Group #41006655 — subgroups

Subgroups — #41006655

Order 26973.

426 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 61 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×#15688531	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582648	C₃	1	3	yes	maximal, complemented
3	#22582639	C₃	1	6	yes	maximal, complemented
4	C₃×#15688519	C₃	1	2	yes	maximal, complemented
5	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688531	—	3	3	no	—
7	#15688531	—	3	6	no	—
8	C₃×#11758608	C₃²	1	2	yes	—
9	C₃×#11758605	—	3	1	no	—
10	C₃×#11758606	—	3	1	no	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
14	#15688519	—	3	2	no	—
15	#15688519	—	3	2	no	—
16	#15688519	—	3	2	no	—
17	#15688519	—	3	9	no	—
18	#15688514	—	3	18	no	—
19	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×#733	—	3	1	no	—
22	C₃₇×#733	—	3	1	no	—
23	C₃₇×#733	—	3	1	no	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	—	3	1	no	—
26	C₃²×C₃₇⋊C₃	—	3	6	no	—
27	3₋¹⁺²×C₃₇	—	3	3	no	—
28	3₋¹⁺²×C₃₇	—	3	6	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×C₉×C₃₇	—	3	3	no	—
32	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
36	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
37	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
38	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
39	C₃²⋊(C₃₇⋊C₃)	H(3,𝔽₃)	1	3	yes	—
40	#11758608	—	3	2	no	—
41	#11758608	—	3	2	no	—
42	#11758608	H(3,𝔽₃)	1	6	yes	—
43	C₃×#66524	—	3	2	no	—
44	#11758606	—	3	9	no	—
45	#11758605	—	3	9	no	—
46	C₉×C₃₇⋊C₃	—	3	9	no	—
47	#10624757	—	37	1	no	maximal, Sylow
48	C₃²×C₃₇	—	9	1	no	—
49	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
51	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₃₇⋊C₃	—	3	3	no	—
53	C₃×C₃₇⋊C₃	—	9	3	no	—
54	C₃×C₃₇⋊C₃	—	9	6	no	—
55	C₃×C₃₇⋊C₃	—	27	6	no	—
56	C₉×C₃₇	—	3	1	no	—
57	C₉×C₃₇	—	3	1	no	—
58	C₉×C₃₇	—	3	1	no	—
59	C₉×C₃₇	—	9	3	no	—
60	#66524	—	3	6	no	—
61	C₃×#737	—	37	1	no	—
62	#6911	—	37	3	no	—
63	#6909	—	37	6	no	—
64	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic, complemented
65	C₃×#732	—	37	2	no	—
66	C₃×C₃₇	—	3	1	no	—
67	C₃×C₃₇	—	3	2	no	—
68	C₃×C₃₇	—	3	1	no	—
69	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
70	C₃₇⋊C₃	—	9	3	no	—
71	C₃₇⋊C₃	—	27	6	no	—
72	#737	—	111	3	no	—
73	#737	—	111	6	no	—
74	C₃×3₋¹⁺²	—	37	2	no	—
75	C₃×3₋¹⁺²	—	111	1	no	—
76	C₃×3₋¹⁺²	—	111	1	no	—
77	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
78	C₃×H(3,𝔽₃)	—	37	1	no	—
79	C₃×H(3,𝔽₃)	—	37	6	no	—
80	C₃²×C₉	—	111	1	no	—
81	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
82	#733	—	3	1	no	—
83	#733	—	3	1	no	—
84	#733	—	3	1	no	—
85	#732	—	111	9	no	—
86	#732	—	111	2	no	—
87	#732	—	111	2	no	—
88	#732	—	111	2	no	—
89	AΣL(1,𝔽₂₇)	—	111	18	no	—
90	C₃₇	#10624757	1	1	yes	characteristic, complemented, Sylow
91	C₃³	—	111	1	no	—
92	C₃³	—	111	6	no	—
93	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
94	3₋¹⁺²	—	3	6	no	—
95	3₋¹⁺²	—	3	3	no	—
96	3₋¹⁺²	—	37	6	no	—
97	3₋¹⁺²	—	111	9	no	—
98	3₋¹⁺²	—	111	9	no	—
99	3₋¹⁺²	—	111	2	no	—
100	3₋¹⁺²	—	111	2	no	—
101	H(3,𝔽₃)	—	37	3	no	—
102	H(3,𝔽₃)	—	111	6	no	—
103	H(3,𝔽₃)	—	111	6	no	—
104	H(3,𝔽₃)	—	111	6	no	—
105	H(3,𝔽₃)	—	111	2	no	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	111	9	no	—
110	C₃×C₉	—	111	2	no	—
111	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
112	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
113	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
114	C₃²	—	9	1	no	—
115	C₃²	—	333	3	no	—
116	C₃²	—	333	6	no	—
117	C₃²	—	111	3	no	—
118	C₃²	—	999	6	no	—
119	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
120	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
121	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
122	C₉	—	3	1	no	—
123	C₉	—	3	1	no	—
124	C₉	—	3	1	no	—
125	C₉	—	9	3	no	—
126	C₉	—	111	6	no	—
127	C₃	—	3	1	no	—
128	C₃	—	3	2	no	—
129	C₃	—	3	1	no	—
130	C₃	—	333	3	no	—
131	C₃	—	999	6	no	—
132	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688531

