Group #41004390 — subgroups

Subgroups — C₃₇×#10624601 #41004390

Order 26973.

232 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
2	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6861	C₃	1	2	yes	maximal, complemented
5	C₃₇×#6861	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃₇×#6859	C₃	1	2	yes	maximal, complemented
7	C₃₇×#6859	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#6856	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#6856	C₃	1	2	yes	maximal, complemented
10	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	2	yes	—
13	C₃²×C₉×C₃₇	—	3	1	no	—
14	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
15	C₃₇×C₉⋊C₉	—	3	1	no	—
16	C₃₇×C₉⋊C₉	—	3	1	no	—
17	C₃₇×C₃²⋊C₉	—	3	1	no	—
18	C₃₇×C₃²⋊C₉	—	3	1	no	—
19	C₃₇×C₃²⋊C₉	—	3	2	no	—
20	C₃₇×C₃²⋊C₉	—	3	2	no	—
21	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₉²×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	—	3	1	no	—
29	C₃³×C₃₇	—	3	2	no	—
30	C₃³×C₃₇	C₃³	1	1	yes	characteristic
31	3₋¹⁺²×C₃₇	—	9	6	no	—
32	H(3,𝔽₃)×C₃₇	—	3	3	no	—
33	H(3,𝔽₃)×C₃₇	—	9	2	no	—
34	C₃×C₉×C₃₇	—	3	3	no	—
35	C₃×C₉×C₃₇	—	3	2	no	—
36	C₃×C₉×C₃₇	—	3	1	no	—
37	C₃×C₉×C₃₇	—	3	1	no	—
38	C₃×C₉×C₃₇	—	3	1	no	—
39	C₃×C₉×C₃₇	—	3	2	no	—
40	C₃×C₉×C₃₇	—	3	2	no	—
41	C₃×C₉×C₃₇	—	9	3	no	—
42	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
43	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
44	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
45	#10624601	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₃²×C₃₇	—	3	2	no	—
47	C₃²×C₃₇	—	3	1	no	—
48	C₃²×C₃₇	—	3	1	no	—
49	C₃²×C₃₇	—	3	3	no	—
50	C₃²×C₃₇	—	9	2	no	—
51	C₃²×C₃₇	—	9	1	no	—
52	C₃²×C₃₇	—	9	2	no	—
53	C₃²×C₃₇	—	9	2	no	—
54	C₃²×C₃₇	—	9	2	no	—
55	C₃²×C₃₇	—	9	2	no	—
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	C₉×C₃₇	—	3	1	no	—
58	C₉×C₃₇	—	3	1	no	—
59	C₉×C₃₇	—	9	2	no	—
60	C₉×C₃₇	—	9	1	no	—
61	C₉×C₃₇	—	9	1	no	—
62	C₉×C₃₇	—	9	2	no	—
63	C₉×C₃₇	—	9	2	no	—
64	C₉×C₃₇	—	9	1	no	—
65	C₉×C₃₇	#732	1	3	yes	—
66	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	2	yes	complemented
67	C₉×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic, complemented
68	#6887	C₃×C₃₇	1	1	yes	characteristic, complemented
69	#6861	C₃×C₃₇	1	2	yes	complemented
70	#6861	C₃×C₃₇	1	1	yes	characteristic, complemented
71	#6859	C₃×C₃₇	1	2	yes	complemented
72	#6859	C₃×C₃₇	1	1	yes	characteristic, complemented
73	#6856	C₃×C₃₇	1	1	yes	characteristic, complemented
74	#6856	C₃×C₃₇	1	2	yes	complemented
75	C₃×C₃₇	—	9	1	no	—
76	C₃×C₃₇	—	9	3	no	—
77	C₃×C₃₇	—	27	2	no	—
78	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
79	C₃×C₃₇	#6908	1	1	yes	characteristic
80	C₃×C₃₇	#6913	1	2	yes	—
81	C₃×3₋¹⁺²	—	3	2	no	—
82	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
83	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	2	yes	—
84	C₃²×C₉	—	3	1	no	—
85	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
86	C₉⋊C₉	—	3	1	no	—
87	C₉⋊C₉	—	3	1	no	—
88	C₃²⋊C₉	—	3	1	no	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	3	2	no	—
91	C₃²⋊C₉	—	3	2	no	—
92	C₃²⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
93	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
94	C₃²⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
95	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
96	C₃²⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
97	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
98	C₉²	—	3	1	no	—
99	C₃₇	#10624601	1	1	yes	characteristic, direct product factor, Sylow
100	C₃³	—	3	1	no	—
101	C₃³	—	3	2	no	—
102	C₃³	C₃³×C₃₇	1	1	yes	characteristic
103	3₋¹⁺²	—	9	6	no	—
104	H(3,𝔽₃)	—	3	3	no	—
105	H(3,𝔽₃)	—	9	2	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	3	1	no	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	3	1	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	2	no	—
112	C₃×C₉	—	3	2	no	—
113	C₃×C₉	—	9	3	no	—
114	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
115	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
116	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
117	C₃²	—	3	1	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	3	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	9	1	no	—
122	C₃²	—	9	2	no	—
123	C₃²	—	9	2	no	—
124	C₃²	—	9	2	no	—
125	C₃²	—	9	2	no	—
126	C₃²	—	9	2	no	—
127	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
128	C₉	—	3	1	no	—
129	C₉	—	3	1	no	—
130	C₉	—	9	1	no	—
131	C₉	—	9	1	no	—
132	C₉	—	9	2	no	—
133	C₉	—	9	2	no	—
134	C₉	—	9	1	no	—
135	C₉	—	9	2	no	—
136	C₉	C₃₇×#732	1	3	yes	—
137	C₃	—	9	1	no	—
138	C₃	—	9	3	no	—
139	C₃	—	27	2	no	—
140	C₃	C₃×C₃₇×#732	1	1	yes	characteristic
141	C₃	C₃₇×#6908	1	1	yes	characteristic
142	C₃	C₃₇×#6913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, complemented

