Group #41006579 — subgroups

Subgroups — #41006579

Order 26973.

456 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22582642	C₃	1	2	yes	maximal, complemented
2	#22582638	C₃	1	4	yes	maximal, complemented
3	#22582637	C₃	1	2	yes	maximal, complemented
4	C₃×#15688518	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#15688515	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582562	C₃	1	2	yes	maximal, complemented
7	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#11758608	C₃²	1	4	yes	complemented
9	C₃×#11758608	C₃²	1	2	yes	complemented
10	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	complemented
11	C₃²×#66524	C₃²	1	1	yes	characteristic, complemented
12	#15688518	—	3	3	no	—
13	#15688515	—	3	3	no	—
14	#15688515	—	3	6	no	—
15	#15688513	—	3	12	no	—
16	#15688512	—	3	6	no	—
17	C₃²⋊#66524	—	3	6	no	—
18	C₃×H(3,𝔽₃)×C₃₇	—	3	6	no	—
19	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
20	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
21	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	2	yes	—
22	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
24	C₃³×C₃₇	—	3	9	no	—
25	C₃³×C₃₇	—	3	1	no	—
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	H(3,𝔽₃)	1	6	yes	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	C₃²×C₃₇⋊C₃	—	3	2	no	—
30	H(3,𝔽₃)×C₃₇	—	3	18	no	—
31	H(3,𝔽₃)×C₃₇	—	3	2	no	—
32	H(3,𝔽₃)×C₃₇	—	3	2	no	—
33	H(3,𝔽₃)×C₃₇	—	3	2	no	—
34	H(3,𝔽₃)×C₃₇	—	3	2	no	—
35	H(3,𝔽₃)×C₃₇	—	3	2	no	—
36	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	3	yes	—
37	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	3	yes	—
38	C₃²⋊(C₃₇⋊C₃)	—	3	4	no	—
39	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
40	#11758608	—	3	4	no	—
41	#11758608	—	3	4	no	—
42	#11758608	—	3	4	no	—
43	#11758608	—	3	4	no	—
44	#11758608	—	3	2	no	—
45	C₃×#66524	—	3	2	no	—
46	C₃×#66524	—	3	4	no	—
47	C₃×#66524	—	3	1	no	—
48	C₃×#66524	—	3	2	no	—
49	C₃×#66524	—	3	1	no	—
50	#10624748	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	6	no	—
52	C₃²×C₃₇	—	3	3	no	—
53	C₃²×C₃₇	—	3	3	no	—
54	C₃²×C₃₇	—	9	1	no	—
55	C₃²×C₃₇	—	9	6	no	—
56	C₃²×C₃₇	—	27	1	no	—
57	C₃²×C₃₇	—	27	1	no	—
58	C₃²×C₃₇	—	27	9	no	—
59	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
60	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	C₃×C₃₇⋊C₃	—	9	2	no	—
63	C₃×C₃₇⋊C₃	—	27	2	no	—
64	#66524	—	9	1	no	—
65	#66524	—	9	4	no	—
66	#66524	—	9	2	no	—
67	H(5,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
68	#6910	—	37	2	no	—
69	#6909	—	37	2	no	—
70	#6909	—	37	4	no	—
71	C₃×#732	—	37	1	no	—
72	C₃×#731	—	37	1	no	—
73	#6893	—	37	2	no	—
74	C₃×C₃₇	—	3	1	no	—
75	C₃×C₃₇	—	3	2	no	—
76	C₃×C₃₇	—	3	1	no	—
77	C₃×C₃₇	—	9	6	no	—
78	C₃×C₃₇	—	9	3	no	—
79	C₃×C₃₇	—	9	3	no	—
80	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
81	C₃₇⋊C₃	—	27	2	no	—
82	C₃×3₋¹⁺²	—	37	4	no	—
83	C₃×3₋¹⁺²	—	37	2	no	—
84	C₃×H(3,𝔽₃)	—	3	6	no	—
85	C₃×H(3,𝔽₃)	—	3	3	no	—
86	C₃×H(3,𝔽₃)	—	3	3	no	—
87	C₃×H(3,𝔽₃)	—	37	2	no	—
88	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	2	yes	—
89	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
90	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
91	C₃²×C₉	—	37	1	no	—
92	#732	—	111	3	no	—
93	#731	—	111	3	no	—
94	#731	—	111	6	no	—
95	AΣL(1,𝔽₂₇)	—	111	12	no	—
96	AΣL(1,𝔽₂₇)	—	111	6	no	—
97	C₃²⋊C₉	—	111	6	no	—
98	C₃₇	#10624748	1	1	yes	characteristic, complemented, Sylow
99	C₃³	—	3	9	no	—
100	C₃³	—	3	1	no	—
101	C₃³	—	3	1	no	—
102	C₃³	—	111	2	no	—
103	C₃³	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
104	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
105	3₋¹⁺²	—	111	4	no	—
106	3₋¹⁺²	—	111	4	no	—
107	3₋¹⁺²	—	111	4	no	—
108	3₋¹⁺²	—	111	4	no	—
109	3₋¹⁺²	—	111	2	no	—
110	H(3,𝔽₃)	—	3	18	no	—
111	H(3,𝔽₃)	—	3	2	no	—
112	H(3,𝔽₃)	—	3	2	no	—
113	H(3,𝔽₃)	—	3	2	no	—
114	H(3,𝔽₃)	—	3	2	no	—
115	H(3,𝔽₃)	—	3	2	no	—
116	H(3,𝔽₃)	—	111	4	no	—
117	H(3,𝔽₃)	—	111	2	no	—
118	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
119	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
120	C₃×C₉	—	111	2	no	—
121	C₃×C₉	—	111	4	no	—
122	C₃×C₉	—	111	1	no	—
123	C₃×C₉	—	111	2	no	—
124	C₃×C₉	—	111	1	no	—
125	C₃²	—	3	6	no	—
126	C₃²	—	3	3	no	—
127	C₃²	—	3	3	no	—
128	C₃²	—	9	1	no	—
129	C₃²	—	9	6	no	—
130	C₃²	—	27	1	no	—
131	C₃²	—	27	1	no	—
132	C₃²	—	27	9	no	—
133	C₃²	—	333	2	no	—
134	C₃²	—	999	2	no	—
135	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
136	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
137	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
138	C₉	—	333	1	no	—
139	C₉	—	333	4	no	—
140	C₉	—	333	2	no	—
141	C₃	—	3	2	no	—
142	C₃	—	3	1	no	—
143	C₃	—	3	1	no	—
144	C₃	—	9	6	no	—
145	C₃	—	9	3	no	—
146	C₃	—	9	3	no	—
147	C₃	—	999	2	no	—
148	C₃	#22582550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582642

