Group #41005718 — subgroups

Subgroups — #41005718

Order 26973.

452 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 52 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	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582552	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582551	C₃	1	2	yes	maximal, complemented
4	#22582446	C₃	1	2	yes	maximal, complemented
5	#22582445	C₃	1	2	yes	maximal, complemented
6	#22582444	C₃	1	4	yes	maximal, complemented
7	C₃₇×#6857	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	—	3	1	no	—
9	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
10	C₃×#11758608	C₃²	1	4	yes	complemented
11	C₃×#11758606	—	3	1	no	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
14	H(3,𝔽₃)×C₃₇⋊C₃	—	3	9	no	—
15	C₃²⋊#66524	C₃²	1	2	yes	complemented
16	#15688502	—	3	4	no	—
17	#15688502	—	3	2	no	—
18	#15688502	—	3	2	no	—
19	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	complemented
20	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic, complemented
22	C₃³×C₃₇	—	3	1	no	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	3	2	no	—
25	C₃²×C₃₇⋊C₃	—	3	3	no	—
26	C₃²×C₃₇⋊C₃	—	3	9	no	—
27	C₃²×C₃₇⋊C₃	—	9	2	no	—
28	C₃²×C₃₇⋊C₃	—	9	1	no	—
29	C₃²×C₃₇⋊C₃	—	9	6	no	—
30	C₃²×C₃₇⋊C₃	—	9	3	no	—
31	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
32	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
33	3₋¹⁺²×C₃₇	—	9	2	no	—
34	H(3,𝔽₃)×C₃₇	—	3	3	no	—
35	C₃×C₉×C₃₇	—	3	1	no	—
36	C₃×C₉×C₃₇	—	3	2	no	—
37	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
38	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
39	#11758608	—	9	4	no	—
40	C₃×#66524	—	3	4	no	—
41	C₃×#66524	—	3	2	no	—
42	#11758606	—	9	3	no	—
43	#10624630	—	37	1	no	maximal, Sylow
44	C₃²×C₃₇	—	3	1	no	—
45	C₃²×C₃₇	—	3	3	no	—
46	C₃²×C₃₇	—	3	2	no	—
47	C₃²×C₃₇	—	3	1	no	—
48	C₃²×C₃₇	—	9	2	no	—
49	C₃²×C₃₇	—	9	1	no	—
50	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₃₇⋊C₃	—	3	2	no	—
52	C₃×C₃₇⋊C₃	—	3	1	no	—
53	C₃×C₃₇⋊C₃	—	3	2	no	—
54	C₃×C₃₇⋊C₃	—	3	6	no	—
55	C₃×C₃₇⋊C₃	—	3	4	no	—
56	C₃×C₃₇⋊C₃	—	3	2	no	—
57	C₃×C₃₇⋊C₃	—	9	9	no	—
58	C₃×C₃₇⋊C₃	—	9	27	no	—
59	C₃×C₃₇⋊C₃	—	9	4	no	—
60	C₃×C₃₇⋊C₃	—	9	2	no	—
61	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
62	C₉×C₃₇	—	9	1	no	—
63	C₉×C₃₇	—	9	2	no	—
64	#66524	—	9	2	no	—
65	#66524	—	9	4	no	—
66	C₃²×H(3,𝔽₃)	—	37	1	no	—
67	#6891	—	37	1	no	—
68	#6891	—	37	2	no	—
69	#6857	—	37	4	no	—
70	#6857	—	37	2	no	—
71	#6857	—	37	2	no	—
72	#6857	C₃₇⋊C₃	1	1	yes	characteristic, complemented
73	C₃×C₃₇	—	9	1	no	—
74	C₃×C₃₇	—	9	3	no	—
75	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
76	C₃×C₃₇	#6909	1	2	yes	—
77	C₃×C₃₇	#6910	1	1	yes	characteristic
78	C₃₇⋊C₃	—	3	3	no	—
79	C₃₇⋊C₃	—	9	2	no	—
80	C₃₇⋊C₃	—	9	6	no	—
81	C₃⁴	—	37	1	no	—
82	C₃⁴	—	111	1	no	—
83	C₃×3₋¹⁺²	—	37	4	no	—
84	C₃×3₋¹⁺²	—	111	1	no	—
85	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	complemented
86	C₃×H(3,𝔽₃)	—	37	2	no	—
87	C₃×H(3,𝔽₃)	—	111	9	no	—
88	C₃×H(3,𝔽₃)	—	111	2	no	—
89	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
90	C₃²⋊C₉	—	37	2	no	—
91	C₃²⋊C₉	—	111	2	no	—
92	C₃²⋊C₉	—	111	4	no	—
93	C₃²⋊C₉	—	111	2	no	—
94	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
95	C₃₇	#10624630	1	1	yes	characteristic, complemented, Sylow
96	C₃³	—	3	1	no	—
97	C₃³	—	37	1	no	—
98	C₃³	—	37	2	no	—
99	C₃³	—	333	2	no	—
100	C₃³	—	333	1	no	—
101	C₃³	—	333	6	no	—
102	C₃³	—	333	3	no	—
103	C₃³	—	111	3	no	—
104	C₃³	—	111	9	no	—
105	C₃³	—	111	2	no	—
106	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
107	3₋¹⁺²	—	9	2	no	—
108	3₋¹⁺²	—	333	3	no	—
109	3₋¹⁺²	—	333	4	no	—
110	H(3,𝔽₃)	—	3	3	no	—
111	H(3,𝔽₃)	—	111	6	no	—
112	H(3,𝔽₃)	—	111	18	no	—
113	C₃×C₉	—	3	2	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	111	2	no	—
116	C₃×C₉	—	111	4	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	3	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	9	2	no	—
122	C₃²	—	9	1	no	—
123	C₃²	—	37	3	no	—
124	C₃²	—	333	9	no	—
125	C₃²	—	333	27	no	—
126	C₃²	—	333	4	no	—
127	C₃²	—	333	2	no	—
128	C₃²	—	111	2	no	—
129	C₃²	—	111	6	no	—
130	C₃²	—	111	4	no	—
131	C₃²	—	111	2	no	—
132	C₃²	—	111	2	no	—
133	C₃²	—	111	1	no	—
134	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
135	C₉	—	9	1	no	—
136	C₉	—	9	2	no	—
137	C₉	—	333	2	no	—
138	C₉	—	333	4	no	—
139	C₃	—	9	1	no	—
140	C₃	—	9	3	no	—
141	C₃	—	333	6	no	—
142	C₃	—	333	2	no	—
143	C₃	—	111	3	no	—
144	C₃	AΣL(1,𝔽₂₇)×C₃₇⋊C₃	1	1	yes	characteristic
145	C₃	#22582635	1	2	yes	—
146	C₃	#22582640	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×H(3,𝔽₃)×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#22582552

