Group #41006826 — subgroups

Subgroups — #41006826

Order 26973.

252 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22582568	C₃	1	2	yes	maximal, complemented
2	#22582556	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²⋊#66524	C₃	1	4	yes	maximal, complemented
5	H(3,𝔽₃)⋊#66524	C₃	1	2	yes	maximal
6	#22582470	C₃	1	2	yes	maximal, complemented
7	C₃×C₃₇×#731	C₃	1	1	yes	maximal, characteristic
8	C₃×#11758608	—	3	1	no	—
9	C₃²×#66524	C₃²	1	2	yes	—
10	C₃²×#66524	C₃²	1	1	yes	characteristic
11	#15688504	—	3	4	no	—
12	#15688504	—	3	2	no	—
13	C₃²⋊#66524	—	3	2	no	—
14	C₃²⋊#66524	C₃²	1	2	yes	—
15	C₃²⋊#66524	C₃²	1	4	yes	—
16	C₉×#66524	—	3	2	no	—
17	C₉×#66524	—	3	1	no	—
18	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
19	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×#731	—	3	3	no	—
22	C₃³×C₃₇	—	3	1	no	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	3₋¹⁺²×C₃₇	—	3	2	no	—
25	3₋¹⁺²×C₃₇	—	3	4	no	—
26	H(3,𝔽₃)×C₃₇	—	3	1	no	—
27	H(3,𝔽₃)×C₃₇	—	3	2	no	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	2	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	complemented
32	#11758608	—	9	3	no	—
33	C₃×#66524	—	3	2	no	—
34	C₃×#66524	—	3	3	no	—
35	C₃×#66524	—	3	6	no	—
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	4	no	—
38	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
39	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
40	#10624785	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	2	no	—
43	C₃²×C₃₇	—	9	1	no	—
44	C₃²×C₃₇	—	9	1	no	—
45	C₃²×C₃₇	—	9	2	no	—
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₃²×C₃₇	#737	1	2	yes	—
49	C₉×C₃₇	—	3	3	no	—
50	C₉×C₃₇	—	9	2	no	—
51	#66524	—	3	2	no	—
52	#66524	—	3	6	no	—
53	#66524	—	9	2	no	—
54	#66524	—	9	4	no	—
55	#66524	#731	1	3	yes	complemented
56	C₃×#731	C₃₇⋊C₃	1	1	yes	characteristic
57	#6894	—	37	2	no	—
58	C₉⋊H(3,𝔽₃)	—	37	1	no	—
59	C₃×C₉²	—	37	1	no	—
60	3₋¹⁺²⋊C₉	—	37	4	no	—
61	H(3,𝔽₃)⋊C₉	—	37	2	no	—
62	#6867	—	37	2	no	—
63	C₃×C₃₇	—	3	1	no	—
64	C₃×C₃₇	—	3	2	no	—
65	C₃×C₃₇	—	27	1	no	—
66	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
67	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
68	C₃×C₃₇	#6908	1	2	yes	—
69	C₃×3₋¹⁺²	—	111	1	no	—
70	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
71	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
72	C₃²×C₉	—	37	1	no	—
73	C₃²×C₉	—	37	2	no	—
74	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
75	#731	—	3	3	no	—
76	C₉⋊C₉	—	111	2	no	—
77	C₉⋊C₉	—	111	4	no	—
78	C₃²⋊C₉	—	37	4	no	—
79	C₃²⋊C₉	—	37	2	no	—
80	C₃²⋊C₉	—	111	2	no	—
81	C₉²	—	111	1	no	—
82	C₉²	—	111	2	no	—
83	C₃₇	#10624785	1	1	yes	characteristic, complemented, Sylow
84	C₃³	—	3	1	no	—
85	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
86	3₋¹⁺²	—	3	2	no	—
87	3₋¹⁺²	—	3	4	no	—
88	3₋¹⁺²	—	333	3	no	—
89	H(3,𝔽₃)	—	3	1	no	—
90	H(3,𝔽₃)	—	3	2	no	—
91	C₃×C₉	—	3	1	no	—
92	C₃×C₉	—	3	2	no	—
93	C₃×C₉	—	3	2	no	—
94	C₃×C₉	—	37	1	no	—
95	C₃×C₉	—	37	2	no	—
96	C₃×C₉	—	111	3	no	—
97	C₃×C₉	—	111	2	no	—
98	C₃×C₉	—	111	4	no	—
99	C₃×C₉	—	111	2	no	—
100	C₃×C₉	—	111	6	no	—
101	C₃×C₉	#11758608	1	3	yes	complemented
102	C₃²	—	3	1	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	9	1	no	—
105	C₃²	—	9	1	no	—
106	C₃²	—	9	2	no	—
107	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
108	C₃²	C₃×#11758608	1	1	yes	characteristic
109	C₃²	#15688530	1	2	yes	—
110	C₉	—	3	3	no	—
111	C₉	—	9	2	no	—
112	C₉	—	37	3	no	—
113	C₉	—	333	2	no	—
114	C₉	—	333	4	no	—
115	C₉	—	111	2	no	—
116	C₉	—	111	6	no	—
117	C₃	—	3	1	no	—
118	C₃	—	3	2	no	—
119	C₃	—	27	1	no	—
120	C₃	#22582556	1	1	yes	characteristic
121	C₃	#731×C₃₇⋊C₃	1	1	yes	characteristic
122	C₃	#22582632	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22582568

C₃

yes

maximal, complemented

#22582556

C₃

yes

maximal, characteristic

C₃×C₉×#66524

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal

#22582470

C₃

yes

maximal, complemented

C₃×C₃₇×#731

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#731

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

yes

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

yes

C₃₇⋊C₃

yes

characteristic

#6894

—

C₉⋊H(3,𝔽₃)

—

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#731

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

#10624785

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

#11758608

yes

complemented

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

108

C₃²

C₃×#11758608

yes

characteristic

109

C₃²

#15688530

yes

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

333

—

114

C₉

—

333

—

115

C₉

—

111

—

116

C₉

—

111

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

#22582556

yes

characteristic

121

C₃

#731×C₃₇⋊C₃

yes

characteristic

122

C₃

#22582632

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#731	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₃×#731	C₃₇⋊C₃
hypocenter	C₃₇	#10624785
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#731

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#731

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#731

N	H ≅ G/N	type
1. #22582568	C₃	semidirect
3. C₃×C₉×#66524	C₃	semidirect
4. 3₋¹⁺²⋊#66524	C₃	semidirect
6. #22582470	C₃	semidirect
31. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
55. #66524	#731	semidirect
83. C₃₇	#10624785	semidirect
101. C₃×C₉	#11758608	semidirect

H ≅ G/N

type

1. #22582568

C₃

semidirect

3. C₃×C₉×#66524

C₃

semidirect

4. 3₋¹⁺²⋊#66524

C₃

semidirect

6. #22582470

C₃

semidirect

31. C₃×C₉×C₃₇

3₋¹⁺²

semidirect

55. #66524

#731

semidirect

83. C₃₇

#10624785

semidirect

101. C₃×C₉

#11758608

semidirect

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

subgroup	index
#22582568	3
#22582556	3
C₃×C₉×#66524	3
3₋¹⁺²⋊#66524	3
H(3,𝔽₃)⋊#66524	3
#22582470	3
C₃×C₃₇×#731	3
#10624785	37

subgroup

index

C₃×C₉×#66524

C₃×C₃₇×#731

Subgroups — #41006826

Conjugacy classes of subgroups

Complements

Maximal subgroups