Group #41008238 — subgroups

Subgroups — #41008238

Order 26973.

948 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 286 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	#22582679	C₃	1	3	yes	maximal, complemented
2	#22582661	C₃	1	18	yes	maximal, complemented
3	#22582643	C₃	1	3	yes	maximal, complemented
4	#22582638	C₃	1	3	yes	maximal, complemented
5	#22582633	C₃	1	9	yes	maximal, complemented
6	C₃×#15688521	C₃	1	2	yes	maximal, complemented
7	C₃×#15688519	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	#15688532	—	3	18	no	—
10	#15688531	C₃²	1	27	yes	complemented
11	C₃×#11758608	C₃²	1	6	yes	—
12	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
14	#15688521	C₃²	1	6	yes	complemented
15	#15688521	C₃²	1	12	yes	complemented
16	#15688519	C₃²	1	3	yes	—
17	#15688519	C₃²	1	6	yes	—
18	#15688516	C₃²	1	27	yes	complemented
19	#15688513	C₃²	1	27	yes	complemented
20	C₃₇×#737	—	3	3	no	—
21	C₃₇×#737	—	3	6	no	—
22	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
24	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	complemented
25	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
26	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
27	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
28	C₃³×C₃₇	C₃³	1	3	yes	—
29	C₃³×C₃₇	C₃³	1	1	yes	characteristic
30	C₃²×C₃₇⋊C₃	—	3	3	no	—
31	3₋¹⁺²×C₃₇	—	3	2	no	—
32	3₋¹⁺²×C₃₇	—	3	1	no	—
33	3₋¹⁺²×C₃₇	—	3	18	no	—
34	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	3	yes	—
35	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	6	yes	—
36	H(3,𝔽₃)×C₃₇	—	3	2	no	—
37	H(3,𝔽₃)×C₃₇	—	3	1	no	—
38	C₃×C₉×C₃₇	—	3	2	no	—
39	C₃×C₉×C₃₇	—	3	1	no	—
40	C₃×C₉×C₃₇	—	3	9	no	—
41	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
42	C₃×C₉×C₃₇	C₃³	1	2	yes	—
43	C₃×C₉×C₃₇	C₃³	1	3	yes	—
44	C₃×C₉×C₃₇	C₃³	1	3	yes	—
45	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
46	C₃²⋊(C₃₇⋊C₃)	C₃³	1	9	yes	—
47	#11758608	—	3	6	no	—
48	#11758608	—	3	6	no	—
49	#11758608	C₃³	1	18	yes	—
50	C₃×#66524	—	3	6	no	—
51	#11758606	—	3	27	no	—
52	#11758605	—	3	27	no	—
53	C₉×C₃₇⋊C₃	—	3	27	no	—
54	#10624936	—	37	1	no	maximal, Sylow
55	C₃²×C₃₇	—	3	3	no	—
56	C₃²×C₃₇	—	9	1	no	—
57	C₃²×C₃₇	—	9	3	no	—
58	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
59	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
61	C₃×C₃₇⋊C₃	—	3	9	no	—
62	C₃×C₃₇⋊C₃	—	27	3	no	—
63	C₉×C₃₇	—	3	1	no	—
64	C₉×C₃₇	—	3	2	no	—
65	C₉×C₃₇	—	3	3	no	—
66	C₉×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
67	#66524	—	3	18	no	—
68	3₋¹⁺⁴	—	37	3	no	—
69	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic, complemented
70	#6913	—	37	18	no	—
71	#6910	—	37	3	no	—
72	#6909	—	37	3	no	—
73	#6908	—	37	9	no	—
74	C₃×#733	—	37	2	no	—
75	C₃×#732	—	37	1	no	—
76	C₃×C₃₇	—	3	1	no	—
77	C₃×C₃₇	—	3	1	no	—
78	C₃×C₃₇	—	3	2	no	—
79	C₃×C₃₇	—	9	3	no	—
80	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
81	C₃₇⋊C₃	—	9	9	no	—
82	#737	—	3	3	no	—
83	#737	—	3	6	no	—
84	#737	—	37	27	no	—
85	#737	—	111	18	no	—
86	C₃×3₋¹⁺²	—	37	6	no	—
87	C₃×3₋¹⁺²	—	111	9	no	—
88	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
89	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
90	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	6	yes	complemented
91	C₃×H(3,𝔽₃)	—	37	3	no	—
92	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
93	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
94	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
95	#733	—	37	12	no	—
96	#733	—	37	6	no	—
97	#732	—	37	6	no	—
98	#732	—	37	3	no	—
99	#731	—	37	27	no	—
100	AΣL(1,𝔽₂₇)	—	37	27	no	—
101	C₃₇	#10624936	1	1	yes	characteristic, complemented, Sylow
102	C₃³	—	111	3	no	—
103	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
104	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
105	3₋¹⁺²	—	3	2	no	—
106	3₋¹⁺²	—	3	1	no	—
107	3₋¹⁺²	—	3	18	no	—
108	3₋¹⁺²	—	37	18	no	—
109	3₋¹⁺²	—	111	6	no	—
110	3₋¹⁺²	—	111	6	no	—
111	3₋¹⁺²	—	111	27	no	—
112	3₋¹⁺²	—	111	27	no	—
113	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
114	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	6	yes	—
115	H(3,𝔽₃)	—	3	2	no	—
116	H(3,𝔽₃)	—	3	1	no	—
117	H(3,𝔽₃)	—	37	9	no	—
118	H(3,𝔽₃)	—	111	6	no	—
119	C₃×C₉	—	3	2	no	—
120	C₃×C₉	—	3	1	no	—
121	C₃×C₉	—	3	9	no	—
122	C₃×C₉	—	111	6	no	—
123	C₃×C₉	—	111	27	no	—
124	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
125	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
126	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
127	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
128	C₃²	—	3	3	no	—
129	C₃²	—	9	1	no	—
130	C₃²	—	9	3	no	—
131	C₃²	—	111	9	no	—
132	C₃²	—	999	3	no	—
133	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
134	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
135	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
136	C₉	—	3	1	no	—
137	C₉	—	3	2	no	—
138	C₉	—	3	3	no	—
139	C₉	—	111	18	no	—
140	C₉	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
141	C₃	—	3	1	no	—
142	C₃	—	3	1	no	—
143	C₃	—	3	2	no	—
144	C₃	—	9	3	no	—
145	C₃	—	333	9	no	—
146	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582679

