Group #41006730 — subgroups

Subgroups — #41006730

Order 26973.

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

normal subgroups only

This group has 326 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×#11758606	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	C₃×#11758606	—	3	3	no	—
9	C₃×#11758606	—	3	3	no	—
10	C₃×#11758606	—	3	3	no	—
11	C₃×#11758606	—	3	3	no	—
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	#15688506	—	3	3	no	—
15	#15688504	—	3	6	no	—
16	C₃²⋊#66524	C₃²	1	6	yes	—
17	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
18	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃³×C₃₇	C₃³	1	1	yes	characteristic
20	C₃²×C₃₇⋊C₃	—	3	1	no	—
21	C₃²×C₃₇⋊C₃	—	3	1	no	—
22	C₃²×C₃₇⋊C₃	—	3	1	no	—
23	C₃²×C₃₇⋊C₃	—	9	1	no	—
24	C₃²×C₃₇⋊C₃	—	9	2	no	—
25	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
26	C₃×C₉×C₃₇	—	3	1	no	—
27	C₃×C₉×C₃₇	—	3	1	no	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	3	no	—
30	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
31	C₃×#66524	—	3	1	no	—
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	6	no	—
38	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
39	C₃×#66524	3₋¹⁺²	1	3	yes	—
40	#11758606	—	3	27	no	—
41	#10624770	—	37	1	no	maximal, Sylow
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	2	no	—
44	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
47	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₃×C₃₇⋊C₃	—	3	3	no	—
49	C₃×C₃₇⋊C₃	—	9	3	no	—
50	C₃×C₃₇⋊C₃	—	9	3	no	—
51	C₃×C₃₇⋊C₃	—	9	1	no	—
52	C₃×C₃₇⋊C₃	—	9	2	no	—
53	C₃×C₃₇⋊C₃	—	9	3	no	—
54	C₉×C₃₇	—	3	3	no	—
55	C₉×C₃₇	—	9	3	no	—
56	#66524	—	3	3	no	—
57	#66524	—	3	3	no	—
58	#66524	—	9	6	no	—
59	C₃²×3₋¹⁺²	—	37	1	no	—
60	#6895	—	37	3	no	—
61	#6868	—	37	2	no	—
62	#6868	C₃₇⋊C₃	1	1	yes	characteristic, complemented
63	#6867	—	37	3	no	—
64	#6866	—	37	3	no	—
65	C₃×C₃₇	—	3	1	no	—
66	C₃×C₃₇	—	3	1	no	—
67	C₃×C₃₇	—	3	1	no	—
68	C₃×C₃₇	#6891	1	1	yes	characteristic
69	C₃×C₃₇	#6913	1	1	yes	characteristic
70	C₃×C₃₇	#6913	1	2	yes	—
71	C₃₇⋊C₃	—	9	3	no	—
72	C₃⁴	—	37	1	no	—
73	C₃×3₋¹⁺²	—	111	3	no	—
74	C₃×3₋¹⁺²	—	111	3	no	—
75	C₃×3₋¹⁺²	—	111	3	no	—
76	C₃×3₋¹⁺²	—	111	3	no	—
77	C₃²×C₉	—	37	1	no	—
78	C₃²×C₉	—	37	1	no	—
79	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
80	C₉⋊C₉	—	111	6	no	—
81	C₉⋊C₉	—	111	3	no	—
82	C₃²⋊C₉	—	37	6	no	—
83	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
84	C₃₇	#10624770	1	1	yes	characteristic, complemented, Sylow
85	C₃³	—	37	3	no	—
86	C₃³	—	333	1	no	—
87	C₃³	—	333	2	no	—
88	C₃³	—	111	1	no	—
89	C₃³	—	111	1	no	—
90	C₃³	—	111	1	no	—
91	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
92	3₋¹⁺²	—	111	27	no	—
93	C₃×C₉	—	3	1	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	3	1	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	37	3	no	—
98	C₃×C₉	—	37	3	no	—
99	C₃×C₉	—	111	6	no	—
100	C₃×C₉	—	111	1	no	—
101	C₃×C₉	—	111	1	no	—
102	C₃×C₉	—	111	1	no	—
103	C₃×C₉	—	111	1	no	—
104	C₃×C₉	—	111	1	no	—
105	C₃×C₉	—	111	1	no	—
106	C₃×C₉	#11758606	1	3	yes	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	333	3	no	—
110	C₃²	—	333	3	no	—
111	C₃²	—	333	3	no	—
112	C₃²	—	333	1	no	—
113	C₃²	—	333	2	no	—
114	C₃²	—	111	3	no	—
115	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
116	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
117	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
118	C₃²	C₃×#11758606	1	1	yes	characteristic
119	C₉	—	3	3	no	—
120	C₉	—	9	3	no	—
121	C₉	—	333	6	no	—
122	C₉	—	111	3	no	—
123	C₉	—	111	3	no	—
124	C₃	—	3	1	no	—
125	C₃	—	3	1	no	—
126	C₃	—	3	1	no	—
127	C₃	—	333	3	no	—
128	C₃	#22582552	1	1	yes	characteristic
129	C₃	#22582658	1	1	yes	characteristic
130	C₃	#22582663	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758606

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

—

—

—

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

—

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

—

C₃×#66524

3₋¹⁺²

yes

—

#11758606

—

#10624770

—

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

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

#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

—

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

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624770

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

—

111

—

102

C₃×C₉

—

111

—

103

C₃×C₉

—

111

—

104

C₃×C₉

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

#11758606

yes

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

333

—

110

C₃²

—

333

—

111

C₃²

—

333

—

112

C₃²

—

333

—

113

C₃²

—

333

—

114

C₃²

—

111

—

115

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

116

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

117

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

118

C₃²

C₃×#11758606

yes

characteristic

119

C₉

—

120

C₉

—

121

C₉

—

333

—

122

C₉

—

111

—

123

C₉

—

111

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

333

—

128

C₃

#22582552

yes

characteristic

129

C₃

#22582658

yes

characteristic

130

C₃

#22582663

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	#10624770
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
62. #6868	C₃₇⋊C₃	semidirect
84. C₃₇	#10624770	semidirect

H ≅ G/N

type

2. #22582575

C₃

semidirect

3. #22582472

C₃

semidirect

4. #22582469

C₃

semidirect

6. C₃₇×#6868

C₃

semidirect

62. #6868

C₃₇⋊C₃

semidirect

84. C₃₇

#10624770

semidirect

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

subgroup	index
C₃²×#11758606	3
#22582575	3
#22582472	3
#22582469	3
#22582467	3
C₃₇×#6868	3
#10624770	37

subgroup

index

C₃₇×#6868

Subgroups — #41006730

Conjugacy classes of subgroups

Complements

Maximal subgroups