Group #14549770 — subgroups

Subgroups — #14549770

Order 2187.

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

normal subgroups only

This group has 815 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.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6891	C₃	1	2	yes	maximal
2	C₃×#6891	C₃	1	1	yes	maximal, characteristic
3	#10624547	C₃	1	1	yes	maximal, characteristic
4	#10624451	C₃	1	9	yes	maximal, complemented
5	C₃⁵	C₃²	1	1	yes	characteristic
6	C₃²×3₋¹⁺²	—	3	2	no	—
7	C₃²×3₋¹⁺²	—	3	1	no	—
8	#6891	—	3	9	no	—
9	#6891	—	3	18	no	—
10	C₃×C₃²⋊C₉	—	3	2	no	—
11	C₃×C₃²⋊C₉	—	3	2	no	—
12	C₃×C₃²⋊C₉	—	3	2	no	—
13	C₃×C₃²⋊C₉	—	3	2	no	—
14	C₃×C₃²⋊C₉	—	3	1	no	—
15	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
17	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
18	C₃⁴	—	3	6	no	—
19	C₃⁴	—	3	3	no	—
20	C₃⁴	—	3	3	no	—
21	C₃⁴	—	9	2	no	—
22	C₃⁴	—	9	2	no	—
23	C₃⁴	—	9	1	no	—
24	C₃⁴	—	9	2	no	—
25	C₃⁴	—	9	1	no	—
26	C₃⁴	—	9	1	no	—
27	C₃⁴	H(3,𝔽₃)	1	2	yes	—
28	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃⁴	C₃³	1	1	yes	characteristic
30	C₃×3₋¹⁺²	—	3	9	no	—
31	C₃×3₋¹⁺²	—	3	18	no	—
32	C₃×3₋¹⁺²	—	9	3	no	—
33	C₃×3₋¹⁺²	—	9	3	no	—
34	C₃×3₋¹⁺²	—	9	6	no	—
35	C₃×3₋¹⁺²	—	9	6	no	—
36	C₃×3₋¹⁺²	—	9	6	no	—
37	C₃×3₋¹⁺²	—	9	3	no	—
38	C₃²×C₉	—	3	3	no	—
39	C₃²×C₉	—	3	6	no	—
40	C₃²×C₉	—	3	3	no	—
41	C₃²⋊C₉	—	3	18	no	—
42	C₃²⋊C₉	—	3	18	no	—
43	C₃²⋊C₉	—	3	18	no	—
44	C₃²⋊C₉	—	3	6	no	—
45	C₃²⋊C₉	—	3	18	no	—
46	C₃²⋊C₉	—	9	6	no	—
47	C₃²⋊C₉	—	9	3	no	—
48	C₃²⋊C₉	—	9	3	no	—
49	C₃²⋊C₉	3₋¹⁺²	1	9	yes	complemented
50	C₃³	—	3	2	no	—
51	C₃³	—	3	2	no	—
52	C₃³	—	3	2	no	—
53	C₃³	—	3	2	no	—
54	C₃³	—	3	2	no	—
55	C₃³	—	3	2	no	—
56	C₃³	—	3	2	no	—
57	C₃³	—	3	2	no	—
58	C₃³	—	3	2	no	—
59	C₃³	—	3	1	no	—
60	C₃³	—	3	2	no	—
61	C₃³	—	3	2	no	—
62	C₃³	—	3	2	no	—
63	C₃³	—	3	2	no	—
64	C₃³	—	3	1	no	—
65	C₃³	—	3	1	no	—
66	C₃³	—	3	1	no	—
67	C₃³	—	3	2	no	—
68	C₃³	—	3	2	no	—
69	C₃³	—	3	2	no	—
70	C₃³	—	3	2	no	—
71	C₃³	—	3	2	no	—
72	C₃³	—	3	1	no	—
73	C₃³	—	3	1	no	—
74	C₃³	—	3	2	no	—
75	C₃³	—	3	2	no	—
76	C₃³	—	3	1	no	—
77	C₃³	—	3	1	no	—
78	C₃³	—	9	9	no	—
79	C₃³	—	9	18	no	—
80	C₃³	—	9	9	no	—
81	C₃³	—	9	18	no	—
82	C₃³	—	9	9	no	—
83	C₃³	—	9	18	no	—
84	C₃³	—	9	9	no	—
85	C₃³	—	9	9	no	—
86	C₃³	—	9	18	no	—
87	C₃³	AΣL(1,𝔽₂₇)	1	6	yes	—
88	C₃³	#731	1	3	yes	—
89	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
90	C₃³	C₃×3₋¹⁺²	1	3	yes	—
91	3₋¹⁺²	—	9	27	no	—
92	3₋¹⁺²	—	9	54	no	—
93	C₃×C₉	—	3	9	no	—
94	C₃×C₉	—	3	18	no	—
95	C₃×C₉	—	9	6	no	—
96	C₃×C₉	—	9	3	no	—
97	C₃×C₉	—	9	3	no	—
98	C₃×C₉	—	9	3	no	—
99	C₃×C₉	—	9	6	no	—
100	C₃×C₉	—	9	6	no	—
101	C₃×C₉	—	9	3	no	—
102	C₃×C₉	—	9	6	no	—
103	C₃×C₉	—	9	3	no	—
104	C₃²	—	3	3	no	—
105	C₃²	—	3	3	no	—
106	C₃²	—	3	3	no	—
107	C₃²	—	3	6	no	—
108	C₃²	—	3	6	no	—
109	C₃²	—	3	6	no	—
110	C₃²	—	3	6	no	—
111	C₃²	—	3	6	no	—
112	C₃²	—	3	6	no	—
113	C₃²	—	3	3	no	—
114	C₃²	—	9	81	no	—
115	C₃²	—	9	2	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	9	2	no	—
118	C₃²	—	9	2	no	—
119	C₃²	—	9	2	no	—
120	C₃²	—	9	1	no	—
121	C₃²	—	9	1	no	—
122	C₃²	—	9	2	no	—
123	C₃²	—	9	2	no	—
124	C₃²	—	9	2	no	—
125	C₃²	—	9	2	no	—
126	C₃²	—	9	2	no	—
127	C₃²	—	9	2	no	—
128	C₃²	—	9	1	no	—
129	C₃²	—	9	1	no	—
130	C₃²	—	9	1	no	—
131	C₃²	—	9	2	no	—
132	C₃²	—	9	2	no	—
133	C₃²	—	9	2	no	—
134	C₃²	—	9	2	no	—
135	C₃²	—	9	2	no	—
136	C₃²	#6891	1	1	yes	characteristic
137	C₃²	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
138	C₃²	C₃×#731	1	1	yes	characteristic
139	C₃²	#6909	1	1	yes	characteristic
140	C₃²	#6910	1	2	yes	—
141	C₃²	#6911	1	1	yes	characteristic
142	C₃²	#6911	1	1	yes	characteristic
143	C₃²	#6912	1	2	yes	—
144	C₃²	#6912	1	2	yes	—
145	C₉	—	9	9	no	—
146	C₉	—	9	18	no	—
147	C₉	—	27	3	no	—
148	C₃	—	9	3	no	—
149	C₃	—	9	3	no	—
150	C₃	—	9	6	no	—
151	C₃	#10624589	1	1	yes	characteristic
152	C₃	#10624610	1	2	yes	—
153	C₃	#10624633	1	2	yes	—
154	C₃	#10624643	1	1	yes	characteristic
155	C₃	#10624667	1	1	yes	characteristic
156	C₃	#10624668	1	2	yes	—
157	C₃	#10624772	1	1	yes	characteristic
158	C₃	#10624778	1	1	yes	characteristic
159	C₃	#10624781	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6891

