Group #41005657 — subgroups

Subgroups — #41005657

Order 26973.

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

normal subgroups only

This group has 200 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	#22582578	C₃	1	1	yes	maximal, characteristic, complemented
2	C₉×#11758608	C₃	1	2	yes	maximal, complemented
3	C₃₇⋊#6862	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582456	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582451	C₃	1	2	yes	maximal
6	#22582450	C₃	1	2	yes	maximal, complemented
7	#22582449	C₃	1	2	yes	maximal, complemented
8	#22582445	C₃	1	1	yes	maximal, characteristic
9	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
10	C₃×#11758608	C₃²	1	2	yes	—
11	#15688504	—	3	2	no	—
12	#15688504	—	3	2	no	—
13	#15688504	—	3	2	no	—
14	#15688504	—	3	1	no	—
15	C₃²⋊#66524	C₃²	1	2	yes	—
16	C₃²⋊#66524	C₃²	1	2	yes	—
17	C₃²⋊#66524	C₃²	1	1	yes	characteristic
18	C₃²⋊#66524	C₃²	1	2	yes	—
19	C₉×#66524	—	3	2	no	—
20	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
21	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
22	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
23	C₃₇×C₃²⋊C₉	—	3	2	no	—
24	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	3₋¹⁺²×C₃₇	—	9	3	no	—
29	H(3,𝔽₃)×C₃₇	—	9	1	no	—
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃×C₉×C₃₇	—	9	1	no	—
33	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
35	#11758608	—	3	6	no	—
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	2	no	—
39	C₃×#66524	—	3	1	no	—
40	C₃×#66524	—	3	2	no	—
41	#10624625	—	37	1	no	maximal, Sylow
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	2	no	—
44	C₃²×C₃₇	—	3	1	no	—
45	C₃²×C₃₇	—	9	1	no	—
46	C₃²×C₃₇	—	9	2	no	—
47	C₃²×C₃₇	—	9	1	no	—
48	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₉×C₃₇	—	3	2	no	—
50	C₉×C₃₇	—	3	1	no	—
51	C₉×C₃₇	—	9	1	no	—
52	C₉×C₃₇	—	9	1	no	—
53	#66524	—	3	6	no	—
54	#66524	—	9	2	no	—
55	#66524	—	9	1	no	—
56	#66524	—	9	2	no	—
57	#66524	—	9	2	no	—
58	#6895	—	37	1	no	—
59	C₉⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
60	C₉×3₋¹⁺²	—	37	2	no	—
61	#6862	—	37	1	no	—
62	#6861	—	37	1	no	—
63	#6859	—	37	2	no	—
64	#6858	—	37	2	no	—
65	#6858	—	37	2	no	—
66	#6857	—	37	1	no	—
67	C₃×C₃₇	—	9	1	no	—
68	C₃×C₃₇	—	27	1	no	—
69	C₃×C₃₇	#6908	1	2	yes	—
70	C₃×C₃₇	#6910	1	1	yes	characteristic
71	C₃×C₃₇	#6913	1	1	yes	characteristic
72	C₃×3₋¹⁺²	—	3	1	no	—
73	C₃×3₋¹⁺²	—	37	2	no	—
74	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
75	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
76	C₉⋊C₉	—	111	2	no	—
77	C₉⋊C₉	—	111	2	no	—
78	C₉⋊C₉	—	111	2	no	—
79	C₉⋊C₉	—	111	1	no	—
80	C₃²⋊C₉	—	3	2	no	—
81	C₃²⋊C₉	—	37	1	no	—
82	C₃²⋊C₉	—	37	2	no	—
83	C₃²⋊C₉	—	37	2	no	—
84	C₃²⋊C₉	—	37	2	no	—
85	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
86	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
87	C₉²	—	111	2	no	—
88	C₃₇	#10624625	1	1	yes	characteristic, complemented, Sylow
89	C₃³	—	3	1	no	—
90	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
91	3₋¹⁺²	—	9	3	no	—
92	3₋¹⁺²	—	111	6	no	—
93	H(3,𝔽₃)	—	9	1	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	3	1	no	—
96	C₃×C₉	—	9	1	no	—
97	C₃×C₉	—	111	2	no	—
98	C₃×C₉	—	111	2	no	—
99	C₃×C₉	—	111	2	no	—
100	C₃×C₉	—	111	1	no	—
101	C₃×C₉	—	111	2	no	—
102	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
103	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
104	C₃²	—	3	2	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	9	2	no	—
108	C₃²	—	9	1	no	—
109	C₃²	—	9	1	no	—
110	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
111	C₉	—	3	1	no	—
112	C₉	—	3	2	no	—
113	C₉	—	9	1	no	—
114	C₉	—	9	1	no	—
115	C₉	—	333	2	no	—
116	C₉	—	333	2	no	—
117	C₉	—	333	2	no	—
118	C₉	—	333	1	no	—
119	C₉	—	111	6	no	—
120	C₃	—	9	1	no	—
121	C₃	—	27	1	no	—
122	C₃	#22582630	1	2	yes	—
123	C₃	#22582642	1	1	yes	characteristic
124	C₃	#22582661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582578

