Group #41005184 — subgroups

Subgroups — #41005184

Order 26973.

758 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 121 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×#11758608	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊#66524	C₃	1	2	yes	maximal, complemented
3	C₃×C₃²⋊#66524	C₃	1	9	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	3	no	—
6	C₃×#11758608	—	3	9	no	—
7	C₃²×#66524	C₃²	1	6	yes	—
8	C₃²×#66524	C₃²	1	3	yes	—
9	C₃²⋊#66524	—	3	6	no	—
10	C₃²⋊#66524	—	3	27	no	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃⁴×C₃₇	C₃²	1	3	yes	—
13	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
14	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	—	3	9	no	—
17	C₃³×C₃₇	—	3	3	no	—
18	C₃³×C₃₇	—	3	3	no	—
19	C₃³×C₃₇	—	3	6	no	—
20	C₃³×C₃₇	—	3	18	no	—
21	C₃³×C₃₇	—	3	6	no	—
22	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
23	C₃³×C₃₇	3₋¹⁺²	1	9	yes	complemented
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	H(3,𝔽₃)×C₃₇	—	3	27	no	—
26	#11758608	—	3	27	no	—
27	C₃×#66524	—	3	6	no	—
28	C₃×#66524	—	3	3	no	—
29	C₃×#66524	—	3	18	no	—
30	C₃×#66524	—	3	6	no	—
31	C₃×#66524	H(3,𝔽₃)	1	9	yes	complemented
32	#10624563	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	3	no	—
34	C₃²×C₃₇	—	3	3	no	—
35	C₃²×C₃₇	—	3	9	no	—
36	C₃²×C₃₇	—	3	27	no	—
37	C₃²×C₃₇	—	3	6	no	—
38	C₃²×C₃₇	—	9	3	no	—
39	C₃²×C₃₇	—	9	18	no	—
40	C₃²×C₃₇	—	9	9	no	—
41	C₃²×C₃₇	—	9	6	no	—
42	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
43	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
45	C₃²×C₃₇	#737	1	6	yes	—
46	#66524	—	3	9	no	—
47	#66524	—	9	6	no	—
48	C₃²×3₋¹⁺²	—	37	1	no	—
49	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×C₃²⋊C₉	—	37	9	no	—
51	C₃×C₃²⋊C₉	—	37	2	no	—
52	C₃×C₃₇	—	9	3	no	—
53	C₃×C₃₇	—	9	9	no	—
54	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
55	C₃×C₃₇	#6891	1	1	yes	characteristic
56	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	9	yes	—
57	C₃×C₃₇	#6893	1	2	yes	—
58	C₃⁴	C₃×C₃₇⋊C₃	1	3	yes	—
59	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
60	C₃×3₋¹⁺²	—	111	9	no	—
61	C₃×3₋¹⁺²	—	111	3	no	—
62	C₃×H(3,𝔽₃)	—	3	9	no	—
63	C₃×H(3,𝔽₃)	—	3	3	no	—
64	C₃²×C₉	—	37	3	no	—
65	C₃²×C₉	—	37	6	no	—
66	C₃²⋊C₉	—	111	27	no	—
67	C₃²⋊C₉	—	111	6	no	—
68	C₃₇	#10624563	1	1	yes	characteristic, complemented, Sylow
69	C₃³	—	3	9	no	—
70	C₃³	—	3	3	no	—
71	C₃³	—	3	3	no	—
72	C₃³	—	3	6	no	—
73	C₃³	—	3	3	no	—
74	C₃³	—	3	18	no	—
75	C₃³	—	3	6	no	—
76	C₃³	#11758608	1	9	yes	complemented
77	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
78	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
79	3₋¹⁺²	—	111	27	no	—
80	H(3,𝔽₃)	—	3	27	no	—
81	C₃×C₉	—	37	9	no	—
82	C₃×C₉	—	111	18	no	—
83	C₃×C₉	—	111	6	no	—
84	C₃×C₉	—	111	6	no	—
85	C₃×C₉	—	111	3	no	—
86	C₃²	—	3	3	no	—
87	C₃²	—	3	9	no	—
88	C₃²	—	3	3	no	—
89	C₃²	—	3	27	no	—
90	C₃²	—	3	6	no	—
91	C₃²	—	9	3	no	—
92	C₃²	—	9	18	no	—
93	C₃²	—	9	9	no	—
94	C₃²	—	9	6	no	—
95	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
96	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
97	C₃²	C₃×#11758608	1	3	yes	—
98	C₃²	#15688530	1	6	yes	—
99	C₉	—	333	6	no	—
100	C₉	—	111	9	no	—
101	C₃	—	9	3	no	—
102	C₃	—	9	9	no	—
103	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
104	C₃	#22582554	1	1	yes	characteristic
105	C₃	#22582556	1	9	yes	—
106	C₃	#22582562	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic, complemented

C₃×C₃²⋊#66524

C₃

yes

maximal, complemented

C₃×C₃²⋊#66524

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)×C₃₇

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃×#11758608

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

—

C₃²⋊#66524

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃⁴×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

3₋¹⁺²

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

H(3,𝔽₃)×C₃₇

—

—

—

—

—

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

#737

yes

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₃₇

#6893

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624563

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

#11758608

yes

complemented

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃²

C₃×#11758608

yes

—

C₃²

#15688530

yes

—

C₉

—

333

—

100

C₉

—

111

—

101

C₃

—

102

C₃

—

103

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

104

C₃

#22582554

yes

characteristic

105

C₃

#22582556

yes

—

106

C₃

#22582562

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624563
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator, socle, cosocle

C₃³×C₃₇

C₃³

Fitting, prime radical

C₃²×H(3,𝔽₃)×C₃₇

C₃

hypercenter

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#11758608	C₃	semidirect
2. C₃×C₃²⋊#66524	C₃	semidirect
3. C₃×C₃²⋊#66524	C₃	semidirect
23. C₃³×C₃₇	3₋¹⁺²	semidirect
31. C₃×#66524	H(3,𝔽₃)	semidirect
68. C₃₇	#10624563	semidirect
76. C₃³	#11758608	semidirect

H ≅ G/N

type

1. C₃²×#11758608

C₃

semidirect

2. C₃×C₃²⋊#66524

C₃

semidirect

3. C₃×C₃²⋊#66524

C₃

semidirect

23. C₃³×C₃₇

3₋¹⁺²

semidirect

31. C₃×#66524

H(3,𝔽₃)

semidirect

68. C₃₇

#10624563

semidirect

76. C₃³

#11758608

semidirect

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

subgroup	index
C₃²×#11758608	3
C₃×C₃²⋊#66524	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624563	37

subgroup

index

C₃²×#11758608

C₃×C₃²⋊#66524

C₃²×H(3,𝔽₃)×C₃₇

#10624563

Subgroups — #41005184

Conjugacy classes of subgroups

Complements

Maximal subgroups