Group #14550417 — subgroups

Subgroups — #14550417

Order 2187.

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

normal subgroups only

This group has 572 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	C₃²×#732	C₃	1	1	yes	maximal, characteristic
2	#10624862	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624698	C₃	1	4	yes	maximal, complemented
4	#10624698	C₃	1	2	yes	maximal, complemented
5	#10624689	C₃	1	2	yes	maximal, complemented
6	C₃×#6867	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624549	C₃	1	2	yes	maximal, complemented
8	C₃²×H(3,𝔽₃)	C₃²	1	2	yes	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	1	yes	characteristic
11	C₃×#732	—	3	3	no	—
12	C₃×#732	—	3	6	no	—
13	C₃×#732	C₉	1	9	yes	complemented
14	C₉×H(3,𝔽₃)	—	3	3	no	—
15	C₉×H(3,𝔽₃)	—	3	6	no	—
16	C₃×C₃²⋊C₉	—	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	2	yes	—
20	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	H(3,𝔽₃)⋊C₉	—	3	6	no	—
22	H(3,𝔽₃)⋊C₉	—	3	12	no	—
23	#6867	—	3	3	no	—
24	#6866	—	3	6	no	—
25	#6855	—	3	6	no	—
26	C₃⁴	—	3	1	no	—
27	C₃⁴	—	3	2	no	—
28	C₃⁴	C₃³	1	1	yes	characteristic
29	C₃×H(3,𝔽₃)	—	3	2	no	—
30	C₃×H(3,𝔽₃)	—	3	4	no	—
31	C₃×H(3,𝔽₃)	—	3	2	no	—
32	C₃×H(3,𝔽₃)	—	3	4	no	—
33	C₃×H(3,𝔽₃)	—	3	2	no	—
34	C₃×H(3,𝔽₃)	—	3	4	no	—
35	C₃×H(3,𝔽₃)	—	3	8	no	—
36	C₃×H(3,𝔽₃)	—	3	4	no	—
37	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
38	C₃×H(3,𝔽₃)	C₃×C₉	1	6	yes	complemented
39	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
40	C₃×H(3,𝔽₃)	3₋¹⁺²	1	6	yes	—
41	C₃²×C₉	—	3	2	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	3	1	no	—
44	C₃²×C₉	—	3	2	no	—
45	C₃²×C₉	—	3	4	no	—
46	C₃²×C₉	—	3	2	no	—
47	C₃²×C₉	—	3	9	no	—
48	C₃²×C₉	—	3	1	no	—
49	C₃²×C₉	—	3	4	no	—
50	C₃²×C₉	—	3	2	no	—
51	C₃²×C₉	C₃×C₉	1	3	yes	complemented
52	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
53	#732	—	3	27	no	—
54	C₃²⋊C₉	—	3	18	no	—
55	C₃²⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	—	3	2	no	—
58	C₃²⋊C₉	—	3	2	no	—
59	C₃²⋊C₉	—	3	4	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	4	no	—
62	C₃²⋊C₉	—	3	4	no	—
63	C₃²⋊C₉	—	3	4	no	—
64	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
65	C₃³	—	3	3	no	—
66	C₃³	—	3	3	no	—
67	C₃³	—	3	6	no	—
68	C₃³	—	3	6	no	—
69	C₃³	—	9	1	no	—
70	C₃³	—	9	2	no	—
71	C₃³	—	9	2	no	—
72	C₃³	—	9	4	no	—
73	C₃³	—	27	1	no	—
74	C₃³	—	27	2	no	—
75	C₃³	—	27	2	no	—
76	C₃³	—	27	4	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	1	yes	characteristic
80	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
81	C₃³	C₃×3₋¹⁺²	1	2	yes	—
82	C₃³	#737	1	4	yes	—
83	C₃³	#737	1	2	yes	—
84	H(3,𝔽₃)	—	3	6	no	—
85	H(3,𝔽₃)	—	3	12	no	—
86	H(3,𝔽₃)	—	3	6	no	—
87	H(3,𝔽₃)	—	3	6	no	—
88	H(3,𝔽₃)	—	3	12	no	—
89	H(3,𝔽₃)	—	9	8	no	—
90	H(3,𝔽₃)	—	9	4	no	—
91	H(3,𝔽₃)	C₃²⋊C₉	1	9	yes	—
92	C₃×C₉	—	3	3	no	—
93	C₃×C₉	—	3	6	no	—
94	C₃×C₉	—	3	3	no	—
95	C₃×C₉	—	3	6	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	9	1	no	—
98	C₃×C₉	—	9	4	no	—
99	C₃×C₉	—	9	2	no	—
100	C₃×C₉	—	9	2	no	—
101	C₃×C₉	—	9	6	no	—
102	C₃×C₉	—	9	12	no	—
103	C₃×C₉	—	9	4	no	—
104	C₃×C₉	—	9	2	no	—
105	C₃×C₉	—	27	1	no	—
106	C₃×C₉	—	27	9	no	—
107	C₃×C₉	—	27	2	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	1	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	4	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	2	no	—
114	C₃²	—	3	9	no	—
115	C₃²	—	9	1	no	—
116	C₃²	—	9	2	no	—
117	C₃²	—	9	2	no	—
118	C₃²	—	9	4	no	—
119	C₃²	—	9	3	no	—
120	C₃²	—	9	6	no	—
121	C₃²	—	9	6	no	—
122	C₃²	—	27	3	no	—
123	C₃²	—	27	6	no	—
124	C₃²	—	27	6	no	—
125	C₃²	—	27	12	no	—
126	C₃²	—	27	2	no	—
127	C₃²	—	27	4	no	—
128	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
129	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
130	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
131	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
132	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
133	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
134	C₃²	#6893	1	2	yes	—
135	C₉	—	9	3	no	—
136	C₉	—	9	6	no	—
137	C₉	—	9	3	no	—
138	C₉	—	27	4	no	—
139	C₉	—	27	2	no	—
140	C₃	—	3	1	no	—
141	C₃	—	3	2	no	—
142	C₃	—	3	1	no	—
143	C₃	—	3	2	no	—
144	C₃	—	3	4	no	—
145	C₃	—	3	2	no	—
146	C₃	—	9	9	no	—
147	C₃	—	27	6	no	—
148	C₃	#10624555	1	1	yes	characteristic
149	C₃	#10624748	1	1	yes	characteristic
150	C₃	#10624753	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#732

