Group #14547325 — subgroups

Subgroups — #14547325

Order 2187.

408 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 56 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	#10624883	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624820	C₃	1	2	yes	maximal, complemented
3	#10624745	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624735	C₃	1	1	yes	maximal, characteristic
5	#10624689	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624614	C₃	1	2	yes	maximal, complemented
7	#10624610	C₃	1	2	yes	maximal, complemented
8	#10624604	C₃	1	2	yes	maximal, complemented
9	#10624563	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×#731	—	3	3	no	—
13	C₃×AΣL(1,𝔽₂₇)	—	3	2	no	—
14	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
15	C₉⋊H(3,𝔽₃)	—	3	3	no	—
16	C₉⋊H(3,𝔽₃)	—	3	6	no	—
17	#6891	—	3	2	no	—
18	#6891	C₃²	1	2	yes	—
19	#6891	C₃²	1	1	yes	characteristic
20	#6891	C₃²	1	2	yes	—
21	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
22	C₃×C₃²⋊C₉	—	3	1	no	—
23	C₃×C₃²⋊C₉	—	3	2	no	—
24	C₃×C₃²⋊C₉	—	3	1	no	—
25	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
28	H(3,𝔽₃)⋊C₉	—	3	6	no	—
29	#6866	—	3	2	no	—
30	#6866	—	3	1	no	—
31	#6861	—	3	6	no	—
32	#6858	—	3	6	no	—
33	#6858	—	3	6	no	—
34	C₃⁴	H(3,𝔽₃)	1	2	yes	—
35	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃⁴	C₃³	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	6	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	9	3	no	—
43	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
44	C₃×H(3,𝔽₃)	—	3	3	no	—
45	C₃×H(3,𝔽₃)	—	3	1	no	—
46	C₃×H(3,𝔽₃)	—	3	2	no	—
47	C₃×H(3,𝔽₃)	—	3	2	no	—
48	C₃×H(3,𝔽₃)	—	3	1	no	—
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃×H(3,𝔽₃)	—	9	2	no	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	3	no	—
53	C₃²×C₉	—	3	2	no	—
54	C₃²×C₉	—	3	1	no	—
55	C₃²×C₉	—	3	2	no	—
56	#731	—	9	9	no	—
57	AΣL(1,𝔽₂₇)	—	9	6	no	—
58	AΣL(1,𝔽₂₇)	—	9	3	no	—
59	C₃²⋊C₉	—	3	6	no	—
60	C₃²⋊C₉	—	3	12	no	—
61	C₃²⋊C₉	—	3	1	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	—	3	2	no	—
64	C₃²⋊C₉	—	3	2	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	1	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	9	3	no	—
77	C₃²⋊C₉	—	9	6	no	—
78	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
79	C₃³	—	3	1	no	—
80	C₃³	—	3	1	no	—
81	C₃³	—	3	2	no	—
82	C₃³	—	3	2	no	—
83	C₃³	—	3	2	no	—
84	C₃³	—	3	2	no	—
85	C₃³	—	3	2	no	—
86	C₃³	—	3	2	no	—
87	C₃³	—	9	1	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	9	1	no	—
90	C₃³	—	9	2	no	—
91	C₃³	—	9	2	no	—
92	C₃³	—	9	4	no	—
93	C₃³	—	9	1	no	—
94	C₃³	#731	1	3	yes	complemented
95	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
96	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
97	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
98	3₋¹⁺²	—	3	3	no	—
99	3₋¹⁺²	—	9	2	no	—
100	3₋¹⁺²	—	9	6	no	—
101	3₋¹⁺²	—	9	3	no	—
102	3₋¹⁺²	—	9	2	no	—
103	3₋¹⁺²	—	9	4	no	—
104	3₋¹⁺²	—	27	2	no	—
105	3₋¹⁺²	—	27	6	no	—
106	H(3,𝔽₃)	—	9	3	no	—
107	H(3,𝔽₃)	—	9	6	no	—
108	H(3,𝔽₃)	—	9	3	no	—
109	H(3,𝔽₃)	—	9	6	no	—
110	H(3,𝔽₃)	—	9	1	no	—
111	H(3,𝔽₃)	—	9	2	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	3	3	no	—
114	C₃×C₉	—	3	6	no	—
115	C₃×C₉	—	9	2	no	—
116	C₃×C₉	—	9	1	no	—
117	C₃×C₉	—	9	2	no	—
118	C₃×C₉	—	9	4	no	—
119	C₃×C₉	—	9	3	no	—
120	C₃×C₉	—	9	2	no	—
121	C₃×C₉	—	9	1	no	—
122	C₃×C₉	—	9	2	no	—
123	C₃×C₉	—	9	2	no	—
124	C₃×C₉	—	27	3	no	—
125	C₃×C₉	—	27	2	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	3	2	no	—
128	C₃²	—	3	3	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	2	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	—	9	3	no	—
137	C₃²	—	9	6	no	—
138	C₃²	—	9	1	no	—
139	C₃²	—	9	2	no	—
140	C₃²	—	9	2	no	—
141	C₃²	—	9	2	no	—
142	C₃²	—	9	4	no	—
143	C₃²	—	27	1	no	—
144	C₃²	—	27	2	no	—
145	C₃²	—	27	2	no	—
146	C₃²	—	27	2	no	—
147	C₃²	—	27	2	no	—
148	C₃²	—	27	1	no	—
149	C₃²	—	27	1	no	—
150	C₃²	—	27	2	no	—
151	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
152	C₃²	C₃×#731	1	1	yes	characteristic
153	C₃²	#6910	1	2	yes	—
154	C₉	—	9	3	no	—
155	C₉	—	9	3	no	—
156	C₉	—	9	6	no	—
157	C₉	—	27	2	no	—
158	C₉	—	27	2	no	—
159	C₉	—	27	1	no	—
160	C₉	—	27	2	no	—
161	C₃	—	3	1	no	—
162	C₃	—	3	2	no	—
163	C₃	—	9	1	no	—
164	C₃	—	9	2	no	—
165	C₃	—	9	3	no	—
166	C₃	—	27	2	no	—
167	C₃	—	81	1	no	—
168	C₃	#10624745	1	1	yes	characteristic
169	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
170	C₃	#10624750	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624883

