Group #14548021 — subgroups

Subgroups — #14548021

Order 2187.

391 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624881	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624816	C₃	1	2	yes	maximal, complemented
3	#10624805	C₃	1	2	yes	maximal, complemented
4	#10624775	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624743	C₃	1	1	yes	maximal, characteristic
6	#10624668	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624661	C₃	1	2	yes	maximal, complemented
8	#10624620	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624594	C₃	1	2	yes	maximal, complemented
10	C₃²×3₋¹⁺²	—	3	1	no	—
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
14	C₃×AΣL(1,𝔽₂₇)	—	3	2	no	—
15	#6893	—	3	6	no	—
16	#6893	—	3	3	no	—
17	#6891	—	3	2	no	—
18	#6891	C₃²	1	2	yes	—
19	#6891	C₃²	1	2	yes	—
20	#6891	C₃²	1	1	yes	characteristic
21	#6891	C₃²	1	1	yes	characteristic
22	#6891	C₃²	1	2	yes	—
23	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	3₋¹⁺²⋊C₉	—	3	6	no	—
26	#6867	—	3	3	no	—
27	#6866	—	3	2	no	—
28	#6866	—	3	4	no	—
29	#6861	—	3	3	no	—
30	#6860	—	3	6	no	—
31	#6859	—	3	3	no	—
32	#6859	—	3	6	no	—
33	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃⁴	H(3,𝔽₃)	1	2	yes	—
35	C₃⁴	C₃³	1	1	yes	characteristic
36	C₃×3₋¹⁺²	—	3	6	no	—
37	C₃×3₋¹⁺²	—	3	1	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	3	1	no	—
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	9	3	no	—
45	C₃×3₋¹⁺²	—	9	9	no	—
46	C₃×H(3,𝔽₃)	—	3	1	no	—
47	C₃×H(3,𝔽₃)	—	3	2	no	—
48	C₃×H(3,𝔽₃)	—	3	1	no	—
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	—	9	3	no	—
52	C₃²×C₉	—	3	3	no	—
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	3	2	no	—
55	AΣL(1,𝔽₂₇)	—	9	3	no	—
56	AΣL(1,𝔽₂₇)	—	9	6	no	—
57	C₃²⋊C₉	—	3	18	no	—
58	C₃²⋊C₉	—	3	2	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	2	no	—
64	C₃²⋊C₉	—	3	2	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	1	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	—	3	2	no	—
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
73	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
74	C₃³	—	3	1	no	—
75	C₃³	—	3	3	no	—
76	C₃³	—	3	2	no	—
77	C₃³	—	9	1	no	—
78	C₃³	—	9	1	no	—
79	C₃³	—	9	2	no	—
80	C₃³	—	9	1	no	—
81	C₃³	—	9	2	no	—
82	C₃³	—	9	2	no	—
83	C₃³	—	9	2	no	—
84	C₃³	—	9	4	no	—
85	C₃³	—	9	1	no	—
86	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
87	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
88	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
89	3₋¹⁺²	—	3	3	no	—
90	3₋¹⁺²	—	9	2	no	—
91	3₋¹⁺²	—	9	18	no	—
92	3₋¹⁺²	—	9	3	no	—
93	3₋¹⁺²	—	9	6	no	—
94	3₋¹⁺²	—	9	6	no	—
95	3₋¹⁺²	—	9	1	no	—
96	3₋¹⁺²	—	9	2	no	—
97	3₋¹⁺²	—	9	6	no	—
98	3₋¹⁺²	—	27	2	no	—
99	H(3,𝔽₃)	—	9	3	no	—
100	H(3,𝔽₃)	—	9	6	no	—
101	H(3,𝔽₃)	—	9	1	no	—
102	H(3,𝔽₃)	—	9	2	no	—
103	H(3,𝔽₃)	—	27	2	no	—
104	H(3,𝔽₃)	—	27	1	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	3	6	no	—
107	C₃×C₉	—	3	3	no	—
108	C₃×C₉	—	9	2	no	—
109	C₃×C₉	—	9	9	no	—
110	C₃×C₉	—	9	1	no	—
111	C₃×C₉	—	9	2	no	—
112	C₃×C₉	—	9	1	no	—
113	C₃×C₉	—	9	2	no	—
114	C₃×C₉	—	9	2	no	—
115	C₃×C₉	—	27	2	no	—
116	C₃²	—	3	1	no	—
117	C₃²	—	3	2	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	3	2	no	—
121	C₃²	—	3	2	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	3	3	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	9	1	no	—
128	C₃²	—	9	2	no	—
129	C₃²	—	9	2	no	—
130	C₃²	—	9	4	no	—
131	C₃²	—	9	2	no	—
132	C₃²	—	27	3	no	—
133	C₃²	—	27	3	no	—
134	C₃²	—	27	6	no	—
135	C₃²	—	27	1	no	—
136	C₃²	—	27	2	no	—
137	C₃²	—	27	1	no	—
138	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
139	C₉	—	9	3	no	—
140	C₉	—	9	6	no	—
141	C₉	—	9	3	no	—
142	C₉	—	27	2	no	—
143	C₉	—	27	1	no	—
144	C₉	—	27	2	no	—
145	C₉	—	27	2	no	—
146	C₃	—	9	1	no	—
147	C₃	—	9	2	no	—
148	C₃	—	9	1	no	—
149	C₃	—	9	3	no	—
150	C₃	—	27	2	no	—
151	C₃	—	81	1	no	—
152	C₃	#10624749	1	1	yes	characteristic
153	C₃	#10624750	1	2	yes	—
154	C₃	#10624754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624881

C₃

yes

maximal, characteristic, complemented

#10624816

C₃

yes

maximal, complemented

#10624805

C₃

yes

maximal, complemented

#10624775

C₃

yes

maximal, characteristic, complemented

#10624743

C₃

yes

maximal, characteristic

#10624668

C₃

yes

maximal, characteristic, complemented

#10624661

C₃

yes

maximal, complemented

#10624620

C₃

yes

maximal, characteristic, complemented

#10624594

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

—

—

—

—

—

—

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

C₃⁴

H(3,𝔽₃)

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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

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

H(3,𝔽₃)

yes

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

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

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

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

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

—

137

C₃²

—

138

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

139

C₉

—

140

C₉

—

141

C₉

—

142

C₉

—

143

C₉

—

144

C₉

—

145

C₉

—

146

C₃

—

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

—

151

C₃

—

152

C₃

#10624749

yes

characteristic

153

C₃

#10624750

yes

—

154

C₃

#10624754

yes

characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624743	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

C₃

G ▷ C₃⁴ ▷ C₃² ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. #10624881	C₃	semidirect
2. #10624816	C₃	semidirect
3. #10624805	C₃	semidirect
4. #10624775	C₃	semidirect
6. #10624668	C₃	semidirect
7. #10624661	C₃	semidirect
8. #10624620	C₃	semidirect
9. #10624594	C₃	semidirect

H ≅ G/N

type

1. #10624881

C₃

semidirect

2. #10624816

C₃

semidirect

3. #10624805

C₃

semidirect

4. #10624775

C₃

semidirect

6. #10624668

C₃

semidirect

7. #10624661

C₃

semidirect

8. #10624620

C₃

semidirect

9. #10624594

C₃

semidirect

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

subgroup	index
#10624881	3
#10624816	3
#10624805	3
#10624775	3
#10624743	3
#10624668	3
#10624661	3
#10624620	3
#10624594	3

subgroup

index

Subgroups — #14548021

Conjugacy classes of subgroups

Complements

Maximal subgroups