C₃

yes

maximal

C₃×#6891

C₃

yes

maximal, characteristic

#10624547

C₃

yes

maximal, characteristic

#10624451

C₃

yes

maximal, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

#6891

—

#6891

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

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

—

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

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

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

AΣL(1,𝔽₂₇)

yes

—

C₃³

#731

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

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

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

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

#6891

yes

characteristic

137

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

138

C₃²

C₃×#731

yes

characteristic

139

C₃²

#6909

yes

characteristic

140

C₃²

#6910

yes

—

141

C₃²

#6911

yes

characteristic

142

C₃²

#6911

yes

characteristic

143

C₃²

#6912

yes

—

144

C₃²

#6912

yes

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₃

—

149

C₃

—

150

C₃

—

151

C₃

#10624589

yes

characteristic

152

C₃

#10624610

yes

—

153

C₃

#10624633

yes

—

154

C₃

#10624643

yes

characteristic

155

C₃

#10624667

yes

characteristic

156

C₃

#10624668

yes

—

157

C₃

#10624772

yes

characteristic

158

C₃

#10624778

yes

characteristic

159

C₃

#10624781

yes

—

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

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃⁵ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #10624451	C₃	semidirect
49. C₃²⋊C₉	3₋¹⁺²	semidirect

H ≅ G/N

type

4. #10624451

C₃

semidirect

49. 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
C₃×#6891	3
#10624547	3
#10624451	3

subgroup

index

C₃×#6891

#10624547

#10624451

Subgroups — #14549770

Conjugacy classes of subgroups

Complements

Maximal subgroups