Group #41004485 — subgroups

Subgroups — C₃×C₃₇×3₋¹⁺²⋊C₉ #41004485

Order 26973.

572 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 188 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	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	C₃₇×3₋¹⁺²⋊C₉	C₃	1	9	yes	maximal, direct product factor
6	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
8	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
9	C₃×3₋¹⁺²×C₃₇	C₉	1	9	yes	complemented
10	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
11	C₃²×C₉×C₃₇	—	3	2	no	—
12	C₃²×C₉×C₃₇	—	3	1	no	—
13	C₃²×C₉×C₃₇	—	3	1	no	—
14	C₃²×C₉×C₃₇	—	3	9	no	—
15	C₃²×C₉×C₃₇	—	3	1	no	—
16	C₃²×C₉×C₃₇	—	3	1	no	—
17	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
18	C₃₇×C₃²⋊C₉	—	3	2	no	—
19	C₃₇×C₃²⋊C₉	—	3	2	no	—
20	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
21	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
22	C₃³×C₃₇	—	3	3	no	—
23	C₃³×C₃₇	—	3	3	no	—
24	C₃³×C₃₇	—	3	3	no	—
25	C₃³×C₃₇	C₃×C₉	1	3	yes	—
26	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
27	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³×C₃₇	3₋¹⁺²	1	6	yes	—
29	C₃³×C₃₇	C₃³	1	1	yes	characteristic
30	3₋¹⁺²×C₃₇	—	3	18	no	—
31	3₋¹⁺²×C₃₇	—	3	6	no	—
32	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	complemented
33	C₃×C₉×C₃₇	—	3	18	no	—
34	C₃×C₉×C₃₇	—	3	9	no	—
35	C₃×C₉×C₃₇	—	3	9	no	—
36	C₃×C₉×C₃₇	—	3	3	no	—
37	C₃×C₉×C₃₇	—	3	3	no	—
38	C₃×C₉×C₃₇	—	3	9	no	—
39	C₃×C₉×C₃₇	—	3	3	no	—
40	C₃×C₉×C₃₇	—	3	3	no	—
41	C₃×C₉×C₃₇	—	9	3	no	—
42	C₃×C₉×C₃₇	—	9	3	no	—
43	C₃×3₋¹⁺²⋊C₉	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₃²×C₃₇	—	3	9	no	—
45	C₃²×C₃₇	—	3	18	no	—
46	C₃²×C₃₇	—	3	2	no	—
47	C₃²×C₃₇	—	3	1	no	—
48	C₃²×C₃₇	—	3	2	no	—
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	2	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	C₃²⋊C₉	1	6	yes	—
53	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
54	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
55	C₃²×C₃₇	#731	1	3	yes	—
56	C₃²×C₃₇	#733	1	3	yes	—
57	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
58	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
60	C₉×C₃₇	—	3	9	no	—
61	C₉×C₃₇	—	3	9	no	—
62	C₉×C₃₇	—	9	3	no	—
63	C₉×C₃₇	—	9	3	no	—
64	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
65	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic
66	C₃×C₃²⋊C₉	C₃×C₃₇	1	1	yes	characteristic
67	C₃×C₃²⋊C₉	C₃×C₃₇	1	1	yes	characteristic
