Group #14547201 — subgroups

Subgroups — #14547201

Order 2187.

474 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 59 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×#731	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6894	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624781	C₃	1	2	yes	maximal, complemented
4	#10624767	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624743	C₃	1	1	yes	maximal, characteristic
6	#10624731	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624610	C₃	1	2	yes	maximal, complemented
8	#10624609	C₃	1	2	yes	maximal, complemented
9	#10624602	C₃	1	2	yes	maximal, complemented
10	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₃×#731	—	3	2	no	—
15	C₃×#731	—	3	1	no	—
16	C₃×#731	—	3	9	no	—
17	C₃×#731	—	3	2	no	—
18	C₃×#731	—	3	1	no	—
19	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
20	#6894	—	3	3	no	—
21	#6894	—	3	6	no	—
22	#6891	C₃²	1	2	yes	—
23	#6891	C₃²	1	2	yes	—
24	#6891	C₃²	1	1	yes	characteristic
25	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
26	C₃×C₉⋊C₉	—	3	2	no	—
27	C₃×C₉⋊C₉	—	3	1	no	—
28	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
29	3₋¹⁺²⋊C₉	—	3	6	no	—
30	#6867	—	3	2	no	—
31	#6867	—	3	1	no	—
32	#6859	—	3	6	no	—
33	#6858	—	3	6	no	—
34	#6857	—	3	6	no	—
35	C₃⁴	—	3	1	no	—
36	C₃⁴	C₃³	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	6	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	3	1	no	—
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	9	3	no	—
46	C₃×3₋¹⁺²	—	9	4	no	—
47	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
48	C₃×H(3,𝔽₃)	—	3	1	no	—
49	C₃×H(3,𝔽₃)	—	3	3	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	—	3	1	no	—
52	C₃×H(3,𝔽₃)	—	3	1	no	—
53	C₃×H(3,𝔽₃)	—	3	2	no	—
54	C₃×H(3,𝔽₃)	—	9	2	no	—
55	C₃²×C₉	—	3	2	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	1	no	—
58	C₃²×C₉	—	3	2	no	—
59	C₃²×C₉	—	3	3	no	—
60	C₃²×C₉	—	3	2	no	—
61	C₃²×C₉	—	9	2	no	—
62	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
63	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
64	#731	—	3	18	no	—
65	#731	—	3	9	no	—
66	#731	—	9	6	no	—
67	#731	—	9	3	no	—
68	AΣL(1,𝔽₂₇)	—	9	9	no	—
69	C₉⋊C₉	—	3	9	no	—
70	C₉⋊C₉	—	3	18	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃²⋊C₉	—	3	2	no	—
74	C₃²⋊C₉	—	3	2	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	1	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	—	3	2	no	—
81	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
82	C₃³	—	3	1	no	—
83	C₃³	—	3	2	no	—
84	C₃³	—	3	3	no	—
85	C₃³	—	9	2	no	—
86	C₃³	—	9	3	no	—
87	C₃³	—	9	1	no	—
88	C₃³	—	9	1	no	—
89	C₃³	—	9	3	no	—
90	C₃³	—	27	2	no	—
91	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
92	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
93	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
94	3₋¹⁺²	—	3	6	no	—
95	3₋¹⁺²	—	3	3	no	—
96	3₋¹⁺²	—	3	6	no	—
97	3₋¹⁺²	—	9	6	no	—
98	3₋¹⁺²	—	9	4	no	—
99	3₋¹⁺²	—	9	6	no	—
100	3₋¹⁺²	—	9	6	no	—
101	3₋¹⁺²	—	9	2	no	—
102	3₋¹⁺²	—	9	4	no	—
103	3₋¹⁺²	—	9	3	no	—
104	3₋¹⁺²	—	27	2	no	—
105	H(3,𝔽₃)	—	3	3	no	—
106	H(3,𝔽₃)	—	9	6	no	—
107	H(3,𝔽₃)	—	9	2	no	—
108	H(3,𝔽₃)	—	9	1	no	—
109	H(3,𝔽₃)	—	9	2	no	—
110	H(3,𝔽₃)	—	9	3	no	—
111	H(3,𝔽₃)	—	27	2	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	2	no	—
114	C₃×C₉	—	3	6	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	3	no	—
117	C₃×C₉	—	3	2	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	2	no	—
120	C₃×C₉	—	9	3	no	—
121	C₃×C₉	—	9	6	no	—
122	C₃×C₉	—	9	6	no	—
123	C₃×C₉	—	9	3	no	—
124	C₃×C₉	—	9	2	no	—
125	C₃×C₉	—	9	6	no	—
126	C₃×C₉	—	9	1	no	—
127	C₃×C₉	—	9	2	no	—
128	C₃×C₉	—	9	2	no	—
129	C₃×C₉	AΣL(1,𝔽₂₇)	1	3	yes	complemented
130	C₃²	—	3	1	no	—
131	C₃²	—	3	1	no	—
132	C₃²	—	3	2	no	—
133	C₃²	—	3	2	no	—
134	C₃²	—	3	2	no	—
135	C₃²	—	3	2	no	—
136	C₃²	—	3	2	no	—
137	C₃²	—	3	2	no	—
138	C₃²	—	9	3	no	—
139	C₃²	—	9	6	no	—
140	C₃²	—	9	3	no	—
141	C₃²	—	27	9	no	—
142	C₃²	—	27	1	no	—
143	C₃²	—	27	2	no	—
144	C₃²	—	27	1	no	—
145	C₃²	—	27	1	no	—
146	C₃²	—	27	2	no	—
147	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
148	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
149	C₃²	#6909	1	2	yes	—
150	C₃²	#6910	1	3	yes	—
151	C₉	—	3	3	no	—
152	C₉	—	9	3	no	—
153	C₉	—	9	2	no	—
154	C₉	—	9	6	no	—
155	C₉	—	9	6	no	—
156	C₉	—	27	1	no	—
157	C₉	—	27	2	no	—
158	C₉	—	27	2	no	—
159	C₃	—	3	3	no	—
160	C₃	—	3	1	no	—
161	C₃	—	3	2	no	—
162	C₃	—	9	2	no	—
163	C₃	—	27	3	no	—
164	C₃	—	81	1	no	—
165	C₃	#10624736	1	1	yes	characteristic
166	C₃	#10624743	1	1	yes	characteristic
167	C₃	#10624756	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#731

