Group #41006709 — subgroups

Subgroups — #41006709

Order 26973.

398 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 67 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×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582575	C₃	1	3	yes	maximal, complemented
3	#22582472	C₃	1	2	yes	maximal, complemented
4	#22582469	C₃	1	3	yes	maximal, complemented
5	#22582467	C₃	1	3	yes	maximal
6	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃²×#66524	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
11	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
12	C₃×C₉×C₃₇⋊C₃	—	3	2	no	—
13	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
14	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
15	#15688506	—	3	3	no	—
16	#15688504	—	3	6	no	—
17	C₃²⋊#66524	C₃²	1	6	yes	—
18	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
19	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	—	3	1	no	—
22	C₃²×C₃₇⋊C₃	—	3	1	no	—
23	C₃²×C₃₇⋊C₃	—	3	1	no	—
24	C₃²×C₃₇⋊C₃	—	3	3	no	—
25	C₃²×C₃₇⋊C₃	—	3	6	no	—
26	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
27	C₃×C₉×C₃₇	—	3	1	no	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	3	no	—
31	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
32	C₃×#66524	—	3	1	no	—
33	C₃×#66524	—	3	1	no	—
34	C₃×#66524	—	3	1	no	—
35	C₃×#66524	—	3	1	no	—
36	C₃×#66524	—	3	1	no	—
37	C₃×#66524	—	3	1	no	—
38	C₃×#66524	—	3	6	no	—
39	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
40	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×#66524	3₋¹⁺²	1	3	yes	—
42	C₉×C₃₇⋊C₃	—	3	18	no	—
43	C₉×C₃₇⋊C₃	—	3	9	no	—
44	#10624766	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	2	no	—
47	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
49	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
50	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
51	C₃×C₃₇⋊C₃	—	3	9	no	—
52	C₃×C₃₇⋊C₃	—	3	9	no	—
53	C₃×C₃₇⋊C₃	—	3	3	no	—
54	C₃×C₃₇⋊C₃	—	3	3	no	—
55	C₃×C₃₇⋊C₃	—	3	6	no	—
56	C₃×C₃₇⋊C₃	—	3	9	no	—
57	C₉×C₃₇	—	3	3	no	—
58	C₉×C₃₇	—	9	3	no	—
59	#66524	—	3	2	no	—
60	#66524	—	3	1	no	—
61	#66524	—	3	3	no	—
62	#66524	—	9	6	no	—
63	C₃³×C₉	—	37	1	no	—
64	#6895	—	37	3	no	—
65	#6868	—	37	2	no	—
66	#6868	C₃₇⋊C₃	1	1	yes	characteristic, complemented
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	#6891	1	1	yes	characteristic
73	C₃×C₃₇	#6908	1	1	yes	characteristic
74	C₃×C₃₇	#6908	1	2	yes	—
75	C₃₇⋊C₃	—	3	9	no	—
76	C₃⁴	—	37	1	no	—
77	C₃²×C₉	—	37	1	no	—
78	C₃²×C₉	—	37	1	no	—
79	C₃²×C₉	—	111	2	no	—
80	C₃²×C₉	—	111	1	no	—
81	C₃²×C₉	—	111	3	no	—
82	C₃²×C₉	—	111	3	no	—
83	C₃²×C₉	—	111	3	no	—
84	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₉⋊C₉	—	111	6	no	—
86	C₉⋊C₉	—	111	3	no	—
87	C₃²⋊C₉	—	37	6	no	—
88	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
89	C₃₇	#10624766	1	1	yes	characteristic, complemented, Sylow
90	C₃³	—	37	3	no	—
91	C₃³	—	111	3	no	—
92	C₃³	—	111	6	no	—
93	C₃³	—	111	1	no	—
94	C₃³	—	111	1	no	—
95	C₃³	—	111	1	no	—
96	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	3	3	no	—
101	C₃×C₉	—	37	2	no	—
102	C₃×C₉	—	37	1	no	—
103	C₃×C₉	—	37	3	no	—
104	C₃×C₉	—	111	18	no	—
105	C₃×C₉	—	111	9	no	—
106	C₃×C₉	—	111	6	no	—
107	C₃×C₉	—	111	1	no	—
108	C₃×C₉	—	111	1	no	—
109	C₃×C₉	—	111	1	no	—
110	C₃×C₉	—	111	1	no	—
111	C₃×C₉	—	111	1	no	—
112	C₃×C₉	—	111	1	no	—
113	C₃×C₉	#11758606	1	3	yes	—
114	C₃²	—	3	1	no	—
115	C₃²	—	3	2	no	—
116	C₃²	—	111	9	no	—
117	C₃²	—	111	9	no	—
118	C₃²	—	111	9	no	—
119	C₃²	—	111	3	no	—
120	C₃²	—	111	3	no	—
121	C₃²	—	111	6	no	—
122	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
123	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
124	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
125	C₃²	C₃×#11758606	1	1	yes	characteristic
126	C₉	—	3	3	no	—
127	C₉	—	9	3	no	—
128	C₉	—	333	6	no	—
129	C₉	—	111	2	no	—
130	C₉	—	111	1	no	—
131	C₉	—	111	3	no	—
132	C₃	—	3	1	no	—
133	C₃	—	3	1	no	—
134	C₃	—	3	1	no	—
135	C₃	—	111	9	no	—
136	C₃	#22582552	1	1	yes	characteristic
137	C₃	#22582631	1	1	yes	characteristic
138	C₃	#22582634	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#22582575

