Group #41006443 — subgroups

Subgroups — #41006443

Order 26973.

216 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 73 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	#22582630	C₃	1	1	yes	maximal, characteristic
2	#22582614	C₃	1	2	yes	maximal, complemented
3	#22582613	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#15688509	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582487	C₃	1	2	yes	maximal, complemented
6	#22582483	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582482	C₃	1	2	yes	maximal, complemented
8	#22582473	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6873	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688531	C₃²	1	2	yes	—
11	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
12	#15688520	C₉	1	2	yes	—
13	#15688518	C₉	1	1	yes	characteristic
14	#15688517	C₉	1	2	yes	—
15	#15688515	C₉	1	1	yes	characteristic
16	#15688514	C₉	1	2	yes	—
17	#15688512	C₉	1	1	yes	characteristic
18	#15688511	—	3	2	no	—
19	#15688511	C₃²	1	2	yes	—
20	#15688511	C₃²	1	2	yes	—
21	#15688510	—	3	2	no	—
22	#15688509	—	3	2	no	—
23	#15688509	—	3	1	no	—
24	#15688509	—	3	2	no	—
25	C₃×#11758601	—	3	1	no	—
26	C₃×#11758601	C₃²	1	1	yes	characteristic
27	C₃×#11758601	C₃²	1	1	yes	characteristic
28	C₂₇×C₃₇⋊C₃	—	3	2	no	—
29	C₃₇×#737	C₃²	1	1	yes	characteristic
30	C₃₇×C₂₇⋊C₃	C₃²	1	2	yes	—
31	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
32	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
33	3₋¹⁺²×C₃₇	—	3	2	no	—
34	3₋¹⁺²×C₃₇	C₃×C₉	1	2	yes	—
35	H(3,𝔽₃)×C₃₇	C₃×C₉	1	1	yes	characteristic
36	C₃×C₉×C₃₇	—	3	1	no	—
37	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
38	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	2	yes	—
39	#11758608	C₃×C₉	1	2	yes	—
40	#11758608	C₃×C₉	1	2	yes	—
41	C₃×#66524	C₃×C₉	1	1	yes	characteristic
42	C₃×#66524	C₃×C₉	1	1	yes	characteristic
43	#11758606	—	3	2	no	—
44	#11758605	—	3	2	no	—
45	C₉×C₃₇⋊C₃	—	3	2	no	—
46	C₉×C₃₇⋊C₃	—	3	1	no	—
47	C₉×C₃₇⋊C₃	—	3	2	no	—
48	C₂₇×C₃₇	—	3	1	no	—
49	C₂₇×C₃₇	—	3	2	no	—
50	#11758601	—	3	2	no	—
51	#11758601	—	3	2	no	—
52	#11758601	—	3	1	no	—
53	#11758601	H(3,𝔽₃)	1	2	yes	—
54	#11758601	H(3,𝔽₃)	1	1	yes	characteristic
55	#10624725	—	37	1	no	maximal, Sylow
56	C₃²×C₃₇	—	3	1	no	—
57	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
58	C₃×C₃₇⋊C₃	—	3	1	no	—
59	C₃×C₃₇⋊C₃	—	3	2	no	—
