Group #41007557 — subgroups

Subgroups — #41007557

Order 26973.

1166 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 292 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×#15688532	C₃	1	3	yes	maximal, complemented
2	C₃³×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582565	C₃	1	6	yes	maximal, complemented
4	#22582561	C₃	1	9	yes	maximal, complemented
5	#22582560	C₃	1	9	yes	maximal, complemented
6	#22582554	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582550	C₃	1	1	yes	maximal, characteristic
8	#22582539	C₃	1	9	yes	maximal, complemented
9	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688532	—	3	27	no	—
11	C₃×#11758608	C₃²	1	3	yes	—
12	C₃×#11758605	C₃²	1	9	yes	complemented
13	C₃×#11758606	C₃²	1	9	yes	complemented
14	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	9	yes	—
15	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
16	C₃²×#66524	—	3	9	no	—
17	C₃²×#66524	C₃²	1	3	yes	—
18	C₃²×#66524	C₃²	1	3	yes	—
19	C₃²×#66524	C₃²	1	3	yes	—
20	C₃²×#66524	C₃²	1	2	yes	—
21	C₃²×#66524	C₃²	1	1	yes	characteristic
22	C₃×C₉×C₃₇⋊C₃	C₃²	1	9	yes	complemented
23	C₃²⋊#66524	C₃²	1	9	yes	—
24	#15688502	C₃²	1	54	yes	complemented
25	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
26	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
27	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
28	C₃³×C₃₇	—	3	3	no	—
29	C₃³×C₃₇	—	3	9	no	—
30	C₃³×C₃₇	C₃³	1	1	yes	characteristic
31	C₃³×C₃₇	C₃³	1	3	yes	—
32	C₃²×C₃₇⋊C₃	C₃³	1	9	yes	—
33	3₋¹⁺²×C₃₇	—	3	9	no	—
34	C₃×C₉×C₃₇	C₃³	1	9	yes	—
35	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
36	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
37	#11758608	—	3	9	no	—
38	C₃×#66524	—	3	6	no	—
39	C₃×#66524	—	3	9	no	—
40	C₃×#66524	—	3	9	no	—
41	C₃×#66524	—	3	9	no	—
42	C₃×#66524	H(3,𝔽₃)	1	9	yes	complemented
43	C₃×#66524	C₃³	1	6	yes	—
44	C₃×#66524	C₃³	1	9	yes	—
45	C₃×#66524	C₃³	1	1	yes	characteristic
46	C₃×#66524	C₃³	1	1	yes	characteristic
47	C₃×#66524	C₃³	1	1	yes	characteristic
48	#11758606	—	3	27	no	—
49	#11758605	—	3	27	no	—
50	C₉×C₃₇⋊C₃	—	3	27	no	—
51	#10624869	—	37	1	no	maximal, Sylow
52	C₃²×C₃₇	—	3	27	no	—
53	C₃²×C₃₇	—	3	3	no	—
54	C₃²×C₃₇	—	3	3	no	—
55	C₃²×C₃₇	—	3	6	no	—
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
57	C₃²×C₃₇	#737	1	9	yes	complemented
58	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
59	C₃×C₃₇⋊C₃	—	3	9	no	—
60	C₃×C₃₇⋊C₃	—	3	27	no	—
61	C₉×C₃₇	—	3	9	no	—
62	#66524	—	3	9	no	—
63	#66524	—	3	6	no	—
64	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
65	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
66	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
67	C₃×#737	—	37	3	no	—
68	C₃³×C₉	—	37	1	no	—
69	#6893	—	37	6	no	—
70	C₉⋊H(3,𝔽₃)	—	37	9	no	—
71	C₉⋊H(3,𝔽₃)	—	37	9	no	—
72	#6891	—	37	1	no	—
73	#6891	C₃₇⋊C₃	1	1	yes	characteristic, complemented
74	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
75	C₉×H(3,𝔽₃)	—	37	9	no	—
76	C₃×C₃₇	—	3	3	no	—
77	C₃×C₃₇	—	3	9	no	—
78	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃×C₃₇	C₃×#737	1	3	yes	—
80	C₃₇⋊C₃	—	9	9	no	—
81	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
82	#737	—	111	27	no	—
83	C₃×3₋¹⁺²	—	37	3	no	—
84	C₃×3₋¹⁺²	—	37	9	no	—
85	C₃×3₋¹⁺²	—	37	9	no	—
86	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
87	C₃×H(3,𝔽₃)	—	37	3	no	—
88	C₃×H(3,𝔽₃)	—	37	9	no	—
89	C₃²×C₉	—	37	1	no	—
90	C₃²×C₉	—	37	3	no	—
91	C₃²×C₉	—	37	3	no	—
92	C₃²×C₉	—	37	3	no	—
93	C₃²×C₉	—	37	2	no	—
94	C₃²×C₉	—	37	9	no	—
95	C₃²×C₉	—	111	9	no	—
96	C₃²⋊C₉	—	37	9	no	—
97	C₃²⋊C₉	—	37	54	no	—
98	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	9	yes	—
99	C₃₇	#10624869	1	1	yes	characteristic, complemented, Sylow
100	C₃³	—	3	3	no	—
101	C₃³	—	3	9	no	—
102	C₃³	—	37	9	no	—
103	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
105	3₋¹⁺²	—	3	9	no	—
106	3₋¹⁺²	—	111	27	no	—
107	3₋¹⁺²	—	111	27	no	—
108	3₋¹⁺²	—	111	9	no	—
109	H(3,𝔽₃)	—	111	9	no	—
110	H(3,𝔽₃)	—	111	27	no	—
111	C₃×C₉	—	37	9	no	—
112	C₃×C₉	—	37	1	no	—
113	C₃×C₉	—	37	1	no	—
114	C₃×C₉	—	37	1	no	—
115	C₃×C₉	—	37	6	no	—
116	C₃×C₉	—	37	9	no	—
117	C₃×C₉	—	111	27	no	—
118	C₃×C₉	—	111	9	no	—
119	C₃×C₉	—	111	9	no	—
120	C₃×C₉	—	111	9	no	—
121	C₃×C₉	—	111	6	no	—
122	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
123	C₃²	—	3	27	no	—
124	C₃²	—	3	3	no	—
125	C₃²	—	3	3	no	—
126	C₃²	—	3	6	no	—
127	C₃²	—	111	9	no	—
128	C₃²	—	111	27	no	—
129	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
130	C₃²	#15688532	1	9	yes	complemented
131	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
132	C₉	—	3	9	no	—
133	C₉	—	37	3	no	—
134	C₉	—	37	3	no	—
135	C₉	—	37	3	no	—
136	C₉	—	111	6	no	—
137	C₉	—	111	9	no	—
138	C₃	—	3	3	no	—
139	C₃	—	3	9	no	—
140	C₃	—	333	9	no	—
141	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
142	C₃	C₃×#15688532	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688532

