Group #41006777 — subgroups

Subgroups — #41006777

Order 26973.

572 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 82 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	#22582554	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)⋊#66524	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	C₃×#11758608	—	3	9	no	—
8	C₃×#11758608	—	3	2	no	—
9	C₃²×#66524	C₃²	1	3	yes	complemented
10	C₃²⋊#66524	—	3	2	no	—
11	C₃²⋊#66524	—	3	2	no	—
12	C₃²⋊#66524	C₃²	1	6	yes	complemented
13	C₃⁴×C₃₇	—	3	1	no	—
14	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
15	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
16	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
17	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
18	C₃³×C₃₇	—	3	9	no	—
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	—	3	3	no	—
21	C₃³×C₃₇	—	3	3	no	—
22	C₃³×C₃₇	—	9	1	no	—
23	C₃³×C₃₇	—	9	1	no	—
24	C₃³×C₃₇	—	9	3	no	—
25	C₃³×C₃₇	—	9	3	no	—
26	C₃³×C₃₇	—	9	3	no	—
27	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
28	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₃₇	C₃³	1	1	yes	characteristic
30	H(3,𝔽₃)×C₃₇	—	3	18	no	—
31	H(3,𝔽₃)×C₃₇	—	3	6	no	—
32	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	9	yes	complemented
33	#11758608	—	3	18	no	—
34	#11758608	—	3	9	no	—
35	#11758608	—	9	6	no	—
36	C₃×#66524	—	3	3	no	—
37	C₃×#66524	—	3	9	no	—
38	C₃×#66524	—	3	6	no	—
39	#10624779	—	37	1	no	maximal, Sylow
40	C₃²×C₃₇	—	3	1	no	—
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	3	1	no	—
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	2	no	—
47	C₃²×C₃₇	—	3	9	no	—
48	C₃²×C₃₇	—	3	2	no	—
49	C₃²×C₃₇	—	3	2	no	—
50	C₃²×C₃₇	—	3	2	no	—
51	C₃²×C₃₇	—	9	9	no	—
52	C₃²×C₃₇	—	9	27	no	—
53	C₃²×C₃₇	—	9	3	no	—
54	C₃²×C₃₇	—	9	3	no	—
55	C₃²×C₃₇	—	9	3	no	—
56	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
57	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
59	#66524	—	3	9	no	—
60	#66524	—	9	6	no	—
61	C₃²×3₋¹⁺²	—	37	1	no	—
62	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
63	#6891	—	37	2	no	—
64	H(3,𝔽₃)⋊C₉	—	37	9	no	—
65	C₃×C₃₇	—	3	3	no	—
66	C₃×C₃₇	—	3	3	no	—
67	C₃×C₃₇	—	9	2	no	—
68	C₃×C₃₇	—	9	9	no	—
69	C₃×C₃₇	#6891	1	1	yes	characteristic
70	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
71	C₃×C₃₇	#6910	1	1	yes	characteristic
72	C₃×C₃₇	#6911	1	1	yes	characteristic
73	C₃⁴	—	3	1	no	—
74	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
75	C₃×3₋¹⁺²	—	111	2	no	—
76	C₃×3₋¹⁺²	—	111	2	no	—
77	C₃×3₋¹⁺²	—	111	1	no	—
78	C₃×3₋¹⁺²	—	111	9	no	—
79	C₃×H(3,𝔽₃)	—	3	2	no	—
80	C₃×H(3,𝔽₃)	—	3	9	no	—
81	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
82	C₃²×C₉	—	37	3	no	—
83	C₃²⋊C₉	—	37	6	no	—
84	C₃²⋊C₉	—	111	2	no	—
85	C₃²⋊C₉	—	111	2	no	—
86	C₃₇	#10624779	1	1	yes	characteristic, complemented, Sylow
87	C₃³	—	3	9	no	—
88	C₃³	—	3	3	no	—
89	C₃³	—	3	3	no	—
90	C₃³	—	3	3	no	—
91	C₃³	—	9	1	no	—
92	C₃³	—	9	1	no	—
93	C₃³	—	9	3	no	—
94	C₃³	—	9	3	no	—
95	C₃³	—	9	3	no	—
96	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
97	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
98	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
99	3₋¹⁺²	—	333	6	no	—
100	3₋¹⁺²	—	111	18	no	—
101	3₋¹⁺²	—	111	9	no	—
102	H(3,𝔽₃)	—	3	18	no	—
103	H(3,𝔽₃)	—	3	6	no	—
104	H(3,𝔽₃)	#11758608	1	9	yes	complemented
105	C₃×C₉	—	111	6	no	—
106	C₃×C₉	—	111	3	no	—
107	C₃×C₉	—	111	9	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	1	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	2	no	—
115	C₃²	—	3	9	no	—
116	C₃²	—	3	2	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	9	9	no	—
120	C₃²	—	9	27	no	—
121	C₃²	—	9	3	no	—
122	C₃²	—	9	3	no	—
123	C₃²	—	9	3	no	—
124	C₃²	#15688512	1	3	yes	—
125	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
126	C₃²	C₃×#11758608	1	3	yes	—
127	C₉	—	333	6	no	—
128	C₉	—	111	9	no	—
129	C₃	—	3	3	no	—
130	C₃	—	3	3	no	—
131	C₃	—	9	2	no	—
132	C₃	—	9	9	no	—
133	C₃	#22582554	1	1	yes	characteristic
134	C₃	C₃×#15688512	1	1	yes	characteristic
135	C₃	#22582642	1	1	yes	characteristic
136	C₃	#22582647	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic, complemented

#22582554

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)×C₃₇

C₃

yes

maximal, characteristic

—

—

—

—

C₃²×#66524

C₃²

yes

—

—

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

—

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

—

H(3,𝔽₃)×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

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

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

yes

characteristic

#6891

—

H(3,𝔽₃)⋊C₉

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

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

—

C₃₇

#10624779

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

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

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

—

111

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

#11758608

yes

complemented

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

#15688512

yes

—

125

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

126

C₃²

C₃×#11758608

yes

—

127

C₉

—

333

—

128

C₉

—

111

—

129

C₃

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

#22582554

yes

characteristic

134

C₃

C₃×#15688512

yes

characteristic

135

C₃

#22582642

yes

characteristic

136

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	#10624779
LCS	G ▷ C₃³×C₃₇ ▷ C₃×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

#10624779

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃²×#11758608	C₃	semidirect
2. #22582554	C₃	semidirect
3. H(3,𝔽₃)⋊#66524	C₃	semidirect
9. C₃²×#66524	C₃²	semidirect
12. C₃²⋊#66524	C₃²	semidirect
32. H(3,𝔽₃)×C₃₇	3₋¹⁺²	semidirect
86. C₃₇	#10624779	semidirect
104. H(3,𝔽₃)	#11758608	semidirect

H ≅ G/N

type

1. C₃²×#11758608

C₃

semidirect

2. #22582554

C₃

semidirect

3. H(3,𝔽₃)⋊#66524

C₃

semidirect

9. C₃²×#66524

C₃²

semidirect

12. C₃²⋊#66524

C₃²

semidirect

32. H(3,𝔽₃)×C₃₇

3₋¹⁺²

semidirect

86. C₃₇

#10624779

semidirect

104. H(3,𝔽₃)

#11758608

semidirect

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

subgroup	index
C₃²×#11758608	3
#22582554	3
H(3,𝔽₃)⋊#66524	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624779	37

subgroup

index

C₃²×H(3,𝔽₃)×C₃₇

Subgroups — #41006777

Conjugacy classes of subgroups

Complements

Maximal subgroups