Group #41006815 — subgroups

Subgroups — #41006815

Order 26973.

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

normal subgroups only

This group has 432 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	C₃³×#66524	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582568	C₃	1	2	yes	maximal, complemented
3	#22582556	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, complemented
6	#22582468	C₃	1	2	yes	maximal, complemented
7	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
8	C₃×#11758608	—	3	1	no	—
9	C₃²×#66524	—	3	2	no	—
10	C₃²×#66524	—	3	1	no	—
11	C₃²×#66524	—	3	9	no	—
12	C₃²×#66524	C₃²	1	2	yes	—
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	#15688504	—	3	4	no	—
15	#15688504	—	3	2	no	—
16	C₃²⋊#66524	—	3	2	no	—
17	C₃²⋊#66524	C₃²	1	2	yes	—
18	C₃²⋊#66524	C₃²	1	4	yes	—
19	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
20	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
21	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
22	C₃₇×AΣL(1,𝔽₂₇)	—	3	3	no	—
23	C₃³×C₃₇	—	3	6	no	—
24	C₃³×C₃₇	—	3	3	no	—
25	C₃³×C₃₇	—	3	1	no	—
26	C₃³×C₃₇	—	3	2	no	—
27	C₃³×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	3₋¹⁺²	1	3	yes	complemented
29	C₃³×C₃₇	C₃³	1	1	yes	characteristic
30	3₋¹⁺²×C₃₇	—	3	2	no	—
31	3₋¹⁺²×C₃₇	—	3	4	no	—
32	H(3,𝔽₃)×C₃₇	—	3	1	no	—
33	H(3,𝔽₃)×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	—	3	2	no	—
35	#11758608	—	9	3	no	—
36	C₃×#66524	—	3	18	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	9	no	—
39	C₃×#66524	—	3	3	no	—
40	C₃×#66524	—	3	6	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	4	no	—
43	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
44	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
45	#10624784	—	37	1	no	maximal, Sylow
46	C₃²×C₃₇	—	3	9	no	—
47	C₃²×C₃₇	—	3	18	no	—
48	C₃²×C₃₇	—	3	3	no	—
49	C₃²×C₃₇	—	3	3	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	6	no	—
52	C₃²×C₃₇	—	3	2	no	—
53	C₃²×C₃₇	—	9	1	no	—
54	C₃²×C₃₇	—	9	1	no	—
55	C₃²×C₃₇	—	9	2	no	—
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
58	C₃²×C₃₇	#737	1	2	yes	—
59	C₉×C₃₇	—	9	2	no	—
60	#66524	—	3	6	no	—
61	#66524	—	3	2	no	—
62	#66524	—	9	2	no	—
63	#66524	—	9	4	no	—
64	#66524	AΣL(1,𝔽₂₇)	1	3	yes	complemented
65	C₃³×C₉	—	37	1	no	—
66	C₃×AΣL(1,𝔽₂₇)	C₃₇⋊C₃	1	1	yes	characteristic
67	#6894	—	37	2	no	—
68	C₉⋊H(3,𝔽₃)	—	37	1	no	—
69	3₋¹⁺²⋊C₉	—	37	4	no	—
70	H(3,𝔽₃)⋊C₉	—	37	2	no	—
71	#6866	—	37	2	no	—
72	C₃×C₃₇	—	3	9	no	—
73	C₃×C₃₇	—	3	1	no	—
74	C₃×C₃₇	—	3	2	no	—
75	C₃×C₃₇	—	27	1	no	—
76	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
78	C₃×C₃₇	#6908	1	2	yes	—
79	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃×3₋¹⁺²	—	111	1	no	—
81	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
82	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
83	C₃²×C₉	—	37	1	no	—
84	C₃²×C₉	—	37	2	no	—
85	C₃²×C₉	—	111	1	no	—
86	C₃²×C₉	—	111	2	no	—
87	C₃²×C₉	—	111	9	no	—
88	AΣL(1,𝔽₂₇)	—	3	3	no	—
89	C₉⋊C₉	—	111	2	no	—
90	C₉⋊C₉	—	111	4	no	—
91	C₃²⋊C₉	—	37	4	no	—
92	C₃²⋊C₉	—	37	2	no	—
93	C₃²⋊C₉	—	111	2	no	—
94	C₃₇	#10624784	1	1	yes	characteristic, complemented, Sylow
95	C₃³	—	3	3	no	—
96	C₃³	—	3	6	no	—
97	C₃³	—	3	1	no	—
98	C₃³	—	3	2	no	—
99	C₃³	—	3	1	no	—
100	C₃³	#11758608	1	3	yes	complemented
101	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
102	3₋¹⁺²	—	3	2	no	—
103	3₋¹⁺²	—	3	4	no	—
104	3₋¹⁺²	—	333	3	no	—
105	H(3,𝔽₃)	—	3	1	no	—
106	H(3,𝔽₃)	—	3	2	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	37	1	no	—
109	C₃×C₉	—	37	2	no	—
110	C₃×C₉	—	111	3	no	—
111	C₃×C₉	—	111	9	no	—
112	C₃×C₉	—	111	2	no	—
113	C₃×C₉	—	111	4	no	—
114	C₃×C₉	—	111	18	no	—
115	C₃×C₉	—	111	2	no	—
116	C₃×C₉	—	111	6	no	—
117	C₃²	—	3	9	no	—
118	C₃²	—	3	18	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	3	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	3	6	no	—
124	C₃²	—	9	1	no	—
125	C₃²	—	9	1	no	—
126	C₃²	—	9	2	no	—
127	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
128	C₃²	C₃×#11758608	1	1	yes	characteristic
129	C₃²	#15688530	1	2	yes	—
130	C₉	—	9	2	no	—
131	C₉	—	37	3	no	—
132	C₉	—	333	2	no	—
133	C₉	—	333	4	no	—
134	C₉	—	111	2	no	—
135	C₉	—	111	6	no	—
136	C₃	—	3	1	no	—
137	C₃	—	3	2	no	—
138	C₃	—	3	9	no	—
139	C₃	—	27	1	no	—
140	C₃	#22582556	1	1	yes	characteristic
141	C₃	AΣL(1,𝔽₂₇)×C₃₇⋊C₃	1	1	yes	characteristic
142	C₃	#22582629	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×#66524

