Group #41006701 — subgroups

Subgroups — #41006701

Order 26973.

1940 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1940 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₃⁴×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582554	C₃	1	2	yes	maximal, complemented
3	#22582467	C₃	1	9	yes	maximal, complemented
4	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	—	3	2	no	—
6	C₃³×C₃₇⋊C₃	—	3	1	no	—
7	C₃³×C₃₇⋊C₃	—	3	9	no	—
8	C₃³×C₃₇⋊C₃	—	3	9	no	—
9	C₃³×C₃₇⋊C₃	—	3	18	no	—
10	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
11	C₃×#11758608	—	3	2	no	—
12	C₃²⋊#66524	—	3	2	no	—
13	C₃²⋊#66524	—	3	2	no	—
14	C₃²⋊#66524	C₃²	1	6	yes	complemented
15	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
16	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
17	C₃₇×C₃²⋊C₉	—	3	2	no	—
18	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	complemented
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	—	3	6	no	—
21	C₃³×C₃₇	—	3	3	no	—
22	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
23	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	—	3	243	no	—
26	C₃²×C₃₇⋊C₃	—	3	18	no	—
27	C₃²×C₃₇⋊C₃	—	3	9	no	—
28	C₃²×C₃₇⋊C₃	—	3	18	no	—
29	C₃²×C₃₇⋊C₃	—	3	18	no	—
30	C₃²×C₃₇⋊C₃	—	3	18	no	—
31	C₃²×C₃₇⋊C₃	—	3	9	no	—
32	C₃²×C₃₇⋊C₃	—	3	18	no	—
33	C₃²×C₃₇⋊C₃	—	3	3	no	—
34	C₃²×C₃₇⋊C₃	—	3	6	no	—
35	C₃²×C₃₇⋊C₃	—	3	9	no	—
36	C₃²×C₃₇⋊C₃	—	3	18	no	—
37	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
38	3₋¹⁺²×C₃₇	—	9	3	no	—
39	C₃×C₉×C₃₇	—	3	3	no	—
40	#11758608	—	9	6	no	—
41	C₃×#66524	—	3	6	no	—
42	#10624765	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	—	3	27	no	—
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	2	no	—
47	C₃²×C₃₇	—	3	2	no	—
48	C₃²×C₃₇	—	3	2	no	—
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	2	no	—
51	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
52	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	6	yes	—
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
55	C₃×C₃₇⋊C₃	—	3	81	no	—
56	C₃×C₃₇⋊C₃	—	3	162	no	—
57	C₃×C₃₇⋊C₃	—	3	54	no	—
58	C₃×C₃₇⋊C₃	—	3	9	no	—
59	C₃×C₃₇⋊C₃	—	3	9	no	—
60	C₃×C₃₇⋊C₃	—	3	18	no	—
61	C₃×C₃₇⋊C₃	—	3	27	no	—
62	C₉×C₃₇	—	9	3	no	—
63	#66524	—	9	6	no	—
64	C₃⁵	—	37	1	no	—
65	#6891	—	37	2	no	—
66	#6891	C₃₇⋊C₃	1	1	yes	characteristic, complemented
67	#6866	—	37	9	no	—
68	C₃×C₃₇	—	3	3	no	—
69	C₃×C₃₇	—	3	6	no	—
70	C₃×C₃₇	—	3	3	no	—
71	C₃×C₃₇	#6891	1	1	yes	characteristic
72	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
73	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
74	C₃₇⋊C₃	—	3	27	no	—
75	C₃⁴	—	37	3	no	—
76	C₃⁴	—	111	9	no	—
77	C₃⁴	—	111	9	no	—
78	C₃⁴	—	111	18	no	—
79	C₃⁴	—	111	1	no	—
80	C₃⁴	—	111	2	no	—
81	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃×3₋¹⁺²	—	3	1	no	—
83	C₃×3₋¹⁺²	—	111	2	no	—
84	C₃²⋊C₉	—	3	2	no	—
85	C₃²⋊C₉	—	37	6	no	—
86	C₃²⋊C₉	—	111	2	no	—
87	C₃²⋊C₉	—	111	2	no	—
88	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
89	C₃₇	#10624765	1	1	yes	characteristic, complemented, Sylow
90	C₃³	—	3	3	no	—
91	C₃³	—	3	6	no	—
92	C₃³	—	3	3	no	—
93	C₃³	—	37	9	no	—
94	C₃³	—	111	9	no	—
95	C₃³	—	111	18	no	—
96	C₃³	—	111	243	no	—
97	C₃³	—	111	9	no	—
98	C₃³	—	111	18	no	—
99	C₃³	—	111	18	no	—
100	C₃³	—	111	18	no	—
101	C₃³	—	111	18	no	—
102	C₃³	—	111	9	no	—
103	C₃³	—	111	18	no	—
104	C₃³	—	111	3	no	—
105	C₃³	—	111	6	no	—
106	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
107	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
108	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
109	3₋¹⁺²	—	9	3	no	—
110	3₋¹⁺²	—	333	6	no	—
111	C₃×C₉	—	3	3	no	—
112	C₃×C₉	—	111	6	no	—
113	C₃²	—	3	27	no	—
114	C₃²	—	3	1	no	—
115	C₃²	—	3	2	no	—
116	C₃²	—	3	2	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	111	27	no	—
122	C₃²	—	111	81	no	—
123	C₃²	—	111	162	no	—
124	C₃²	—	111	54	no	—
125	C₃²	—	111	9	no	—
126	C₃²	—	111	9	no	—
127	C₃²	—	111	18	no	—
128	C₃²	#15688512	1	3	yes	—
129	C₃²	#15688514	1	6	yes	—
130	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
131	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
132	C₉	—	9	3	no	—
133	C₉	—	333	6	no	—
134	C₃	—	3	3	no	—
135	C₃	—	3	6	no	—
136	C₃	—	3	3	no	—
137	C₃	—	111	27	no	—
138	C₃	#22582551	1	1	yes	characteristic
139	C₃	C₃×#15688512	1	1	yes	characteristic
140	C₃	C₃×#15688514	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#22582554

