Group #41008524 — subgroups

Subgroups — #41008524

Order 26973.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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×#15688541	C₃	1	2	yes	maximal, complemented
2	C₃×#15688541	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582702	C₃	1	9	yes	maximal, complemented
4	C₃×#15688532	C₃	1	1	yes	maximal, characteristic
5	#15688541	—	3	2	no	—
6	#15688541	—	3	2	no	—
7	#15688541	—	3	1	no	—
8	#15688541	—	3	2	no	—
9	#15688541	C₃²	1	6	yes	complemented
10	C₃×#11758603	—	3	2	no	—
11	C₃×#11758603	C₃²	1	3	yes	complemented
12	#15688532	—	3	2	no	—
13	#15688532	—	3	2	no	—
14	#15688532	—	3	2	no	—
15	#15688532	C₉	1	6	yes	—
16	#15688532	C₃²	1	3	yes	—
17	C₃×#11758608	C₉	1	1	yes	characteristic
18	C₃×#11758605	—	3	1	no	—
19	C₃×#11758606	—	3	1	no	—
20	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
21	C₃²×#66524	C₃²	1	1	yes	characteristic
22	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
23	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
24	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	—	3	1	no	—
26	3₋¹⁺²×C₃₇	—	3	2	no	—
27	3₋¹⁺²×C₃₇	C₃×C₉	1	3	yes	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
30	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	3	yes	—
31	#11758608	—	3	2	no	—
32	#11758608	C₃×C₉	1	3	yes	—
33	C₃×#66524	—	3	1	no	—
34	C₃×#66524	C₃×C₉	1	2	yes	—
35	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
37	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
38	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
39	C₃×#66524	3₋¹⁺²	1	2	yes	—
40	C₃×#66524	3₋¹⁺²	1	2	yes	—
41	C₃×#66524	C₃³	1	1	yes	characteristic
42	#11758606	—	3	6	no	—
43	#11758606	—	3	3	no	—
44	#11758605	—	3	6	no	—
45	#11758605	—	3	3	no	—
46	C₉×C₃₇⋊C₃	—	3	6	no	—
47	C₉×C₃₇⋊C₃	—	3	3	no	—
48	#11758603	—	3	6	no	—
49	#11758603	—	3	3	no	—
50	#10624704	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	9	1	no	—
52	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
53	C₃²×C₃₇	C₃²⋊C₉	1	2	yes	—
54	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
55	C₃×C₃₇⋊C₃	—	3	3	no	—
56	C₃×C₃₇⋊C₃	—	9	3	no	—
57	C₉×C₃₇	—	3	3	no	—
58	#66524	—	3	3	no	—
59	#66524	C₃²⋊C₉	1	2	yes	—
60	#66524	C₃²⋊C₉	1	2	yes	—
61	#66524	C₃²⋊C₉	1	2	yes	—
62	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
64	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
65	C₃×#737	—	37	1	no	—
66	C₃×C₂₇⋊C₃	—	37	1	no	—
67	C₃×C₂₇⋊C₃	—	37	2	no	—
68	#6873	—	37	9	no	—
69	C₃×C₃₇	—	3	1	no	—
70	C₃×C₃₇	—	3	2	no	—
71	C₃×C₃₇	—	3	1	no	—
72	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
73	C₃₇⋊C₃	—	9	3	no	—
74	#737	—	37	6	no	—
75	#737	—	37	3	no	—
76	#737	—	111	2	no	—
77	#737	—	111	2	no	—
78	#737	—	111	2	no	—
79	C₃×3₋¹⁺²	—	37	1	no	—
80	C₃×3₋¹⁺²	—	111	1	no	—
81	C₃×3₋¹⁺²	—	111	1	no	—
82	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic
83	C₃×H(3,𝔽₃)	—	37	1	no	—
84	C₃²×C₉	—	37	1	no	—
85	C₃²×C₉	—	111	1	no	—
86	C₂₇⋊C₃	—	37	6	no	—
87	C₂₇⋊C₃	—	111	2	no	—
88	C₂₇⋊C₃	—	111	1	no	—
89	C₂₇⋊C₃	—	111	2	no	—
90	C₂₇⋊C₃	—	111	2	no	—
91	C₃×C₂₇	—	37	3	no	—
92	C₃×C₂₇	—	111	2	no	—
93	C₃₇	#10624704	1	1	yes	characteristic, complemented, Sylow
94	C₃³	—	111	1	no	—
95	C₃³	C₃×#66526	1	1	yes	characteristic
96	3₋¹⁺²	—	3	2	no	—
97	3₋¹⁺²	—	37	3	no	—
98	3₋¹⁺²	—	111	6	no	—
99	3₋¹⁺²	—	111	3	no	—
100	3₋¹⁺²	—	111	6	no	—
101	3₋¹⁺²	—	111	3	no	—
102	3₋¹⁺²	—	111	2	no	—
103	3₋¹⁺²	C₃×#66526	1	3	yes	—
104	H(3,𝔽₃)	—	37	3	no	—
105	H(3,𝔽₃)	—	111	2	no	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	37	2	no	—
108	C₃×C₉	—	37	1	no	—
109	C₃×C₉	—	37	1	no	—
110	C₃×C₉	—	37	1	no	—
111	C₃×C₉	—	37	2	no	—
112	C₃×C₉	—	37	2	no	—
113	C₃×C₉	—	37	2	no	—
114	C₃×C₉	—	37	1	no	—
115	C₃×C₉	—	111	3	no	—
116	C₃×C₉	—	111	6	no	—
117	C₃×C₉	—	111	1	no	—
118	C₂₇	—	111	3	no	—
119	C₂₇	—	111	6	no	—
120	C₃²	—	9	1	no	—
121	C₃²	—	333	3	no	—
122	C₃²	—	111	3	no	—
123	C₃²	C₃²⋊#66526	1	1	yes	characteristic
124	C₃²	C₃²⋊#66526	1	2	yes	—
125	C₃²	C₃²×#66526	1	1	yes	characteristic
126	C₉	—	3	3	no	—
127	C₉	—	37	1	no	—
128	C₉	—	37	2	no	—
129	C₉	—	37	2	no	—
130	C₉	—	37	2	no	—
131	C₉	—	37	1	no	—
132	C₉	—	37	1	no	—
133	C₉	—	111	3	no	—
134	C₃	—	3	1	no	—
135	C₃	—	3	2	no	—
136	C₃	—	3	1	no	—
137	C₃	—	333	3	no	—
138	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688541