C₃

yes

maximal, characteristic, complemented

#22582568

C₃

yes

maximal, complemented

#22582556

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²⋊#66524

C₃

yes

maximal, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, complemented

#22582468

C₃

yes

maximal, complemented

C₃×C₃₇×AΣL(1,𝔽₂₇)

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃²×#66524

—

C₃²×#66524

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃₇×AΣL(1,𝔽₂₇)

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

3₋¹⁺²

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

—

yes

—

yes

—

C₃²×C₃₇

—

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

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

—

—

—

—

AΣL(1,𝔽₂₇)

yes

complemented

C₃³×C₉

—

C₃×AΣL(1,𝔽₂₇)

C₃₇⋊C₃

yes

characteristic

#6894

—

C₉⋊H(3,𝔽₃)

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6866

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₇

#6908

yes

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

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

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃₇

#10624784

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

100

C₃³

#11758608

yes

complemented

101

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

333

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃×C₉

—

111

—

113

C₃×C₉

—

111

—

114

C₃×C₉

—

111

—

115

C₃×C₉

—

111

—

116

C₃×C₉

—

111

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

128

C₃²

C₃×#11758608

yes

characteristic

129

C₃²

#15688530

yes

—

130

C₉

—

131

C₉

—

132

C₉

—

333

—

133

C₉

—

333

—

134

C₉

—

111

—

135

C₉

—

111

—

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

#22582556

yes

characteristic

141

C₃

AΣL(1,𝔽₂₇)×C₃₇⋊C₃

yes

characteristic

142

C₃

#22582629

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃×AΣL(1,𝔽₂₇)	C₃₇⋊C₃
hypocenter	C₃₇	#10624784
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×AΣL(1,𝔽₂₇)

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting, prime radical

C₃×C₃₇×AΣL(1,𝔽₂₇)

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

hypercenter

C₃×AΣL(1,𝔽₂₇)

C₃₇⋊C₃

hypocenter

C₃₇

#10624784

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×AΣL(1,𝔽₂₇)

N	H ≅ G/N	type
1. C₃³×#66524	C₃	semidirect
2. #22582568	C₃	semidirect
3. #22582556	C₃	semidirect
4. 3₋¹⁺²⋊#66524	C₃	semidirect
5. H(3,𝔽₃)⋊#66524	C₃	semidirect
6. #22582468	C₃	semidirect
28. C₃³×C₃₇	3₋¹⁺²	semidirect
64. #66524	AΣL(1,𝔽₂₇)	semidirect
94. C₃₇	#10624784	semidirect
100. C₃³	#11758608	semidirect

H ≅ G/N

type

1. C₃³×#66524

C₃

semidirect

2. #22582568

C₃

semidirect

3. #22582556

C₃

semidirect

4. 3₋¹⁺²⋊#66524

C₃

semidirect

5. H(3,𝔽₃)⋊#66524

C₃

semidirect

6. #22582468

C₃

semidirect

28. C₃³×C₃₇

3₋¹⁺²

semidirect

64. #66524

AΣL(1,𝔽₂₇)

semidirect

94. C₃₇

#10624784

semidirect

100. C₃³

#11758608

semidirect

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

subgroup	index
C₃³×#66524	3
#22582568	3
#22582556	3
3₋¹⁺²⋊#66524	3
H(3,𝔽₃)⋊#66524	3
#22582468	3
C₃×C₃₇×AΣL(1,𝔽₂₇)	3
#10624784	37

subgroup

index

C₃³×#66524

C₃×C₃₇×AΣL(1,𝔽₂₇)

Subgroups — #41006815

Conjugacy classes of subgroups

Complements

Maximal subgroups