Group #14547616 — subgroups

Subgroups — #14547616

Order 2187.

608 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 53 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	#10624883	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624778	C₃	1	2	yes	maximal, complemented
3	#10624769	C₃	1	2	yes	maximal, complemented
4	#10624741	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624602	C₃	1	4	yes	maximal, complemented
6	#10624591	C₃	1	2	yes	maximal, complemented
7	#10624562	C₃	1	1	yes	maximal, characteristic
8	C₃²×H(3,𝔽₃)	—	3	3	no	—
9	C₃²×H(3,𝔽₃)	C₃²	1	2	yes	—
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×#731	—	3	3	no	—
13	C₉⋊H(3,𝔽₃)	—	3	3	no	—
14	C₉⋊H(3,𝔽₃)	—	3	6	no	—
15	#6891	—	3	3	no	—
16	#6891	C₃²	1	4	yes	—
17	#6891	C₃²	1	2	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
19	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
20	H(3,𝔽₃)⋊C₉	—	3	6	no	—
21	#6866	—	3	6	no	—
22	#6859	—	3	12	no	—
23	#6857	—	3	6	no	—
24	C₃⁴	—	3	2	no	—
25	C₃⁴	—	3	1	no	—
26	C₃⁴	H(3,𝔽₃)	1	3	yes	—
27	C₃⁴	C₃³	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	3	9	no	—
29	C₃×3₋¹⁺²	—	3	2	no	—
30	C₃×3₋¹⁺²	—	3	4	no	—
31	C₃×H(3,𝔽₃)	—	3	2	no	—
32	C₃×H(3,𝔽₃)	—	3	1	no	—
33	C₃×H(3,𝔽₃)	—	3	1	no	—
34	C₃×H(3,𝔽₃)	—	3	2	no	—
35	C₃×H(3,𝔽₃)	—	3	2	no	—
36	C₃×H(3,𝔽₃)	—	3	4	no	—
37	C₃×H(3,𝔽₃)	—	9	3	no	—
38	C₃×H(3,𝔽₃)	—	9	6	no	—
39	C₃×H(3,𝔽₃)	—	9	3	no	—
40	C₃×H(3,𝔽₃)	—	9	9	no	—
41	C₃×H(3,𝔽₃)	—	9	18	no	—
42	C₃×H(3,𝔽₃)	—	9	9	no	—
43	C₃²×C₉	—	3	2	no	—
44	C₃²×C₉	H(3,𝔽₃)	1	3	yes	complemented
45	#731	—	9	9	no	—
46	C₃²⋊C₉	—	3	18	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	4	no	—
52	C₃²⋊C₉	—	3	4	no	—
53	C₃²⋊C₉	—	3	4	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₃²⋊C₉	—	3	4	no	—
56	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	complemented
57	C₃³	—	3	3	no	—
58	C₃³	—	3	3	no	—
59	C₃³	—	3	6	no	—
60	C₃³	—	3	3	no	—
61	C₃³	—	3	3	no	—
62	C₃³	—	3	6	no	—
63	C₃³	—	9	1	no	—
64	C₃³	—	9	2	no	—
65	C₃³	—	9	3	no	—
66	C₃³	—	9	6	no	—
67	C₃³	—	9	6	no	—
68	C₃³	—	9	2	no	—
69	C₃³	—	9	6	no	—
70	C₃³	—	9	6	no	—
71	C₃³	—	9	4	no	—
72	C₃³	—	9	1	no	—
73	C₃³	—	9	2	no	—
74	C₃³	—	9	6	no	—
75	C₃³	—	9	3	no	—
76	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
78	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	3₋¹⁺²	—	9	6	no	—
80	3₋¹⁺²	—	27	9	no	—
81	3₋¹⁺²	—	27	4	no	—
82	H(3,𝔽₃)	—	3	3	no	—
83	H(3,𝔽₃)	—	3	6	no	—
84	H(3,𝔽₃)	—	9	6	no	—
85	H(3,𝔽₃)	—	9	12	no	—
86	H(3,𝔽₃)	—	9	3	no	—
87	H(3,𝔽₃)	—	9	6	no	—
88	H(3,𝔽₃)	—	9	2	no	—
89	H(3,𝔽₃)	—	9	4	no	—
90	H(3,𝔽₃)	—	9	81	no	—
91	C₃×C₉	—	3	3	no	—
92	C₃×C₉	—	3	6	no	—
93	C₃×C₉	—	9	3	no	—
94	C₃×C₉	—	9	2	no	—
95	C₃×C₉	—	9	4	no	—
96	C₃×C₉	—	27	2	no	—
97	C₃²	—	3	1	no	—
98	C₃²	—	3	2	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	3	4	no	—
105	C₃²	—	9	9	no	—
106	C₃²	—	9	9	no	—
107	C₃²	—	9	18	no	—
108	C₃²	—	9	3	no	—
109	C₃²	—	9	3	no	—
110	C₃²	—	9	3	no	—
111	C₃²	—	9	6	no	—
112	C₃²	—	9	6	no	—
113	C₃²	—	27	3	no	—
114	C₃²	—	27	6	no	—
115	C₃²	—	27	6	no	—
116	C₃²	—	27	12	no	—
117	C₃²	—	27	1	no	—
118	C₃²	—	27	2	no	—
119	C₃²	—	27	2	no	—
120	C₃²	—	27	2	no	—
121	C₃²	—	27	2	no	—
122	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
123	C₃²	#6910	1	3	yes	—
124	C₉	—	9	3	no	—
125	C₉	—	9	6	no	—
126	C₉	—	27	2	no	—
127	C₉	—	27	4	no	—
128	C₃	—	3	3	no	—
129	C₃	—	9	1	no	—
130	C₃	—	9	2	no	—
131	C₃	—	27	3	no	—
132	C₃	—	27	3	no	—
133	C₃	—	27	6	no	—
134	C₃	#10624736	1	1	yes	characteristic
135	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
136	C₃	#10624749	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624883