C₃

yes

maximal, complemented

C₃×#15688541

C₃

yes

maximal, characteristic, complemented

#22582702

C₃

yes

maximal, complemented

C₃×#15688532

C₃

yes

maximal, characteristic

—

—

—

—

C₃²

yes

complemented

C₃×#11758603

—

C₃×#11758603

C₃²

yes

—

—

—

C₉

yes

—

#15688532

C₃²

yes

—

C₃×#11758608

C₉

yes

characteristic

C₃×#11758605

—

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

#11758608

—

#11758608

C₃×C₉

yes

—

C₃×#66524

—

C₃×#66524

C₃×C₉

yes

—

yes

yes

—

C₃×#66524

3₋¹⁺²

yes

characteristic

C₃×#66524

3₋¹⁺²

yes

characteristic

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

3₋¹⁺²

yes

—

C₃×#66524

C₃³

yes

—

—

—

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃×H(3,𝔽₃)

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

characteristic

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

#6873

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃₇⋊C₃

—

#737

—

#737

—

#737

—

111

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

#66526

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃₇

#10624704

yes

characteristic, complemented, Sylow

C₃³

—

111

—

C₃³

C₃×#66526

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

—

111

—

102

3₋¹⁺²

—

111

—

103

3₋¹⁺²

C₃×#66526

yes

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

—

111

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

111

—

116

C₃×C₉

—

111

—

117

C₃×C₉

—

111

—

118

C₂₇

—

111

—

119

C₂₇

—

111

—

120

C₃²

—

121

C₃²

—

333

—

122

C₃²

—

111

—

123

C₃²

C₃²⋊#66526

yes

characteristic

124

C₃²

C₃²⋊#66526

yes

—

125

C₃²

C₃²×#66526

yes

characteristic

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

111

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

—

333

—

138

C₃

C₃×C₃²⋊#66526

yes

characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
socle	C₃×C₃₇	C₃×C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	#10624704
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃

C₃×C₃²⋊#66526

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃²

C₃²×#66526

Fitting

C₃×3₋¹⁺²×C₃₇

C₉

socle

C₃×C₃₇

C₃×C₃²⋊C₉

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

C₃³

C₃×3₋¹⁺²

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#15688541	C₃	semidirect
2. C₃×#15688541	C₃	semidirect
3. #22582702	C₃	semidirect
9. #15688541	C₃²	semidirect
11. C₃×#11758603	C₃²	semidirect
93. C₃₇	#10624704	semidirect

H ≅ G/N

type

1. C₃×#15688541

C₃

semidirect

2. C₃×#15688541

C₃

semidirect

3. #22582702

C₃

semidirect

9. #15688541

C₃²

semidirect

11. C₃×#11758603

C₃²

semidirect

93. C₃₇

#10624704

semidirect

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

subgroup	index
C₃×#15688541	3
#22582702	3
C₃×#15688532	3
#10624704	37

subgroup

index

Subgroups — #41008524

Conjugacy classes of subgroups

Complements

Maximal subgroups