Group #14548309 — subgroups

Subgroups — #14548309

Order 2187.

1168 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624823	C₃	1	2	yes	maximal, complemented
3	#10624781	C₃	1	2	yes	maximal, complemented
4	#10624779	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624779	C₃	1	2	yes	maximal, complemented
6	#10624777	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624742	C₃	1	1	yes	maximal, characteristic
8	#10624633	C₃	1	2	yes	maximal, complemented
9	C₃×#6857	C₃	1	1	yes	maximal, characteristic, complemented
10	3₋¹⁺⁴	—	3	27	no	—
11	C₃×#737	—	3	3	no	—
12	C₃×#737	—	3	6	no	—
13	C₃²×3₋¹⁺²	—	3	3	no	—
14	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
17	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
18	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
19	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
20	#6891	C₃²	1	2	yes	—
21	#6891	C₃²	1	2	yes	—
22	#6891	C₃²	1	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	3₋¹⁺²⋊C₉	—	3	6	no	—
25	3₋¹⁺²⋊C₉	—	3	6	no	—
26	H(3,𝔽₃)⋊C₉	—	3	3	no	—
27	H(3,𝔽₃)⋊C₉	—	3	3	no	—
28	H(3,𝔽₃)⋊C₉	—	3	6	no	—
29	#6860	—	3	6	no	—
30	#6857	—	3	3	no	—
31	C₃⁴	—	3	1	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	C₃³	1	1	yes	characteristic
34	#737	—	3	54	no	—
35	#737	—	3	162	no	—
36	#737	—	3	27	no	—
37	C₃×3₋¹⁺²	—	3	18	no	—
38	C₃×3₋¹⁺²	—	3	81	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	3	2	no	—
43	C₃×3₋¹⁺²	—	3	18	no	—
44	C₃×3₋¹⁺²	—	3	9	no	—
45	C₃×3₋¹⁺²	—	3	2	no	—
46	C₃×3₋¹⁺²	—	3	1	no	—
47	C₃×3₋¹⁺²	—	3	2	no	—
48	C₃×3₋¹⁺²	—	3	2	no	—
49	C₃×3₋¹⁺²	—	3	2	no	—
50	C₃×3₋¹⁺²	—	9	3	no	—
51	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
52	C₃×H(3,𝔽₃)	—	3	2	no	—
53	C₃×H(3,𝔽₃)	—	3	9	no	—
54	C₃×H(3,𝔽₃)	—	3	2	no	—
55	C₃×H(3,𝔽₃)	—	3	1	no	—
56	C₃×H(3,𝔽₃)	—	9	3	no	—
57	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
58	C₃²×C₉	—	3	9	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	1	no	—
61	C₃²×C₉	—	3	1	no	—
62	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
63	AΣL(1,𝔽₂₇)	—	9	9	no	—
64	C₃²⋊C₉	—	3	1	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	—	3	2	no	—
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	3	2	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	C₃³	—	3	9	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	3	3	no	—
78	C₃³	—	9	1	no	—
79	C₃³	—	9	1	no	—
80	C₃³	—	9	1	no	—
81	C₃³	—	9	3	no	—
82	C₃³	—	9	1	no	—
83	C₃³	—	9	3	no	—
84	C₃³	—	27	1	no	—
85	C₃³	—	27	1	no	—
86	C₃³	—	27	1	no	—
87	C₃³	—	27	1	no	—
88	C₃³	—	27	1	no	—
89	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
90	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
91	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
92	3₋¹⁺²	—	3	18	no	—
93	3₋¹⁺²	—	3	18	no	—
94	3₋¹⁺²	—	3	162	no	—
95	3₋¹⁺²	—	3	18	no	—
96	3₋¹⁺²	—	3	18	no	—
97	3₋¹⁺²	—	3	9	no	—
98	3₋¹⁺²	—	3	6	no	—
99	3₋¹⁺²	—	9	4	no	—
100	3₋¹⁺²	—	9	3	no	—
101	3₋¹⁺²	—	9	6	no	—
102	3₋¹⁺²	—	9	6	no	—
103	3₋¹⁺²	—	27	2	no	—
104	H(3,𝔽₃)	—	3	18	no	—
105	H(3,𝔽₃)	—	3	6	no	—
106	H(3,𝔽₃)	—	3	6	no	—
107	H(3,𝔽₃)	—	9	3	no	—
108	H(3,𝔽₃)	—	9	4	no	—
109	H(3,𝔽₃)	AΣL(1,𝔽₂₇)	1	9	yes	—
110	C₃×C₉	—	3	81	no	—
111	C₃×C₉	—	3	3	no	—
112	C₃×C₉	—	3	6	no	—
113	C₃×C₉	—	3	9	no	—
114	C₃×C₉	—	3	18	no	—
115	C₃×C₉	—	9	1	no	—
116	C₃×C₉	—	9	3	no	—
117	C₃×C₉	—	9	1	no	—
118	C₃×C₉	—	9	2	no	—
119	C₃×C₉	—	9	2	no	—
120	C₃×C₉	—	9	2	no	—
121	C₃×C₉	—	27	1	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	3	9	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	2	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	2	no	—
131	C₃²	—	3	1	no	—
132	C₃²	—	3	2	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	9	3	no	—
135	C₃²	—	9	3	no	—
136	C₃²	—	9	3	no	—
137	C₃²	—	9	3	no	—
138	C₃²	—	27	3	no	—
139	C₃²	—	27	9	no	—
140	C₃²	—	27	9	no	—
141	C₃²	—	27	1	no	—
142	C₃²	—	27	1	no	—
143	C₃²	—	27	1	no	—
144	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
145	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
146	C₉	—	3	9	no	—
147	C₉	—	3	18	no	—
148	C₉	—	9	3	no	—
149	C₉	—	27	2	no	—
150	C₉	—	27	1	no	—
151	C₉	—	27	2	no	—
152	C₉	—	27	2	no	—
153	C₃	—	3	3	no	—
154	C₃	—	9	2	no	—
155	C₃	—	9	1	no	—
156	C₃	—	9	9	no	—
157	C₃	—	27	3	no	—
158	C₃	#10624730	1	1	yes	characteristic
159	C₃	#10624752	1	1	yes	characteristic
160	C₃	#10624752	1	1	yes	characteristic
161	C₃	#10624753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic, complemented

