Group #41005185 — subgroups

Subgroups — #41005185

Order 26973.

758 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 112 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×#11758608	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#15688502	C₃	1	9	yes	maximal, complemented
5	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
7	C₃×#11758608	—	3	3	no	—
8	C₃×#11758608	—	3	9	no	—
9	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
10	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
11	C₃²×#66524	C₃²	1	3	yes	—
12	C₃²×#66524	C₃²	1	3	yes	—
13	C₃²⋊#66524	—	3	3	no	—
14	#15688502	—	3	27	no	—
15	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
17	C₃₇×C₃²⋊C₉	—	3	3	no	—
18	C₃³×C₃₇	—	3	3	no	—
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	—	3	3	no	—
21	C₃³×C₃₇	—	3	3	no	—
22	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	C₃²×C₃₇⋊C₃	—	3	9	no	—
25	C₃²×C₃₇⋊C₃	—	3	3	no	—
26	C₃²×C₃₇⋊C₃	—	3	6	no	—
27	C₃²×C₃₇⋊C₃	—	3	9	no	—
28	C₃²×C₃₇⋊C₃	—	3	9	no	—
29	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
30	C₃×C₉×C₃₇	—	3	3	no	—
31	C₃×C₉×C₃₇	—	3	9	no	—
32	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
33	#11758608	—	3	27	no	—
34	C₃×#66524	—	3	6	no	—
35	C₃×#66524	—	3	3	no	—
36	C₃×#66524	—	3	9	no	—
37	C₃×#66524	—	3	3	no	—
38	C₃×#66524	H(3,𝔽₃)	1	9	yes	complemented
39	#10624563	—	37	1	no	maximal, Sylow
40	C₃²×C₃₇	—	3	3	no	—
41	C₃²×C₃₇	—	3	3	no	—
42	C₃²×C₃₇	—	3	3	no	—
43	C₃²×C₃₇	—	3	3	no	—
44	C₃²×C₃₇	—	9	9	no	—
45	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
48	C₃²×C₃₇	#737	1	3	yes	—
49	C₃²×C₃₇	#737	1	3	yes	—
50	C₃×C₃₇⋊C₃	—	3	9	no	—
51	C₃×C₃₇⋊C₃	—	3	27	no	—
52	C₃×C₃₇⋊C₃	—	9	3	no	—
53	C₃×C₃₇⋊C₃	—	9	18	no	—
54	C₃×C₃₇⋊C₃	—	9	3	no	—
55	C₃×C₃₇⋊C₃	—	9	3	no	—
56	C₉×C₃₇	—	9	3	no	—
57	#66524	—	3	9	no	—
58	#66524	—	9	3	no	—
59	C₃²×3₋¹⁺²	—	37	1	no	—
60	C₃²×H(3,𝔽₃)	—	37	1	no	—
61	C₃×C₃²⋊C₉	—	37	1	no	—
62	C₃×C₃²⋊C₉	—	37	9	no	—
63	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
64	C₃×C₃₇	—	9	3	no	—
65	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
66	C₃×C₃₇	#6891	1	1	yes	characteristic
67	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	9	yes	—
68	C₃×C₃₇	#6893	1	1	yes	characteristic
69	C₃×C₃₇	#6893	1	1	yes	characteristic
70	C₃₇⋊C₃	—	9	9	no	—
71	C₃⁴	—	37	3	no	—
72	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃×3₋¹⁺²	—	111	9	no	—
74	C₃×3₋¹⁺²	—	111	3	no	—
75	C₃×H(3,𝔽₃)	—	111	9	no	—
76	C₃×H(3,𝔽₃)	—	111	3	no	—
77	C₃²×C₉	—	37	3	no	—
78	C₃²×C₉	—	37	3	no	—
79	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
80	C₃²⋊C₉	—	3	3	no	—
81	C₃²⋊C₉	—	111	27	no	—
82	C₃²⋊C₉	—	111	3	no	—
83	C₃₇	#10624563	1	1	yes	characteristic, complemented, Sylow
84	C₃³	—	3	3	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	3	3	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	37	9	no	—
89	C₃³	—	111	9	no	—
90	C₃³	—	111	3	no	—
91	C₃³	—	111	6	no	—
92	C₃³	—	111	9	no	—
93	C₃³	—	111	9	no	—
94	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
95	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
96	3₋¹⁺²	—	111	27	no	—
97	H(3,𝔽₃)	—	111	27	no	—
98	C₃×C₉	—	3	9	no	—
99	C₃×C₉	—	3	3	no	—
100	C₃×C₉	—	37	9	no	—
101	C₃×C₉	—	111	9	no	—
102	C₃×C₉	—	111	3	no	—
103	C₃×C₉	—	111	6	no	—
104	C₃×C₉	—	111	3	no	—
105	C₃²	—	3	3	no	—
106	C₃²	—	3	3	no	—
107	C₃²	—	3	3	no	—
108	C₃²	—	3	3	no	—
109	C₃²	—	9	9	no	—
110	C₃²	—	333	3	no	—
111	C₃²	—	333	18	no	—
112	C₃²	—	333	3	no	—
113	C₃²	—	333	3	no	—
114	C₃²	—	111	9	no	—
115	C₃²	—	111	27	no	—
116	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
117	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
118	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
119	C₃²	#15688531	1	3	yes	—
120	C₃²	#15688532	1	3	yes	—
121	C₉	—	9	3	no	—
122	C₉	—	333	3	no	—
123	C₉	—	111	9	no	—
124	C₃	—	9	3	no	—
125	C₃	—	333	9	no	—
126	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
127	C₃	#22582551	1	1	yes	characteristic
128	C₃	#22582557	1	9	yes	—
129	C₃	#22582564	1	1	yes	characteristic
130	C₃	#22582565	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal, characteristic, complemented

