Group #41007017 — subgroups

Subgroups — #41007017

Order 26973.

222 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 70 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×#15688520	C₃	1	1	yes	maximal, characteristic
2	#22582571	C₃	1	1	yes	maximal, characteristic
3	#22582571	C₃	1	1	yes	maximal, characteristic
4	C₃×#15688505	C₃	1	1	yes	maximal, characteristic
5	#22582481	C₃	1	2	yes	maximal
6	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal
7	#22582478	C₃	1	2	yes	maximal
8	#22582471	C₃	1	2	yes	maximal
9	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
10	C₃×#11758608	C₃²	1	1	yes	characteristic
11	C₃×#11758608	C₃²	1	1	yes	characteristic
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	2	yes	—
14	#15688520	—	3	3	no	—
15	#15688505	—	3	2	no	—
16	#15688505	—	3	1	no	—
17	#15688505	—	3	1	no	—
18	#15688505	—	3	1	no	—
19	#15688504	—	3	2	no	—
20	#15688504	—	3	2	no	—
21	C₃²⋊#66524	C₃²	1	2	yes	—
22	C₃²⋊#66524	C₃²	1	2	yes	—
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×C₉⋊C₉	—	3	2	no	—
26	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₉²×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	3₋¹⁺²×C₃₇	—	3	2	no	—
30	3₋¹⁺²×C₃₇	3₋¹⁺²	1	3	yes	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃×C₉×C₃₇	—	3	3	no	—
33	C₃×C₉×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
35	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
36	#11758608	—	3	1	no	—
37	#11758608	—	3	1	no	—
38	#11758608	—	3	2	no	—
39	#11758608	—	3	2	no	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	1	no	—
43	C₃×#66524	—	3	1	no	—
44	C₃×#66524	—	3	2	no	—
45	C₃×#66524	—	3	1	no	—
46	C₃×#66524	—	3	1	no	—
47	C₃×#66524	—	3	2	no	—
48	C₃×#66524	—	9	2	no	—
49	#10624799	—	37	1	no	maximal, Sylow
50	C₃²×C₃₇	—	3	2	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
54	C₃²×C₃₇	#737	1	2	yes	—
55	C₉×C₃₇	—	3	3	no	—
56	C₉×C₃₇	—	3	2	no	—
57	C₉×C₃₇	—	9	2	no	—
58	C₉×C₃₇	#733	1	3	yes	complemented
59	#66524	—	9	2	no	—
60	#66524	—	9	2	no	—
61	#66524	—	9	2	no	—
62	#66524	—	9	1	no	—
63	#66524	—	9	1	no	—
64	#66524	—	9	1	no	—
65	C₃×#733	—	37	1	no	—
66	#6894	—	37	1	no	—
67	#6894	—	37	1	no	—
68	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
69	C₃×C₉⋊C₉	—	37	1	no	—
70	3₋¹⁺²⋊C₉	—	37	2	no	—
71	3₋¹⁺²⋊C₉	—	37	2	no	—
72	3₋¹⁺²⋊C₉	—	37	2	no	—
73	#6868	—	37	2	no	—
74	C₃×C₃₇	—	3	1	no	—
75	C₃×C₃₇	—	3	2	no	—
76	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
78	C₃×C₃₇	#6913	1	2	yes	—
79	C₃×3₋¹⁺²	—	37	1	no	—
80	C₃×3₋¹⁺²	—	37	1	no	—
81	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃²×C₉	—	37	2	no	—
83	C₃²×C₉	—	37	1	no	—
84	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	#733	—	111	3	no	—
86	C₉⋊C₉	—	3	2	no	—
87	C₉⋊C₉	—	111	2	no	—
88	C₉⋊C₉	—	111	2	no	—
89	C₉⋊C₉	—	111	2	no	—
90	C₉⋊C₉	—	111	1	no	—
91	C₉⋊C₉	—	111	1	no	—
92	C₉⋊C₉	—	111	1	no	—
93	C₃²⋊C₉	—	37	2	no	—
94	C₃²⋊C₉	—	37	2	no	—
95	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
96	C₉²	—	3	1	no	—
97	C₃₇	#10624799	1	1	yes	characteristic, complemented, Sylow
98	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
99	3₋¹⁺²	—	3	2	no	—
100	3₋¹⁺²	—	111	1	no	—
101	3₋¹⁺²	—	111	1	no	—
102	3₋¹⁺²	—	111	2	no	—
103	3₋¹⁺²	—	111	2	no	—
104	3₋¹⁺²	#11758605	1	3	yes	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	333	2	no	—
109	C₃×C₉	—	111	2	no	—
110	C₃×C₉	—	111	2	no	—
111	C₃×C₉	—	111	1	no	—
112	C₃×C₉	—	111	2	no	—
113	C₃×C₉	—	111	1	no	—
114	C₃×C₉	—	111	1	no	—
115	C₃×C₉	—	111	1	no	—
116	C₃×C₉	—	111	2	no	—
117	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
118	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
119	C₃²	—	3	1	no	—
120	C₃²	—	3	2	no	—
121	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
122	C₃²	C₃×#11758605	1	1	yes	characteristic
123	C₃²	#15688531	1	2	yes	—
124	C₉	—	3	3	no	—
125	C₉	—	3	2	no	—
126	C₉	—	9	2	no	—
127	C₉	—	333	1	no	—
128	C₉	—	333	1	no	—
129	C₉	—	333	1	no	—
130	C₉	—	333	2	no	—
131	C₉	—	333	2	no	—
132	C₉	—	333	2	no	—
133	C₉	#15688520	1	3	yes	complemented
134	C₃	—	3	1	no	—
135	C₃	—	3	2	no	—
136	C₃	#22582558	1	1	yes	characteristic
137	C₃	C₃×#15688520	1	1	yes	characteristic
138	C₃	#22582655	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688520

