁₃×C₂₇	C₃²	semidirect
20. C₃×#6872	C₁₃	direct (N × H)
41. C₃×C₂₇⋊C₃	C₃×C₁₃	semidirect
42. C₃²×C₂₇	C₃×C₁₃	semidirect
43. #6872	C₃×C₁₃	direct (N × H)
65. C₂₇⋊C₃	C₃²×C₁₃	semidirect
69. C₃×C₂₇	C₃²×C₁₃	semidirect
73. C₃×C₁₃	#6872	direct (N × H)
94. C₁₃	C₃×#6872	direct (N × H)
109. C₃	C₁₃×#6872	direct (N × H)

H ≅ G/N

type

2. C₃×C₁₃×C₂₇⋊C₃

C₃

semidirect

3. C₃²×C₁₃×C₂₇

C₃

semidirect

4. C₁₃×#6872

C₃

direct (N × H)

15. C₁₃×C₂₇⋊C₃

C₃²

semidirect

19. C₃×C₁₃×C₂₇

C₃²

semidirect

20. C₃×#6872

C₁₃

direct (N × H)

41. C₃×C₂₇⋊C₃

C₃×C₁₃

semidirect

42. C₃²×C₂₇

C₃×C₁₃

semidirect

43. #6872

C₃×C₁₃

direct (N × H)

65. C₂₇⋊C₃

C₃²×C₁₃

semidirect

69. C₃×C₂₇

C₃²×C₁₃

semidirect

73. C₃×C₁₃

#6872

direct (N × H)

94. C₁₃

C₃×#6872

direct (N × H)

109. C₃

C₁₃×#6872

direct (N × H)

Subgroups — C₃×C₁₃×#6872 #23744852

Conjugacy classes of subgroups

Complements

Subgroups — C3×C13×#6872 #23744852

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₁₃×#6872 #23744852