Group #14550721 — subgroups

Subgroups — #14550721

Order 2187.

1412 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×#6866	C₃	1	3	yes	maximal
3	#10624556	C₃	1	3	yes	maximal, complemented
4	#10624466	C₃	1	2	yes	maximal, complemented
5	#10624466	C₃	1	3	yes	maximal, complemented
6	#10624466	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃⁵	C₃²	1	1	yes	characteristic
8	C₃²×3₋¹⁺²	—	3	3	no	—
9	C₃²×3₋¹⁺²	—	3	27	no	—
10	C₃²×3₋¹⁺²	—	3	3	no	—
11	C₃²×3₋¹⁺²	—	3	3	no	—
12	C₃²×3₋¹⁺²	—	3	3	no	—
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₃³×C₉	C₃²	1	1	yes	characteristic
15	C₃³×C₉	C₃²	1	1	yes	characteristic
16	C₃×C₉⋊C₉	—	3	3	no	—
17	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
19	#6866	—	3	9	no	—
20	#6855	—	3	6	no	—
21	C₃⁴	—	3	9	no	—
22	C₃⁴	—	3	1	no	—
23	C₃⁴	—	3	1	no	—
24	C₃⁴	—	3	1	no	—
25	C₃⁴	—	3	3	no	—
26	C₃⁴	—	3	6	no	—
27	C₃⁴	—	9	1	no	—
28	C₃⁴	—	9	3	no	—
29	C₃⁴	—	9	2	no	—
30	C₃⁴	3₋¹⁺²	1	3	yes	—
31	C₃⁴	C₃³	1	1	yes	characteristic
32	C₃×3₋¹⁺²	—	3	81	no	—
33	C₃×3₋¹⁺²	—	3	81	no	—
34	C₃×3₋¹⁺²	—	3	81	no	—
35	C₃×3₋¹⁺²	—	3	27	no	—
36	C₃×3₋¹⁺²	—	3	54	no	—
37	C₃×3₋¹⁺²	—	3	27	no	—
38	C₃²×C₉	—	3	1	no	—
39	C₃²×C₉	—	3	1	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	—	3	1	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	3	1	no	—
44	C₃²×C₉	—	3	1	no	—
45	C₃²×C₉	—	3	1	no	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	—	3	9	no	—
48	C₃²×C₉	—	3	9	no	—
49	C₃²×C₉	—	3	9	no	—
50	C₃²×C₉	—	3	3	no	—
51	C₃²×C₉	—	3	6	no	—
52	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
53	C₃²×C₉	3₋¹⁺²	1	3	yes	—
54	C₃²×C₉	3₋¹⁺²	1	3	yes	—
55	C₉⋊C₉	—	9	9	no	—
56	C₃²⋊C₉	—	3	6	no	—
57	C₃²⋊C₉	—	3	6	no	—
58	C₃²⋊C₉	—	3	12	no	—
59	C₃²⋊C₉	3₋¹⁺²	1	9	yes	complemented
60	C₃³	—	3	1	no	—
61	C₃³	—	3	3	no	—
62	C₃³	—	3	2	no	—
63	C₃³	—	3	3	no	—
64	C₃³	—	3	9	no	—
65	C₃³	—	3	9	no	—
66	C₃³	—	3	9	no	—
67	C₃³	—	3	3	no	—
68	C₃³	—	3	9	no	—
69	C₃³	—	3	9	no	—
70	C₃³	—	9	3	no	—
71	C₃³	—	9	3	no	—
72	C₃³	—	9	9	no	—
73	C₃³	—	9	18	no	—
74	C₃³	—	9	18	no	—
75	C₃³	—	9	36	no	—
76	C₃³	—	9	3	no	—
77	C₃³	—	9	3	no	—
78	C₃³	—	9	3	no	—
79	C₃³	—	9	3	no	—
80	C₃³	—	9	1	no	—
81	C₃³	—	9	3	no	—
82	C₃³	—	9	2	no	—
83	C₃³	—	9	3	no	—
84	C₃³	—	9	3	no	—
85	C₃³	—	9	2	no	—
86	C₃³	—	9	1	no	—
87	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
88	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
89	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
90	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
91	C₃³	C₃×3₋¹⁺²	1	3	yes	—
92	C₃³	#737	1	6	yes	—
93	3₋¹⁺²	—	3	243	no	—
94	C₃×C₉	—	3	9	no	—
95	C₃×C₉	—	3	9	no	—
96	C₃×C₉	—	3	9	no	—
97	C₃×C₉	—	3	9	no	—
98	C₃×C₉	—	3	9	no	—
99	C₃×C₉	—	3	9	no	—
100	C₃×C₉	—	3	9	no	—
101	C₃×C₉	—	3	3	no	—
102	C₃×C₉	—	3	3	no	—
103	C₃×C₉	—	3	3	no	—
104	C₃×C₉	—	3	6	no	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	3	9	no	—
107	C₃×C₉	—	3	9	no	—
108	C₃×C₉	—	3	9	no	—
109	C₃×C₉	—	3	9	no	—
110	C₃×C₉	—	9	3	no	—
111	C₃×C₉	—	9	3	no	—
112	C₃×C₉	—	9	3	no	—
113	C₃×C₉	—	9	6	no	—
114	C₃×C₉	—	9	6	no	—
115	C₃×C₉	—	9	6	no	—
116	C₃×C₉	—	9	6	no	—
117	C₃×C₉	AΣL(1,𝔽₂₇)	1	9	yes	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	3	1	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	3	3	no	—
125	C₃²	—	3	6	no	—
126	C₃²	—	3	6	no	—
127	C₃²	—	3	12	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	1	no	—
131	C₃²	—	3	9	no	—
132	C₃²	—	9	27	no	—
133	C₃²	—	9	27	no	—
134	C₃²	—	9	18	no	—
135	C₃²	—	9	9	no	—
136	C₃²	—	9	3	no	—
137	C₃²	—	9	3	no	—
138	C₃²	—	9	9	no	—
139	C₃²	—	9	18	no	—
140	C₃²	—	9	3	no	—
141	C₃²	#6891	1	1	yes	characteristic
142	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
143	C₃²	#6894	1	3	yes	—
144	C₃²	#6896	1	3	yes	—
145	C₃²	#6899	1	2	yes	—
146	C₃²	#6900	1	1	yes	characteristic
147	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
148	C₃²	#6913	1	1	yes	characteristic
149	C₃²	#6913	1	3	yes	—
150	C₃²	#6913	1	2	yes	—
151	C₉	—	3	9	no	—
152	C₉	—	3	9	no	—
153	C₉	—	3	9	no	—
154	C₉	—	9	9	no	—
155	C₉	—	27	6	no	—
156	C₃	—	3	3	no	—
157	C₃	—	3	3	no	—
158	C₃	—	3	2	no	—
159	C₃	—	3	1	no	—
160	C₃	—	9	9	no	—
161	C₃	#10624565	1	1	yes	characteristic
162	C₃	#10624765	1	1	yes	characteristic
163	C₃	#10624774	1	1	yes	characteristic
164	C₃	#10624774	1	1	yes	characteristic
165	C₃	#10624794	1	3	yes	—
166	C₃	#10624798	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×#6866