C₃

yes

maximal, complemented

#22582638

C₃

yes

maximal, complemented

#22582637

C₃

yes

maximal, complemented

C₃×#15688518

C₃

yes

maximal, characteristic, complemented

C₃×#15688515

C₃

yes

maximal, characteristic, complemented

#22582562

C₃

yes

maximal, complemented

C₃₇×H(5,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

C₃²

yes

complemented

C₃×#11758608

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

complemented

C₃²×#66524

C₃²

yes

characteristic, complemented

—

—

—

—

—

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₃₇

H(3,𝔽₃)

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

#732

—

111

—

#731

—

111

—

#731

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624748

yes

characteristic, complemented, Sylow

C₃³

—

100

C₃³

—

101

C₃³

—

102

C₃³

—

111

—

103

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

104

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

105

3₋¹⁺²

—

111

—

106

3₋¹⁺²

—

111

—

107

3₋¹⁺²

—

111

—

108

3₋¹⁺²

—

111

—

109

3₋¹⁺²

—

111

—

110

H(3,𝔽₃)

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

—

113

H(3,𝔽₃)

—

114

H(3,𝔽₃)

—

115

H(3,𝔽₃)

—

116

H(3,𝔽₃)

—

111

—

117

H(3,𝔽₃)

—

111

—

118

H(3,𝔽₃)

C₃²⋊(C₃₇⋊C₃)

yes

—

119

H(3,𝔽₃)

C₃²⋊(C₃₇⋊C₃)

yes

—

120

C₃×C₉

—

111

—

121

C₃×C₉

—

111

—

122

C₃×C₉

—

111

—

123

C₃×C₉

—

111

—

124

C₃×C₉

—

111

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

333

—

134

C₃²

—

999

—

135

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

136

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

137

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

138

C₉

—

333

—

139

C₉

—

333

—

140

C₉

—

333

—

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

—

146

C₃

—

147

C₃

—

999

—

148

C₃

#22582550

yes

characteristic

	subgroup	G/H
center	C₃	#22582550
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×H(5,𝔽₃)	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624748
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ H(5,𝔽₃)

subgroup

G/H

center

C₃

#22582550

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×H(5,𝔽₃)

C₃

socle

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ H(5,𝔽₃)

N	H ≅ G/N	type
1. #22582642	C₃	semidirect
2. #22582638	C₃	semidirect
3. #22582637	C₃	semidirect
4. C₃×#15688518	C₃	semidirect
5. C₃×#15688515	C₃	semidirect
6. #22582562	C₃	semidirect
7. C₃₇×H(5,𝔽₃)	C₃	semidirect
8. C₃×#11758608	C₃²	semidirect
9. C₃×#11758608	C₃²	semidirect
10. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
11. C₃²×#66524	C₃²	semidirect
67. H(5,𝔽₃)	C₃₇⋊C₃	semidirect
98. C₃₇	#10624748	semidirect

H ≅ G/N

type

1. #22582642

C₃

semidirect

2. #22582638

C₃

semidirect

3. #22582637

C₃

semidirect

4. C₃×#15688518

C₃

semidirect

5. C₃×#15688515

C₃

semidirect

6. #22582562

C₃

semidirect

7. C₃₇×H(5,𝔽₃)

C₃

semidirect

8. C₃×#11758608

C₃²

semidirect

9. C₃×#11758608

C₃²

semidirect

10. C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

semidirect

11. C₃²×#66524

C₃²

semidirect

67. H(5,𝔽₃)

C₃₇⋊C₃

semidirect

98. C₃₇

#10624748

semidirect

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

subgroup	index
#22582642	3
#22582638	3
#22582637	3
C₃×#15688518	3
C₃×#15688515	3
#22582562	3
C₃₇×H(5,𝔽₃)	3
#10624748	37

subgroup

index

C₃₇×H(5,𝔽₃)

Subgroups — #41006579

Conjugacy classes of subgroups

Complements

Maximal subgroups