68	3₋¹⁺²⋊C₉	C₃×C₃₇	1	9	yes	direct product factor
69	C₃×C₃₇	—	3	6	no	—
70	C₃×C₃₇	—	3	1	no	—
71	C₃×C₃₇	—	3	2	no	—
72	C₃×C₃₇	3₋¹⁺²⋊C₉	1	9	yes	direct product factor
73	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
74	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
75	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
76	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
77	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
78	C₃×3₋¹⁺²	—	3	2	no	—
79	C₃×3₋¹⁺²	—	3	6	no	—
80	C₃×3₋¹⁺²	C₉×C₃₇	1	9	yes	complemented
81	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
82	C₃²×C₉	—	3	2	no	—
83	C₃²×C₉	—	3	1	no	—
84	C₃²×C₉	—	3	1	no	—
85	C₃²×C₉	—	3	9	no	—
86	C₃²×C₉	—	3	1	no	—
87	C₃²×C₉	—	3	1	no	—
88	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
89	C₃²⋊C₉	—	3	2	no	—
90	C₃²⋊C₉	—	3	2	no	—
91	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
92	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
93	C₃₇	C₃×3₋¹⁺²⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
94	C₃³	—	3	3	no	—
95	C₃³	—	3	3	no	—
96	C₃³	—	3	3	no	—
97	C₃³	C₃×C₉×C₃₇	1	3	yes	—
98	C₃³	H(3,𝔽₃)×C₃₇	1	2	yes	—
99	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
100	C₃³	3₋¹⁺²×C₃₇	1	6	yes	—
101	C₃³	C₃³×C₃₇	1	1	yes	characteristic
102	3₋¹⁺²	—	3	18	no	—
103	3₋¹⁺²	—	3	6	no	—
104	3₋¹⁺²	C₃×C₉×C₃₇	1	9	yes	complemented
105	C₃×C₉	—	3	18	no	—
106	C₃×C₉	—	3	9	no	—
107	C₃×C₉	—	3	9	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	3	3	no	—
110	C₃×C₉	—	3	9	no	—
111	C₃×C₉	—	3	3	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	9	3	no	—
114	C₃×C₉	—	9	3	no	—
115	C₃²	—	3	9	no	—
116	C₃²	—	3	18	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	2	no	—
122	C₃²	—	3	1	no	—
123	C₃²	C₃₇×C₃²⋊C₉	1	6	yes	—
124	C₃²	C₃₇×C₃²⋊C₉	1	3	yes	—
125	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	3	yes	—
126	C₃²	C₃₇×#731	1	3	yes	—
127	C₃²	C₃₇×#733	1	3	yes	—
128	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
129	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
130	C₃²	C₃×3₋¹⁺²×C₃₇	1	2	yes	—
131	C₉	—	3	9	no	—
132	C₉	—	3	9	no	—
133	C₉	—	9	3	no	—
134	C₉	—	9	3	no	—
135	C₃	—	3	6	no	—
136	C₃	—	3	1	no	—
137	C₃	—	3	2	no	—
138	C₃	C₃₇×3₋¹⁺²⋊C₉	1	9	yes	direct product factor
139	C₃	C₃×C₃₇×C₃²⋊C₉	1	1	yes	characteristic
140	C₃	C₃×C₃₇×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
141	C₃	C₃×C₃₇×#731	1	1	yes	characteristic
142	C₃	C₃×C₃₇×#733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic

