Group #41005731 — subgroups

Subgroups — #41005731

Order 26973.

452 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 58 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	#22582554	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582554	C₃	1	2	yes	maximal, complemented
3	#22582451	C₃	1	9	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	2	no	—
6	C₃×#11758608	—	3	1	no	—
7	C₃²⋊#66524	—	3	2	no	—
8	C₃²⋊#66524	—	3	2	no	—
9	C₃²⋊#66524	—	3	2	no	—
10	C₃²⋊#66524	C₃²	1	3	yes	complemented
11	C₃²⋊#66524	C₃²	1	6	yes	complemented
12	C₃⁴×C₃₇	—	3	1	no	—
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
15	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
16	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
17	C₃³×C₃₇	—	3	3	no	—
18	C₃³×C₃₇	—	3	9	no	—
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	—	9	2	no	—
21	C₃³×C₃₇	—	9	1	no	—
22	C₃³×C₃₇	—	9	6	no	—
23	C₃³×C₃₇	—	9	3	no	—
24	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
25	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	H(3,𝔽₃)×C₃₇	—	3	9	no	—
28	H(3,𝔽₃)×C₃₇	—	3	18	no	—
29	#11758608	—	9	6	no	—
30	#11758608	—	9	3	no	—
31	C₃×#66524	—	3	3	no	—
32	C₃×#66524	—	3	6	no	—
33	#10624632	—	37	1	no	maximal, Sylow
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	—	3	1	no	—
36	C₃²×C₃₇	—	3	2	no	—
37	C₃²×C₃₇	—	3	1	no	—
38	C₃²×C₃₇	—	3	9	no	—
39	C₃²×C₃₇	—	3	4	no	—
40	C₃²×C₃₇	—	3	2	no	—
41	C₃²×C₃₇	—	3	2	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	9	9	no	—
44	C₃²×C₃₇	—	9	27	no	—
45	C₃²×C₃₇	—	9	6	no	—
46	C₃²×C₃₇	—	9	3	no	—
47	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	#66524	—	9	3	no	—
50	#66524	—	9	6	no	—
51	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
52	#6891	—	37	1	no	—
53	#6891	—	37	2	no	—
54	#6859	—	37	9	no	—
55	C₃×C₃₇	—	3	3	no	—
56	C₃×C₃₇	—	9	2	no	—
57	C₃×C₃₇	—	9	1	no	—
58	C₃×C₃₇	—	9	9	no	—
59	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
60	C₃×C₃₇	#6910	1	2	yes	—
61	C₃×C₃₇	#6911	1	1	yes	characteristic
62	C₃⁴	—	3	1	no	—
63	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
64	C₃×3₋¹⁺²	—	111	2	no	—
65	C₃×3₋¹⁺²	—	111	1	no	—
66	C₃×H(3,𝔽₃)	—	3	9	no	—
67	C₃×H(3,𝔽₃)	—	3	2	no	—
68	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
69	C₃²⋊C₉	—	37	3	no	—
70	C₃²⋊C₉	—	37	6	no	—
71	C₃²⋊C₉	—	111	2	no	—
72	C₃²⋊C₉	—	111	2	no	—
73	C₃²⋊C₉	—	111	2	no	—
74	C₃₇	#10624632	1	1	yes	characteristic, complemented, Sylow
75	C₃³	—	3	3	no	—
76	C₃³	—	3	9	no	—
77	C₃³	—	3	3	no	—
78	C₃³	—	9	2	no	—
79	C₃³	—	9	1	no	—
80	C₃³	—	9	6	no	—
81	C₃³	—	9	3	no	—
82	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
83	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
84	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
85	3₋¹⁺²	—	333	6	no	—
86	3₋¹⁺²	—	333	3	no	—
87	H(3,𝔽₃)	—	3	9	no	—
88	H(3,𝔽₃)	—	3	18	no	—
89	C₃×C₉	—	111	3	no	—
90	C₃×C₉	—	111	6	no	—
91	C₃²	—	3	2	no	—
92	C₃²	—	3	1	no	—
93	C₃²	—	3	2	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	3	9	no	—
96	C₃²	—	3	4	no	—
97	C₃²	—	3	2	no	—
98	C₃²	—	3	2	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	9	9	no	—
101	C₃²	—	9	27	no	—
102	C₃²	—	9	6	no	—
103	C₃²	—	9	3	no	—
104	C₃²	#15688513	1	3	yes	—
105	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
106	C₉	—	333	3	no	—
107	C₉	—	333	6	no	—
108	C₃	—	3	3	no	—
109	C₃	—	9	2	no	—
110	C₃	—	9	1	no	—
111	C₃	—	9	9	no	—
112	C₃	C₃×#15688513	1	1	yes	characteristic
113	C₃	#22582642	1	2	yes	—
114	C₃	#22582647	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582554

C₃

yes

maximal, characteristic, complemented

#22582554

C₃

yes

maximal, complemented

#22582451

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)×C₃₇

C₃

yes

maximal, characteristic

—

—

—

—

—

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

complemented

C₃⁴×C₃₇

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

#66524

—

#66524

—

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

yes

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₇

#6910

yes

—

C₃×C₃₇

#6911

yes

characteristic

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624632

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

#15688513

yes

—

105

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

106

C₉

—

333

—

107

C₉

—

333

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

C₃×#15688513

yes

characteristic

113

C₃

#22582642

yes

—

114

C₃

#22582647

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624632
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting, prime radical

C₃²×H(3,𝔽₃)×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

hypercenter

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

hypocenter

C₃₇

#10624632

LCS

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. #22582554	C₃	semidirect
2. #22582554	C₃	semidirect
3. #22582451	C₃	semidirect
10. C₃²⋊#66524	C₃²	semidirect
11. C₃²⋊#66524	C₃²	semidirect
74. C₃₇	#10624632	semidirect

H ≅ G/N

type

1. #22582554

C₃

semidirect

2. #22582554

C₃

semidirect

3. #22582451

C₃

semidirect

10. C₃²⋊#66524

C₃²

semidirect

11. C₃²⋊#66524

C₃²

semidirect

74. C₃₇

#10624632

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22582554	3
#22582451	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624632	37

subgroup

index

#22582554

#22582451

C₃²×H(3,𝔽₃)×C₃₇

#10624632

Subgroups — #41005731

Conjugacy classes of subgroups

Complements

Maximal subgroups