C₉×H(3,𝔽₃)×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃₇×#6887

C₃

yes

maximal, characteristic, complemented

C₃₇×#6861

C₃

yes

maximal, complemented

C₃₇×#6861

C₃

yes

maximal, characteristic, complemented

C₃₇×#6859

C₃

yes

maximal, complemented

C₃₇×#6859

C₃

yes

maximal, characteristic, complemented

C₃₇×#6856

C₃

yes

maximal, characteristic, complemented

C₃₇×#6856

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₃₇

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

characteristic

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₉⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₉²×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

3₋¹⁺²×C₃₇

—

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×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

yes

C₃×C₉×C₃₇

yes

C₃×C₉×C₃₇

yes

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×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

#732

yes

—

C₉⋊H(3,𝔽₃)

C₃×C₃₇

yes

complemented

C₉×H(3,𝔽₃)

C₃×C₃₇

yes

characteristic, complemented

#6887

C₃×C₃₇

yes

characteristic, complemented

#6861

C₃×C₃₇

yes

complemented

#6861

C₃×C₃₇

yes

characteristic, complemented

#6859

C₃×C₃₇

yes

complemented

#6859

C₃×C₃₇

yes

characteristic, complemented

#6856

C₃×C₃₇

yes

characteristic, complemented

#6856

C₃×C₃₇

yes

complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₃²×C₃₇

yes

characteristic

C₃×H(3,𝔽₃)

C₃²×C₃₇

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₃₇

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₃₇

yes

characteristic

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₃²⋊C₉

C₃²×C₃₇

yes

characteristic

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₃²⋊C₉

C₃²×C₃₇

yes

characteristic

C₃²⋊C₉

C₃²×C₃₇

yes

—

C₉²

—

C₃₇

#10624601

yes

characteristic, direct product factor, Sylow

100

C₃³

—

101

C₃³

—

102

C₃³

C₃³×C₃₇

yes

characteristic

103

3₋¹⁺²

—

104

H(3,𝔽₃)