C₃

yes

maximal, complemented

#22582661

C₃

yes

maximal, complemented

#22582643

C₃

yes

maximal, complemented

#22582638

C₃

yes

maximal, complemented

#22582633

C₃

yes

maximal, complemented

C₃×#15688521

C₃

yes

maximal, complemented

C₃×#15688519

C₃

yes

maximal, characteristic, complemented

C₃×C₃₇×#737

C₃

yes

maximal, characteristic, complemented

#15688532

—

#15688531

C₃²

yes

complemented

C₃×#11758608

C₃²

yes

—

3₋¹⁺²×C₃₇⋊C₃

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

#15688521

C₃²

yes

complemented

#15688521

C₃²

yes

complemented

#15688519

C₃²

yes

—

#15688519

C₃²

yes

—

#15688516

C₃²

yes

complemented

#15688513

C₃²

yes

complemented

C₃₇×#737

—

C₃₇×#737

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

H(3,𝔽₃)

yes

—

3₋¹⁺²×C₃₇

H(3,𝔽₃)

yes

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

C₃³

yes

—

#11758608

—

#11758608

—

#11758608

C₃³

yes

—

C₃×#66524

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

#10624936

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃×H(3,𝔽₃)

yes

—

#66524

—

3₋¹⁺⁴

—

C₃×#737

C₃₇⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×H(3,𝔽₃)

yes

characteristic

C₃₇⋊C₃

—

—

—

—

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

—

—

—

—

100

AΣL(1,𝔽₂₇)

—

101

C₃₇

#10624936

yes

characteristic, complemented, Sylow

102

C₃³

—

111

—

103

C₃³

C₃²×C₃₇⋊C₃

yes

—

104

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

105

3₋¹⁺²

—

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

—

111

—

112

3₋¹⁺²

—

111

—

113

3₋¹⁺²

C₃²⋊(C₃₇⋊C₃)

yes

—

114

3₋¹⁺²

C₃²⋊(C₃₇⋊C₃)

yes

—

115

H(3,𝔽₃)

—

116

H(3,𝔽₃)

—

117

H(3,𝔽₃)

—

118

H(3,𝔽₃)

—

111

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

111

—

123

C₃×C₉

—

111

—

124

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

125

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

126

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

127

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

111

—

132

C₃²

—

999

—

133

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

134

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

135

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

111

—

140

C₉

C₃×C₃²⋊(C₃₇⋊C₃)

yes

—

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

—

333

—

146

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃×C₃₇×#737	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624936
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

subgroup

G/H

center

C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃²×C₃₇

C₃⁴

Frattini

C₃²

C₃³×C₃₇⋊C₃

Fitting

C₃×C₃₇×#737

C₃

socle

C₃×C₃₇

C₃²×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

N	H ≅ G/N	type
1. #22582679	C₃	semidirect
2. #22582661	C₃	semidirect
3. #22582643	C₃	semidirect
4. #22582638	C₃	semidirect
5. #22582633	C₃	semidirect
6. C₃×#15688521	C₃	semidirect
7. C₃×#15688519	C₃	semidirect
8. C₃×C₃₇×#737	C₃	semidirect
10. #15688531	C₃²	semidirect
14. #15688521	C₃²	semidirect
15. #15688521	C₃²	semidirect
18. #15688516	C₃²	semidirect
19. #15688513	C₃²	semidirect
24. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
27. C₃²×C₉×C₃₇	C₃²	semidirect
69. C₃×#737	C₃₇⋊C₃	semidirect
90. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
94. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
101. C₃₇	#10624936	semidirect

H ≅ G/N

type

1. #22582679

C₃

semidirect

2. #22582661

C₃

semidirect

3. #22582643

C₃

semidirect

4. #22582638

C₃

semidirect

5. #22582633

C₃

semidirect

6. C₃×#15688521

C₃

semidirect

7. C₃×#15688519

C₃

semidirect

8. C₃×C₃₇×#737

C₃

semidirect

10. #15688531

C₃²

semidirect

14. #15688521

C₃²

semidirect

15. #15688521

C₃²

semidirect

18. #15688516

C₃²

semidirect

19. #15688513

C₃²

semidirect

24. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

27. C₃²×C₉×C₃₇

C₃²

semidirect

69. C₃×#737

C₃₇⋊C₃

semidirect

90. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

94. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

101. C₃₇

#10624936

semidirect

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

subgroup	index
#22582679	3
#22582661	3
#22582643	3
#22582638	3
#22582633	3
C₃×#15688521	3
C₃×#15688519	3
C₃×C₃₇×#737	3
#10624936	37

subgroup

index

C₃×C₃₇×#737

Subgroups — #41008238

Conjugacy classes of subgroups

Complements

Maximal subgroups