C₃

yes

maximal, characteristic, complemented

C₉×#11758608

C₃

yes

maximal, complemented

C₃₇⋊#6862

C₃

yes

maximal, characteristic, complemented

#22582456

C₃

yes

maximal, characteristic, complemented

#22582451

C₃

yes

maximal

#22582450

C₃

yes

maximal, complemented

#22582449

C₃

yes

maximal, complemented

#22582445

C₃

yes

maximal, characteristic

C₃₇×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₃×3₋¹⁺²×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

yes

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

C₉⋊H(3,𝔽₃)

C₃₇⋊C₃

yes

characteristic

C₉×3₋¹⁺²

—

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6908

yes

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉²

—

111

—

C₃₇

#10624625

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

—

111

—

102

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

103

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

333

—

116

C₉

—

333

—

117

C₉

—

333

—

118

C₉

—

333

—

119

C₉

—

111

—

120

C₃

—

121

C₃

—

122

C₃

#22582630

yes

—

123

C₃

#22582642

yes

characteristic

124

C₃

#22582661

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×C₉⋊H(3,𝔽₃)	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₉⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624625
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉⋊H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×C₉⋊H(3,𝔽₃)

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃×H(3,𝔽₃)×C₃₇

C₃²

hypercenter

C₉⋊H(3,𝔽₃)

C₃₇⋊C₃

hypocenter

C₃₇

#10624625

LCS

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉⋊H(3,𝔽₃)

N	H ≅ G/N	type
1. #22582578	C₃	semidirect
2. C₉×#11758608	C₃	semidirect
3. C₃₇⋊#6862	C₃	semidirect
4. #22582456	C₃	semidirect
6. #22582450	C₃	semidirect
7. #22582449	C₃	semidirect
88. C₃₇	#10624625	semidirect

H ≅ G/N

type

1. #22582578

C₃

semidirect

2. C₉×#11758608

C₃

semidirect

3. C₃₇⋊#6862

C₃

semidirect

4. #22582456

C₃

semidirect

6. #22582450

C₃

semidirect

7. #22582449

C₃

semidirect

88. C₃₇

#10624625

semidirect

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

subgroup	index
#22582578	3
C₉×#11758608	3
C₃₇⋊#6862	3
#22582456	3
#22582451	3
#22582450	3
#22582449	3
#22582445	3
C₃₇×C₉⋊H(3,𝔽₃)	3
#10624625	37

subgroup

index

C₃₇⋊#6862

C₃₇×C₉⋊H(3,𝔽₃)

Subgroups — #41005657

Conjugacy classes of subgroups

Complements

Maximal subgroups