Group #41008227 — subgroups

Subgroups — #41008227

Order 26973.

948 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 268 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	#22582680	C₃	1	3	yes	maximal, complemented
2	C₃×#15688532	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582658	C₃	1	9	yes	maximal, complemented
4	#22582657	C₃	1	9	yes	maximal, complemented
5	#22582652	C₃	1	2	yes	maximal, complemented
6	#22582637	C₃	1	3	yes	maximal, complemented
7	#22582631	C₃	1	9	yes	maximal, complemented
8	C₃×#15688515	C₃	1	3	yes	maximal, complemented
9	C₃₇×#6911	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688532	—	3	9	no	—
11	#15688530	—	3	18	no	—
12	#15688530	C₃²	1	27	yes	complemented
13	C₃×#11758608	—	3	9	no	—
14	C₃×#11758608	C₃²	1	6	yes	—
15	C₃×#11758608	C₃²	1	1	yes	characteristic
16	C₃×#11758605	C₃²	1	3	yes	complemented
17	C₃×#11758606	C₃²	1	3	yes	complemented
18	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
19	C₃²×#66524	C₃²	1	1	yes	characteristic
20	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	complemented
21	#15688521	C₃²	1	18	yes	complemented
22	#15688515	C₃²	1	9	yes	complemented
23	#15688515	C₃²	1	9	yes	complemented
24	#15688515	C₃²	1	9	yes	complemented
25	#15688512	C₃²	1	27	yes	complemented
26	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
27	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
28	C₃₇×#732	C₃²	1	9	yes	—
29	C₃³×C₃₇	—	3	3	no	—
30	C₃³×C₃₇	C₃³	1	1	yes	characteristic
31	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
32	3₋¹⁺²×C₃₇	—	3	3	no	—
33	H(3,𝔽₃)×C₃₇	—	3	6	no	—
34	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
35	C₃×C₉×C₃₇	C₃³	1	3	yes	—
36	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
37	#11758608	—	3	6	no	—
38	#11758608	—	3	6	no	—
39	#11758608	—	3	54	no	—
40	#11758608	H(3,𝔽₃)	1	9	yes	—
41	#11758608	C₃³	1	18	yes	—
42	C₃×#66524	—	3	3	no	—
43	C₃×#66524	—	3	6	no	—
44	C₃×#66524	—	3	27	no	—
45	C₃×#66524	C₃³	1	1	yes	characteristic
46	C₃×#66524	C₃³	1	1	yes	characteristic
47	C₃×#66524	C₃³	1	1	yes	characteristic
48	C₃×#66524	C₃³	1	3	yes	—
49	#11758606	—	3	9	no	—
50	#11758605	—	3	9	no	—
51	C₉×C₃₇⋊C₃	—	3	9	no	—
52	#10624935	—	37	1	no	maximal, Sylow
53	C₃²×C₃₇	—	3	9	no	—
54	C₃²×C₃₇	—	9	1	no	—
55	C₃²×C₃₇	—	27	3	no	—
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
57	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
58	C₃×C₃₇⋊C₃	—	3	3	no	—
59	C₃×C₃₇⋊C₃	—	9	3	no	—
60	C₉×C₃₇	—	3	3	no	—
61	#66524	—	3	1	no	—
62	#66524	—	3	1	no	—
63	#66524	—	3	1	no	—
64	#66524	—	3	18	no	—
65	#66524	C₃×H(3,𝔽₃)	1	9	yes	—
66	3₋¹⁺⁴	—	37	3	no	—
67	C₃×#737	—	37	1	no	—
68	#6913	—	37	9	no	—
69	#6913	—	37	9	no	—
70	#6912	—	37	2	no	—
71	#6911	C₃₇⋊C₃	1	1	yes	characteristic, complemented
72	#6909	—	37	3	no	—
73	#6908	—	37	9	no	—
74	C₃×#731	—	37	3	no	—
75	C₃×C₃₇	—	3	1	no	—
76	C₃×C₃₇	—	3	3	no	—
77	C₃×C₃₇	—	9	9	no	—
78	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃₇⋊C₃	—	9	3	no	—
80	#737	—	37	27	no	—
81	#737	—	111	18	no	—
82	#737	—	111	9	no	—
83	C₃×3₋¹⁺²	—	37	1	no	—
84	C₃×3₋¹⁺²	—	37	3	no	—
85	C₃×3₋¹⁺²	—	37	3	no	—
86	C₃×3₋¹⁺²	—	37	6	no	—
87	C₃×3₋¹⁺²	—	111	9	no	—
88	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
89	C₃×H(3,𝔽₃)	—	37	1	no	—
90	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
91	C₃²×C₉	—	37	1	no	—
92	C₃²×C₉	—	37	3	no	—
93	#733	—	37	18	no	—
94	#732	C₃×C₃₇⋊C₃	1	9	yes	—
95	#731	—	37	9	no	—
96	#731	—	37	9	no	—
97	#731	—	37	9	no	—
98	AΣL(1,𝔽₂₇)	—	37	27	no	—
99	C₃₇	#10624935	1	1	yes	characteristic, complemented, Sylow
100	C₃³	—	3	3	no	—
101	C₃³	—	37	3	no	—
102	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
103	3₋¹⁺²	—	3	3	no	—
104	3₋¹⁺²	—	37	9	no	—
105	3₋¹⁺²	—	37	18	no	—
106	3₋¹⁺²	—	111	9	no	—
107	3₋¹⁺²	—	111	9	no	—
108	3₋¹⁺²	—	111	54	no	—
109	3₋¹⁺²	—	111	6	no	—
110	3₋¹⁺²	—	111	6	no	—
111	H(3,𝔽₃)	—	3	6	no	—
112	H(3,𝔽₃)	—	111	3	no	—
113	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	—
114	C₃×C₉	—	37	3	no	—
115	C₃×C₉	—	37	1	no	—
116	C₃×C₉	—	37	1	no	—
117	C₃×C₉	—	37	1	no	—
118	C₃×C₉	—	111	9	no	—
119	C₃×C₉	—	111	27	no	—
120	C₃×C₉	—	111	3	no	—
121	C₃×C₉	—	111	6	no	—
122	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
123	C₃²	—	3	9	no	—
124	C₃²	—	9	1	no	—
125	C₃²	—	27	3	no	—
126	C₃²	—	333	3	no	—
127	C₃²	—	111	3	no	—
128	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
129	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
130	C₉	—	3	3	no	—
131	C₉	—	37	9	no	—
132	C₉	—	111	1	no	—
133	C₉	—	111	1	no	—
134	C₉	—	111	1	no	—
135	C₉	—	111	18	no	—
136	C₃	—	3	3	no	—
137	C₃	—	3	1	no	—
138	C₃	—	9	9	no	—
139	C₃	—	333	3	no	—
140	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582680

