Group #41005291 — subgroups

Subgroups — #41005291

Order 26973.

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

normal subgroups only

This group has 334 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×#15688530	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582548	C₃	1	2	yes	maximal, complemented
3	#22582539	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582455	C₃	1	2	yes	maximal, complemented
5	#22582450	C₃	1	2	yes	maximal, complemented
6	#22582446	C₃	1	2	yes	maximal, complemented
7	#22582445	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582442	C₃	1	1	yes	maximal, characteristic
9	C₃₇×#6859	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688530	—	3	9	no	—
11	C₃×#11758608	—	3	2	no	—
12	C₃×#11758608	C₃²	1	2	yes	—
13	C₃×#11758608	C₃²	1	2	yes	—
14	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
15	C₃²×#66524	—	3	1	no	—
16	C₃²×#66524	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
18	#15688504	—	3	4	no	—
19	C₃²⋊#66524	C₃²	1	1	yes	characteristic
20	C₃²⋊#66524	C₃²	1	2	yes	—
21	#15688502	—	3	2	no	—
22	C₉×#66524	—	3	2	no	—
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
24	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	C₃²×C₃₇⋊C₃	—	3	1	no	—
29	H(3,𝔽₃)×C₃₇	—	3	3	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
33	#11758608	—	3	18	no	—
34	#11758608	—	3	6	no	—
35	#11758608	—	3	6	no	—
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	9	no	—
39	C₃×#66524	—	3	3	no	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
43	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
44	C₉×C₃₇⋊C₃	—	9	3	no	—
45	#10624583	—	37	1	no	maximal, Sylow
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	3	3	no	—
48	C₃²×C₃₇	—	3	2	no	—
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	9	1	no	—
51	C₃²×C₃₇	—	9	2	no	—
52	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×C₃₇⋊C₃	—	3	3	no	—
54	C₃×C₃₇⋊C₃	—	9	1	no	—
55	C₃×C₃₇⋊C₃	—	9	2	no	—
56	C₉×C₃₇	—	9	1	no	—
57	C₉×C₃₇	—	9	2	no	—
58	#66524	—	3	6	no	—
59	#66524	—	3	2	no	—
60	#66524	—	3	6	no	—
61	#66524	—	9	1	no	—
62	#66524	—	9	2	no	—
63	#66524	AΣL(1,𝔽₂₇)	1	3	yes	—
64	C₃×#737	—	37	1	no	—
65	C₉×3₋¹⁺²	—	37	2	no	—
66	C₉×H(3,𝔽₃)	—	37	1	no	—
67	#6860	—	37	2	no	—
68	#6859	C₃₇⋊C₃	1	1	yes	characteristic, complemented
69	#6858	—	37	2	no	—
70	#6857	—	37	1	no	—
71	#6857	—	37	2	no	—
72	#6856	—	37	1	no	—
73	C₃×C₃₇	—	9	1	no	—
74	C₃×C₃₇	—	9	3	no	—
75	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
76	C₃×C₃₇	#6908	1	1	yes	characteristic
77	C₃×C₃₇	#6908	1	2	yes	—
78	C₃₇⋊C₃	—	9	3	no	—
79	#737	—	111	9	no	—
80	C₃×3₋¹⁺²	—	37	2	no	—
81	C₃×3₋¹⁺²	—	37	2	no	—
82	C₃×3₋¹⁺²	—	111	2	no	—
83	C₃×H(3,𝔽₃)	—	37	1	no	—
84	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₃²×C₉	—	37	1	no	—
86	C₃²×C₉	—	111	1	no	—
87	C₃²×C₉	—	111	1	no	—
88	C₉⋊C₉	—	111	4	no	—
89	C₃²⋊C₉	—	37	2	no	—
90	C₃²⋊C₉	—	37	1	no	—
91	C₃²⋊C₉	—	111	2	no	—
92	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
93	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
94	C₉²	—	111	2	no	—
95	C₃₇	#10624583	1	1	yes	characteristic, complemented, Sylow
96	C₃³	—	3	1	no	—
97	C₃³	—	111	1	no	—
98	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
99	3₋¹⁺²	—	111	18	no	—
100	3₋¹⁺²	—	111	6	no	—
101	3₋¹⁺²	—	111	6	no	—
102	H(3,𝔽₃)	—	3	3	no	—
103	H(3,𝔽₃)	—	111	3	no	—
104	C₃×C₉	—	3	2	no	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	37	1	no	—
107	C₃×C₉	—	37	2	no	—
108	C₃×C₉	—	333	3	no	—
109	C₃×C₉	—	111	3	no	—
110	C₃×C₉	—	111	9	no	—
111	C₃×C₉	—	111	2	no	—
112	C₃×C₉	—	111	1	no	—
113	C₃×C₉	—	111	2	no	—
114	C₃×C₉	—	111	2	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	3	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	2	no	—
119	C₃²	—	9	1	no	—
120	C₃²	—	9	2	no	—
121	C₃²	—	333	1	no	—
122	C₃²	—	333	2	no	—
123	C₃²	—	111	3	no	—
124	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
125	C₉	—	9	1	no	—
126	C₉	—	9	2	no	—
127	C₉	—	37	3	no	—
128	C₉	—	333	1	no	—
129	C₉	—	333	2	no	—
130	C₉	—	111	2	no	—
131	C₉	—	111	6	no	—
132	C₉	—	111	6	no	—
133	C₃	—	9	1	no	—
134	C₃	—	9	3	no	—
135	C₃	—	333	3	no	—
136	C₃	AΣL(1,𝔽₂₇)×C₃₇⋊C₃	1	1	yes	characteristic
137	C₃	#22582631	1	1	yes	characteristic
138	C₃	#22582632	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688530