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal, characteristic, complemented

C₃×#15688502

C₃

yes

maximal, complemented

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃×#11758608

—

C₃×#11758608

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃×C₃²⋊(C₃₇⋊C₃)

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

C₃²⋊#66524

—

#15688502

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

#737

yes

—

C₃²×C₃₇

#737

yes

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

#66524

—

#66524

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

#6891

yes

characteristic

C₃×C₃₇

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃×H(3,𝔽₃)

—

111

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624563

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

111

—

102

C₃×C₉

—

111

—

103

C₃×C₉

—

111

—

104

C₃×C₉

—

111

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

333

—

111

C₃²

—

333

—

112

C₃²

—

333

—

113

C₃²

—

333

—

114

C₃²

—

111

—

115

C₃²

—

111

—

116

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

117

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

118

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

119

C₃²

#15688531

yes

—

120

C₃²

#15688532

yes

—

121

C₉

—

122

C₉

—

333

—

123

C₉

—

111

—

124

C₃

—

125

C₃

—

333

—

126

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

127

C₃

#22582551

yes

characteristic

128

C₃

#22582557

yes

—

129

C₃

#22582564

yes

characteristic

130

C₃

#22582565

yes

characteristic

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624563
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₃²⋊C₉

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator, socle, cosocle

C₃³×C₃₇

C₃³

Fitting

C₃×C₃₇×C₃²⋊C₉

C₃

prime radical

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

hypercenter

C₃×C₃²⋊C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
1. C₃²×#11758608	C₃	semidirect
3. C₃×C₃²⋊#66524	C₃	semidirect
4. C₃×#15688502	C₃	semidirect
5. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
29. C₃²×C₃₇⋊C₃	3₋¹⁺²	semidirect
38. C₃×#66524	H(3,𝔽₃)	semidirect
63. C₃×C₃²⋊C₉	C₃₇⋊C₃	semidirect
83. C₃₇	#10624563	semidirect

H ≅ G/N

type

1. C₃²×#11758608

C₃

semidirect

3. C₃×C₃²⋊#66524

C₃

semidirect

4. C₃×#15688502

C₃

semidirect

5. C₃×C₃₇×C₃²⋊C₉

C₃

semidirect

29. C₃²×C₃₇⋊C₃

3₋¹⁺²

semidirect

38. C₃×#66524

H(3,𝔽₃)

semidirect

63. C₃×C₃²⋊C₉

C₃₇⋊C₃

semidirect

83. C₃₇

#10624563

semidirect

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

subgroup	index
C₃²×#11758608	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
C₃×C₃²⋊#66524	3
C₃×#15688502	3
C₃×C₃₇×C₃²⋊C₉	3
#10624563	37

subgroup

index

C₃²×#11758608

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃×C₃²⋊#66524

C₃×#15688502

C₃×C₃₇×C₃²⋊C₉

#10624563

Subgroups — #41005185

Conjugacy classes of subgroups

Complements

Maximal subgroups