Group #41008481 — subgroups

Subgroups — #41008481

Order 26973.

428 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 102 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×#15688542	C₃	1	3	yes	maximal, complemented
2	#22582690	C₃	1	6	yes	maximal, complemented
3	C₃²⋊#11758603	C₃	1	3	yes	maximal, complemented
4	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
5	#15688542	—	3	9	no	—
6	C₃×#11758603	C₃²	1	9	yes	complemented
7	C₃³×C₃₇⋊C₃	C₉	1	1	yes	characteristic
8	C₃²×#66524	C₉	1	1	yes	characteristic
9	C₃²×#66524	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
11	C₃×C₉×C₃₇⋊C₃	—	3	6	no	—
12	C₃×C₉×C₃₇⋊C₃	C₉	1	2	yes	—
13	C₃×C₉×C₃₇⋊C₃	C₉	1	2	yes	—
14	C₃×C₉×C₃₇⋊C₃	C₉	1	2	yes	—
15	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
16	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
17	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic
18	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	—	3	3	no	—
20	C₃²×C₃₇⋊C₃	—	3	3	no	—
21	C₃²×C₃₇⋊C₃	—	3	6	no	—
22	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	C₃×C₉	1	2	yes	—
24	C₃×C₉×C₃₇	—	3	3	no	—
25	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
26	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
27	C₃×#66524	—	3	3	no	—
28	C₃×#66524	—	3	1	no	—
29	C₃×#66524	—	3	2	no	—
30	C₃×#66524	C₃×C₉	1	2	yes	—
31	C₃×#66524	C₃×C₉	1	2	yes	—
32	C₃×#66524	C₃×C₉	1	1	yes	characteristic
33	C₃×#66524	C₃³	1	1	yes	characteristic
34	C₉×C₃₇⋊C₃	—	3	6	no	—
35	C₉×C₃₇⋊C₃	—	3	6	no	—
36	C₉×C₃₇⋊C₃	—	3	6	no	—
37	C₉×C₃₇⋊C₃	—	3	2	no	—
38	C₉×C₃₇⋊C₃	—	3	4	no	—
39	C₉×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
40	C₉×C₃₇⋊C₃	3₋¹⁺²	1	6	yes	—
41	#11758603	—	3	9	no	—
42	#10624683	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	C₂₇⋊C₃	1	3	yes	complemented
46	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
47	C₃×C₃₇⋊C₃	—	3	27	no	—
48	C₃×C₃₇⋊C₃	—	3	2	no	—
49	C₃×C₃₇⋊C₃	—	3	2	no	—
50	C₃×C₃₇⋊C₃	—	3	2	no	—
51	C₃×C₃₇⋊C₃	—	3	1	no	—
52	C₃×C₃₇⋊C₃	—	3	2	no	—
53	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	3	yes	—
54	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	6	yes	complemented
55	C₉×C₃₇	—	3	2	no	—
56	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
57	#66524	—	3	2	no	—
58	#66524	—	3	2	no	—
59	#66524	C₃²⋊C₉	1	3	yes	—
60	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	#66524	C₃×3₋¹⁺²	1	2	yes	—
62	C₃³×C₉	—	37	1	no	—
63	C₃×C₂₇⋊C₃	—	37	3	no	—
64	C₃²⋊C₂₇	—	37	3	no	—
65	C₃²⋊C₂₇	—	37	6	no	—
66	C₃×C₃₇	—	3	3	no	—
67	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
68	C₃×C₃₇	C₃×C₂₇⋊C₃	1	2	yes	—
69	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
70	C₃₇⋊C₃	—	3	3	no	—
71	C₃₇⋊C₃	—	3	6	no	—
72	C₃⁴	—	37	1	no	—
73	C₃²×C₉	—	37	1	no	—
74	C₃²×C₉	—	37	1	no	—
75	C₃²×C₉	—	37	1	no	—
76	C₃²×C₉	—	37	2	no	—
77	C₃²×C₉	—	37	2	no	—
78	C₃²×C₉	—	37	2	no	—
79	C₃²×C₉	—	37	2	no	—
80	C₃²×C₉	—	111	3	no	—
81	C₃²×C₉	—	111	6	no	—
82	C₃²×C₉	#66526	1	1	yes	characteristic
83	C₂₇⋊C₃	—	111	9	no	—
84	C₃×C₂₇	—	37	9	no	—
85	C₃₇	#10624683	1	1	yes	characteristic, complemented, Sylow
86	C₃³	—	37	1	no	—
87	C₃³	—	37	2	no	—
88	C₃³	—	111	3	no	—
89	C₃³	—	111	6	no	—
90	C₃³	—	111	3	no	—
91	C₃³	C₃×#66526	1	1	yes	characteristic
92	C₃×C₉	—	3	3	no	—
93	C₃×C₉	—	37	1	no	—
94	C₃×C₉	—	37	1	no	—
95	C₃×C₉	—	37	3	no	—
96	C₃×C₉	—	37	6	no	—
97	C₃×C₉	—	37	2	no	—
98	C₃×C₉	—	37	2	no	—
99	C₃×C₉	—	111	2	no	—
100	C₃×C₉	—	111	6	no	—
101	C₃×C₉	—	111	4	no	—
102	C₃×C₉	—	111	6	no	—
103	C₃×C₉	—	111	1	no	—
104	C₃×C₉	—	111	3	no	—
105	C₃×C₉	—	111	2	no	—
106	C₃×C₉	—	111	6	no	—
107	C₃×C₉	C₃×#66526	1	2	yes	—
108	C₃×C₉	C₃×#66526	1	1	yes	characteristic
109	C₂₇	—	111	9	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	37	3	no	—
113	C₃²	—	37	6	no	—
114	C₃²	—	111	27	no	—
115	C₃²	—	111	2	no	—
116	C₃²	—	111	2	no	—
117	C₃²	—	111	2	no	—
118	C₃²	—	111	1	no	—
119	C₃²	—	111	2	no	—
120	C₃²	#15688542	1	3	yes	complemented
121	C₃²	C₃²×#66526	1	1	yes	characteristic
122	C₉	—	3	2	no	—
123	C₉	—	37	1	no	—
124	C₉	—	37	3	no	—
125	C₉	—	37	2	no	—
126	C₉	—	111	2	no	—
127	C₉	—	111	2	no	—
128	C₉	C₃²⋊#66526	1	3	yes	—
129	C₃	—	3	3	no	—
130	C₃	—	111	3	no	—
131	C₃	—	111	6	no	—
132	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic
133	C₃	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	1	2	yes	—
134	C₃	C₃×#15688542	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688542