C₃

yes

maximal, characteristic

#10624862

C₃

yes

maximal, characteristic, complemented

#10624698

C₃

yes

maximal, complemented

#10624698

C₃

yes

maximal, complemented

#10624689

C₃

yes

maximal, complemented

C₃×#6867

C₃

yes

maximal, characteristic, complemented

#10624549

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

C₉

yes

complemented

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

#6866

—

#6855

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₉

yes

—

C₃×H(3,𝔽₃)

C₃×C₉

yes

complemented

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃×H(3,𝔽₃)

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

C₃×C₉

yes

complemented

C₃²×C₉

H(3,𝔽₃)

yes

—

#732

—

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

C₃²×C₉

yes

characteristic

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

—

C₃³

#737

yes

—

C₃³

#737

yes

—

—

—

—

—

—

—

—

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

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

C₃×C₃²⋊C₉

yes

characteristic

129

C₃²

C₉×H(3,𝔽₃)

yes

—

130

C₃²

C₉×H(3,𝔽₃)

yes

characteristic

131

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

132

C₃²

C₉⋊H(3,𝔽₃)

yes

—

133

C₃²

C₉⋊H(3,𝔽₃)

yes

—

134

C₃²

#6893

yes

—

135

C₉

—

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₃

—

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

—

146

C₃

—

147

C₃

—

148

C₃

#10624555

yes

characteristic

149

C₃

#10624748

yes

characteristic

150

C₃

#10624753

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃²×#732	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃²×#732

C₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624862	C₃	semidirect
3. #10624698	C₃	semidirect
4. #10624698	C₃	semidirect
5. #10624689	C₃	semidirect
6. C₃×#6867	C₃	semidirect
7. #10624549	C₃	semidirect
13. C₃×#732	C₉	semidirect
38. C₃×H(3,𝔽₃)	C₃×C₉	semidirect
51. C₃²×C₉	C₃×C₉	semidirect

H ≅ G/N

type

2. #10624862

C₃

semidirect

3. #10624698

C₃

semidirect

4. #10624698

C₃

semidirect

5. #10624689

C₃

semidirect

6. C₃×#6867

C₃

semidirect

7. #10624549

C₃

semidirect

13. C₃×#732

C₉

semidirect

38. C₃×H(3,𝔽₃)

C₃×C₉

semidirect

51. C₃²×C₉

C₃×C₉

semidirect

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

subgroup	index
C₃²×#732	3
#10624862	3
#10624698	3
#10624689	3
C₃×#6867	3
#10624549	3

subgroup

index

C₃²×#732

Subgroups — #14550417

Conjugacy classes of subgroups

Complements

Maximal subgroups