Group #41006702 — subgroups

Subgroups — #41006702

Order 26973.

1940 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 1940 conjugacy classes of subgroups — too many to draw. Only its 76 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	C₃⁴×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582552	C₃	1	3	yes	maximal, complemented
3	#22582468	C₃	1	2	yes	maximal, complemented
4	#22582467	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6866	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	—	3	3	no	—
7	C₃³×C₃₇⋊C₃	—	3	3	no	—
8	C₃³×C₃₇⋊C₃	—	3	6	no	—
9	C₃³×C₃₇⋊C₃	—	3	27	no	—
10	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
11	C₃³×C₃₇⋊C₃	C₃²	1	2	yes	—
12	C₃×#11758606	—	3	3	no	—
13	C₃²⋊#66524	C₃²	1	6	yes	complemented
14	#15688502	—	3	6	no	—
15	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
16	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	complemented
17	C₃³×C₃₇	—	3	9	no	—
18	C₃³×C₃₇	—	3	1	no	—
19	C₃³×C₃₇	—	3	2	no	—
20	C₃³×C₃₇	3₋¹⁺²	1	3	yes	complemented
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	C₃²×C₃₇⋊C₃	—	3	81	no	—
23	C₃²×C₃₇⋊C₃	—	3	162	no	—
24	C₃²×C₃₇⋊C₃	—	3	27	no	—
25	C₃²×C₃₇⋊C₃	—	3	54	no	—
26	C₃²×C₃₇⋊C₃	—	3	2	no	—
27	C₃²×C₃₇⋊C₃	—	3	2	no	—
28	C₃²×C₃₇⋊C₃	—	3	2	no	—
29	C₃²×C₃₇⋊C₃	—	3	1	no	—
30	C₃²×C₃₇⋊C₃	—	3	2	no	—
31	C₃²×C₃₇⋊C₃	—	3	27	no	—
32	C₃²×C₃₇⋊C₃	—	3	9	no	—
33	C₃²×C₃₇⋊C₃	—	3	18	no	—
34	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
35	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	6	yes	complemented
36	C₃×C₉×C₃₇	—	3	3	no	—
37	C₃×#66524	—	3	6	no	—
38	#11758606	—	9	9	no	—
39	#10624765	—	37	1	no	maximal, Sylow
40	C₃²×C₃₇	—	3	9	no	—
41	C₃²×C₃₇	—	3	18	no	—
42	C₃²×C₃₇	—	3	3	no	—
43	C₃²×C₃₇	—	3	9	no	—
44	C₃²×C₃₇	—	3	3	no	—
45	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
46	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
47	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₃×C₃₇⋊C₃	—	3	18	no	—
49	C₃×C₃₇⋊C₃	—	3	18	no	—
50	C₃×C₃₇⋊C₃	—	3	243	no	—
51	C₃×C₃₇⋊C₃	—	3	18	no	—
52	C₃×C₃₇⋊C₃	—	3	9	no	—
53	C₃×C₃₇⋊C₃	—	3	3	no	—
54	C₃×C₃₇⋊C₃	—	3	6	no	—
55	C₃×C₃₇⋊C₃	—	3	18	no	—
56	C₃×C₃₇⋊C₃	—	3	6	no	—
57	C₃×C₃₇⋊C₃	—	3	18	no	—
58	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	9	yes	—
59	C₉×C₃₇	—	9	3	no	—
60	#66524	—	9	6	no	—
61	C₃⁵	—	37	1	no	—
62	#6891	—	37	3	no	—
63	#6866	—	37	2	no	—
64	#6866	—	37	6	no	—
65	#6866	C₃₇⋊C₃	1	1	yes	characteristic, complemented
66	C₃×C₃₇	—	3	9	no	—
67	C₃×C₃₇	—	3	2	no	—
68	C₃×C₃₇	—	3	1	no	—
69	C₃×C₃₇	#6891	1	1	yes	characteristic
70	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
71	C₃₇⋊C₃	—	3	18	no	—
72	C₃₇⋊C₃	—	3	9	no	—
73	C₃⁴	—	37	2	no	—
74	C₃⁴	—	37	1	no	—
75	C₃⁴	—	111	3	no	—
76	C₃⁴	—	111	6	no	—
77	C₃⁴	—	111	27	no	—
78	C₃⁴	—	111	3	no	—
79	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	C₃×3₋¹⁺²	—	111	3	no	—
81	C₃²⋊C₉	—	37	6	no	—
82	C₃²⋊C₉	—	111	6	no	—
83	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
84	C₃₇	#10624765	1	1	yes	characteristic, complemented, Sylow
85	C₃³	—	3	9	no	—
86	C₃³	—	3	1	no	—
87	C₃³	—	3	2	no	—
88	C₃³	—	37	3	no	—
89	C₃³	—	37	6	no	—
90	C₃³	—	111	9	no	—
91	C₃³	—	111	27	no	—
92	C₃³	—	111	18	no	—
93	C₃³	—	111	81	no	—
94	C₃³	—	111	162	no	—
95	C₃³	—	111	27	no	—
96	C₃³	—	111	54	no	—
97	C₃³	—	111	2	no	—
98	C₃³	—	111	2	no	—
99	C₃³	—	111	2	no	—
100	C₃³	—	111	2	no	—
101	C₃³	—	111	1	no	—
102	C₃³	#11758606	1	3	yes	complemented
103	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
104	3₋¹⁺²	—	333	9	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	111	6	no	—
107	C₃²	—	3	9	no	—
108	C₃²	—	3	18	no	—
109	C₃²	—	3	3	no	—
110	C₃²	—	3	3	no	—
111	C₃²	—	3	9	no	—
112	C₃²	—	37	9	no	—
113	C₃²	—	111	18	no	—
114	C₃²	—	111	18	no	—
115	C₃²	—	111	18	no	—
116	C₃²	—	111	9	no	—
117	C₃²	—	111	18	no	—
118	C₃²	—	111	243	no	—
119	C₃²	—	111	3	no	—
120	C₃²	—	111	6	no	—
121	C₃²	—	111	6	no	—
122	C₃²	—	111	18	no	—
123	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
124	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
125	C₃²	C₃×#11758606	1	1	yes	characteristic
126	C₉	—	9	3	no	—
127	C₉	—	333	6	no	—
128	C₃	—	3	2	no	—
129	C₃	—	3	1	no	—
130	C₃	—	3	9	no	—
131	C₃	—	111	9	no	—
132	C₃	—	111	18	no	—
133	C₃	#22582552	1	1	yes	characteristic
134	C₃	AΣL(1,𝔽₂₇)×C₃₇⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#22582552

