Group #41008158 — subgroups

Subgroups — C₃×#22582643 #41008158

Order 26973.

1088 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 292 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	C₃²×#11758608	C₃	1	2	yes	maximal, complemented
2	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#22582643	C₃	1	27	yes	maximal, direct product factor
4	C₃×#15688516	C₃	1	9	yes	maximal, complemented
5	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	—	3	1	no	—
7	C₃×#11758608	—	3	2	no	—
8	C₃×#11758608	—	3	2	no	—
9	C₃×#11758608	—	3	12	no	—
10	C₃×#11758608	C₃²	1	6	yes	complemented
11	C₃×#11758608	C₃²	1	18	yes	complemented
12	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
14	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
15	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	9	yes	—
16	C₃²×#66524	—	3	2	no	—
17	#15688516	C₃²	1	81	yes	complemented
18	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
20	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
21	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
22	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	complemented
23	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
24	C₃³×C₃₇	—	9	3	no	—
25	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
26	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃³×C₃₇	H(3,𝔽₃)	1	6	yes	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	C₃³×C₃₇	C₃³	1	3	yes	—
30	C₃²×C₃₇⋊C₃	—	3	9	no	—
31	C₃²×C₃₇⋊C₃	—	3	3	no	—
32	C₃²×C₃₇⋊C₃	—	9	9	no	—
33	3₋¹⁺²×C₃₇	—	3	18	no	—
34	3₋¹⁺²×C₃₇	—	3	9	no	—
35	C₃×C₉×C₃₇	—	3	6	no	—
36	C₃×C₉×C₃₇	—	3	3	no	—
37	C₃×C₉×C₃₇	C₃³	1	9	yes	complemented
38	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
39	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
40	C₃²⋊(C₃₇⋊C₃)	C₃³	1	9	yes	—
41	#11758608	—	3	18	no	—
42	#11758608	—	3	12	no	—
43	#11758608	—	3	18	no	—
44	#11758608	C₃³	1	18	yes	complemented
45	C₃×#66524	—	3	18	no	—
46	C₃×#66524	—	3	6	no	—
47	C₃×#6910	—	37	1	no	maximal, Sylow
48	C₃²×C₃₇	—	3	3	no	—
49	C₃²×C₃₇	—	3	6	no	—
50	C₃²×C₃₇	—	3	3	no	—
51	C₃²×C₃₇	—	9	6	no	—
52	C₃²×C₃₇	—	9	3	no	—
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	6	yes	—
54	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
55	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
58	C₃×C₃₇⋊C₃	—	3	9	no	—
59	C₃×C₃₇⋊C₃	—	9	27	no	—
60	C₃×C₃₇⋊C₃	—	9	9	no	—
61	C₉×C₃₇	—	3	9	no	—
62	#66524	—	3	18	no	—
63	C₃²×3₋¹⁺²	—	37	2	no	—
64	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
65	C₃²×H(3,𝔽₃)	—	37	1	no	—
66	#6910	—	37	27	no	—
67	C₃×#731	—	37	9	no	—
68	C₃×C₃₇	—	3	3	no	—
69	C₃×C₃₇	—	3	6	no	—
70	C₃×C₃₇	—	3	2	no	—
71	C₃×C₃₇	—	3	1	no	—
72	C₃×C₃₇	#6910	1	3	yes	complemented
73	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃₇⋊C₃	—	9	9	no	—
75	C₃⁴	—	111	1	no	—
76	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
77	C₃×3₋¹⁺²	—	3	6	no	—
78	C₃×3₋¹⁺²	—	3	2	no	—
79	C₃×3₋¹⁺²	—	3	1	no	—
80	C₃×3₋¹⁺²	—	37	6	no	—
81	C₃×3₋¹⁺²	—	37	18	no	—
82	C₃×3₋¹⁺²	—	111	12	no	—
83	C₃×3₋¹⁺²	—	111	2	no	—
84	C₃×3₋¹⁺²	—	111	2	no	—
85	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	complemented
86	C₃×H(3,𝔽₃)	—	37	3	no	—
87	C₃×H(3,𝔽₃)	—	37	9	no	—
88	C₃×H(3,𝔽₃)	—	111	6	no	—
89	C₃×H(3,𝔽₃)	—	111	2	no	—
90	C₃²×C₉	—	111	2	no	—
91	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
92	#731	—	37	81	no	—
93	C₃₇	C₃×#6910	1	1	yes	characteristic, complemented, Sylow
94	C₃³	—	9	3	no	—
95	C₃³	—	333	9	no	—
96	C₃³	—	111	9	no	—
97	C₃³	—	111	3	no	—
98	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
99	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
100	C₃³	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
101	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
102	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
103	3₋¹⁺²	—	3	18	no	—
104	3₋¹⁺²	—	3	9	no	—
105	3₋¹⁺²	—	37	18	no	—
106	3₋¹⁺²	—	111	12	no	—
107	3₋¹⁺²	—	111	18	no	—
108	3₋¹⁺²	—	111	18	no	—
109	H(3,𝔽₃)	—	37	9	no	—
110	H(3,𝔽₃)	—	111	6	no	—
111	H(3,𝔽₃)	—	111	18	no	—
112	C₃×C₉	—	3	6	no	—
113	C₃×C₉	—	3	3	no	—
114	C₃×C₉	—	111	18	no	—
115	C₃×C₉	—	111	6	no	—
116	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	complemented
117	C₃²	—	3	3	no	—
118	C₃²	—	3	6	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	9	6	no	—
121	C₃²	—	9	3	no	—
122	C₃²	—	333	27	no	—
123	C₃²	—	333	9	no	—
124	C₃²	—	111	9	no	—
125	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
126	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
127	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
128	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
129	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
130	C₉	—	3	9	no	—
131	C₉	—	111	18	no	—
132	C₃	—	3	3	no	—
133	C₃	—	3	6	no	—
134	C₃	—	3	2	no	—
135	C₃	—	3	1	no	—
136	C₃	—	333	9	no	—
137	C₃	#22582643	1	3	yes	direct product factor
138	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

