Group #41005097 — subgroups

Subgroups — #41005097

Order 26973.

2150 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 2150 conjugacy classes of subgroups — too many to draw. Only its 202 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	C₃⁴×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#66524	C₃	1	2	yes	maximal, complemented
3	C₃×#15688502	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	—	3	3	no	—
6	C₃³×C₃₇⋊C₃	—	3	3	no	—
7	C₃³×C₃₇⋊C₃	—	3	3	no	—
8	C₃³×C₃₇⋊C₃	—	3	9	no	—
9	C₃³×C₃₇⋊C₃	—	3	18	no	—
10	C₃³×C₃₇⋊C₃	C₉	1	9	yes	complemented
11	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
12	C₃²×#66524	C₃²	1	6	yes	complemented
13	C₃²⋊#66524	—	3	6	no	—
14	#15688502	—	3	27	no	—
15	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
17	C₃₇×C₃²⋊C₉	—	3	3	no	—
18	C₃³×C₃₇	—	3	3	no	—
19	C₃³×C₃₇	—	3	6	no	—
20	C₃³×C₃₇	C₃×C₉	1	3	yes	—
21	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
22	C₃³×C₃₇	3₋¹⁺²	1	3	yes	—
23	C₃³×C₃₇	3₋¹⁺²	1	3	yes	—
24	C₃³×C₃₇	C₃³	1	1	yes	characteristic
25	C₃²×C₃₇⋊C₃	—	3	27	no	—
26	C₃²×C₃₇⋊C₃	—	3	81	no	—
27	C₃²×C₃₇⋊C₃	—	3	81	no	—
28	C₃²×C₃₇⋊C₃	—	3	54	no	—
29	C₃²×C₃₇⋊C₃	—	3	27	no	—
30	C₃²×C₃₇⋊C₃	—	3	6	no	—
31	C₃²×C₃₇⋊C₃	—	3	3	no	—
32	C₃²×C₃₇⋊C₃	—	3	6	no	—
33	C₃²×C₃₇⋊C₃	—	3	6	no	—
34	C₃²×C₃₇⋊C₃	—	3	3	no	—
35	C₃²×C₃₇⋊C₃	—	3	3	no	—
36	C₃²×C₃₇⋊C₃	—	3	81	no	—
37	C₃²×C₃₇⋊C₃	C₃×C₉	1	9	yes	—
38	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	9	yes	—
39	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	18	yes	—
40	C₃×C₉×C₃₇	—	3	3	no	—
41	C₃×C₉×C₃₇	—	3	9	no	—
42	C₃×#66524	—	3	18	no	—
43	C₃×#66524	—	3	6	no	—
44	#10624547	—	37	1	no	maximal, Sylow
45	C₃²×C₃₇	—	3	9	no	—
46	C₃²×C₃₇	—	3	3	no	—
47	C₃²×C₃₇	—	3	6	no	—
48	C₃²×C₃₇	—	3	9	no	—
49	C₃²×C₃₇	—	3	9	no	—
50	C₃²×C₃₇	C₃²⋊C₉	1	9	yes	complemented
51	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
52	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
55	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
56	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₃×C₃₇⋊C₃	—	3	243	no	—
58	C₃×C₃₇⋊C₃	—	3	18	no	—
59	C₃×C₃₇⋊C₃	—	3	9	no	—
60	C₃×C₃₇⋊C₃	—	3	9	no	—
61	C₃×C₃₇⋊C₃	—	3	9	no	—
62	C₃×C₃₇⋊C₃	—	3	9	no	—
63	C₃×C₃₇⋊C₃	—	3	18	no	—
64	C₃×C₃₇⋊C₃	—	3	18	no	—
65	C₃×C₃₇⋊C₃	—	3	18	no	—
66	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	27	yes	complemented
67	C₉×C₃₇	—	9	3	no	—
68	#66524	—	9	6	no	—
69	C₃⁵	—	37	1	no	—
70	C₃×C₃²⋊C₉	—	37	2	no	—
71	C₃×C₃²⋊C₉	—	37	9	no	—
