Group #41006723 — subgroups

Subgroups — #41006723

Order 26973.

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

normal subgroups only

This group has 464 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×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	#22582552	C₃	1	3	yes	maximal, complemented
3	#22582468	C₃	1	2	yes	maximal, complemented
4	#22582467	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6866	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃³×C₃₇⋊C₃	C₃²	1	2	yes	—
8	C₃×#11758606	—	3	3	no	—
9	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
10	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
12	C₃²⋊#66524	C₃²	1	6	yes	—
13	#15688502	—	3	6	no	—
14	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
15	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
16	C₃³×C₃₇	—	3	1	no	—
17	C₃³×C₃₇	—	3	2	no	—
18	C₃³×C₃₇	—	9	3	no	—
19	C₃³×C₃₇	3₋¹⁺²	1	3	yes	complemented
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	—	3	2	no	—
22	C₃²×C₃₇⋊C₃	—	3	2	no	—
23	C₃²×C₃₇⋊C₃	—	3	2	no	—
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	C₃²×C₃₇⋊C₃	—	3	2	no	—
26	C₃²×C₃₇⋊C₃	—	9	3	no	—
27	C₃²×C₃₇⋊C₃	—	9	6	no	—
28	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
29	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	6	yes	complemented
30	C₃×C₉×C₃₇	—	3	3	no	—
31	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
32	C₃×#66524	—	3	6	no	—
33	#11758606	—	9	9	no	—
34	#10624769	—	37	1	no	maximal, Sylow
35	C₃²×C₃₇	—	3	3	no	—
36	C₃²×C₃₇	—	3	3	no	—
37	C₃²×C₃₇	—	9	3	no	—
38	C₃²×C₃₇	—	9	6	no	—
39	C₃²×C₃₇	—	9	3	no	—
40	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
42	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
43	C₃×C₃₇⋊C₃	—	3	3	no	—
44	C₃×C₃₇⋊C₃	—	3	6	no	—
45	C₃×C₃₇⋊C₃	—	3	3	no	—
46	C₃×C₃₇⋊C₃	—	3	6	no	—
47	C₃×C₃₇⋊C₃	—	9	6	no	—
48	C₃×C₃₇⋊C₃	—	9	3	no	—
49	C₃×C₃₇⋊C₃	—	9	6	no	—
50	C₃×C₃₇⋊C₃	—	9	6	no	—
51	C₃×C₃₇⋊C₃	—	9	6	no	—
52	C₃×C₃₇⋊C₃	—	9	6	no	—
53	C₉×C₃₇	—	9	3	no	—
54	#66524	—	9	6	no	—
55	C₃²×H(3,𝔽₃)	—	37	1	no	—
56	#6891	—	37	3	no	—
57	#6866	—	37	2	no	—
58	#6866	—	37	6	no	—
59	#6866	C₃₇⋊C₃	1	1	yes	characteristic, complemented
60	C₃×C₃₇	—	3	2	no	—
61	C₃×C₃₇	—	3	1	no	—
62	C₃×C₃₇	—	9	3	no	—
63	C₃×C₃₇	#6891	1	1	yes	characteristic
64	C₃×C₃₇	#6909	1	3	yes	—
65	C₃₇⋊C₃	—	9	3	no	—
66	C₃₇⋊C₃	—	9	6	no	—
67	C₃⁴	—	37	2	no	—
68	C₃⁴	—	37	1	no	—
69	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
70	C₃×3₋¹⁺²	—	111	3	no	—
71	C₃×H(3,𝔽₃)	—	111	3	no	—
72	C₃×H(3,𝔽₃)	—	111	6	no	—
73	C₃×H(3,𝔽₃)	—	111	3	no	—
74	C₃²⋊C₉	—	37	6	no	—
75	C₃²⋊C₉	—	111	6	no	—
76	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
77	C₃₇	#10624769	1	1	yes	characteristic, complemented, Sylow
78	C₃³	—	3	1	no	—
79	C₃³	—	3	2	no	—
80	C₃³	—	9	3	no	—
81	C₃³	—	37	6	no	—
82	C₃³	—	37	3	no	—
83	C₃³	—	333	6	no	—
84	C₃³	—	333	3	no	—
85	C₃³	—	111	2	no	—
86	C₃³	—	111	2	no	—
87	C₃³	—	111	2	no	—
88	C₃³	—	111	2	no	—
89	C₃³	—	111	1	no	—
90	C₃³	#11758606	1	3	yes	complemented
91	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
92	3₋¹⁺²	—	333	9	no	—
93	H(3,𝔽₃)	—	111	27	no	—
94	C₃×C₉	—	3	3	no	—
95	C₃×C₉	—	111	6	no	—
96	C₃²	—	3	3	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	9	3	no	—
99	C₃²	—	9	6	no	—
100	C₃²	—	9	3	no	—
101	C₃²	—	333	6	no	—
102	C₃²	—	333	6	no	—
103	C₃²	—	333	6	no	—
104	C₃²	—	333	6	no	—
105	C₃²	—	333	3	no	—
106	C₃²	—	333	6	no	—
107	C₃²	—	111	6	no	—
108	C₃²	—	111	3	no	—
109	C₃²	—	111	3	no	—
110	C₃²	—	111	6	no	—
111	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
112	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
113	C₃²	C₃×#11758606	1	1	yes	characteristic
114	C₉	—	9	3	no	—
115	C₉	—	333	6	no	—
116	C₃	—	3	2	no	—
117	C₃	—	3	1	no	—
118	C₃	—	9	3	no	—
119	C₃	—	333	6	no	—
120	C₃	—	333	3	no	—
121	C₃	#22582552	1	1	yes	characteristic
122	C₃	#22582636	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

