Group #41006727 — subgroups

Subgroups — #41006727

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×#11758605	C₃	1	1	yes	maximal, characteristic
2	#22582578	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×#6895	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#11758605	—	3	3	no	—
9	C₃×#11758605	—	3	3	no	—
10	C₃×#11758605	—	3	3	no	—
11	C₃×#11758605	—	3	3	no	—
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	#15688504	—	3	6	no	—
15	C₃²⋊#66524	C₃²	1	6	yes	—
16	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
17	C₃₇×C₉⋊C₉	—	3	3	no	—
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	3	no	—
24	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
25	C₃×C₉×C₃₇	—	3	1	no	—
26	C₃×C₉×C₃₇	—	3	1	no	—
27	C₃×C₉×C₃₇	—	3	1	no	—
28	C₃×C₉×C₃₇	—	3	3	no	—
29	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
30	C₃×#66524	—	3	1	no	—
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	6	no	—
37	C₃×#66524	3₋¹⁺²	1	3	yes	—
38	C₃×#66524	3₋¹⁺²	1	3	yes	—
39	#11758605	—	3	27	no	—
40	#10624770	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	—	3	3	no	—
42	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
44	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
45	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₃×C₃₇⋊C₃	—	3	3	no	—
47	C₃×C₃₇⋊C₃	—	9	3	no	—
48	C₃×C₃₇⋊C₃	—	9	3	no	—
49	C₃×C₃₇⋊C₃	—	9	3	no	—
50	C₃×C₃₇⋊C₃	—	9	3	no	—
51	C₉×C₃₇	—	3	3	no	—
52	C₉×C₃₇	—	9	3	no	—
53	#66524	—	3	3	no	—
54	#66524	—	3	3	no	—
55	#66524	—	9	6	no	—
56	C₃²×3₋¹⁺²	—	37	1	no	—
57	#6895	—	37	2	no	—
58	#6895	C₃₇⋊C₃	1	1	yes	characteristic, complemented
59	#6868	—	37	3	no	—
60	#6867	—	37	3	no	—
61	#6866	—	37	3	no	—
62	C₃×C₃₇	—	3	1	no	—
63	C₃×C₃₇	—	3	1	no	—
64	C₃×C₃₇	—	3	1	no	—
65	C₃×C₃₇	#6891	1	1	yes	characteristic
66	C₃×C₃₇	#6913	1	3	yes	—
67	C₃₇⋊C₃	—	9	3	no	—
68	C₃⁴	—	37	1	no	—
69	C₃×3₋¹⁺²	—	111	3	no	—
70	C₃×3₋¹⁺²	—	111	3	no	—
71	C₃×3₋¹⁺²	—	111	3	no	—
72	C₃×3₋¹⁺²	—	111	3	no	—
73	C₃²×C₉	—	37	1	no	—
74	C₃²×C₉	—	37	1	no	—
75	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
76	C₉⋊C₉	—	3	3	no	—
77	C₉⋊C₉	—	111	6	no	—
78	C₃²⋊C₉	—	37	6	no	—
79	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
80	C₃₇	#10624770	1	1	yes	characteristic, complemented, Sylow
81	C₃³	—	37	3	no	—
82	C₃³	—	333	3	no	—
83	C₃³	—	111	1	no	—
84	C₃³	—	111	1	no	—
85	C₃³	—	111	1	no	—
86	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
87	3₋¹⁺²	—	111	27	no	—
88	C₃×C₉	—	3	1	no	—
89	C₃×C₉	—	3	1	no	—
90	C₃×C₉	—	3	1	no	—
91	C₃×C₉	—	3	3	no	—
92	C₃×C₉	—	37	3	no	—
93	C₃×C₉	—	37	3	no	—
94	C₃×C₉	—	111	6	no	—
95	C₃×C₉	—	111	1	no	—
96	C₃×C₉	—	111	1	no	—
97	C₃×C₉	—	111	1	no	—
98	C₃×C₉	—	111	1	no	—
99	C₃×C₉	—	111	1	no	—
100	C₃×C₉	—	111	1	no	—
101	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
102	C₃²	—	3	3	no	—
103	C₃²	—	333	3	no	—
104	C₃²	—	333	3	no	—
105	C₃²	—	333	3	no	—
106	C₃²	—	333	3	no	—
107	C₃²	—	111	3	no	—
108	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
109	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
110	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
111	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
112	C₉	—	3	3	no	—
113	C₉	—	9	3	no	—
114	C₉	—	333	6	no	—
115	C₉	—	111	3	no	—
116	C₉	—	111	3	no	—
117	C₃	—	3	1	no	—
118	C₃	—	3	1	no	—
119	C₃	—	3	1	no	—
120	C₃	—	333	3	no	—
121	C₃	#22582551	1	1	yes	characteristic
122	C₃	#22582655	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758605

C₃

yes

maximal, characteristic

#22582578

C₃

yes

maximal, complemented

#22582471

C₃

yes

maximal, complemented

#22582469

C₃

yes

maximal, complemented

#22582467

C₃

yes

maximal

C₃₇×#6895

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

—

—

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

#15688504

—

C₃²⋊#66524

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

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

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

—

3₋¹⁺²

yes

—

C₃×#66524

3₋¹⁺²

yes

—

#11758605

—

#10624770

—

maximal, Sylow

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

—

C₉×C₃₇

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

#6895

—

#6895

C₃₇⋊C₃

yes

characteristic, complemented

#6868

—

#6867

—

#6866

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

#6913

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

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624770

yes

characteristic, complemented, Sylow

C₃³

—

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

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

102

C₃²

—

103

C₃²

—

333

—

104

C₃²

—

333

—

105

C₃²

—

333

—

106

C₃²

—

333

—

107

C₃²

—

111

—

108

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

109

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

110

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

111

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

112

C₉

—

113

C₉

—

114

C₉

—

333

—

115

C₉

—

111

—

116

C₉

—

111

—

117

C₃

—

118

C₃

—

119

C₃

—

120

C₃

—

333

—

121

C₃

#22582551

yes

characteristic

122

C₃

#22582655

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6895	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6895	C₃₇⋊C₃
hypocenter	C₃₇	#10624770
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6895

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6895

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 ◁ C₃²×C₉ ◁ #6895

N	H ≅ G/N	type
2. #22582578	C₃	semidirect
3. #22582471	C₃	semidirect
4. #22582469	C₃	semidirect
6. C₃₇×#6895	C₃	semidirect
58. #6895	C₃₇⋊C₃	semidirect
80. C₃₇	#10624770	semidirect

H ≅ G/N

type

2. #22582578

C₃

semidirect

3. #22582471

C₃

semidirect

4. #22582469

C₃

semidirect

6. C₃₇×#6895

C₃

semidirect

58. #6895

C₃₇⋊C₃

semidirect

80. C₃₇

#10624770

semidirect

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

subgroup	index
C₃²×#11758605	3
#22582578	3
#22582471	3
#22582469	3
#22582467	3
C₃₇×#6895	3
#10624770	37

subgroup

index

C₃₇×#6895

Subgroups — #41006727

Conjugacy classes of subgroups

Complements

Maximal subgroups