Group #14550448 — subgroups

Subgroups — #14550448

Order 2187.

599 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 128 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×#733	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₃×3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal
4	C₃×3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal
5	C₃×#6868	C₃	1	2	yes	maximal
6	#10624558	C₃	1	2	yes	maximal
7	C₃×#6855	C₃	1	1	yes	maximal, characteristic
8	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
9	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	1	yes	characteristic
11	C₃³×C₉	C₃²	1	2	yes	—
12	C₃³×C₉	C₃²	1	1	yes	characteristic
13	C₃×#733	—	3	3	no	—
14	C₃×#733	—	3	6	no	—
15	C₃×#733	C₉	1	9	yes	complemented
16	C₉×3₋¹⁺²	—	3	3	no	—
17	C₉×3₋¹⁺²	—	3	6	no	—
18	C₃×C₉⋊C₉	—	3	2	no	—
19	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
20	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
21	C₃×C₉²	—	3	1	no	—
22	3₋¹⁺²⋊C₉	—	3	6	no	—
23	3₋¹⁺²⋊C₉	—	3	12	no	—
24	#6868	—	3	6	no	—
25	#6855	—	3	2	no	—
26	#6855	—	3	1	no	—
27	#6855	—	3	4	no	—
28	#6855	—	3	2	no	—
29	C₃⁴	C₃³	1	1	yes	characteristic
30	C₃×3₋¹⁺²	—	3	2	no	—
31	C₃×3₋¹⁺²	—	3	2	no	—
32	C₃×3₋¹⁺²	—	3	4	no	—
33	C₃×3₋¹⁺²	—	3	4	no	—
34	C₃×3₋¹⁺²	—	3	4	no	—
35	C₃×3₋¹⁺²	—	3	2	no	—
36	C₃×3₋¹⁺²	—	3	4	no	—
37	C₃×3₋¹⁺²	—	3	8	no	—
38	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
39	C₃×3₋¹⁺²	C₃×C₉	1	3	yes	—
40	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
41	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	3	2	no	—
44	C₃²×C₉	—	3	1	no	—
45	C₃²×C₉	—	3	1	no	—
46	C₃²×C₉	—	3	2	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	—	3	2	no	—
49	C₃²×C₉	—	3	2	no	—
50	C₃²×C₉	—	3	2	no	—
51	C₃²×C₉	—	3	2	no	—
52	C₃²×C₉	—	3	4	no	—
53	C₃²×C₉	—	3	2	no	—
54	C₃²×C₉	—	3	3	no	—
55	C₃²×C₉	—	3	6	no	—
56	C₃²×C₉	—	3	2	no	—
57	C₃²×C₉	—	3	4	no	—
58	C₃²×C₉	—	3	4	no	—
59	C₃²×C₉	—	9	2	no	—
60	C₃²×C₉	—	9	4	no	—
61	C₃²×C₉	C₃×C₉	1	3	yes	—
62	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
63	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
64	#733	—	3	27	no	—
65	C₉⋊C₉	—	3	18	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	—	3	4	no	—
69	C₃²⋊C₉	—	3	4	no	—
70	C₃²⋊C₉	—	3	4	no	—
71	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
72	C₉²	—	3	9	no	—
73	C₃³	—	3	2	no	—
74	C₃³	—	3	4	no	—
75	C₃³	—	3	1	no	—
76	C₃³	—	3	2	no	—
77	C₃³	C₃²×C₉	1	1	yes	characteristic
78	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
80	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
81	C₃³	C₃×3₋¹⁺²	1	2	yes	—
82	C₃³	#737	1	4	yes	—
83	C₃³	#737	1	2	yes	—
84	3₋¹⁺²	—	3	6	no	—
85	3₋¹⁺²	—	3	12	no	—
86	3₋¹⁺²	—	3	6	no	—
87	3₋¹⁺²	—	3	6	no	—
88	3₋¹⁺²	—	3	12	no	—
89	3₋¹⁺²	—	9	8	no	—
90	3₋¹⁺²	—	9	4	no	—
91	3₋¹⁺²	C₃²⋊C₉	1	9	yes	—
92	C₃×C₉	—	3	2	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	3	3	no	—
96	C₃×C₉	—	3	6	no	—
97	C₃×C₉	—	3	4	no	—
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	3	6	no	—
100	C₃×C₉	—	3	3	no	—
101	C₃×C₉	—	3	6	no	—
102	C₃×C₉	—	3	6	no	—
103	C₃×C₉	—	3	2	no	—
104	C₃×C₉	—	3	2	no	—
105	C₃×C₉	—	3	6	no	—
106	C₃×C₉	—	9	3	no	—
107	C₃×C₉	—	9	6	no	—
108	C₃×C₉	—	9	4	no	—
109	C₃×C₉	—	9	4	no	—
110	C₃×C₉	—	9	1	no	—
111	C₃×C₉	—	9	2	no	—
112	C₃×C₉	—	9	2	no	—
113	C₃×C₉	—	9	4	no	—
114	C₃×C₉	—	9	2	no	—
115	C₃×C₉	—	9	4	no	—
116	C₃×C₉	—	9	2	no	—
117	C₃×C₉	—	9	2	no	—
118	C₃×C₉	—	9	6	no	—
119	C₃×C₉	—	9	12	no	—
120	C₃×C₉	—	9	8	no	—
121	C₃×C₉	—	9	4	no	—
122	C₃×C₉	—	9	4	no	—
123	C₃×C₉	#733	1	3	yes	complemented
124	C₃²	—	3	2	no	—
125	C₃²	—	3	4	no	—
126	C₃²	—	3	4	no	—
127	C₃²	—	3	8	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	2	no	—
130	C₃²	—	3	2	no	—
131	C₃²	—	3	4	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	3	2	no	—
134	C₃²	—	3	2	no	—
135	C₃²	—	3	4	no	—
136	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
137	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
138	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
139	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
140	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
141	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
142	C₃²	#6893	1	2	yes	—
143	C₃²	C₃×#733	1	1	yes	characteristic
144	C₃²	#6908	1	2	yes	—
145	C₃²	#6912	1	2	yes	—
146	C₃²	#6913	1	4	yes	—
147	C₉	—	3	9	no	—
148	C₉	—	3	3	no	—
149	C₉	—	9	3	no	—
150	C₉	—	9	6	no	—
151	C₉	—	9	2	no	—
152	C₉	—	9	6	no	—
153	C₉	—	9	6	no	—
154	C₉	—	27	4	no	—
155	C₃	—	3	1	no	—
156	C₃	—	3	2	no	—
157	C₃	—	3	4	no	—
158	C₃	—	3	2	no	—
159	C₃	#10624555	1	1	yes	characteristic
160	C₃	C₉×#733	1	1	yes	characteristic
161	C₃	#10624723	1	2	yes	—
162	C₃	#10624744	1	1	yes	characteristic
163	C₃	#10624746	1	2	yes	—
164	C₃	#10624799	1	2	yes	—
165	C₃	#10624802	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#733

