Group #41008520 — subgroups

Subgroups — C₃×#22582702 #41008520

Order 26973.

320 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₃²×#11758603	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582702	C₃	1	9	yes	maximal, direct product factor
4	C₃×#15688532	C₃	1	1	yes	maximal, characteristic
5	#15688541	—	3	4	no	—
6	#15688541	C₃²	1	6	yes	complemented
7	C₃×#11758603	—	3	1	no	—
8	C₃×#11758603	—	3	2	no	—
9	C₃×#11758603	—	3	2	no	—
10	C₃×#11758603	C₃²	1	3	yes	complemented
11	#15688532	—	3	2	no	—
12	#15688532	—	3	2	no	—
13	#15688532	—	3	2	no	—
14	#15688532	C₉	1	6	yes	—
15	#15688532	C₃²	1	3	yes	—
16	C₃×#11758608	C₉	1	1	yes	characteristic
17	C₃×#11758605	—	3	1	no	—
18	C₃×#11758606	—	3	1	no	—
19	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
20	C₃²×#66524	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
22	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
23	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	3₋¹⁺²×C₃₇	—	3	2	no	—
26	3₋¹⁺²×C₃₇	C₃×C₉	1	3	yes	—
27	C₃×C₉×C₃₇	—	3	1	no	—
28	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
29	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	3	yes	—
30	#11758608	—	3	2	no	—
31	#11758608	C₃×C₉	1	3	yes	—
32	C₃×#66524	—	3	1	no	—
33	C₃×#66524	C₃×C₉	1	2	yes	—
34	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
36	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
37	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
38	C₃×#66524	3₋¹⁺²	1	2	yes	—
39	C₃×#66524	3₋¹⁺²	1	2	yes	—
40	C₃×#66524	C₃³	1	1	yes	characteristic
41	#11758606	—	3	6	no	—
42	#11758606	—	3	3	no	—
43	#11758605	—	3	6	no	—
44	#11758605	—	3	3	no	—
45	C₉×C₃₇⋊C₃	—	3	6	no	—
46	C₉×C₃₇⋊C₃	—	3	3	no	—
47	#11758603	—	3	3	no	—
48	#11758603	—	3	6	no	—
49	C₃×#6873	—	37	1	no	maximal, Sylow
50	C₃²×C₃₇	—	3	3	no	—
51	C₃²×C₃₇	C₃²⋊C₉	1	2	yes	—
52	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
53	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
54	C₃×C₃₇⋊C₃	—	3	3	no	—
55	C₃×C₃₇⋊C₃	—	9	3	no	—
56	C₉×C₃₇	—	3	3	no	—
57	#66524	—	3	3	no	—
58	#66524	C₃²⋊C₉	1	2	yes	—
59	#66524	C₃²⋊C₉	1	2	yes	—
60	#66524	C₃²⋊C₉	1	2	yes	—
61	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
63	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃×#737	—	37	1	no	—
65	C₃×C₂₇⋊C₃	—	37	2	no	—
66	C₃²×C₂₇	—	37	1	no	—
67	#6873	—	37	9	no	—
68	C₃×C₃₇	—	3	2	no	—
69	C₃×C₃₇	—	3	1	no	—
70	C₃×C₃₇	#6873	1	3	yes	complemented
71	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
72	C₃₇⋊C₃	—	9	3	no	—
73	#737	—	37	6	no	—
74	#737	—	37	3	no	—
75	#737	—	111	2	no	—
76	#737	—	111	2	no	—
77	#737	—	111	2	no	—
78	C₃×3₋¹⁺²	—	37	1	no	—
79	C₃×3₋¹⁺²	—	111	1	no	—
80	C₃×3₋¹⁺²	—	111	1	no	—
81	C₃×3₋¹⁺²	#66526	1	1	yes	characteristic
82	C₃×H(3,𝔽₃)	—	37	1	no	—
83	C₃²×C₉	—	37	1	no	—
84	C₃²×C₉	—	111	1	no	—
85	C₂₇⋊C₃	—	37	6	no	—
86	C₂₇⋊C₃	—	111	4	no	—
87	C₃×C₂₇	—	37	3	no	—
88	C₃×C₂₇	—	111	2	no	—
89	C₃×C₂₇	—	111	1	no	—
90	C₃×C₂₇	—	111	2	no	—
91	C₃₇	C₃×#6873	1	1	yes	characteristic, complemented, Sylow
92	C₃³	—	111	1	no	—
93	C₃³	C₃×#66526	1	1	yes	characteristic
94	3₋¹⁺²	—	3	2	no	—
95	3₋¹⁺²	—	37	3	no	—
96	3₋¹⁺²	—	111	6	no	—
97	3₋¹⁺²	—	111	3	no	—
98	3₋¹⁺²	—	111	6	no	—
99	3₋¹⁺²	—	111	3	no	—
100	3₋¹⁺²	—	111	2	no	—
101	3₋¹⁺²	C₃×#66526	1	3	yes	—
102	H(3,𝔽₃)	—	37	3	no	—
103	H(3,𝔽₃)	—	111	2	no	—
104	C₃×C₉	—	3	1	no	—
105	C₃×C₉	—	37	2	no	—
106	C₃×C₉	—	37	2	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	1	no	—
113	C₃×C₉	—	111	3	no	—
114	C₃×C₉	—	111	6	no	—
115	C₃×C₉	—	111	1	no	—
116	C₂₇	—	111	3	no	—
117	C₂₇	—	111	6	no	—
118	C₃²	—	3	3	no	—
119	C₃²	—	333	3	no	—
120	C₃²	—	111	3	no	—
121	C₃²	C₃²⋊#66526	1	2	yes	—
122	C₃²	C₃²⋊#66526	1	1	yes	characteristic
123	C₃²	C₃²×#66526	1	1	yes	characteristic
124	C₉	—	3	3	no	—
125	C₉	—	37	1	no	—
126	C₉	—	37	2	no	—
127	C₉	—	37	2	no	—
128	C₉	—	37	2	no	—
129	C₉	—	37	1	no	—
130	C₉	—	37	1	no	—
131	C₉	—	111	3	no	—
132	C₃	—	3	2	no	—
133	C₃	—	3	1	no	—
134	C₃	—	333	3	no	—
135	C₃	#22582702	1	3	yes	direct product factor
136	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688541