C₃

yes

maximal, complemented

C₃³×#66524

C₃

yes

maximal, characteristic, complemented

#22582565

C₃

yes

maximal, complemented

#22582561

C₃

yes

maximal, complemented

#22582560

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, characteristic, complemented

#22582550

C₃

yes

maximal, characteristic

#22582539

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic, complemented

#15688532

—

C₃×#11758608

C₃²

yes

—

C₃×#11758605

C₃²

yes

complemented

C₃×#11758606

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

#15688502

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

yes

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624869

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

#737

yes

complemented

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

—

#66524

C₃×H(3,𝔽₃)

yes

—

#66524

C₃×H(3,𝔽₃)

yes

—

#66524

C₃×H(3,𝔽₃)

yes

—

C₃×#737

—

C₃³×C₉

—

#6893

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic, complemented

C₃²⋊H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

C₃×#737

yes

—

C₃₇⋊C₃

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624869

yes

characteristic, complemented, Sylow

100

C₃³

—

101

C₃³

—

102

C₃³

—

103

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

104

C₃³

C₃²×C₃₇⋊C₃

yes

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

111

—

107

3₋¹⁺²

—

111

—

108

3₋¹⁺²

—

111

—

109

H(3,𝔽₃)

—

111

—

110

H(3,𝔽₃)

—

111

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

111

—

118

C₃×C₉

—

111

—

119

C₃×C₉

—

111

—

120

C₃×C₉

—

111

—

121

C₃×C₉

—

111

—

122

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

111

—

128

C₃²

—

111

—

129

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

130

C₃²

#15688532

yes

complemented

131

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

—

111

—

137

C₉

—

111

—

138

C₃

—

139

C₃

—

140

C₃

—

333

—

141

C₃

C₃×H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

142

C₃

C₃×#15688532

yes

—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6891	C₃
prime radical	#22582550	C₃
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624869
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃₇×#6891

C₃

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
1. C₃×#15688532	C₃	semidirect
2. C₃³×#66524	C₃	semidirect
3. #22582565	C₃	semidirect
4. #22582561	C₃	semidirect
5. #22582560	C₃	semidirect
6. #22582554	C₃	semidirect
8. #22582539	C₃	semidirect
9. C₃₇×#6891	C₃	semidirect
12. C₃×#11758605	C₃²	semidirect
13. C₃×#11758606	C₃²	semidirect
22. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
24. #15688502	C₃²	semidirect
42. C₃×#66524	H(3,𝔽₃)	semidirect
57. C₃²×C₃₇	#737	semidirect
73. #6891	C₃₇⋊C₃	semidirect
99. C₃₇	#10624869	semidirect
130. C₃²	#15688532	semidirect

H ≅ G/N

type

1. C₃×#15688532

C₃

semidirect

2. C₃³×#66524

C₃

semidirect

3. #22582565

C₃

semidirect

4. #22582561

C₃

semidirect

5. #22582560

C₃

semidirect

6. #22582554

C₃

semidirect

8. #22582539

C₃

semidirect

9. C₃₇×#6891

C₃

semidirect

12. C₃×#11758605

C₃²

semidirect

13. C₃×#11758606

C₃²

semidirect

22. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

24. #15688502

C₃²

semidirect

42. C₃×#66524

H(3,𝔽₃)

semidirect

57. C₃²×C₃₇

#737

semidirect

73. #6891

C₃₇⋊C₃

semidirect

99. C₃₇

#10624869

semidirect

130. C₃²

#15688532

semidirect

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

subgroup	index
C₃×#15688532	3
C₃³×#66524	3
#22582565	3
#22582561	3
#22582560	3
#22582554	3
#22582550	3
#22582539	3
C₃₇×#6891	3
#10624869	37

subgroup

index

C₃³×#66524

C₃₇×#6891

Subgroups — #41007557

Conjugacy classes of subgroups

Complements

Maximal subgroups