C₃

yes

maximal, complemented

C₃×#15688532

C₃

yes

maximal, characteristic, complemented

#22582658

C₃

yes

maximal, complemented

#22582657

C₃

yes

maximal, complemented

#22582652

C₃

yes

maximal, complemented

#22582637

C₃

yes

maximal, complemented

#22582631

C₃

yes

maximal, complemented

C₃×#15688515

C₃

yes

maximal, complemented

C₃₇×#6911

C₃

yes

maximal, characteristic, complemented

#15688532

—

#15688530

—

#15688530

C₃²

yes

complemented

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

characteristic

C₃×#11758605

C₃²

yes

complemented

C₃×#11758606

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

complemented

#15688521

C₃²

yes

complemented

#15688515

C₃²

yes

complemented

#15688515

C₃²

yes

complemented

#15688515

C₃²

yes

complemented

#15688512

C₃²

yes

complemented

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃₇×#732

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

yes

—

#11758608

C₃³

yes

—

—

—

—

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624935

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

—

—

—

—

C₃×H(3,𝔽₃)

yes

—

3₋¹⁺⁴

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

#6909

—

#6908

—

C₃×#731

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

#737

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

#733

—

#732

C₃×C₃₇⋊C₃

yes

—

#731

—

#731

—

#731

—

AΣL(1,𝔽₂₇)

—

C₃₇

#10624935

yes

characteristic, complemented, Sylow

100

C₃³

—

101

C₃³

—

102

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

111

—

107

3₋¹⁺²

—

111

—

108

3₋¹⁺²

—

111

—

109

3₋¹⁺²

—

111

—

110

3₋¹⁺²

—

111

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

—

111

—

113

H(3,𝔽₃)

C₃²×C₃₇⋊C₃

yes

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

111

—

119

C₃×C₉

—

111

—

120

C₃×C₉

—

111

—

121

C₃×C₉

—

111

—

122

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

333

—

127

C₃²

—

111

—

128

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

129

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

130

C₉

—

131

C₉

—

132

C₉

—

111

—

133

C₉

—

111

—

134

C₉

—

111

—

135

C₉

—

111

—

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

—

333

—

140

C₃

C₃×H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃₇×#6911	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	#6911	C₃₇⋊C₃
hypocenter	C₃₇	#10624935
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

subgroup

G/H

center

C₃

C₃×H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃₇×#6911

C₃

socle

C₃×C₃₇

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ #6911

N	H ≅ G/N	type
1. #22582680	C₃	semidirect
2. C₃×#15688532	C₃	semidirect
3. #22582658	C₃	semidirect
4. #22582657	C₃	semidirect
5. #22582652	C₃	semidirect
6. #22582637	C₃	semidirect
7. #22582631	C₃	semidirect
8. C₃×#15688515	C₃	semidirect
9. C₃₇×#6911	C₃	semidirect
12. #15688530	C₃²	semidirect
16. C₃×#11758605	C₃²	semidirect
17. C₃×#11758606	C₃²	semidirect
20. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
21. #15688521	C₃²	semidirect
22. #15688515	C₃²	semidirect
23. #15688515	C₃²	semidirect
24. #15688515	C₃²	semidirect
25. #15688512	C₃²	semidirect
71. #6911	C₃₇⋊C₃	semidirect
99. C₃₇	#10624935	semidirect

H ≅ G/N

type

1. #22582680

C₃

semidirect

2. C₃×#15688532

C₃

semidirect

3. #22582658

C₃

semidirect

4. #22582657

C₃

semidirect

5. #22582652

C₃

semidirect

6. #22582637

C₃

semidirect

7. #22582631

C₃

semidirect

8. C₃×#15688515

C₃

semidirect

9. C₃₇×#6911

C₃

semidirect

12. #15688530

C₃²

semidirect

16. C₃×#11758605

C₃²

semidirect

17. C₃×#11758606

C₃²

semidirect

20. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

21. #15688521

C₃²

semidirect

22. #15688515

C₃²

semidirect

23. #15688515

C₃²

semidirect

24. #15688515

C₃²

semidirect

25. #15688512

C₃²

semidirect

71. #6911

C₃₇⋊C₃

semidirect

99. C₃₇

#10624935

semidirect

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

subgroup	index
#22582680	3
C₃×#15688532	3
#22582658	3
#22582657	3
#22582652	3
#22582637	3
#22582631	3
C₃×#15688515	3
C₃₇×#6911	3
#10624935	37

subgroup

index

C₃₇×#6911

Subgroups — #41008227

Conjugacy classes of subgroups

Complements

Maximal subgroups