Group #14550675 — subgroups

Subgroups — #14550675

Order 2187.

728 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 80 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×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
2	#10624883	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624699	C₃	1	4	yes	maximal, complemented
4	#10624698	C₃	1	2	yes	maximal, complemented
5	#10624690	C₃	1	2	yes	maximal, complemented
6	C₃×#6866	C₃	1	2	yes	maximal, complemented
7	C₃×#6866	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
12	C₃×AΣL(1,𝔽₂₇)	—	3	6	no	—
13	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
14	C₉⋊H(3,𝔽₃)	—	3	3	no	—
15	C₉⋊H(3,𝔽₃)	—	3	6	no	—
16	#6891	—	3	3	no	—
17	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
20	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
21	3₋¹⁺²⋊C₉	—	3	12	no	—
22	H(3,𝔽₃)⋊C₉	—	3	6	no	—
23	#6867	—	3	6	no	—
24	#6866	—	3	3	no	—
25	#6866	—	3	6	no	—
26	C₃⁴	—	3	1	no	—
27	C₃⁴	—	3	2	no	—
28	C₃⁴	—	3	1	no	—
29	C₃⁴	—	3	2	no	—
30	C₃⁴	—	3	2	no	—
31	C₃⁴	—	3	4	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	—	9	3	no	—
34	C₃⁴	—	9	6	no	—
35	C₃⁴	3₋¹⁺²	1	3	yes	complemented
36	C₃⁴	C₃³	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	4	no	—
38	C₃×3₋¹⁺²	—	3	9	no	—
39	C₃×3₋¹⁺²	—	3	4	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	4	no	—
44	C₃×3₋¹⁺²	—	3	4	no	—
45	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	complemented
46	C₃×H(3,𝔽₃)	—	3	1	no	—
47	C₃×H(3,𝔽₃)	—	3	2	no	—
48	C₃×H(3,𝔽₃)	—	3	2	no	—
49	C₃×H(3,𝔽₃)	—	3	4	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
52	C₃²×C₉	—	3	2	no	—
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	3	2	no	—
55	C₃²×C₉	—	3	2	no	—
56	C₃²×C₉	—	3	4	no	—
57	AΣL(1,𝔽₂₇)	—	3	27	no	—
58	C₃²⋊C₉	—	3	18	no	—
59	C₃²⋊C₉	—	3	2	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	—	3	4	no	—
64	C₃²⋊C₉	—	3	4	no	—
65	C₃²⋊C₉	—	3	4	no	—
66	C₃²⋊C₉	—	3	4	no	—
67	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
68	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
69	C₃³	—	3	3	no	—
70	C₃³	—	3	3	no	—
71	C₃³	—	3	3	no	—
72	C₃³	—	3	6	no	—
73	C₃³	—	9	1	no	—
74	C₃³	—	9	2	no	—
75	C₃³	—	9	6	no	—
76	C₃³	—	9	3	no	—
77	C₃³	—	9	3	no	—
78	C₃³	—	9	12	no	—
79	C₃³	—	9	6	no	—
80	C₃³	—	9	6	no	—
81	C₃³	—	9	9	no	—
82	C₃³	—	9	18	no	—
83	C₃³	—	9	18	no	—
84	C₃³	—	9	36	no	—
85	C₃³	—	9	1	no	—
86	C₃³	—	9	2	no	—
87	C₃³	—	9	4	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	9	2	no	—
90	C₃³	—	9	1	no	—
91	C₃³	—	27	1	no	—
92	C₃³	—	27	2	no	—
93	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
95	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
96	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
97	C₃³	C₃×3₋¹⁺²	1	2	yes	—
98	C₃³	#737	1	4	yes	—
99	C₃³	#737	1	2	yes	—
100	3₋¹⁺²	—	3	6	no	—
101	3₋¹⁺²	—	3	12	no	—
102	3₋¹⁺²	—	9	4	no	—
103	3₋¹⁺²	—	9	4	no	—
104	3₋¹⁺²	—	9	4	no	—
105	3₋¹⁺²	—	27	9	no	—
106	H(3,𝔽₃)	—	3	3	no	—
107	H(3,𝔽₃)	—	3	6	no	—
108	H(3,𝔽₃)	—	9	4	no	—
109	H(3,𝔽₃)	—	9	2	no	—
110	C₃×C₉	—	3	6	no	—
111	C₃×C₉	—	3	3	no	—
112	C₃×C₉	—	3	6	no	—
113	C₃×C₉	—	9	2	no	—
114	C₃×C₉	—	9	2	no	—
115	C₃×C₉	—	9	2	no	—
116	C₃×C₉	—	9	4	no	—
117	C₃×C₉	—	9	2	no	—
118	C₃×C₉	—	27	1	no	—
119	C₃×C₉	—	27	2	no	—
120	C₃×C₉	—	27	2	no	—
121	C₃×C₉	—	27	4	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	2	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	3	4	no	—
127	C₃²	—	3	2	no	—
128	C₃²	—	9	1	no	—
129	C₃²	—	9	2	no	—
130	C₃²	—	9	2	no	—
131	C₃²	—	9	4	no	—
132	C₃²	—	9	9	no	—
133	C₃²	—	9	9	no	—
134	C₃²	—	9	18	no	—
135	C₃²	—	9	18	no	—
136	C₃²	—	9	36	no	—
137	C₃²	—	9	18	no	—
138	C₃²	—	9	3	no	—
139	C₃²	—	9	3	no	—
140	C₃²	—	9	6	no	—
141	C₃²	—	9	3	no	—
142	C₃²	—	9	3	no	—
143	C₃²	—	9	6	no	—
144	C₃²	—	27	3	no	—
145	C₃²	—	27	6	no	—
146	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
147	C₃²	#6891	1	1	yes	characteristic
148	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
149	C₃²	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
150	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
151	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
152	C₃²	#6893	1	2	yes	—
153	C₉	—	9	6	no	—
154	C₉	—	9	6	no	—
155	C₉	—	9	3	no	—
156	C₉	—	27	4	no	—
157	C₉	—	27	2	no	—
158	C₃	—	3	1	no	—
159	C₃	—	3	2	no	—
160	C₃	—	3	1	no	—
161	C₃	—	3	2	no	—
162	C₃	—	3	4	no	—
163	C₃	—	3	2	no	—
164	C₃	—	9	9	no	—
165	C₃	—	27	3	no	—
166	C₃	#10624563	1	1	yes	characteristic
167	C₃	#10624761	1	1	yes	characteristic
168	C₃	#10624762	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×AΣL(1,𝔽₂₇)

