Group #41006295 — subgroups

Subgroups — C₃×#22582482 #41006295

Order 26973.

320 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 112 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×#15688531	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688511	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×#11758601	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582482	C₃	1	9	yes	maximal, direct product factor
5	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688531	—	3	2	no	—
7	#15688531	—	3	2	no	—
8	#15688531	—	3	2	no	—
9	#15688531	C₉	1	6	yes	—
10	#15688531	C₃²	1	3	yes	—
11	C₃×#11758608	C₉	1	2	yes	—
12	C₃×#11758605	—	3	1	no	—
13	C₃×#11758606	—	3	1	no	—
14	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
15	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
16	#15688511	—	3	2	no	—
17	#15688511	C₃²	1	3	yes	complemented
18	C₃×#11758601	—	3	2	no	—
19	C₃×#11758601	—	3	1	no	—
20	C₃×#11758601	—	3	1	no	—
21	C₃×#11758601	C₃²	1	3	yes	complemented
22	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
23	C₃₇×C₂₇⋊C₃	—	3	2	no	—
24	C₃₇×C₂₇⋊C₃	C₃²	1	3	yes	complemented
25	C₃×C₂₇×C₃₇	—	3	1	no	—
26	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	3	1	no	—
28	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
29	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
30	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
32	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
33	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
34	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
35	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
36	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
37	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	3	yes	—
38	#11758608	—	3	4	no	—
39	#11758608	C₃×C₉	1	6	yes	—
40	C₃×#66524	—	3	2	no	—
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	C₂₇×C₃₇	—	3	3	no	—
48	#11758601	—	3	3	no	—
49	#11758601	—	3	3	no	—
50	C₃×#6872	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	3	no	—
52	C₃²×C₃₇	C₃²⋊C₉	1	2	yes	—
53	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
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	C₃²⋊C₉	1	2	yes	—
58	C₉×C₃₇	C₃²⋊C₉	1	2	yes	—
59	C₉×C₃₇	C₃²⋊C₉	1	2	yes	—
60	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
62	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
63	#66524	—	3	6	no	—
64	C₃×#737	—	37	1	no	—
65	C₃×C₂₇⋊C₃	—	37	1	no	—
66	C₃×C₂₇⋊C₃	C₃₇⋊C₃	1	1	yes	characteristic, complemented
67	C₃²×C₂₇	—	37	1	no	—
68	#6872	—	37	9	no	—
69	C₃×C₃₇	—	3	2	no	—
70	C₃×C₃₇	—	3	1	no	—
71	C₃×C₃₇	#6872	1	3	yes	complemented
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	2	no	—
80	C₃×3₋¹⁺²	—	111	1	no	—
81	C₃×3₋¹⁺²	—	111	1	no	—
82	C₃×H(3,𝔽₃)	—	37	1	no	—
83	C₃²×C₉	—	111	1	no	—
84	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₂₇⋊C₃	—	3	2	no	—
86	C₂₇⋊C₃	—	37	3	no	—
87	C₂₇⋊C₃	—	111	2	no	—
88	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	3	yes	complemented
89	C₃×C₂₇	—	3	1	no	—
90	C₃×C₂₇	—	37	3	no	—
91	C₃×C₂₇	—	111	2	no	—
92	C₃×C₂₇	—	111	1	no	—
93	C₃×C₂₇	—	111	1	no	—
94	C₃₇	C₃×#6872	1	1	yes	characteristic, complemented, Sylow
95	C₃³	—	111	1	no	—
96	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
97	3₋¹⁺²	—	37	6	no	—
98	3₋¹⁺²	—	111	4	no	—
99	3₋¹⁺²	—	111	6	no	—
100	3₋¹⁺²	—	111	6	no	—
101	3₋¹⁺²	—	111	3	no	—
102	3₋¹⁺²	—	111	3	no	—
103	H(3,𝔽₃)	—	37	3	no	—
104	H(3,𝔽₃)	—	111	2	no	—
105	C₃×C₉	—	111	2	no	—
106	C₃×C₉	—	111	6	no	—
107	C₃×C₉	—	111	3	no	—
108	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
109	C₃×C₉	#11758605	1	1	yes	characteristic
110	C₃×C₉	#11758605	1	2	yes	—
111	C₃×C₉	#11758606	1	1	yes	characteristic
112	C₃×C₉	#11758606	1	2	yes	—
113	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
114	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
115	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
116	C₂₇	—	3	3	no	—
117	C₂₇	—	111	3	no	—
118	C₂₇	—	111	3	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	333	3	no	—
121	C₃²	—	111	3	no	—
122	C₃²	#15688502	1	2	yes	—
123	C₃²	#15688502	1	1	yes	characteristic
124	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
125	C₉	—	111	6	no	—
126	C₉	#15688502	1	2	yes	—
127	C₉	#15688502	1	2	yes	—
128	C₉	#15688502	1	2	yes	—
129	C₉	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
130	C₉	C₃×#11758606	1	1	yes	characteristic
131	C₉	C₃×#11758605	1	1	yes	characteristic
132	C₃	—	3	2	no	—
133	C₃	—	3	1	no	—
134	C₃	—	333	3	no	—
135	C₃	#22582482	1	3	yes	direct product factor
136	C₃	C₃×#15688502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688531