C₃

yes

maximal

#10624556

C₃

yes

maximal, complemented

#10624466

C₃

yes

maximal, complemented

#10624466

C₃

yes

maximal, complemented

#10624466

C₃

yes

maximal, characteristic, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

#6866

—

#6855

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

3₋¹⁺²

yes

—

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

—

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

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

3₋¹⁺²

yes

complemented

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

3₋¹⁺²

—

243

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

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

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

AΣL(1,𝔽₂₇)

yes

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

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

#6891

yes

characteristic

142

C₃²

C₉⋊H(3,𝔽₃)

yes

—

143

C₃²

#6894

yes

—

144

C₃²

#6896

yes

—

145

C₃²

#6899

yes

—

146

C₃²

#6900

yes

characteristic

147

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

148

C₃²

#6913

yes

characteristic

149

C₃²

#6913

yes

—

150

C₃²

#6913

yes

—

151

C₉

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₃

—

157

C₃

—

158

C₃

—

159

C₃

—

160

C₃

—

161

C₃

#10624565

yes

characteristic

162

C₃

#10624765

yes

characteristic

163

C₃

#10624774

yes

characteristic

164

C₃

#10624774

yes

characteristic

165

C₃

#10624794

yes

—

166

C₃

#10624798

yes

—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
3. #10624556	C₃	semidirect
4. #10624466	C₃	semidirect
5. #10624466	C₃	semidirect
6. #10624466	C₃	semidirect
59. C₃²⋊C₉	3₋¹⁺²	semidirect

H ≅ G/N

type

3. #10624556

C₃

semidirect

4. #10624466

C₃

semidirect

5. #10624466

C₃

semidirect

6. #10624466

C₃

semidirect

59. C₃²⋊C₉

3₋¹⁺²

semidirect

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃×#6866	3
#10624556	3
#10624466	3

subgroup

index

C₃³×3₋¹⁺²

C₃×#6866

#10624556

#10624466

Subgroups — #14550721

Conjugacy classes of subgroups

Complements

Maximal subgroups