C₃

yes

maximal, characteristic, complemented

C₃×#6894

C₃

yes

maximal, characteristic, complemented

#10624781

C₃

yes

maximal, complemented

#10624767

C₃

yes

maximal, characteristic, complemented

#10624743

C₃

yes

maximal, characteristic

#10624731

C₃

yes

maximal, characteristic, complemented

#10624610

C₃

yes

maximal, complemented

#10624609

C₃

yes

maximal, complemented

#10624602

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

—

—

—

C₃×AΣL(1,𝔽₂₇)

—

#6894

—

#6894

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

—

—

—

—

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

yes

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

—

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

—

110

H(3,𝔽₃)

—

111

H(3,𝔽₃)

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

—

129

C₃×C₉

AΣL(1,𝔽₂₇)

yes

complemented

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

148

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

149

C₃²

#6909

yes

—

150

C₃²

#6910

yes

—

151

C₉

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

—

159

C₃

—

160

C₃

—

161

C₃

—

162

C₃

—

163

C₃

—

164

C₃

—

165

C₃

#10624736

yes

characteristic

166

C₃

#10624743

yes

characteristic

167

C₃

#10624756

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624743	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

C₃

G ▷ C₃⁴ ▷ C₃² ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃²×#731	C₃	semidirect
2. C₃×#6894	C₃	semidirect
3. #10624781	C₃	semidirect
4. #10624767	C₃	semidirect
6. #10624731	C₃	semidirect
7. #10624610	C₃	semidirect
8. #10624609	C₃	semidirect
9. #10624602	C₃	semidirect
129. C₃×C₉	AΣL(1,𝔽₂₇)	semidirect

H ≅ G/N

type

1. C₃²×#731

C₃

semidirect

2. C₃×#6894

C₃

semidirect

3. #10624781

C₃

semidirect

4. #10624767

C₃

semidirect

6. #10624731

C₃

semidirect

7. #10624610

C₃

semidirect

8. #10624609

C₃

semidirect

9. #10624602

C₃

semidirect

129. C₃×C₉

AΣL(1,𝔽₂₇)

semidirect

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

subgroup	index
C₃²×#731	3
C₃×#6894	3
#10624781	3
#10624767	3
#10624743	3
#10624731	3
#10624610	3
#10624609	3
#10624602	3

subgroup

index

C₃²×#731

Subgroups — #14547201

Conjugacy classes of subgroups

Complements

Maximal subgroups