C₃

yes

maximal, characteristic

C₃×#15688511

C₃

yes

maximal, characteristic, complemented

C₃²×#11758601

C₃

yes

maximal, characteristic, complemented

#22582482

C₃

yes

maximal, direct product factor

C₃×C₃₇×C₂₇⋊C₃

C₃

yes

maximal, characteristic, complemented

—

—

—

C₉

yes

—

#15688531

C₃²

yes

—

C₃×#11758608

C₉

yes

—

C₃×#11758605

—

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

#15688511

—

#15688511

C₃²

yes

—

—

—

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₂₇⋊C₃

—

C₃₇×C₂₇⋊C₃

C₃²

yes

complemented

C₃×C₂₇×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

#11758608

—

#11758608

C₃×C₉

yes

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₂₇×C₃₇

—

—

—

—

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

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

#66524

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

C₃₇⋊C₃

yes

characteristic, complemented

C₃²×C₂₇

—

#6872

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

#6872

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×H(3,𝔽₃)

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

C₃×C₃₇⋊C₃

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₃₇

C₃×#6872

yes

characteristic, complemented, Sylow

C₃³

—

111

—

C₃³

C₉×C₃₇⋊C₃

yes

characteristic

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

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃×C₉

C₉×C₃₇⋊C₃

yes

—

109

C₃×C₉

#11758605

yes

characteristic

110

C₃×C₉

#11758605

yes

—

111

C₃×C₉

#11758606

yes

characteristic

112

C₃×C₉

#11758606

yes

—

113

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

114

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

115

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

116

C₂₇

—

117

C₂₇

—

111

—

118

C₂₇

—

111

—

119

C₃²

—

120

C₃²

—

333

—

121

C₃²

—

111

—

122

C₃²

#15688502

yes

—

123

C₃²

#15688502

yes

characteristic

124

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

125

C₉

—

111

—

126

C₉

#15688502

yes

—

127

C₉

#15688502

yes

—

128

C₉

#15688502

yes

—

129

C₉

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

130

C₉

C₃×#11758606

yes

characteristic

131

C₉

C₃×#11758605

yes

characteristic

132

C₃

—

133

C₃

—

134

C₃

—

333

—

135

C₃

#22582482

yes

direct product factor

136

C₃

C₃×#15688502

yes

characteristic

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₂₇⋊C₃	C₃
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃×C₂₇⋊C₃	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6872
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₂₇⋊C₃

subgroup

G/H

center

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃×C₃₇×C₂₇⋊C₃

C₃

socle

C₃²×C₃₇

C₃²⋊C₉

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

cosocle

C₃×C₉×C₃₇

C₃³

hypercenter

C₃×C₂₇⋊C₃

C₃₇⋊C₃

hypocenter

C₃₇

C₃×#6872

LCS

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₂₇⋊C₃

N	H ≅ G/N	type
2. C₃×#15688511	C₃	semidirect
3. C₃²×#11758601	C₃	semidirect
4. #22582482	C₃	direct (N × H)
5. C₃×C₃₇×C₂₇⋊C₃	C₃	semidirect
17. #15688511	C₃²	semidirect
21. C₃×#11758601	C₃²	semidirect
24. C₃₇×C₂₇⋊C₃	C₃²	semidirect
66. C₃×C₂₇⋊C₃	C₃₇⋊C₃	semidirect
71. C₃×C₃₇	#6872	semidirect
88. C₂₇⋊C₃	C₃×C₃₇⋊C₃	semidirect
94. C₃₇	C₃×#6872	semidirect
135. C₃	#22582482	direct (N × H)

H ≅ G/N

type

2. C₃×#15688511

C₃

semidirect

3. C₃²×#11758601

C₃

semidirect

4. #22582482

C₃

direct (N × H)

5. C₃×C₃₇×C₂₇⋊C₃

C₃

semidirect

17. #15688511

C₃²

semidirect

21. C₃×#11758601

C₃²

semidirect

24. C₃₇×C₂₇⋊C₃

C₃²

semidirect

66. C₃×C₂₇⋊C₃

C₃₇⋊C₃

semidirect

71. C₃×C₃₇

#6872

semidirect

88. C₂₇⋊C₃

C₃×C₃₇⋊C₃

semidirect

94. C₃₇

C₃×#6872

semidirect

135. C₃

#22582482

direct (N × H)

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

subgroup	index
C₃×#15688531	3
C₃×#15688511	3
C₃²×#11758601	3
#22582482	3
C₃×C₃₇×C₂₇⋊C₃	3
C₃×#6872	37

subgroup

index

C₃×C₃₇×C₂₇⋊C₃

Subgroups — C₃×#22582482 #41006295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582482 #41006295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582482 #41006295