Group #41005722 — subgroups

Subgroups — #41005722

Order 26973.

452 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 58 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	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	#22582554	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582554	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582444	C₃	1	9	yes	maximal, complemented
5	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	—	3	1	no	—
7	C₃×#11758608	—	3	1	no	—
8	C₃×#11758608	C₃²	1	3	yes	complemented
9	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
10	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
12	C₃²⋊#66524	—	3	2	no	—
13	C₃²⋊#66524	—	3	1	no	—
14	C₃²⋊#66524	—	3	2	no	—
15	C₃²⋊#66524	C₃²	1	3	yes	complemented
16	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	complemented
18	C₃₇×C₃²⋊C₉	—	3	2	no	—
19	C₃₇×C₃²⋊C₉	—	3	1	no	—
20	C₃³×C₃₇	—	3	3	no	—
21	C₃³×C₃₇	—	9	1	no	—
22	C₃³×C₃₇	—	9	1	no	—
23	C₃³×C₃₇	—	9	1	no	—
24	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
25	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	3	9	no	—
28	C₃²×C₃₇⋊C₃	—	3	3	no	—
29	C₃²×C₃₇⋊C₃	—	9	3	no	—
30	C₃²×C₃₇⋊C₃	—	9	3	no	—
31	C₃²×C₃₇⋊C₃	—	9	3	no	—
32	3₋¹⁺²×C₃₇	—	9	3	no	—
33	C₃×C₉×C₃₇	—	3	3	no	—
34	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
35	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
36	#11758608	—	9	3	no	—
37	#11758608	—	9	3	no	—
38	C₃×#66524	—	3	3	no	—
39	C₃×#66524	—	3	3	no	—
40	#10624630	—	37	1	no	maximal, Sylow
41	C₃²×C₃₇	—	3	1	no	—
42	C₃²×C₃₇	—	3	1	no	—
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	3	1	no	—
48	C₃²×C₃₇	—	3	2	no	—
49	C₃²×C₃₇	—	3	2	no	—
50	C₃²×C₃₇	—	3	2	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	9	9	no	—
53	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
54	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃×C₃₇⋊C₃	—	3	9	no	—
56	C₃×C₃₇⋊C₃	—	9	27	no	—
57	C₃×C₃₇⋊C₃	—	9	3	no	—
58	C₃×C₃₇⋊C₃	—	9	3	no	—
59	C₃×C₃₇⋊C₃	—	9	3	no	—
60	C₉×C₃₇	—	9	3	no	—
61	#66524	—	9	3	no	—
62	#66524	—	9	3	no	—
63	C₃²×H(3,𝔽₃)	—	37	1	no	—
64	#6891	—	37	1	no	—
65	#6891	—	37	1	no	—
66	#6891	C₃₇⋊C₃	1	1	yes	characteristic, complemented
67	#6857	—	37	9	no	—
68	C₃×C₃₇	—	3	3	no	—
69	C₃×C₃₇	—	9	2	no	—
70	C₃×C₃₇	—	9	1	no	—
71	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
72	C₃×C₃₇	#6909	1	1	yes	characteristic
73	C₃×C₃₇	#6909	1	1	yes	characteristic
74	C₃×C₃₇	#6910	1	1	yes	characteristic
75	C₃₇⋊C₃	—	9	9	no	—
76	C₃⁴	—	111	1	no	—
77	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
78	C₃×3₋¹⁺²	—	37	3	no	—
79	C₃×3₋¹⁺²	—	111	1	no	—
80	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	complemented
81	C₃×H(3,𝔽₃)	—	37	3	no	—
82	C₃×H(3,𝔽₃)	—	111	9	no	—
83	C₃×H(3,𝔽₃)	—	111	2	no	—
84	C₃²⋊C₉	—	3	2	no	—
85	C₃²⋊C₉	—	3	1	no	—
86	C₃²⋊C₉	—	37	3	no	—
87	C₃²⋊C₉	—	111	2	no	—
88	C₃²⋊C₉	—	111	2	no	—
89	C₃²⋊C₉	—	111	1	no	—
90	C₃₇	#10624630	1	1	yes	characteristic, complemented, Sylow
91	C₃³	—	3	3	no	—
92	C₃³	—	9	1	no	—
93	C₃³	—	9	1	no	—
94	C₃³	—	9	1	no	—
95	C₃³	—	333	3	no	—
96	C₃³	—	333	3	no	—
97	C₃³	—	333	3	no	—
98	C₃³	—	111	9	no	—
99	C₃³	—	111	3	no	—
100	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
101	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
102	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
103	3₋¹⁺²	—	9	3	no	—
104	3₋¹⁺²	—	333	3	no	—
105	3₋¹⁺²	—	333	3	no	—
106	H(3,𝔽₃)	—	111	9	no	—
107	H(3,𝔽₃)	—	111	18	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	111	3	no	—
110	C₃×C₉	—	111	3	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	2	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	9	9	no	—
123	C₃²	—	333	27	no	—
124	C₃²	—	333	3	no	—
125	C₃²	—	333	3	no	—
126	C₃²	—	333	3	no	—
127	C₃²	—	111	9	no	—
128	C₃²	#15688514	1	3	yes	—
129	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
130	C₉	—	9	3	no	—
131	C₉	—	333	3	no	—
132	C₉	—	333	3	no	—
133	C₃	—	3	3	no	—
134	C₃	—	9	2	no	—
135	C₃	—	9	1	no	—
136	C₃	—	333	9	no	—
137	C₃	C₃×#15688514	1	1	yes	characteristic
138	C₃	#22582638	1	1	yes	characteristic
139	C₃	#22582639	1	1	yes	characteristic
140	C₃	#22582644	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