C₃

yes

maximal, characteristic

#22582571

C₃

yes

maximal, characteristic

#22582571

C₃

yes

maximal, characteristic

C₃×#15688505

C₃

yes

maximal, characteristic

#22582481

C₃

yes

maximal

3₋¹⁺²⋊#66524

C₃

yes

maximal

#22582478

C₃

yes

maximal

#22582471

C₃

yes

maximal

C₉×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

C₃×#11758608

C₃²

yes

characteristic

C₃×#11758608

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃²×#66524

C₃²

yes

—

—

—

—

—

—

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

yes

—

—

—

—

—

—

—

—

—

C₉×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic

C₃×C₉⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

#6868

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#733

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

C₃₇

#10624799

yes

characteristic, complemented, Sylow

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

100

3₋¹⁺²

—

111

—

101

3₋¹⁺²

—

111

—

102

3₋¹⁺²

—

111

—

103

3₋¹⁺²

—

111

—

104

3₋¹⁺²

#11758605

yes

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

333

—

109

C₃×C₉

—

111

—

110

C₃×C₉

—

111

—

111

C₃×C₉

—

111

—

112

C₃×C₉

—

111

—

113

C₃×C₉

—

111

—

114

C₃×C₉

—

111

—

115

C₃×C₉

—

111

—

116

C₃×C₉

—

111

—

117

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

118

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

119

C₃²

—

120

C₃²

—

121

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

122

C₃²

C₃×#11758605

yes

characteristic

123

C₃²

#15688531

yes

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₉

—

333

—

128

C₉

—

333

—

129

C₉

—

333

—

130

C₉

—

333

—

131

C₉

—

333

—

132

C₉

—

333

—

133

C₉

#15688520

yes

complemented

134

C₃

—

135

C₃

—

136

C₃

#22582558

yes

characteristic

137

C₃

C₃×#15688520

yes

characteristic

138

C₃

#22582655

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, prime radical, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₉×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₉×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624799
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, prime radical, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₉×3₋¹⁺²×C₃₇

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

hypercenter

C₉×3₋¹⁺²

C₃₇⋊C₃

hypocenter

C₃₇

#10624799

LCS

G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₉×3₋¹⁺²

N	H ≅ G/N	type
58. C₉×C₃₇	#733	semidirect
97. C₃₇	#10624799	semidirect
133. C₉	#15688520	semidirect

H ≅ G/N

type

58. C₉×C₃₇

#733

semidirect

97. C₃₇

#10624799

semidirect

133. C₉

#15688520

semidirect

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

subgroup	index
C₃×#15688520	3
#22582571	3
C₃×#15688505	3
#22582481	3
3₋¹⁺²⋊#66524	3
#22582478	3
#22582471	3
C₉×3₋¹⁺²×C₃₇	3
#10624799	37

subgroup

index

C₉×3₋¹⁺²×C₃₇

Subgroups — #41007017

Conjugacy classes of subgroups

Complements

Maximal subgroups