Group #41007396 — subgroups

Subgroups — #41007396

Order 26973.

198 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 52 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	C₃×#15688521	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688519	C₃	1	1	yes	maximal, characteristic
3	#22582538	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582521	C₃	1	9	yes	maximal, complemented
5	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#11758608	—	3	1	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
8	#15688521	—	3	1	no	—
9	#15688521	—	3	1	no	—
10	#15688521	C₃²	1	3	yes	complemented
11	#15688519	—	3	1	no	—
12	#15688519	—	3	1	no	—
13	#15688519	C₃²	1	3	yes	—
14	#15688506	C₃²	1	3	yes	complemented
15	#15688505	—	3	1	no	—
16	C₃²⋊#66524	—	3	1	no	—
17	C₉×#66524	—	3	1	no	—
18	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃₇×#733	—	3	1	no	—
21	C₃₇×#733	—	3	1	no	—
22	C₃₇×#733	C₃²	1	3	yes	complemented
23	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	9	1	no	—
25	3₋¹⁺²×C₃₇	—	3	3	no	—
26	3₋¹⁺²×C₃₇	—	9	1	no	—
27	3₋¹⁺²×C₃₇	—	9	1	no	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	—	9	1	no	—
32	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
35	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
36	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
37	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
38	#11758608	—	3	3	no	—
39	#11758608	—	9	1	no	—
40	#11758608	—	9	1	no	—
41	C₃×#66524	—	3	3	no	—
42	C₃×#66524	—	9	1	no	—
43	#10624848	—	37	1	no	maximal, Sylow
44	C₃²×C₃₇	—	9	1	no	—
45	C₃²×C₃₇	#732	1	1	yes	characteristic
46	C₃²×C₃₇	#732	1	1	yes	characteristic
47	C₃²×C₃₇	#732	1	1	yes	characteristic
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₃₇⋊C₃	—	9	3	no	—
50	C₃×C₃₇⋊C₃	—	27	3	no	—
51	C₉×C₃₇	—	3	1	no	—
52	C₉×C₃₇	—	3	1	no	—
53	C₉×C₃₇	—	3	1	no	—
54	C₉×C₃₇	—	9	3	no	—
55	#66524	—	9	3	no	—
56	#66524	—	9	3	no	—
57	C₃×#733	—	37	1	no	—
58	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic, complemented
59	C₃×#732	—	37	1	no	—
60	#6887	—	37	1	no	—
61	#6883	—	37	9	no	—
62	C₃×C₃₇	—	3	1	no	—
63	C₃×C₃₇	—	3	1	no	—
64	C₃×C₃₇	—	3	1	no	—
65	C₃×C₃₇	—	3	1	no	—
66	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
67	C₃₇⋊C₃	—	27	3	no	—
68	C₃×3₋¹⁺²	—	3	1	no	—
69	C₃×3₋¹⁺²	—	111	1	no	—
70	C₃×H(3,𝔽₃)	—	111	1	no	—
71	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
72	#733	—	3	1	no	—
73	#733	—	3	1	no	—
74	#733	—	37	3	no	—
75	#733	—	111	1	no	—
76	#733	—	111	1	no	—
77	#733	C₃×C₃₇⋊C₃	1	3	yes	complemented
78	#732	—	37	3	no	—
79	#732	—	111	1	no	—
80	#732	—	111	1	no	—
81	C₉⋊C₉	—	37	3	no	—
82	C₉⋊C₉	—	111	1	no	—
83	C₃²⋊C₉	—	111	1	no	—
84	C₉²	—	111	1	no	—
85	C₃₇	#10624848	1	1	yes	characteristic, complemented, Sylow
86	C₃³	—	333	1	no	—
87	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
88	3₋¹⁺²	—	3	3	no	—
89	3₋¹⁺²	—	9	1	no	—
90	3₋¹⁺²	—	9	1	no	—
91	3₋¹⁺²	—	333	1	no	—
92	3₋¹⁺²	—	333	1	no	—
93	3₋¹⁺²	—	111	3	no	—
94	H(3,𝔽₃)	—	333	1	no	—
95	H(3,𝔽₃)	—	333	1	no	—
96	H(3,𝔽₃)	—	111	3	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	9	1	no	—
101	C₃×C₉	—	333	1	no	—
102	C₃×C₉	—	111	3	no	—
103	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
104	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
105	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
106	C₃²	—	9	1	no	—
107	C₃²	—	333	3	no	—
108	C₃²	—	999	3	no	—
109	C₃²	#15688518	1	1	yes	characteristic
110	C₃²	#15688518	1	1	yes	characteristic
111	C₃²	#15688518	1	1	yes	characteristic
112	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
113	C₉	—	3	1	no	—
114	C₉	—	3	1	no	—
115	C₉	—	3	1	no	—
116	C₉	—	9	3	no	—
117	C₉	—	333	3	no	—
118	C₉	—	333	3	no	—
119	C₃	—	3	1	no	—
120	C₃	—	3	1	no	—
121	C₃	—	3	1	no	—
122	C₃	—	3	1	no	—
123	C₃	—	999	3	no	—
124	C₃	C₃×#15688518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688521

