Group #41005375 — subgroups

Subgroups — #41005375

Order 26973.

260 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22582554	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic
4	#22582458	C₃	1	9	yes	maximal, complemented
5	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	—	3	1	no	—
7	C₃²×#66524	—	3	1	no	—
8	C₃²×#66524	—	3	1	no	—
9	C₃²⋊#66524	—	3	1	no	—
10	C₃²⋊#66524	—	3	1	no	—
11	C₃²⋊#66524	—	3	1	no	—
12	C₃²⋊#66524	—	3	1	no	—
13	C₃²⋊#66524	—	3	1	no	—
14	C₃²⋊#66524	—	3	1	no	—
15	C₃²⋊#66524	C₃²	1	3	yes	—
16	C₃²⋊#66524	C₃²	1	3	yes	—
17	C₃²⋊#66524	C₃²	1	3	yes	—
18	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
20	C₃₇×C₃²⋊C₉	—	3	1	no	—
21	C₃₇×C₃²⋊C₉	—	3	1	no	—
22	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
23	C₃³×C₃₇	—	3	3	no	—
24	C₃³×C₃₇	—	3	3	no	—
25	C₃³×C₃₇	—	9	1	no	—
26	C₃³×C₃₇	—	9	1	no	—
27	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₃₇	C₃³	1	1	yes	characteristic
31	3₋¹⁺²×C₃₇	—	9	3	no	—
32	C₃×C₉×C₃₇	—	3	3	no	—
33	#11758608	—	9	3	no	—
34	C₃×#66524	—	3	3	no	—
35	C₃×#66524	—	3	3	no	—
36	C₃×#66524	—	3	3	no	—
37	C₃×#66524	—	9	3	no	—
38	C₃×#66524	—	9	3	no	—
39	#10624597	—	37	1	no	maximal, Sylow
40	C₃²×C₃₇	—	3	1	no	—
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	3	1	no	—
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	3	1	no	—
48	C₃²×C₃₇	—	3	1	no	—
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	3	1	no	—
53	C₃²×C₃₇	—	3	1	no	—
54	C₃²×C₃₇	—	9	9	no	—
55	C₃²×C₃₇	#731	1	3	yes	—
56	C₃²×C₃₇	#732	1	3	yes	—
57	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₉×C₃₇	—	9	3	no	—
59	#66524	—	9	3	no	—
60	#66524	—	9	3	no	—
61	#66524	—	9	3	no	—
62	#6891	—	37	1	no	—
63	#6891	C₃₇⋊C₃	1	1	yes	characteristic
64	C₃×C₃²⋊C₉	—	37	1	no	—
65	C₃×C₃²⋊C₉	—	37	1	no	—
66	#6861	—	37	9	no	—
67	C₃×C₃₇	—	3	3	no	—
68	C₃×C₃₇	—	3	3	no	—
69	C₃×C₃₇	—	9	1	no	—
70	C₃×C₃₇	—	9	1	no	—
71	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
72	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
73	C₃×C₃₇	#6912	1	1	yes	characteristic
74	C₃×C₃₇	#6912	1	1	yes	characteristic
75	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
76	C₃×3₋¹⁺²	—	3	1	no	—
77	C₃×3₋¹⁺²	—	111	1	no	—
78	C₃²×C₉	—	111	1	no	—
79	C₃²×C₉	—	111	1	no	—
80	C₃²⋊C₉	—	3	1	no	—
81	C₃²⋊C₉	—	3	1	no	—
82	C₃²⋊C₉	—	37	3	no	—
83	C₃²⋊C₉	—	37	3	no	—
84	C₃²⋊C₉	—	37	3	no	—
85	C₃²⋊C₉	—	111	1	no	—
86	C₃²⋊C₉	—	111	1	no	—
87	C₃²⋊C₉	—	111	1	no	—
88	C₃²⋊C₉	—	111	1	no	—
89	C₃²⋊C₉	—	111	1	no	—
90	C₃²⋊C₉	—	111	1	no	—
91	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
92	C₃₇	#10624597	1	1	yes	characteristic, complemented, Sylow
93	C₃³	—	3	3	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	9	1	no	—
96	C₃³	—	9	1	no	—
97	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
98	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
99	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
100	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
101	3₋¹⁺²	—	9	3	no	—
102	3₋¹⁺²	—	333	3	no	—
103	C₃×C₉	—	3	3	no	—
104	C₃×C₉	—	333	3	no	—
105	C₃×C₉	—	333	3	no	—
106	C₃×C₉	—	111	3	no	—
107	C₃×C₉	—	111	3	no	—
108	C₃×C₉	—	111	3	no	—
109	C₃²	—	3	1	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	1	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	9	9	no	—
124	C₃²	#15688517	1	3	yes	—
125	C₃²	#15688518	1	3	yes	—
126	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
127	C₉	—	9	3	no	—
128	C₉	—	333	3	no	—
129	C₉	—	333	3	no	—
130	C₉	—	333	3	no	—
131	C₃	—	3	3	no	—
132	C₃	—	3	3	no	—
133	C₃	—	9	1	no	—
134	C₃	—	9	1	no	—
135	C₃	C₃×#15688517	1	1	yes	characteristic
136	C₃	C₃×#15688518	1	1	yes	characteristic
137	C₃	#22582651	1	1	yes	characteristic
138	C₃	#22582652	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582554

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic

#22582458

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃²×#66524

—

C₃²×#66524

—

—

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

yes

C₃³×C₃₇

yes

C₃³×C₃₇

yes

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

#731

yes

—

C₃²×C₃₇

#732

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

#6861

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624597

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

100

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

101

3₋¹⁺²

—

102

3₋¹⁺²

—

333

—

103

C₃×C₉

—

104

C₃×C₉

—

333

—

105

C₃×C₉

—

333

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃×C₉

—

111

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

#15688517

yes

—

125

C₃²

#15688518

yes

—

126

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

127

C₉

—

128

C₉

—

333

—

129

C₉

—

333

—

130

C₉

—

333

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

C₃×#15688517

yes

characteristic

136

C₃

C₃×#15688518

yes

characteristic

137

C₃

#22582651

yes

characteristic

138

C₃

#22582652

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624597
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6891

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃⁴×C₃₇

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
4. #22582458	C₃	semidirect
92. C₃₇	#10624597	semidirect

H ≅ G/N

type

4. #22582458

C₃

semidirect

92. C₃₇

#10624597

semidirect

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

subgroup	index
#22582554	3
C₃×C₃²⋊#66524	3
#22582458	3
C₃₇×#6891	3
#10624597	37

subgroup

index

#22582554

C₃×C₃²⋊#66524

#22582458

C₃₇×#6891

#10624597

Subgroups — #41005375

Conjugacy classes of subgroups

Complements

Maximal subgroups