C₃³×C₉×C₃₇

C₃

yes

maximal, characteristic

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic

C₃₇×3₋¹⁺²⋊C₉

C₃

yes

maximal, direct product factor

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

—

C₃×3₋¹⁺²×C₃₇

C₉

yes

complemented

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₇

3₋¹⁺²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

—

3₋¹⁺²×C₃₇

C₃×C₉

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×3₋¹⁺²⋊C₉

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₇

#731

yes

—

C₃²×C₃₇

#733

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₃²×3₋¹⁺²

C₃×C₃₇

yes

characteristic

C₃³×C₉

C₃×C₃₇

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₃₇

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₃₇

yes

characteristic

3₋¹⁺²⋊C₉

C₃×C₃₇

yes

direct product factor

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

3₋¹⁺²⋊C₉

yes

direct product factor

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃⁴

C₃²×C₃₇

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₉×C₃₇

yes

complemented

C₃×3₋¹⁺²

C₃²×C₃₇

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₃₇

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₃₇

C₃×3₋¹⁺²⋊C₉

yes

characteristic, direct product factor, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₉×C₃₇

yes

—

C₃³

H(3,𝔽₃)×C₃₇

yes

—

C₃³

H(3,𝔽₃)×C₃₇

yes

characteristic

100

C₃³

3₋¹⁺²×C₃₇

yes

—

101

C₃³

C₃³×C₃₇

yes

characteristic

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

C₃×C₉×C₃₇

yes

complemented

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

C₃₇×C₃²⋊C₉

yes

—

124

C₃²

C₃₇×C₃²⋊C₉

yes

—

125

C₃²

C₃₇×AΣL(1,𝔽₂₇)

yes

—

126

C₃²

C₃₇×#731

yes

—

127

C₃²

C₃₇×#733

yes

—

128

C₃²

C₃²×C₉×C₃₇

yes

characteristic

129

C₃²

C₃×H(3,𝔽₃)×C₃₇

yes

characteristic

130

C₃²

C₃×3₋¹⁺²×C₃₇

yes

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₃

—

136

C₃

—

137

C₃

—

138

C₃

C₃₇×3₋¹⁺²⋊C₉

yes

direct product factor

139

C₃

C₃×C₃₇×C₃²⋊C₉

yes

characteristic

140

C₃

C₃×C₃₇×AΣL(1,𝔽₂₇)

yes

characteristic

141

C₃

C₃×C₃₇×#731

yes

characteristic

142

C₃

C₃×C₃₇×#733

yes

characteristic

	subgroup	G/H
center, socle	C₃³×C₃₇	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₃₇
Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

subgroup

G/H

center, socle

C₃³×C₃₇

H(3,𝔽₃)

commutator

C₃²

C₃²×C₉×C₃₇

Frattini, cosocle

C₃³

C₃³×C₃₇

prime radical

C₃⁴×C₃₇

C₃²

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

N	H ≅ G/N	type
5. C₃₇×3₋¹⁺²⋊C₉	C₃	direct (N × H)
9. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
32. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
43. C₃×3₋¹⁺²⋊C₉	C₃₇	direct (N × H)
68. 3₋¹⁺²⋊C₉	C₃×C₃₇	direct (N × H)
72. C₃×C₃₇	3₋¹⁺²⋊C₉	direct (N × H)
80. C₃×3₋¹⁺²	C₉×C₃₇	semidirect
93. C₃₇	C₃×3₋¹⁺²⋊C₉	direct (N × H)
104. 3₋¹⁺²	C₃×C₉×C₃₇	semidirect
138. C₃	C₃₇×3₋¹⁺²⋊C₉	direct (N × H)

H ≅ G/N

type

5. C₃₇×3₋¹⁺²⋊C₉

C₃

direct (N × H)

9. C₃×3₋¹⁺²×C₃₇

C₉

semidirect

32. 3₋¹⁺²×C₃₇

C₃×C₉

semidirect

43. C₃×3₋¹⁺²⋊C₉

C₃₇

direct (N × H)

68. 3₋¹⁺²⋊C₉

C₃×C₃₇

direct (N × H)

72. C₃×C₃₇

3₋¹⁺²⋊C₉

direct (N × H)

80. C₃×3₋¹⁺²

C₉×C₃₇

semidirect

93. C₃₇

C₃×3₋¹⁺²⋊C₉

direct (N × H)

104. 3₋¹⁺²

C₃×C₉×C₃₇

semidirect

138. C₃

C₃₇×3₋¹⁺²⋊C₉

direct (N × H)

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

subgroup	index
C₃²×3₋¹⁺²×C₃₇	3
C₃³×C₉×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
C₃₇×3₋¹⁺²⋊C₉	3
C₃×3₋¹⁺²⋊C₉	37

subgroup

index

C₃²×3₋¹⁺²×C₃₇

C₃³×C₉×C₃₇

C₃×C₃₇×C₃²⋊C₉

C₃₇×3₋¹⁺²⋊C₉

C₃×3₋¹⁺²⋊C₉

Subgroups — C₃×C₃₇×3₋¹⁺²⋊C₉ #41004485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C37×3−1+2⋊C9 #41004485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×3₋¹⁺²⋊C₉ #41004485