Group #23744855 — subgroups

Subgroups — C₁₃×#10624704 #23744855

Order 9477.

312 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 146 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	C₃×C₁₃×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#6873	C₃	1	9	yes	maximal, complemented
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	2	no	—
15	C₁₃×C₂₇⋊C₃	—	3	2	no	—
16	C₁₃×C₂₇⋊C₃	—	3	2	no	—
17	C₁₃×C₂₇⋊C₃	—	3	1	no	—
18	C₁₃×C₂₇⋊C₃	C₃²	1	6	yes	complemented
19	C₃×C₁₃×C₂₇	—	3	2	no	—
20	C₃×C₁₃×C₂₇	C₃²	1	3	yes	complemented
21	#10624704	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃³×C₁₃	—	3	1	no	—
23	C₃³×C₁₃	C₃×C₉	1	1	yes	characteristic
24	C₁₃×3₋¹⁺²	—	3	4	no	—
25	C₁₃×3₋¹⁺²	—	3	12	no	—
26	C₁₃×3₋¹⁺²	—	3	6	no	—
27	C₁₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
28	C₁₃×H(3,𝔽₃)	—	3	2	no	—
29	C₁₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
30	C₃×C₉×C₁₃	—	3	2	no	—
31	C₃×C₉×C₁₃	—	3	6	no	—
32	C₃×C₉×C₁₃	—	3	3	no	—
33	C₃×C₉×C₁₃	C₃×C₉	1	2	yes	—
34	C₃×C₉×C₁₃	H(3,𝔽₃)	1	2	yes	—
35	C₃×C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃×C₉×C₁₃	3₋¹⁺²	1	2	yes	—
37	C₃×C₉×C₁₃	3₋¹⁺²	1	4	yes	—
38	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
39	C₁₃×C₂₇	—	3	6	no	—
40	C₁₃×C₂₇	—	3	3	no	—
41	C₃×#737	C₃×C₁₃	1	1	yes	characteristic
42	C₃×C₂₇⋊C₃	C₃×C₁₃	1	2	yes	complemented
43	C₃×C₂₇⋊C₃	C₃×C₁₃	1	1	yes	characteristic, complemented
44	#6873	C₃×C₁₃	1	9	yes	complemented
45	C₃²×C₁₃	—	3	3	no	—
46	C₃²×C₁₃	—	9	1	no	—
47	C₃²×C₁₃	—	9	3	no	—
48	C₃²×C₁₃	C₃²⋊C₉	1	1	yes	characteristic
49	C₃²×C₁₃	C₃²⋊C₉	1	2	yes	—
50	C₃²×C₁₃	C₃²×C₉	1	1	yes	characteristic
51	C₉×C₁₃	—	3	6	no	—
52	C₉×C₁₃	C₃²⋊C₉	1	2	yes	—
53	C₉×C₁₃	C₃²⋊C₉	1	4	yes	—
54	C₉×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₉×C₁₃	C₃×3₋¹⁺²	1	2	yes	—
56	#737	—	3	2	no	—
57	#737	—	3	4	no	—
58	#737	C₉×C₁₃	1	6	yes	—
59	#737	C₃²×C₁₃	1	3	yes	—
60	C₃×3₋¹⁺²	—	3	2	no	—
61	C₃×3₋¹⁺²	C₉×C₁₃	1	2	yes	—
62	C₃×H(3,𝔽₃)	C₉×C₁₃	1	1	yes	characteristic
63	C₃²×C₉	—	3	1	no	—
64	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
65	C₂₇⋊C₃	—	3	2	no	—
66	C₂₇⋊C₃	—	3	2	no	—
67	C₂₇⋊C₃	—	3	2	no	—
68	C₂₇⋊C₃	—	3	1	no	—
69	C₂₇⋊C₃	C₃²×C₁₃	1	6	yes	complemented
70	C₃×C₂₇	—	3	2	no	—
71	C₃×C₂₇	C₃²×C₁₃	1	3	yes	complemented
72	C₃×C₁₃	—	3	1	no	—
73	C₃×C₁₃	—	3	2	no	—
74	C₃×C₁₃	—	3	1	no	—
75	C₃×C₁₃	—	9	3	no	—
76	C₃×C₁₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
77	C₃³	—	3	1	no	—
78	C₃³	C₃×C₉×C₁₃	1	1	yes	characteristic
79	3₋¹⁺²	—	3	4	no	—
80	3₋¹⁺²	—	3	12	no	—
81	3₋¹⁺²	—	3	6	no	—
82	3₋¹⁺²	C₃×C₉×C₁₃	1	6	yes	—
83	H(3,𝔽₃)	—	3	2	no	—
84	H(3,𝔽₃)	C₃×C₉×C₁₃	1	3	yes	—
85	C₃×C₉	—	3	2	no	—
86	C₃×C₉	—	3	6	no	—
87	C₃×C₉	—	3	3	no	—
88	C₃×C₉	C₃×C₉×C₁₃	1	2	yes	—
89	C₃×C₉	C₁₃×H(3,𝔽₃)	1	2	yes	—
90	C₃×C₉	C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
91	C₃×C₉	C₁₃×3₋¹⁺²	1	2	yes	—
92	C₃×C₉	C₁₃×3₋¹⁺²	1	4	yes	—
93	C₃×C₉	C₃³×C₁₃	1	1	yes	characteristic
94	C₂₇	—	3	6	no	—
95	C₂₇	—	3	3	no	—
96	C₁₃	#10624704	1	1	yes	characteristic, direct product factor, Sylow
97	C₃²	—	3	3	no	—
98	C₃²	—	9	1	no	—
99	C₃²	—	9	3	no	—
100	C₃²	C₁₃×C₃²⋊C₉	1	1	yes	characteristic
101	C₃²	C₁₃×C₃²⋊C₉	1	2	yes	—
102	C₃²	C₃²×C₉×C₁₃	1	1	yes	characteristic
103	C₉	—	3	6	no	—
104	C₉	C₁₃×C₃²⋊C₉	1	2	yes	—
105	C₉	C₁₃×C₃²⋊C₉	1	4	yes	—
106	C₉	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
107	C₉	C₃×C₁₃×3₋¹⁺²	1	2	yes	—
108	C₃	—	3	1	no	—
109	C₃	—	3	2	no	—
110	C₃	—	3	1	no	—
111	C₃	—	9	3	no	—
112	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

