Group #41006247 — subgroups

Subgroups — #41006247

Order 26973.

362 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 112 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×#11758606	C₃	1	1	yes	maximal, characteristic
2	#22582538	C₃	1	2	yes	maximal, complemented
3	#22582471	C₃	1	3	yes	maximal, complemented
4	#22582469	C₃	1	3	yes	maximal, complemented
5	#22582467	C₃	1	3	yes	maximal
6	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#11758606	—	3	1	no	—
9	C₃×#11758606	—	3	2	no	—
10	C₃×#11758606	—	3	3	no	—
11	C₃×#11758606	—	3	3	no	—
12	C₃×#11758606	C₉	1	9	yes	complemented
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	#15688506	—	3	2	no	—
16	#15688505	—	3	2	no	—
17	C₃²⋊#66524	C₃²	1	6	yes	—
18	C₉×#66524	—	3	2	no	—
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃₇×C₉⋊C₉	—	3	2	no	—
21	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
22	C₉²×C₃₇	—	3	1	no	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	C₃²×C₃₇⋊C₃	—	3	1	no	—
26	C₃²×C₃₇⋊C₃	—	3	1	no	—
27	C₃²×C₃₇⋊C₃	—	9	3	no	—
28	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
29	C₃×C₉×C₃₇	—	3	3	no	—
30	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
31	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
33	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
34	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
35	C₃×#66524	—	3	1	no	—
36	C₃×#66524	—	3	1	no	—
37	C₃×#66524	—	3	1	no	—
38	C₃×#66524	—	3	1	no	—
39	C₃×#66524	—	3	1	no	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	6	no	—
42	C₃×#66524	C₃×C₉	1	3	yes	—
43	C₃×#66524	C₃×C₉	1	3	yes	—
44	#11758606	—	3	9	no	—
45	#11758606	—	3	18	no	—
46	#10624692	—	37	1	no	maximal, Sylow
47	C₃²×C₃₇	—	3	3	no	—
48	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
49	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
51	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
52	C₃×C₃₇⋊C₃	—	3	3	no	—
53	C₃×C₃₇⋊C₃	—	9	3	no	—
54	C₃×C₃₇⋊C₃	—	9	3	no	—
55	C₃×C₃₇⋊C₃	—	9	3	no	—
56	C₃×C₃₇⋊C₃	—	9	3	no	—
57	C₉×C₃₇	—	9	3	no	—
58	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
59	C₉×C₃₇	C₃²⋊C₉	1	6	yes	—
60	#66524	—	3	3	no	—
61	#66524	—	3	3	no	—
62	#66524	—	9	6	no	—
63	C₃²×3₋¹⁺²	—	37	1	no	—
64	#6887	—	37	2	no	—
65	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
66	#6868	—	37	3	no	—
67	#6867	—	37	3	no	—
68	#6866	—	37	3	no	—
69	C₃×C₃₇	—	3	1	no	—
70	C₃×C₃₇	—	3	1	no	—
71	C₃×C₃₇	—	3	1	no	—
72	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
73	C₃×C₃₇	#6913	1	3	yes	—
74	C₃₇⋊C₃	—	9	3	no	—
75	C₃⁴	—	37	1	no	—
76	C₃×3₋¹⁺²	—	37	9	no	—
77	C₃×3₋¹⁺²	—	111	3	no	—
78	C₃×3₋¹⁺²	—	111	3	no	—
79	C₃×3₋¹⁺²	—	111	1	no	—
80	C₃×3₋¹⁺²	—	111	2	no	—
81	C₃²×C₉	—	37	1	no	—
82	C₃²×C₉	—	37	1	no	—
83	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
84	C₉⋊C₉	—	3	2	no	—
85	C₉⋊C₉	—	111	2	no	—
86	C₉⋊C₉	—	111	2	no	—
87	C₃²⋊C₉	—	37	6	no	—
88	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
89	C₉²	—	3	1	no	—
90	C₉²	—	111	2	no	—
91	C₃₇	#10624692	1	1	yes	characteristic, complemented, Sylow
92	C₃³	—	37	3	no	—
93	C₃³	—	333	3	no	—
94	C₃³	—	111	1	no	—
95	C₃³	—	111	1	no	—
96	C₃³	—	111	1	no	—
97	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
98	3₋¹⁺²	—	111	9	no	—
99	3₋¹⁺²	—	111	18	no	—
100	C₃×C₉	—	3	3	no	—
101	C₃×C₉	—	37	3	no	—
102	C₃×C₉	—	37	3	no	—
103	C₃×C₉	—	111	6	no	—
104	C₃×C₉	—	111	1	no	—
105	C₃×C₉	—	111	1	no	—
106	C₃×C₉	—	111	1	no	—
107	C₃×C₉	—	111	1	no	—
108	C₃×C₉	—	111	1	no	—
109	C₃×C₉	—	111	1	no	—
110	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
111	C₃×C₉	#11758605	1	3	yes	—
112	C₃×C₉	#11758606	1	3	yes	—
113	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
114	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
115	C₃²	—	3	3	no	—
116	C₃²	—	333	3	no	—
117	C₃²	—	333	3	no	—
118	C₃²	—	333	3	no	—
119	C₃²	—	333	3	no	—
120	C₃²	—	111	3	no	—
121	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
122	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
123	C₃²	C₃×#11758606	1	1	yes	characteristic
124	C₃²	C₃×#11758605	1	1	yes	characteristic
125	C₉	—	9	3	no	—
126	C₉	—	333	6	no	—
127	C₉	—	111	3	no	—
128	C₉	—	111	3	no	—
129	C₉	#15688502	1	3	yes	—
130	C₉	#15688502	1	6	yes	—
131	C₃	—	3	1	no	—
132	C₃	—	3	1	no	—
133	C₃	—	3	1	no	—
134	C₃	—	333	3	no	—
135	C₃	C₃×#15688502	1	1	yes	characteristic
136	C₃	#22582656	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