#22582552

C₃

yes

maximal, complemented

#22582468

C₃

yes

maximal, complemented

#22582467

C₃

yes

maximal, complemented

C₃₇×#6866

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

characteristic

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊#66524

C₃²

yes

—

#15688502

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

3₋¹⁺²

yes

complemented

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

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

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

C₃²×H(3,𝔽₃)

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

#6909

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624769

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

#11758606

yes

complemented

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

333

—

102

C₃²

—

333

—

103

C₃²

—

333

—

104

C₃²

—

333

—

105

C₃²

—

333

—

106

C₃²

—

333

—

107

C₃²

—

111

—

108

C₃²

—

111

—

109

C₃²

—

111

—

110

C₃²

—

111

—

111

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

112

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

113

C₃²

C₃×#11758606

yes

characteristic

114

C₉

—

115

C₉

—

333

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

—

333

—

120

C₃

—

333

—

121

C₃

#22582552

yes

characteristic

122

C₃

#22582636

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6866	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
hypercenter	#6866	C₃₇⋊C₃
hypocenter	C₃₇	#10624769
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6866

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6866

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6866

N	H ≅ G/N	type
2. #22582552	C₃	semidirect
3. #22582468	C₃	semidirect
4. #22582467	C₃	semidirect
5. C₃₇×#6866	C₃	semidirect
19. C₃³×C₃₇	3₋¹⁺²	semidirect
29. C₃²×C₃₇⋊C₃	3₋¹⁺²	semidirect
59. #6866	C₃₇⋊C₃	semidirect
77. C₃₇	#10624769	semidirect
90. C₃³	#11758606	semidirect

H ≅ G/N

type

2. #22582552

C₃

semidirect

3. #22582468

C₃

semidirect

4. #22582467

C₃

semidirect

5. C₃₇×#6866

C₃

semidirect

19. C₃³×C₃₇

3₋¹⁺²

semidirect

29. C₃²×C₃₇⋊C₃

3₋¹⁺²

semidirect

59. #6866

C₃₇⋊C₃

semidirect

77. C₃₇

#10624769

semidirect

90. C₃³

#11758606

semidirect

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

subgroup	index
C₃²×C₃²⋊(C₃₇⋊C₃)	3
#22582552	3
#22582468	3
#22582467	3
C₃₇×#6866	3
#10624769	37

subgroup

index

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃₇×#6866

Subgroups — #41006723

Conjugacy classes of subgroups

Complements

Maximal subgroups