#22582643

C₃

yes

maximal, direct product factor

C₃×#15688516

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

—

—

—

C₃²

yes

complemented

C₃×#11758608

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃²×#66524

—

#15688516

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

complemented

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

C₃³

yes

—

—

—

—

C₃³

yes

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

—

#6910

—

C₃×#731

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6910

yes

complemented

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

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

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

complemented

#731

—

C₃₇

C₃×#6910

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

100

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

101

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

102

C₃³

C₃²×C₃₇⋊C₃

yes

—

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

H(3,𝔽₃)

—

110

H(3,𝔽₃)

—

111

—

111

H(3,𝔽₃)

—

111

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

111

—

115

C₃×C₉

—

111

—

116

C₃×C₉

C₃²×C₃₇⋊C₃

yes

complemented

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

333

—

123

C₃²

—

333

—

124

C₃²

—

111

—

125

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

126

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

127

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

128

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

129

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

130

C₉

—

131

C₉

—

111

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

333

—

137

C₃

#22582643

yes

direct product factor

138

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6910
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

hypocenter

C₃₇

C₃×#6910

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. C₃²×#11758608	C₃	semidirect
3. #22582643	C₃	direct (N × H)
4. C₃×#15688516	C₃	semidirect
5. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
10. C₃×#11758608	C₃²	semidirect
11. C₃×#11758608	C₃²	semidirect
17. #15688516	C₃²	semidirect
22. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
23. C₃²×C₉×C₃₇	C₃²	semidirect
37. C₃×C₉×C₃₇	C₃³	semidirect
44. #11758608	C₃³	semidirect
64. C₃²×3₋¹⁺²	C₃₇⋊C₃	semidirect
72. C₃×C₃₇	#6910	semidirect
85. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
91. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
93. C₃₇	C₃×#6910	semidirect
116. C₃×C₉	C₃²×C₃₇⋊C₃	semidirect
137. C₃	#22582643	direct (N × H)

H ≅ G/N

type

1. C₃²×#11758608

C₃

semidirect

3. #22582643

C₃

direct (N × H)

4. C₃×#15688516

C₃

semidirect

5. C₃²×3₋¹⁺²×C₃₇

C₃

semidirect

10. C₃×#11758608

C₃²

semidirect

11. C₃×#11758608

C₃²

semidirect

17. #15688516

C₃²

semidirect

22. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

23. C₃²×C₉×C₃₇

C₃²

semidirect

37. C₃×C₉×C₃₇

C₃³

semidirect

44. #11758608

C₃³

semidirect

64. C₃²×3₋¹⁺²

C₃₇⋊C₃

semidirect

72. C₃×C₃₇

#6910

semidirect

85. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

91. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

93. C₃₇

C₃×#6910

semidirect

116. C₃×C₉

C₃²×C₃₇⋊C₃

semidirect

137. C₃

#22582643

direct (N × H)

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

subgroup	index
C₃²×#11758608	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
#22582643	3
C₃×#15688516	3
C₃²×3₋¹⁺²×C₃₇	3
C₃×#6910	37

subgroup

index

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃²×3₋¹⁺²×C₃₇

Subgroups — C₃×#22582643 #41008158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582643 #41008158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582643 #41008158