#10624823

C₃

yes

maximal, complemented

#10624781

C₃

yes

maximal, complemented

#10624779

C₃

yes

maximal, characteristic, complemented

#10624779

C₃

yes

maximal, complemented

#10624777

C₃

yes

maximal, characteristic, complemented

#10624742

C₃

yes

maximal, characteristic

#10624633

C₃

yes

maximal, complemented

C₃×#6857

C₃

yes

maximal, characteristic, complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×AΣL(1,𝔽₂₇)

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6860

—

#6857

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

#737

—

#737

—

162

—

#737

—

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

—

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)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

162

—

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

H(3,𝔽₃)

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

—

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

AΣL(1,𝔽₂₇)

yes

—

110

C₃×C₉

—

111

C₃×C₉

—

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

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

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

C₃²⋊H(3,𝔽₃)

yes

characteristic

145

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

146

C₉

—

147

C₉

—

148

C₉

—

149

C₉

—

150

C₉

—

151

C₉

—

152

C₉

—

153

C₃

—

154

C₃

—

155

C₃

—

156

C₃

—

157

C₃

—

158

C₃

#10624730

yes

characteristic

159

C₃

#10624752

yes

characteristic

160

C₃

#10624752

yes

characteristic

161

C₃

#10624753

yes

characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624742	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. C₃×3₋¹⁺⁴	C₃	semidirect
2. #10624823	C₃	semidirect
3. #10624781	C₃	semidirect
4. #10624779	C₃	semidirect
5. #10624779	C₃	semidirect
6. #10624777	C₃	semidirect
8. #10624633	C₃	semidirect
9. C₃×#6857	C₃	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺⁴

C₃

semidirect

2. #10624823

C₃

semidirect

3. #10624781

C₃

semidirect

4. #10624779

C₃

semidirect

5. #10624779

C₃

semidirect

6. #10624777

C₃

semidirect

8. #10624633

C₃

semidirect

9. C₃×#6857

C₃

semidirect

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

subgroup	index
C₃×3₋¹⁺⁴	3
#10624823	3
#10624781	3
#10624779	3
#10624777	3
#10624742	3
#10624633	3
C₃×#6857	3

subgroup

index

C₃×3₋¹⁺⁴

Subgroups — #14548309

Conjugacy classes of subgroups

Complements

Maximal subgroups