C₃

yes

maximal, characteristic, complemented

C₃×#15688519

C₃

yes

maximal, characteristic

#22582538

C₃

yes

maximal, characteristic, complemented

#22582521

C₃

yes

maximal, complemented

C₃×C₃₇×#733

C₃

yes

maximal, characteristic, complemented

C₃×#11758608

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

#15688521

—

#15688521

—

#15688521

C₃²

yes

—

—

C₃²

yes

—

#15688506

C₃²

yes

—

—

—

C₃×3₋¹⁺²×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#733

—

C₃₇×#733

—

C₃₇×#733

C₃²

yes

complemented

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

yes

C₃²×C₃₇

yes

C₃²×C₃₇

yes

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×#732

—

#6887

—

#6883

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×#732

yes

characteristic

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

—

—

—

111

—

#733

—

111

—

#733

C₃×C₃₇⋊C₃

yes

complemented

#732

—

#732

—

111

—

#732

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₃₇

#10624848

yes

characteristic, complemented, Sylow

C₃³

—

333

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

333

—

102

C₃×C₉

—

111

—

103

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

104

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

105

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

106

C₃²

—

107

C₃²

—

333

—

108

C₃²

—

999

—

109

C₃²

#15688518

yes

characteristic

110

C₃²

#15688518

yes

characteristic

111

C₃²

#15688518

yes

characteristic

112

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

333

—

118

C₉

—

333

—

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

999

—

124

C₃

C₃×#15688518

yes

characteristic

	subgroup	G/H
center	C₃	C₃×#15688518
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃×C₃₇	C₃×#732
prime radical	C₃×#15688519	C₃
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624848
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#733

subgroup

G/H

center

C₃

C₃×#15688518

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×#733

C₃

C₃×C₃₇

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#733

N	H ≅ G/N	type
1. C₃×#15688521	C₃	semidirect
3. #22582538	C₃	semidirect
4. #22582521	C₃	semidirect
5. C₃×C₃₇×#733	C₃	semidirect
10. #15688521	C₃²	semidirect
14. #15688506	C₃²	semidirect
22. C₃₇×#733	C₃²	semidirect
58. C₃×#733	C₃₇⋊C₃	semidirect
77. #733	C₃×C₃₇⋊C₃	semidirect
85. C₃₇	#10624848	semidirect

H ≅ G/N

type

1. C₃×#15688521

C₃

semidirect

3. #22582538

C₃

semidirect

4. #22582521

C₃

semidirect

5. C₃×C₃₇×#733

C₃

semidirect

10. #15688521

C₃²

semidirect

14. #15688506

C₃²

semidirect

22. C₃₇×#733

C₃²

semidirect

58. C₃×#733

C₃₇⋊C₃

semidirect

77. #733

C₃×C₃₇⋊C₃

semidirect

85. C₃₇

#10624848

semidirect

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

subgroup	index
C₃×#15688521	3
C₃×#15688519	3
#22582538	3
#22582521	3
C₃×C₃₇×#733	3
#10624848	37

subgroup

index

C₃×C₃₇×#733

Subgroups — #41007396

Conjugacy classes of subgroups

Complements

Maximal subgroups