C₃

yes

maximal, complemented

C₃²×#11758603

C₃

yes

maximal, characteristic, complemented

#22582702

C₃

yes

maximal, direct product factor

C₃×#15688532

C₃

yes

maximal, characteristic

#15688541

—

#15688541

C₃²

yes

—

—

—

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

—

C₃²×C₃₇

C₃²⋊C₉

yes

characteristic

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

—

C₃²×C₂₇

—

#6873

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6873

yes

complemented

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

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃₇

C₃×#6873

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

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

C₃×#66526

yes

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

111

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

111

—

114

C₃×C₉

—

111

—

115

C₃×C₉

—

111

—

116

C₂₇

—

111

—

117

C₂₇

—

111

—

118

C₃²

—

119

C₃²

—

333

—

120

C₃²

—

111

—

121

C₃²

C₃²⋊#66526

yes

—

122

C₃²

C₃²⋊#66526

yes

characteristic

123

C₃²

C₃²×#66526

yes

characteristic

124

C₉

—

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

111

—

132

C₃

—

133

C₃

—

134

C₃

—

333

—

135

C₃

#22582702

yes

direct product factor

136

C₃

C₃×C₃²⋊#66526

yes

characteristic

	subgroup	G/H
center	C₃²	C₃²⋊#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting	C₃×3₋¹⁺²×C₃₇	C₉
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃×3₋¹⁺²	#66526
hypocenter	C₃₇	C₃×#6873
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃²⋊#66526

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃²

C₃²×#66526

Fitting

C₃×3₋¹⁺²×C₃₇

C₉

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

C₃³

C₃×3₋¹⁺²

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#15688541	C₃	semidirect
2. C₃²×#11758603	C₃	semidirect
3. #22582702	C₃	direct (N × H)
6. #15688541	C₃²	semidirect
10. C₃×#11758603	C₃²	semidirect
70. C₃×C₃₇	#6873	semidirect
91. C₃₇	C₃×#6873	semidirect
135. C₃	#22582702	direct (N × H)

H ≅ G/N

type

1. C₃×#15688541

C₃

semidirect

2. C₃²×#11758603

C₃

semidirect

3. #22582702

C₃

direct (N × H)

6. #15688541

C₃²

semidirect

10. C₃×#11758603

C₃²

semidirect

70. C₃×C₃₇

#6873

semidirect

91. C₃₇

C₃×#6873

semidirect

135. C₃

#22582702

direct (N × H)

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

subgroup	index
C₃×#15688541	3
C₃²×#11758603	3
#22582702	3
C₃×#15688532	3
C₃×#6873	37

subgroup

index

Subgroups — C₃×#22582702 #41008520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582702 #41008520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582702 #41008520