C₃

yes

maximal, complemented

#22582468

C₃

yes

maximal, complemented

#22582467

C₃

yes

maximal, complemented

C₃₇×#6866

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃×#11758606

—

C₃²⋊#66524

C₃²

yes

complemented

#15688502

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

3₋¹⁺²

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

162

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

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

3₋¹⁺²

yes

complemented

C₃×C₉×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

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

243

—

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

—

#66524

—

C₃⁵

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃₇

#10624765

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

162

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

100

C₃³

—

111

—

101

C₃³

—

111

—

102

C₃³

#11758606

yes

complemented

103

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

104

3₋¹⁺²

—

333

—

105

C₃×C₉

—

106

C₃×C₉

—

111

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

111

—

114

C₃²

—

111

—

115

C₃²

—

111

—

116

C₃²

—

111

—

117

C₃²

—

111

—

118

C₃²

—

111

243

—

119

C₃²

—

111

—

120

C₃²

—

111

—

121

C₃²

—

111

—

122

C₃²

—

111

—

123

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

124

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

125

C₃²

C₃×#11758606

yes

characteristic

126

C₉

—

127

C₉

—

333

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

—

111

—

132

C₃

—

111

—

133

C₃

#22582552

yes

characteristic

134

C₃

AΣL(1,𝔽₂₇)×C₃₇⋊C₃

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6866	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇⋊C₃	C₃
hypercenter	#6866	C₃₇⋊C₃
hypocenter	C₃₇	#10624765
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6866

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6866

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇⋊C₃

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6866

N	H ≅ G/N	type
2. #22582552	C₃	semidirect
3. #22582468	C₃	semidirect
4. #22582467	C₃	semidirect
5. C₃₇×#6866	C₃	semidirect
13. C₃²⋊#66524	C₃²	semidirect
16. C₃₇×C₃²⋊C₉	C₃²	semidirect
20. C₃³×C₃₇	3₋¹⁺²	semidirect
35. C₃²×C₃₇⋊C₃	3₋¹⁺²	semidirect
65. #6866	C₃₇⋊C₃	semidirect
83. C₃²⋊C₉	C₃×C₃₇⋊C₃	semidirect
84. C₃₇	#10624765	semidirect
102. C₃³	#11758606	semidirect

H ≅ G/N

type

2. #22582552

C₃

semidirect

3. #22582468

C₃

semidirect

4. #22582467

C₃

semidirect

5. C₃₇×#6866

C₃

semidirect

13. C₃²⋊#66524

C₃²

semidirect

16. C₃₇×C₃²⋊C₉

C₃²

semidirect

20. C₃³×C₃₇

3₋¹⁺²

semidirect

35. C₃²×C₃₇⋊C₃

3₋¹⁺²

semidirect

65. #6866

C₃₇⋊C₃

semidirect

83. C₃²⋊C₉

C₃×C₃₇⋊C₃

semidirect

84. C₃₇

#10624765

semidirect

102. C₃³

#11758606

semidirect

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

subgroup	index
C₃⁴×C₃₇⋊C₃	3
#22582552	3
#22582468	3
#22582467	3
C₃₇×#6866	3
#10624765	37

subgroup

index

C₃⁴×C₃₇⋊C₃

C₃₇×#6866

Subgroups — #41006702

Conjugacy classes of subgroups

Complements

Maximal subgroups