C₃

yes

maximal, characteristic, complemented

#10624778

C₃

yes

maximal, complemented

#10624769

C₃

yes

maximal, complemented

#10624741

C₃

yes

maximal, characteristic, complemented

#10624602

C₃

yes

maximal, complemented

#10624591

C₃

yes

maximal, complemented

#10624562

C₃

yes

maximal, characteristic

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#731

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

H(3,𝔽₃)⋊C₉

—

#6866

—

#6859

—

#6857

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

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)

—

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

H(3,𝔽₃)

yes

complemented

#731

—

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

complemented

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

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

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

C₃²⋊H(3,𝔽₃)

yes

characteristic

123

C₃²

#6910

yes

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₉

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

#10624736

yes

characteristic

135

C₃

UT(4,𝔽₃)

yes

characteristic

136

C₃

#10624749

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624562	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. #10624883	C₃	semidirect
2. #10624778	C₃	semidirect
3. #10624769	C₃	semidirect
4. #10624741	C₃	semidirect
5. #10624602	C₃	semidirect
6. #10624591	C₃	semidirect
44. C₃²×C₉	H(3,𝔽₃)	semidirect
56. C₃²⋊C₉	H(3,𝔽₃)	semidirect

H ≅ G/N

type

1. #10624883

C₃

semidirect

2. #10624778

C₃

semidirect

3. #10624769

C₃

semidirect

4. #10624741

C₃

semidirect

5. #10624602

C₃

semidirect

6. #10624591

C₃

semidirect

44. C₃²×C₉

H(3,𝔽₃)

semidirect

56. C₃²⋊C₉

H(3,𝔽₃)

semidirect

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

subgroup	index
#10624883	3
#10624778	3
#10624769	3
#10624741	3
#10624602	3
#10624591	3
#10624562	3

subgroup

index

Subgroups — #14547616

Conjugacy classes of subgroups

Complements

Maximal subgroups