C₃

yes

maximal, characteristic

#10624883

C₃

yes

maximal, characteristic, complemented

#10624699

C₃

yes

maximal, complemented

#10624698

C₃

yes

maximal, complemented

#10624690

C₃

yes

maximal, complemented

C₃×#6866

C₃

yes

maximal, complemented

C₃×#6866

C₃

yes

maximal, characteristic, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

#6866

—

#6866

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

3₋¹⁺²

yes

complemented

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

3₋¹⁺²

yes

complemented

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

—

AΣL(1,𝔽₂₇)

—

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

H(3,𝔽₃)

yes

—

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×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

#737

yes

—

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

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

—

145

C₃²

—

146

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

147

C₃²

#6891

yes

characteristic

148

C₃²

C₉⋊H(3,𝔽₃)

yes

—

149

C₃²

C₉⋊H(3,𝔽₃)

yes

characteristic

150

C₃²

C₉⋊H(3,𝔽₃)

yes

—

151

C₃²

C₉⋊H(3,𝔽₃)

yes

—

152

C₃²

#6893

yes

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₃

—

159

C₃

—

160

C₃

—

161

C₃

—

162

C₃

—

163

C₃

—

164

C₃

—

165

C₃

—

166

C₃

#10624563

yes

characteristic

167

C₃

#10624761

yes

characteristic

168

C₃

#10624762

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃²×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃²×AΣL(1,𝔽₂₇)

C₃

LCS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624883	C₃	semidirect
3. #10624699	C₃	semidirect
4. #10624698	C₃	semidirect
5. #10624690	C₃	semidirect
6. C₃×#6866	C₃	semidirect
7. C₃×#6866	C₃	semidirect
35. C₃⁴	3₋¹⁺²	semidirect
45. C₃×3₋¹⁺²	3₋¹⁺²	semidirect

H ≅ G/N

type

2. #10624883

C₃

semidirect

3. #10624699

C₃

semidirect

4. #10624698

C₃

semidirect

5. #10624690

C₃

semidirect

6. C₃×#6866

C₃

semidirect

7. C₃×#6866

C₃

semidirect

35. C₃⁴

3₋¹⁺²

semidirect

45. C₃×3₋¹⁺²

3₋¹⁺²

semidirect

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

subgroup	index
C₃²×AΣL(1,𝔽₂₇)	3
#10624883	3
#10624699	3
#10624698	3
#10624690	3
C₃×#6866	3

subgroup

index

C₃²×AΣL(1,𝔽₂₇)

Subgroups — #14550675

Conjugacy classes of subgroups

Complements

Maximal subgroups