C₃

yes

maximal, complemented

#22582690

C₃

yes

maximal, complemented

C₃²⋊#11758603

C₃

yes

maximal, complemented

C₃²×C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#15688542

—

C₃×#11758603

C₃²

yes

complemented

C₃³×C₃₇⋊C₃

C₉

yes

characteristic

C₃²×#66524

C₉

yes

characteristic

C₃²×#66524

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

C₉

yes

—

C₃×C₉×C₃₇⋊C₃

C₉

yes

—

C₃×C₉×C₃₇⋊C₃

C₉

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₉

yes

characteristic

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

—

—

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

characteristic

C₃×#66524

C₃³

yes

characteristic

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

#11758603

—

#10624683

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₂₇⋊C₃

yes

complemented

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃²⋊C₉

yes

—

C₃×C₃₇⋊C₃

C₂₇⋊C₃

yes

complemented

C₉×C₃₇

—

C₉×C₃₇

C₃²⋊C₉

yes

—

#66524

—

#66524

—

#66524

C₃²⋊C₉

yes

—

#66524

C₃×H(3,𝔽₃)

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

—

C₃³×C₉

—

C₃×C₂₇⋊C₃

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

C₃×C₂₇⋊C₃

yes

—

C₃×C₃₇

C₃×C₂₇⋊C₃

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

—

111

—

C₃²×C₉

#66526

yes

characteristic

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃₇

#10624683

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃×#66526

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

100

C₃×C₉

—

111

—

101

C₃×C₉

—

111

—

102

C₃×C₉

—

111

—

103

C₃×C₉

—

111

—

104

C₃×C₉

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

C₃×#66526

yes

—

108

C₃×C₉

C₃×#66526

yes

characteristic

109

C₂₇

—

111

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

111

—

115

C₃²

—

111

—

116

C₃²

—

111

—

117

C₃²

—

111

—

118

C₃²

—

111

—

119

C₃²

—

111

—

120

C₃²

#15688542

yes

complemented

121

C₃²

C₃²×#66526

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

111

—

127

C₉

—

111

—

128

C₉

C₃²⋊#66526

yes

—

129

C₃

—

130

C₃

—

111

—

131

C₃

—

111

—

132

C₃

C₃×C₃²⋊#66526

yes

characteristic

133

C₃

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

yes

—

134

C₃

C₃×#15688542

yes

characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃²×C₉×C₃₇	C₉
prime radical	C₃³×C₃₇⋊C₃	C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₃²×C₉	#66526
hypocenter	C₃₇	#10624683
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉

subgroup

G/H

center, Frattini

C₃²

C₃²×#66526

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃²×C₉×C₃₇

C₉

prime radical

C₃³×C₃₇⋊C₃

C₉

C₃³

C₃²×C₉

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃² ◁ C₃²×C₉

N	H ≅ G/N	type
1. C₃×#15688542	C₃	semidirect
2. #22582690	C₃	semidirect
3. C₃²⋊#11758603	C₃	semidirect
6. C₃×#11758603	C₃²	semidirect
45. C₃²×C₃₇	C₂₇⋊C₃	semidirect
54. C₃×C₃₇⋊C₃	C₂₇⋊C₃	semidirect
85. C₃₇	#10624683	semidirect
120. C₃²	#15688542	semidirect

H ≅ G/N

type

1. C₃×#15688542

C₃

semidirect

2. #22582690

C₃

semidirect

3. C₃²⋊#11758603

C₃

semidirect

6. C₃×#11758603

C₃²

semidirect

45. C₃²×C₃₇

C₂₇⋊C₃

semidirect

54. C₃×C₃₇⋊C₃

C₂₇⋊C₃

semidirect

85. C₃₇

#10624683

semidirect

120. C₃²

#15688542

semidirect

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

subgroup	index
C₃×#15688542	3
#22582690	3
C₃²⋊#11758603	3
C₃²×C₉×C₃₇⋊C₃	3
#10624683	37

subgroup

index

C₃²×C₉×C₃₇⋊C₃

Subgroups — #41008481

Conjugacy classes of subgroups

Complements

Maximal subgroups