C₃

yes

maximal, characteristic, complemented

#22582648

C₃

yes

maximal, complemented

#22582639

C₃

yes

maximal, complemented

C₃×#15688519

C₃

yes

maximal, complemented

C₃×C₃₇×#733

C₃

yes

maximal, characteristic, complemented

#15688531

—

#15688531

—

C₃×#11758608

C₃²

yes

—

C₃×#11758605

—

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

—

—

—

—

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#733

—

C₃₇×#733

—

C₃₇×#733

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

H(3,𝔽₃)

yes

—

—

—

yes

—

C₃×#66524

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624757

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×#732

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

—

—

—

111

—

#732

—

111

—

#732

—

111

—

#732

—

111

—

AΣL(1,𝔽₂₇)

—

111

—

C₃₇

#10624757

yes

characteristic, complemented, Sylow

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

100

3₋¹⁺²

—

111

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

111

—

103

H(3,𝔽₃)

—

111

—

104

H(3,𝔽₃)

—

111

—

105

H(3,𝔽₃)

—

111

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

111

—

110

C₃×C₉

—

111

—

111

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

112

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

113

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

114

C₃²

—

115

C₃²

—

333

—

116

C₃²

—

333

—

117

C₃²

—

111

—

118

C₃²

—

999

—

119

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

120

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

121

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

111

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

—

333

—

131

C₃

—

999

—

132

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624757
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#733

C₃

socle

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
1. C₃×#15688531	C₃	semidirect
2. #22582648	C₃	semidirect
3. #22582639	C₃	semidirect
4. C₃×#15688519	C₃	semidirect
5. C₃×C₃₇×#733	C₃	semidirect
64. C₃×#733	C₃₇⋊C₃	semidirect
90. C₃₇	#10624757	semidirect

H ≅ G/N

type

1. C₃×#15688531

C₃

semidirect

2. #22582648

C₃

semidirect

3. #22582639

C₃

semidirect

4. C₃×#15688519

C₃

semidirect

5. C₃×C₃₇×#733

C₃

semidirect

64. C₃×#733

C₃₇⋊C₃

semidirect

90. C₃₇

#10624757

semidirect

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

subgroup	index
C₃×#15688531	3
#22582648	3
#22582639	3
C₃×#15688519	3
C₃×C₃₇×#733	3
#10624757	37

subgroup

index

C₃×C₃₇×#733

Subgroups — #41006655

Conjugacy classes of subgroups

Complements

Maximal subgroups