—

105

H(3,𝔽₃)

—

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

H(3,𝔽₃)×C₃₇

yes

characteristic

115

C₃×C₉

H(3,𝔽₃)×C₃₇

yes

characteristic

116

C₃×C₉

H(3,𝔽₃)×C₃₇

yes

characteristic

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

C₃×H(3,𝔽₃)×C₃₇

yes

characteristic

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

C₃₇×#732

yes

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

C₃×C₃₇×#732

yes

characteristic

141

C₃

C₃₇×#6908

yes

characteristic

142

C₃

C₃₇×#6913

yes

—

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

subgroup

G/H

center, socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₃₇

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

N	H ≅ G/N	type
1. C₃₇×C₉⋊H(3,𝔽₃)	C₃	semidirect
2. C₉×H(3,𝔽₃)×C₃₇	C₃	semidirect
3. C₃₇×#6887	C₃	semidirect
4. C₃₇×#6861	C₃	semidirect
5. C₃₇×#6861	C₃	semidirect
6. C₃₇×#6859	C₃	semidirect
7. C₃₇×#6859	C₃	semidirect
8. C₃₇×#6856	C₃	semidirect
9. C₃₇×#6856	C₃	semidirect
45. #10624601	C₃₇	direct (N × H)
66. C₉⋊H(3,𝔽₃)	C₃×C₃₇	semidirect
67. C₉×H(3,𝔽₃)	C₃×C₃₇	semidirect
68. #6887	C₃×C₃₇	semidirect
69. #6861	C₃×C₃₇	semidirect
70. #6861	C₃×C₃₇	semidirect
71. #6859	C₃×C₃₇	semidirect
72. #6859	C₃×C₃₇	semidirect
73. #6856	C₃×C₃₇	semidirect
74. #6856	C₃×C₃₇	semidirect
99. C₃₇	#10624601	direct (N × H)

H ≅ G/N

type

1. C₃₇×C₉⋊H(3,𝔽₃)

C₃

semidirect

2. C₉×H(3,𝔽₃)×C₃₇

C₃

semidirect

3. C₃₇×#6887

C₃

semidirect

4. C₃₇×#6861

C₃

semidirect

5. C₃₇×#6861

C₃

semidirect

6. C₃₇×#6859

C₃

semidirect

7. C₃₇×#6859

C₃

semidirect

8. C₃₇×#6856

C₃

semidirect

9. C₃₇×#6856

C₃

semidirect

45. #10624601

C₃₇

direct (N × H)

66. C₉⋊H(3,𝔽₃)

C₃×C₃₇

semidirect

67. C₉×H(3,𝔽₃)

C₃×C₃₇

semidirect

68. #6887

C₃×C₃₇

semidirect

69. #6861

C₃×C₃₇

semidirect

70. #6861

C₃×C₃₇

semidirect

71. #6859

C₃×C₃₇

semidirect

72. #6859

C₃×C₃₇

semidirect

73. #6856

C₃×C₃₇

semidirect

74. #6856

C₃×C₃₇

semidirect

99. C₃₇

#10624601

direct (N × H)

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

subgroup	index
C₃₇×C₉⋊H(3,𝔽₃)	3
C₉×H(3,𝔽₃)×C₃₇	3
C₃₇×#6887	3
C₃₇×#6861	3
C₃₇×#6859	3
C₃₇×#6856	3
#10624601	37

subgroup

index

C₃₇×C₉⋊H(3,𝔽₃)

C₉×H(3,𝔽₃)×C₃₇

C₃₇×#6887

C₃₇×#6861

C₃₇×#6859

C₃₇×#6856

#10624601

Subgroups — C₃₇×#10624601 #41004390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#10624601 #41004390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624601 #41004390