#22582554

C₃

yes

maximal, characteristic, complemented

#22582554

C₃

yes

maximal, characteristic, complemented

#22582444

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃×#11758608

—

C₃×#11758608

C₃²

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

—

—

—

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

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

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×H(3,𝔽₃)

—

#6891

—

#6891

—

#6891

C₃₇⋊C₃

yes

characteristic, complemented

#6857

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624630

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

—

111

—

C₃³

—

111

—

100

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

101

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

102

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

103

3₋¹⁺²

—

104

3₋¹⁺²

—

333

—

105

3₋¹⁺²

—

333

—

106

H(3,𝔽₃)

—

111

—

107

H(3,𝔽₃)

—

111

—

108

C₃×C₉

—

109

C₃×C₉

—

111

—

110

C₃×C₉

—

111

—

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

—

333

—

124

C₃²

—

333

—

125

C₃²

—

333

—

126

C₃²

—

333

—

127

C₃²

—

111

—

128

C₃²

#15688514

yes

—

129

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

130

C₉

—

131

C₉

—

333

—

132

C₉

—

333

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

333

—

137

C₃

C₃×#15688514

yes

characteristic

138

C₃

#22582638

yes

characteristic

139

C₃

#22582639

yes

characteristic

140

C₃

#22582644

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	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624630
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

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
2. #22582554	C₃	semidirect
3. #22582554	C₃	semidirect
4. #22582444	C₃	semidirect
5. C₃₇×#6891	C₃	semidirect
8. C₃×#11758608	C₃²	semidirect
15. C₃²⋊#66524	C₃²	semidirect
17. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
66. #6891	C₃₇⋊C₃	semidirect
80. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
90. C₃₇	#10624630	semidirect

H ≅ G/N

type

2. #22582554

C₃

semidirect

3. #22582554

C₃

semidirect

4. #22582444

C₃

semidirect

5. C₃₇×#6891

C₃

semidirect

8. C₃×#11758608

C₃²

semidirect

15. C₃²⋊#66524

C₃²

semidirect

17. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

66. #6891

C₃₇⋊C₃

semidirect

80. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

90. C₃₇

#10624630

semidirect

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

subgroup	index
C₃²×C₃²⋊(C₃₇⋊C₃)	3
#22582554	3
#22582444	3
C₃₇×#6891	3
#10624630	37

subgroup

index

C₃²×C₃²⋊(C₃₇⋊C₃)

#22582554

#22582444

C₃₇×#6891

#10624630

Subgroups — #41005722

Conjugacy classes of subgroups

Complements

Maximal subgroups