72	C₃×C₃²⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
73	C₃×C₃₇	—	3	9	no	—
74	C₃×C₃₇	C₃×C₃²⋊C₉	1	3	yes	—
75	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
76	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃×C₃₇	#6891	1	2	yes	—
78	C₃×C₃₇	#6891	1	3	yes	—
79	C₃×C₃₇	#6891	1	3	yes	—
80	C₃₇⋊C₃	—	3	27	no	—
81	C₃⁴	—	37	3	no	—
82	C₃⁴	—	37	9	no	—
83	C₃⁴	—	111	9	no	—
84	C₃⁴	—	111	18	no	—
85	C₃⁴	—	111	3	no	—
86	C₃⁴	—	111	3	no	—
87	C₃⁴	—	111	3	no	—
88	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
89	C₃²×C₉	—	37	6	no	—
90	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
91	C₃²⋊C₉	—	3	3	no	—
92	C₃²⋊C₉	—	111	27	no	—
93	C₃²⋊C₉	—	111	6	no	—
94	C₃₇	#10624547	1	1	yes	characteristic, complemented, Sylow
95	C₃³	—	3	3	no	—
96	C₃³	—	3	6	no	—
97	C₃³	—	37	9	no	—
98	C₃³	—	37	9	no	—
99	C₃³	—	37	18	no	—
100	C₃³	—	111	81	no	—
101	C₃³	—	111	27	no	—
102	C₃³	—	111	54	no	—
103	C₃³	—	111	81	no	—
104	C₃³	—	111	81	no	—
105	C₃³	—	111	27	no	—
106	C₃³	—	111	6	no	—
107	C₃³	—	111	3	no	—
108	C₃³	—	111	6	no	—
109	C₃³	—	111	6	no	—
110	C₃³	—	111	3	no	—
111	C₃³	—	111	3	no	—
112	C₃³	C₉×C₃₇⋊C₃	1	3	yes	—
113	C₃³	#11758605	1	3	yes	—
114	C₃³	#11758606	1	3	yes	—
115	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
116	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
117	C₃×C₉	—	3	9	no	—
118	C₃×C₉	—	3	3	no	—
119	C₃×C₉	—	111	18	no	—
120	C₃×C₉	—	111	6	no	—
121	C₃²	—	3	9	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	3	6	no	—
124	C₃²	—	3	9	no	—
125	C₃²	—	3	9	no	—
126	C₃²	—	37	27	no	—
127	C₃²	—	111	243	no	—
128	C₃²	—	111	18	no	—
129	C₃²	—	111	9	no	—
130	C₃²	—	111	9	no	—
131	C₃²	—	111	18	no	—
132	C₃²	—	111	9	no	—
133	C₃²	—	111	18	no	—
134	C₃²	—	111	9	no	—
135	C₃²	—	111	18	no	—
136	C₃²	#15688502	1	9	yes	complemented
137	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
138	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
139	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
140	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	6	yes	—
141	C₃²	C₃×#11758606	1	1	yes	characteristic
142	C₃²	C₃×#11758605	1	1	yes	characteristic
143	C₉	—	9	3	no	—
144	C₉	—	333	6	no	—
145	C₃	—	3	9	no	—
146	C₃	—	111	27	no	—
147	C₃	C₃²⋊C₉×C₃₇⋊C₃	1	3	yes	—
148	C₃	C₃×#15688502	1	1	yes	characteristic
149	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
150	C₃	#22582551	1	2	yes	—
151	C₃	#22582552	1	3	yes	—
152	C₃	#22582553	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₃₇⋊C₃