C₃

yes

maximal, characteristic, complemented

#22582548

C₃

yes

maximal, complemented

#22582539

C₃

yes

maximal, characteristic, complemented

#22582455

C₃

yes

maximal, complemented

#22582450

C₃

yes

maximal, complemented

#22582446

C₃

yes

maximal, complemented

#22582445

C₃

yes

maximal, characteristic, complemented

#22582442

C₃

yes

maximal, characteristic

C₃₇×#6859

C₃

yes

maximal, characteristic, complemented

#15688530

—

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

characteristic

C₃²×#66524

—

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

#15688504

—

C₃²⋊#66524

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

—

#15688502

—

C₉×#66524

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

—

—

—

—

yes

—

C₃×#66524

H(3,𝔽₃)

yes

characteristic

C₉×C₃₇⋊C₃

—

#10624583

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

AΣL(1,𝔽₂₇)

yes

—

C₃×#737

—

C₉×3₋¹⁺²

—

C₉×H(3,𝔽₃)

—

#6860

—

#6859

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃₇⋊C₃

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉²

—

111

—

C₃₇

#10624583

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

—

111

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

111

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

333

—

109

C₃×C₉

—

111

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃×C₉

—

111

—

113

C₃×C₉

—

111

—

114

C₃×C₉

—

111

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

333

—

122

C₃²

—

333

—

123

C₃²

—

111

—

124

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

333

—

129

C₉

—

333

—

130

C₉

—

111

—

131

C₉

—

111

—

132

C₉

—

111

—

133

C₃

—

134

C₃

—

135

C₃

—

333

—

136

C₃

AΣL(1,𝔽₂₇)×C₃₇⋊C₃

yes

characteristic

137

C₃

#22582631

yes

characteristic

138

C₃

#22582632

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6859	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	#22582442	C₃
hypercenter	#6859	C₃₇⋊C₃
hypocenter	C₃₇	#10624583
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6859

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6859

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6859

N	H ≅ G/N	type
1. C₃×#15688530	C₃	semidirect
2. #22582548	C₃	semidirect
3. #22582539	C₃	semidirect
4. #22582455	C₃	semidirect
5. #22582450	C₃	semidirect
6. #22582446	C₃	semidirect
7. #22582445	C₃	semidirect
9. C₃₇×#6859	C₃	semidirect
68. #6859	C₃₇⋊C₃	semidirect
95. C₃₇	#10624583	semidirect

H ≅ G/N

type

1. C₃×#15688530

C₃

semidirect

2. #22582548

C₃

semidirect

3. #22582539

C₃

semidirect

4. #22582455

C₃

semidirect

5. #22582450

C₃

semidirect

6. #22582446

C₃

semidirect

7. #22582445

C₃

semidirect

9. C₃₇×#6859

C₃

semidirect

68. #6859

C₃₇⋊C₃

semidirect

95. C₃₇

#10624583

semidirect

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

subgroup	index
C₃×#15688530	3
#22582548	3
#22582539	3
#22582455	3
#22582450	3
#22582446	3
#22582445	3
#22582442	3
C₃₇×#6859	3
#10624583	37

subgroup

index

C₃₇×#6859

Subgroups — #41005291

Conjugacy classes of subgroups

Complements

Maximal subgroups