C₃

yes

maximal, characteristic, complemented

#10624820

C₃

yes

maximal, complemented

#10624745

C₃

yes

maximal, characteristic, complemented

#10624735

C₃

yes

maximal, characteristic

#10624689

C₃

yes

maximal, characteristic, complemented

#10624614

C₃

yes

maximal, complemented

#10624610

C₃

yes

maximal, complemented

#10624604

C₃

yes

maximal, complemented

#10624563

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

H(3,𝔽₃)⋊C₉

—

—

—

—

—

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

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

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

—

#731

—

AΣL(1,𝔽₂₇)

—

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

—

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

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

#731

yes

complemented

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

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

3₋¹⁺²

—

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

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

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

—

150

C₃²

—

151

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

152

C₃²

C₃×#731

yes

characteristic

153

C₃²

#6910

yes

—

154

C₉

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

—

159

C₉

—

160

C₉

—

161

C₃

—

162

C₃

—

163

C₃

—

164

C₃

—

165

C₃

—

166

C₃

—

167

C₃

—

168

C₃

#10624745

yes

characteristic

169

C₃

UT(4,𝔽₃)

yes

characteristic

170

C₃

#10624750

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624735	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. #10624883	C₃	semidirect
2. #10624820	C₃	semidirect
3. #10624745	C₃	semidirect
5. #10624689	C₃	semidirect
6. #10624614	C₃	semidirect
7. #10624610	C₃	semidirect
8. #10624604	C₃	semidirect
9. #10624563	C₃	semidirect
94. C₃³	#731	semidirect

H ≅ G/N

type

1. #10624883

C₃

semidirect

2. #10624820

C₃

semidirect

3. #10624745

C₃

semidirect

5. #10624689

C₃

semidirect

6. #10624614

C₃

semidirect

7. #10624610

C₃

semidirect

8. #10624604

C₃

semidirect

9. #10624563

C₃

semidirect

94. C₃³

#731

semidirect

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

subgroup	index
#10624883	3
#10624820	3
#10624745	3
#10624735	3
#10624689	3
#10624614	3
#10624610	3
#10624604	3
#10624563	3

subgroup

index

Subgroups — #14547325

Conjugacy classes of subgroups

Complements

Maximal subgroups