C₃

yes

maximal, characteristic, complemented

#22582551

C₃

yes

maximal, complemented

#22582446

C₃

yes

maximal, complemented

#22582445

C₃

yes

maximal, complemented

#22582444

C₃

yes

maximal, complemented

C₃₇×#6857

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

complemented

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

H(3,𝔽₃)×C₃₇⋊C₃

—

C₃²⋊#66524

C₃²

yes

—

—

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic, complemented

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

AΣL(1,𝔽₂₇)

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×H(3,𝔽₃)

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₇

#6909

yes

—

C₃×C₃₇

#6910

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic, complemented

C₃₇

#10624630

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

100

C₃³

—

333

—

101

C₃³

—

333

—

102

C₃³

—

333

—

103

C₃³

—

111

—

104

C₃³

—

111

—

105

C₃³

—

111

—

106

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

107

3₋¹⁺²

—

108

3₋¹⁺²

—

333

—

109

3₋¹⁺²

—

333

—

110

H(3,𝔽₃)

—

111

H(3,𝔽₃)

—

111

—

112

H(3,𝔽₃)

—

111

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

111

—

116

C₃×C₉

—

111

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

333

—

125

C₃²

—

333

—

126

C₃²

—

333

—

127

C₃²

—

333

—

128

C₃²

—

111

—

129

C₃²

—

111

—

130

C₃²

—

111

—

131

C₃²

—

111

—

132

C₃²

—

111

—

133

C₃²

—

111

—

134

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

135

C₉

—

136

C₉

—

137

C₉

—

333

—

138

C₉

—

333

—

139

C₃

—

140

C₃

—

141

C₃

—

333

—

142

C₃

—

333

—

143

C₃

—

111

—

144

C₃

AΣL(1,𝔽₂₇)×C₃₇⋊C₃

yes

characteristic

145

C₃

#22582635

yes

—

146

C₃

#22582640

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6857	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃
hypercenter	#6857	C₃₇⋊C₃
hypocenter	C₃₇	#10624630
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6857

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6857

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇⋊C₃

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6857

N	H ≅ G/N	type
2. #22582552	C₃	semidirect
3. #22582551	C₃	semidirect
4. #22582446	C₃	semidirect
5. #22582445	C₃	semidirect
6. #22582444	C₃	semidirect
7. C₃₇×#6857	C₃	semidirect
10. C₃×#11758608	C₃²	semidirect
15. C₃²⋊#66524	C₃²	semidirect
19. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
21. C₃₇×C₃²⋊C₉	C₃²	semidirect
72. #6857	C₃₇⋊C₃	semidirect
85. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
94. C₃²⋊C₉	C₃×C₃₇⋊C₃	semidirect
95. C₃₇	#10624630	semidirect

H ≅ G/N

type

2. #22582552

C₃

semidirect

3. #22582551

C₃

semidirect

4. #22582446

C₃

semidirect

5. #22582445

C₃

semidirect

6. #22582444

C₃

semidirect

7. C₃₇×#6857

C₃

semidirect

10. C₃×#11758608

C₃²

semidirect

15. C₃²⋊#66524

C₃²

semidirect

19. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

21. C₃₇×C₃²⋊C₉

C₃²

semidirect

72. #6857

C₃₇⋊C₃

semidirect

85. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

94. C₃²⋊C₉

C₃×C₃₇⋊C₃

semidirect

95. C₃₇

#10624630

semidirect

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

subgroup	index
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
#22582552	3
#22582551	3
#22582446	3
#22582445	3
#22582444	3
C₃₇×#6857	3
#10624630	37

subgroup

index

C₃×H(3,𝔽₃)×C₃₇⋊C₃

C₃₇×#6857

Subgroups — #41005718

Conjugacy classes of subgroups

Complements

Maximal subgroups