C₃

yes

maximal, complemented

#22582472

C₃

yes

maximal, complemented

#22582469

C₃

yes

maximal, complemented

#22582467

C₃

yes

maximal

C₃₇×#6868

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

#15688506

—

#15688504

—

C₃²⋊#66524

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

yes

—

yes

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

#10624766

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

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

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

C₃³×C₉

—

#6895

—

#6868

—

#6868

C₃₇⋊C₃

yes

characteristic, complemented

#6867

—

#6866

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624766

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

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

—

111

—

111

C₃×C₉

—

111

—

112

C₃×C₉

—

111

—

113

C₃×C₉

#11758606

yes

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

111

—

117

C₃²

—

111

—

118

C₃²

—

111

—

119

C₃²

—

111

—

120

C₃²

—

111

—

121

C₃²

—

111

—

122

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

123

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

124

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

125

C₃²

C₃×#11758606

yes

characteristic

126

C₉

—

127

C₉

—

128

C₉

—

333

—

129

C₉

—

111

—

130

C₉

—

111

—

131

C₉

—

111

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

111

—

136

C₃

#22582552

yes

characteristic

137

C₃

#22582631

yes

characteristic

138

C₃

#22582634

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624766
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6868

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6868

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6868

N	H ≅ G/N	type
2. #22582575	C₃	semidirect
3. #22582472	C₃	semidirect
4. #22582469	C₃	semidirect
6. C₃₇×#6868	C₃	semidirect
66. #6868	C₃₇⋊C₃	semidirect
89. C₃₇	#10624766	semidirect

H ≅ G/N

type

2. #22582575

C₃

semidirect

3. #22582472

C₃

semidirect

4. #22582469

C₃

semidirect

6. C₃₇×#6868

C₃

semidirect

66. #6868

C₃₇⋊C₃

semidirect

89. C₃₇

#10624766

semidirect

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

subgroup	index
C₃²×C₉×C₃₇⋊C₃	3
#22582575	3
#22582472	3
#22582469	3
#22582467	3
C₃₇×#6868	3
#10624766	37

subgroup

index

C₃²×C₉×C₃₇⋊C₃

C₃₇×#6868

Subgroups — #41006709

Conjugacy classes of subgroups

Complements

Maximal subgroups