60	C₃×C₃₇⋊C₃	—	9	1	no	—
61	C₉×C₃₇	—	3	2	no	—
62	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	C₉×C₃₇	#737	1	2	yes	—
64	#66524	—	3	1	no	—
65	#66524	—	3	1	no	—
66	#66524	—	3	2	no	—
67	#66524	—	3	2	no	—
68	#6908	—	37	1	no	—
69	#6903	—	37	1	no	—
70	#6903	—	37	2	no	—
71	C₃×C₂₇⋊C₃	—	37	1	no	—
72	#6873	—	37	2	no	—
73	#6873	C₃₇⋊C₃	1	1	yes	characteristic, complemented
74	#6872	—	37	1	no	—
75	#6872	—	37	2	no	—
76	#6869	—	37	2	no	—
77	C₃×C₃₇	—	3	1	no	—
78	C₃×C₃₇	—	9	1	no	—
79	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
80	C₃₇⋊C₃	—	9	2	no	—
81	C₃₇⋊C₃	—	9	1	no	—
82	#737	—	37	2	no	—
83	#737	C₃×C₃₇⋊C₃	1	1	yes	characteristic
84	C₃²×C₉	—	37	1	no	—
85	#733	—	37	2	no	—
86	#732	—	37	1	no	—
87	#731	—	37	2	no	—
88	#731	—	37	1	no	—
89	AΣL(1,𝔽₂₇)	—	37	2	no	—
90	AΣL(1,𝔽₂₇)	—	37	1	no	—
91	C₂₇⋊C₃	—	37	2	no	—
92	C₂₇⋊C₃	—	37	2	no	—
93	C₂₇⋊C₃	—	111	2	no	—
94	C₂₇⋊C₃	—	111	1	no	—
95	C₂₇⋊C₃	—	111	2	no	—
96	C₂₇⋊C₃	—	111	2	no	—
97	C₂₇⋊C₃	—	111	2	no	—
98	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	2	yes	—
99	C₃×C₂₇	—	37	1	no	—
100	C₃×C₂₇	—	37	1	no	—
101	C₃×C₂₇	—	111	2	no	—
102	C₃×C₂₇	—	111	1	no	—
103	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃₇	#10624725	1	1	yes	characteristic, complemented, Sylow
105	C₃³	—	37	1	no	—
106	3₋¹⁺²	—	3	2	no	—
107	3₋¹⁺²	—	37	2	no	—
108	3₋¹⁺²	—	37	2	no	—
109	3₋¹⁺²	—	111	2	no	—
110	3₋¹⁺²	—	111	2	no	—
111	3₋¹⁺²	C₉×C₃₇⋊C₃	1	2	yes	—
112	H(3,𝔽₃)	—	37	2	no	—
113	H(3,𝔽₃)	C₉×C₃₇⋊C₃	1	1	yes	characteristic
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	37	1	no	—
116	C₃×C₉	—	37	1	no	—
117	C₃×C₉	—	111	2	no	—
118	C₃×C₉	—	111	1	no	—
119	C₃×C₉	—	111	2	no	—
120	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
121	C₂₇	—	3	2	no	—
122	C₂₇	—	3	1	no	—
123	C₂₇	—	37	2	no	—
124	C₂₇	—	37	1	no	—
125	C₂₇	—	111	1	no	—
126	C₂₇	—	111	2	no	—
127	C₂₇	—	111	2	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	333	1	no	—
130	C₃²	—	111	1	no	—
131	C₃²	—	111	2	no	—
132	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
133	C₉	—	3	2	no	—
134	C₉	—	111	1	no	—
135	C₉	—	111	1	no	—
136	C₉	—	111	2	no	—
137	C₉	—	111	2	no	—
138	C₉	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
139	C₉	#15688532	1	2	yes	—
140	C₃	—	3	1	no	—
141	C₃	—	9	1	no	—
142	C₃	—	333	1	no	—
143	C₃	—	333	2	no	—
144	C₃	#22582539	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582630