C₃

yes

maximal, characteristic

C₃×C₉×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₃×#6868

C₃

yes

maximal

#10624558

C₃

yes

maximal

C₃×#6855

C₃

yes

maximal, characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

—

—

C₉

yes

complemented

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₉²

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

—

—

—

—

—

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×C₉

yes

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

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

C₃×C₉

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

#733

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

#737

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃²⋊C₉

yes

—

C₃×C₉

—

C₃×C₉

—

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

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

#733

yes

complemented

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

C₃×C₃²⋊C₉

yes

characteristic

137

C₃²

C₉×H(3,𝔽₃)

yes

—

138

C₃²

C₉×H(3,𝔽₃)

yes

characteristic

139

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

140

C₃²

C₉⋊H(3,𝔽₃)

yes

—

141

C₃²

C₉⋊H(3,𝔽₃)

yes

—

142

C₃²

#6893

yes

—

143

C₃²

C₃×#733

yes

characteristic

144

C₃²

#6908

yes

—

145

C₃²

#6912

yes

—

146

C₃²

#6913

yes

—

147

C₉

—

148

C₉

—

149

C₉

—

150

C₉

—

151

C₉

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₃

—

156

C₃

—

157

C₃

—

158

C₃

—

159

C₃

#10624555

yes

characteristic

160

C₃

C₉×#733

yes

characteristic

161

C₃

#10624723

yes

—

162

C₃

#10624744

yes

characteristic

163

C₃

#10624746

yes

—

164

C₃

#10624799

yes

—

165

C₃

#10624802

yes

—

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

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, prime radical, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
15. C₃×#733	C₉	semidirect
123. C₃×C₉	#733	semidirect

H ≅ G/N

type

15. C₃×#733

C₉

semidirect

123. C₃×C₉

#733

semidirect

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

subgroup	index
C₃²×#733	3
C₃×C₉×3₋¹⁺²	3
C₃×3₋¹⁺²⋊C₉	3
C₃×#6868	3
#10624558	3
C₃×#6855	3

subgroup

index

C₃²×#733

C₃×C₉×3₋¹⁺²

C₃×3₋¹⁺²⋊C₉

C₃×#6868

#10624558

C₃×#6855

Subgroups — #14550448

Conjugacy classes of subgroups

Complements

Maximal subgroups