C₃

yes

maximal, characteristic

C₃×C₃²⋊#66524

C₃

yes

maximal, complemented

C₃×#15688502

C₃

yes

maximal, complemented

C₃×C₃₇×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

—

C₃³×C₃₇⋊C₃

C₉

yes

complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

C₃²×#66524

C₃²

yes

complemented

C₃²⋊#66524

—

#15688502

—

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃²×C₉×C₃₇

C₃²

yes

complemented

C₃₇×C₃²⋊C₉

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃×C₉

yes

—

C₃³×C₃₇

H(3,𝔽₃)

yes

—

C₃³×C₃₇

3₋¹⁺²

yes

—

C₃³×C₃₇

3₋¹⁺²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

C₃²×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

complemented

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

—

243

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃²⋊C₉

yes

complemented

C₉×C₃₇

—

#66524

—

C₃⁵

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₃₇

#6891

yes

—

C₃×C₃₇

#6891

yes

—

C₃×C₃₇

#6891

yes

—

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

—

111

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃₇

#10624547

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

100

C₃³

—

111

—

101

C₃³

—

111

—

102

C₃³

—

111

—

103

C₃³

—

111

—

104

C₃³

—

111

—

105

C₃³

—

111

—

106

C₃³

—

111

—

107

C₃³

—

111

—

108

C₃³

—

111

—

109

C₃³

—

111

—

110

C₃³

—

111

—

111

C₃³

—

111

—

112

C₃³

C₉×C₃₇⋊C₃

yes

—

113

C₃³

#11758605

yes

—

114

C₃³

#11758606

yes

—

115

C₃³

C₃²⋊(C₃₇⋊C₃)

yes

—

116

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

111

—

120

C₃×C₉

—

111

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

111

243

—

128

C₃²

—

111

—

129

C₃²

—

111

—

130

C₃²

—

111

—

131

C₃²

—

111

—

132

C₃²

—

111

—

133

C₃²

—

111

—

134

C₃²

—

111

—

135

C₃²

—

111

—

136

C₃²

#15688502

yes

complemented

137

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

138

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

—

139

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

140

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

141

C₃²

C₃×#11758606

yes

characteristic

142

C₃²

C₃×#11758605

yes

characteristic

143

C₉

—

144

C₉

—

333

—

145

C₃

—

146

C₃

—

111

—

147

C₃

C₃²⋊C₉×C₃₇⋊C₃

yes

—

148

C₃

C₃×#15688502

yes

characteristic

149

C₃

(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

150

C₃

#22582551

yes

—

151

C₃

#22582552

yes

—

152

C₃

#22582553

yes

—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×C₃₇×C₃²⋊C₉	C₃
socle, cosocle	C₃³×C₃₇	C₃³
prime radical	C₃⁴×C₃₇⋊C₃	C₃
hypercenter	C₃×C₃²⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624547
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₃²⋊C₉

subgroup

G/H

center, Frattini

C₃³

C₃²×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Fitting

C₃×C₃₇×C₃²⋊C₉

C₃

socle, cosocle

C₃³×C₃₇

C₃³

prime radical

C₃⁴×C₃₇⋊C₃

C₃

hypercenter

C₃×C₃²⋊C₉

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃³ ◁ C₃×C₃²⋊C₉

N	H ≅ G/N	type
2. C₃×C₃²⋊#66524	C₃	semidirect
3. C₃×#15688502	C₃	semidirect
4. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
10. C₃³×C₃₇⋊C₃	C₉	semidirect
12. C₃²×#66524	C₃²	semidirect
16. C₃²×C₉×C₃₇	C₃²	semidirect
50. C₃²×C₃₇	C₃²⋊C₉	semidirect
66. C₃×C₃₇⋊C₃	C₃²⋊C₉	semidirect
72. C₃×C₃²⋊C₉	C₃₇⋊C₃	semidirect
90. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
94. C₃₇	#10624547	semidirect
136. C₃²	#15688502	semidirect

H ≅ G/N

type

2. C₃×C₃²⋊#66524

C₃

semidirect

3. C₃×#15688502

C₃

semidirect

4. C₃×C₃₇×C₃²⋊C₉

C₃

semidirect

10. C₃³×C₃₇⋊C₃

C₉

semidirect

12. C₃²×#66524

C₃²

semidirect

16. C₃²×C₉×C₃₇

C₃²

semidirect

50. C₃²×C₃₇

C₃²⋊C₉

semidirect

66. C₃×C₃₇⋊C₃

C₃²⋊C₉

semidirect

72. C₃×C₃²⋊C₉

C₃₇⋊C₃

semidirect

90. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

94. C₃₇

#10624547

semidirect

136. C₃²

#15688502

semidirect

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

subgroup	index
C₃⁴×C₃₇⋊C₃	3
C₃×C₃²⋊#66524	3
C₃×#15688502	3
C₃×C₃₇×C₃²⋊C₉	3
#10624547	37

subgroup

index

C₃⁴×C₃₇⋊C₃

C₃×C₃²⋊#66524

C₃×#15688502

C₃×C₃₇×C₃²⋊C₉

#10624547

Subgroups — #41005097

Conjugacy classes of subgroups

Complements

Maximal subgroups