C₃

yes

maximal, characteristic

#22582614

C₃

yes

maximal, complemented

#22582613

C₃

yes

maximal, characteristic, complemented

C₃×#15688509

C₃

yes

maximal, characteristic, complemented

#22582487

C₃

yes

maximal, complemented

#22582483

C₃

yes

maximal, characteristic, complemented

#22582482

C₃

yes

maximal, complemented

#22582473

C₃

yes

maximal, complemented

C₃₇×#6873

C₃

yes

maximal, characteristic, complemented

#15688531

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688520

C₉

yes

—

#15688518

C₉

yes

characteristic

#15688517

C₉

yes

—

#15688515

C₉

yes

characteristic

#15688514

C₉

yes

—

#15688512

C₉

yes

characteristic

#15688511

—

#15688511

C₃²

yes

—

#15688511

C₃²

yes

—

—

—

—

—

—

C₃²

yes

characteristic

C₃×#11758601

C₃²

yes

characteristic

C₂₇×C₃₇⋊C₃

—

C₃₇×#737

C₃²

yes

characteristic

C₃₇×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇×C₃₇

C₃²

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₃×C₉

yes

—

H(3,𝔽₃)×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

C₃×C₉

yes

—

#11758608

C₃×C₉

yes

—

#11758608

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃×C₉

yes

characteristic

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₂₇×C₃₇

—

C₂₇×C₃₇

—

—

—

—

yes

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

#737

yes

—

—

—

—

—

—

—

—

C₃×C₂₇⋊C₃

—

#6873

—

#6873

C₃₇⋊C₃

yes

characteristic, complemented

#6872

—

#6872

—

#6869

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

#737

—

#737

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

C₃×C₃₇⋊C₃

yes

—

C₃×C₂₇

—

100

C₃×C₂₇

—

101

C₃×C₂₇

—

111

—

102

C₃×C₂₇

—

111

—

103

C₃×C₂₇

C₃×C₃₇⋊C₃

yes

characteristic

104

C₃₇

#10624725

yes

characteristic, complemented, Sylow

105

C₃³

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

111

—

110

3₋¹⁺²

—

111

—

111

3₋¹⁺²

C₉×C₃₇⋊C₃

yes

—

112

H(3,𝔽₃)

—

113

H(3,𝔽₃)

C₉×C₃₇⋊C₃

yes

characteristic

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

111

—

118

C₃×C₉

—

111

—

119

C₃×C₉

—

111

—

120

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

121

C₂₇

—

122

C₂₇

—

123

C₂₇

—

124

C₂₇

—

125

C₂₇

—

111

—

126

C₂₇

—

111

—

127

C₂₇

—

111

—

128

C₃²

—

129

C₃²

—

333

—

130

C₃²

—

111

—

131

C₃²

—

111

—

132

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

133

C₉

—

134

C₉

—

111

—

135

C₉

—

111

—

136

C₉

—

111

—

137

C₉

—

111

—

138

C₉

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

139

C₉

#15688532

yes

—

140

C₃

—

141

C₃

—

142

C₃

—

333

—

143

C₃

—

333

—

144

C₃

#22582539

yes

characteristic

	subgroup	G/H
center	C₉	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6873	C₃
socle	C₃×C₃₇	C₉×H(3,𝔽₃)
prime radical	#22582630	C₃
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	#6873	C₃₇⋊C₃
hypocenter	C₃₇	#10624725
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ #6873

subgroup

G/H

center

C₉

H(3,𝔽₃)×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6873

C₃

socle

C₃×C₃₇

C₉×H(3,𝔽₃)

prime radical

#22582630

C₃

cosocle

C₃×C₉×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₉ ◁ C₃×C₉ ◁ #6873

N	H ≅ G/N	type
2. #22582614	C₃	semidirect
3. #22582613	C₃	semidirect
4. C₃×#15688509	C₃	semidirect
5. #22582487	C₃	semidirect
6. #22582483	C₃	semidirect
7. #22582482	C₃	semidirect
8. #22582473	C₃	semidirect
9. C₃₇×#6873	C₃	semidirect
73. #6873	C₃₇⋊C₃	semidirect
104. C₃₇	#10624725	semidirect

H ≅ G/N

type

2. #22582614

C₃

semidirect

3. #22582613

C₃

semidirect

4. C₃×#15688509

C₃

semidirect

5. #22582487

C₃

semidirect

6. #22582483

C₃

semidirect

7. #22582482

C₃

semidirect

8. #22582473

C₃

semidirect

9. C₃₇×#6873

C₃

semidirect

73. #6873

C₃₇⋊C₃

semidirect

104. C₃₇

#10624725

semidirect

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

subgroup	index
#22582630	3
#22582614	3
#22582613	3
C₃×#15688509	3
#22582487	3
#22582483	3
#22582482	3
#22582473	3
C₃₇×#6873	3
#10624725	37

subgroup

index

C₃₇×#6873

Subgroups — #41006443

Conjugacy classes of subgroups

Complements

Maximal subgroups