C₃

yes

maximal, complemented

#22582467

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

—

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

243

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

complemented

3₋¹⁺²×C₃₇

—

C₃×C₉×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

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

162

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

C₃⁵

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic, complemented

#6866

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃₇

#10624765

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

243

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

100

C₃³

—

111

—

101

C₃³

—

111

—

102

C₃³

—

111

—

103

C₃³

—

111

—

104

C₃³

—

111

—

105

C₃³

—

111

—

106

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

107

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

108

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

109

3₋¹⁺²

—

110

3₋¹⁺²

—

333

—

111

C₃×C₉

—

112

C₃×C₉

—

111

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

111

—

122

C₃²

—

111

—

123

C₃²

—

111

162

—

124

C₃²

—

111

—

125

C₃²

—

111

—

126

C₃²

—

111

—

127

C₃²

—

111

—

128

C₃²

#15688512

yes

—

129

C₃²

#15688514

yes

—

130

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

131

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

132

C₉

—

133

C₉

—

333

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

—

111

—

138

C₃

#22582551

yes

characteristic

139

C₃

C₃×#15688512

yes

characteristic

140

C₃

C₃×#15688514

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇⋊C₃	C₃
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624765
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6891

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇⋊C₃

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
2. #22582554	C₃	semidirect
3. #22582467	C₃	semidirect
4. C₃₇×#6891	C₃	semidirect
14. C₃²⋊#66524	C₃²	semidirect
18. C₃₇×C₃²⋊C₉	C₃²	semidirect
37. C₃²×C₃₇⋊C₃	3₋¹⁺²	semidirect
66. #6891	C₃₇⋊C₃	semidirect
88. C₃²⋊C₉	C₃×C₃₇⋊C₃	semidirect
89. C₃₇	#10624765	semidirect

H ≅ G/N

type

2. #22582554

C₃

semidirect

3. #22582467

C₃

semidirect

4. C₃₇×#6891

C₃

semidirect

14. C₃²⋊#66524

C₃²

semidirect

18. C₃₇×C₃²⋊C₉

C₃²

semidirect

37. C₃²×C₃₇⋊C₃

3₋¹⁺²

semidirect

66. #6891

C₃₇⋊C₃

semidirect

88. C₃²⋊C₉

C₃×C₃₇⋊C₃

semidirect

89. C₃₇

#10624765

semidirect

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

subgroup	index
C₃⁴×C₃₇⋊C₃	3
#22582554	3
#22582467	3
C₃₇×#6891	3
#10624765	37

subgroup

index

C₃⁴×C₃₇⋊C₃

#22582554

#22582467

C₃₇×#6891

#10624765

Subgroups — #41006701

Conjugacy classes of subgroups

Complements

Maximal subgroups