C₃×C₁₃×C₂₇⋊C₃

C₃

yes

maximal, characteristic, complemented

C₁₃×#6873

C₃

yes

maximal, complemented

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

—

C₁₃×C₂₇⋊C₃

—

C₁₃×C₂₇⋊C₃

—

C₁₃×C₂₇⋊C₃

C₃²

yes

complemented

C₃×C₁₃×C₂₇

—

C₃×C₁₃×C₂₇

C₃²

yes

complemented

#10624704

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

C₃×C₂₇⋊C₃

C₃×C₁₃

yes

characteristic, complemented

#6873

C₃×C₁₃

yes

complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

C₃²⋊C₉

yes

characteristic

C₃²×C₁₃

C₃²⋊C₉

yes

—

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

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

C₃²×C₁₃

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

C₃²×C₁₃

yes

complemented

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

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

#10624704

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₁₃×C₃²⋊C₉

yes

characteristic

101

C₃²

C₁₃×C₃²⋊C₉

yes

—

102

C₃²

C₃²×C₉×C₁₃

yes

characteristic

103

C₉

—

104

C₉

C₁₃×C₃²⋊C₉

yes

—

105

C₉

C₁₃×C₃²⋊C₉

yes

—

106

C₉

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

107

C₉

C₃×C₁₃×3₋¹⁺²

yes

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

C₃×C₁₃×C₃²⋊C₉

yes

characteristic

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

subgroup

G/H

center

C₉×C₁₃

C₃×H(3,𝔽₃)

commutator

C₃²

C₃²×C₉×C₁₃

Frattini, cosocle

C₃×C₉

C₃³×C₁₃

socle

C₃×C₁₃

C₃×C₃²⋊C₉

prime radical

C₃×C₁₃×H(3,𝔽₃)

C₉

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₉×C₁₃ ◁ C₃²×C₉×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₃×C₁₃×C₂₇⋊C₃	C₃	semidirect
3. C₃×C₁₃×C₂₇⋊C₃	C₃	semidirect
4. C₁₃×#6873	C₃	semidirect
18. C₁₃×C₂₇⋊C₃	C₃²	semidirect
20. C₃×C₁₃×C₂₇	C₃²	semidirect
21. #10624704	C₁₃	direct (N × H)
42. C₃×C₂₇⋊C₃	C₃×C₁₃	semidirect
43. C₃×C₂₇⋊C₃	C₃×C₁₃	semidirect
44. #6873	C₃×C₁₃	semidirect
69. C₂₇⋊C₃	C₃²×C₁₃	semidirect
71. C₃×C₂₇	C₃²×C₁₃	semidirect
96. C₁₃	#10624704	direct (N × H)

H ≅ G/N

type

2. C₃×C₁₃×C₂₇⋊C₃

C₃

semidirect

3. C₃×C₁₃×C₂₇⋊C₃

C₃

semidirect

4. C₁₃×#6873

C₃

semidirect

18. C₁₃×C₂₇⋊C₃

C₃²

semidirect

20. C₃×C₁₃×C₂₇

C₃²

semidirect

21. #10624704

C₁₃

direct (N × H)

42. C₃×C₂₇⋊C₃

C₃×C₁₃

semidirect

43. C₃×C₂₇⋊C₃

C₃×C₁₃

semidirect

44. #6873

C₃×C₁₃

semidirect

69. C₂₇⋊C₃

C₃²×C₁₃

semidirect

71. C₃×C₂₇

C₃²×C₁₃

semidirect

96. C₁₃

#10624704

direct (N × H)

Subgroups — C₁₃×#10624704 #23744855

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624704 #23744855

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624704 #23744855