C₃

yes

maximal, characteristic

#22582538

C₃

yes

maximal, complemented

#22582471

C₃

yes

maximal, complemented

#22582469

C₃

yes

maximal, complemented

#22582467

C₃

yes

maximal

C₃₇×#6887

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

—

C₉

yes

complemented

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

—

C₃²

yes

—

C₉×#66524

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

#6887

—

#6887

C₃₇⋊C₃

yes

characteristic, complemented

#6868

—

#6867

—

#6866

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

#6913

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

C₉²

—

111

—

C₃₇

#10624692

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

111

—

104

C₃×C₉

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃×C₉

—

111

—

109

C₃×C₉

—

111

—

110

C₃×C₉

C₉×C₃₇⋊C₃

yes

—

111

C₃×C₉

#11758605

yes

—

112

C₃×C₉

#11758606

yes

—

113

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

114

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

115

C₃²

—

116

C₃²

—

333

—

117

C₃²

—

333

—

118

C₃²

—

333

—

119

C₃²

—

333

—

120

C₃²

—

111

—

121

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

122

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

123

C₃²

C₃×#11758606

yes

characteristic

124

C₃²

C₃×#11758605

yes

characteristic

125

C₉

—

126

C₉

—

333

—

127

C₉

—

111

—

128

C₉

—

111

—

129

C₉

#15688502

yes

—

130

C₉

#15688502

yes

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

333

—

135

C₃

C₃×#15688502

yes

characteristic

136

C₃

#22582656

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
cosocle	C₃³×C₃₇	C₃³
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624692
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6887

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6887

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃³×C₃₇⋊C₃

C₃²

cosocle

C₃³×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6887

N	H ≅ G/N	type
2. #22582538	C₃	semidirect
3. #22582471	C₃	semidirect
4. #22582469	C₃	semidirect
6. C₃₇×#6887	C₃	semidirect
12. C₃×#11758606	C₉	semidirect
65. #6887	C₃₇⋊C₃	semidirect
91. C₃₇	#10624692	semidirect

H ≅ G/N

type

2. #22582538

C₃

semidirect

3. #22582471

C₃

semidirect

4. #22582469

C₃

semidirect

6. C₃₇×#6887

C₃

semidirect

12. C₃×#11758606

C₉

semidirect

65. #6887

C₃₇⋊C₃

semidirect

91. C₃₇

#10624692

semidirect

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

subgroup	index
C₃²×#11758606	3
#22582538	3
#22582471	3
#22582469	3
#22582467	3
C₃₇×#6887	3
#10624692	37

subgroup

index

C₃₇×#6887

Subgroups — #41006247

Conjugacy classes of subgroups

Complements

Maximal subgroups