Group #14548262 — subgroups

Subgroups — #14548262

Order 2187.

430 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10624862	C₃	1	1	yes	maximal, characteristic
2	#10624822	C₃	1	2	yes	maximal, complemented
3	#10624782	C₃	1	2	yes	maximal, complemented
4	#10624779	C₃	1	1	yes	maximal, characteristic
5	#10624779	C₃	1	1	yes	maximal, characteristic
6	#10624778	C₃	1	1	yes	maximal, characteristic
7	#10624744	C₃	1	2	yes	maximal, complemented
8	#10624648	C₃	1	2	yes	maximal, complemented
9	C₃×#6860	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃×#733	—	3	6	no	—
14	#6891	C₃²	1	2	yes	—
15	#6891	C₃²	1	2	yes	—
16	#6891	C₃²	1	2	yes	—
17	C₉×H(3,𝔽₃)	—	3	3	no	—
18	C₉×H(3,𝔽₃)	—	3	6	no	—
19	C₃×C₃²⋊C₉	—	3	3	no	—
20	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	3₋¹⁺²⋊C₉	—	3	6	no	—
24	3₋¹⁺²⋊C₉	—	3	6	no	—
25	H(3,𝔽₃)⋊C₉	—	3	3	no	—
26	H(3,𝔽₃)⋊C₉	—	3	3	no	—
27	H(3,𝔽₃)⋊C₉	—	3	3	no	—
28	#6861	—	3	6	no	—
29	#6860	—	3	3	no	—
30	C₃⁴	—	3	1	no	—
31	C₃⁴	C₃³	1	1	yes	characteristic
32	C₃×3₋¹⁺²	—	3	2	no	—
33	C₃×3₋¹⁺²	—	3	2	no	—
34	C₃×3₋¹⁺²	—	3	1	no	—
35	C₃×3₋¹⁺²	—	3	1	no	—
36	C₃×3₋¹⁺²	—	3	2	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	9	3	no	—
40	C₃×3₋¹⁺²	—	9	3	no	—
41	C₃×H(3,𝔽₃)	—	3	2	no	—
42	C₃×H(3,𝔽₃)	—	3	9	no	—
43	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
44	C₃²×C₉	—	3	9	no	—
45	C₃²×C₉	—	3	1	no	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	—	3	1	no	—
49	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
50	#733	—	9	18	no	—
51	C₃²⋊C₉	—	3	18	no	—
52	C₃²⋊C₉	—	3	2	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₃²⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	1	no	—
57	C₃²⋊C₉	—	3	2	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	2	no	—
67	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
68	C₃³	—	3	9	no	—
69	C₃³	—	3	3	no	—
70	C₃³	—	3	3	no	—
71	C₃³	—	9	1	no	—
72	C₃³	—	9	1	no	—
73	C₃³	—	9	1	no	—
74	C₃³	—	27	1	no	—
75	C₃³	—	27	1	no	—
76	C₃³	—	27	1	no	—
77	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
79	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
80	3₋¹⁺²	—	3	6	no	—
81	3₋¹⁺²	—	3	6	no	—
82	3₋¹⁺²	—	9	4	no	—
83	3₋¹⁺²	—	9	4	no	—
84	3₋¹⁺²	—	9	3	no	—
85	3₋¹⁺²	—	9	3	no	—
86	3₋¹⁺²	—	9	6	no	—
87	3₋¹⁺²	—	9	6	no	—
88	3₋¹⁺²	—	9	6	no	—
89	H(3,𝔽₃)	—	3	18	no	—
90	H(3,𝔽₃)	—	3	6	no	—
91	H(3,𝔽₃)	#733	1	9	yes	complemented
92	C₃×C₉	—	3	6	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	9	1	no	—
95	C₃×C₉	—	9	1	no	—
96	C₃×C₉	—	9	3	no	—
97	C₃×C₉	—	9	3	no	—
98	C₃×C₉	—	9	6	no	—
99	C₃×C₉	—	9	1	no	—
100	C₃×C₉	—	9	2	no	—
101	C₃×C₉	—	9	2	no	—
102	C₃×C₉	—	9	2	no	—
103	C₃×C₉	—	27	1	no	—
104	C₃×C₉	—	27	9	no	—
105	C₃×C₉	—	27	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	9	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	3	2	no	—
115	C₃²	—	3	1	no	—
116	C₃²	—	3	2	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	9	3	no	—
119	C₃²	—	9	3	no	—
120	C₃²	—	9	3	no	—
121	C₃²	—	27	3	no	—
122	C₃²	—	27	9	no	—
123	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
124	C₃²	C₃×#733	1	3	yes	—
125	C₉	—	9	6	no	—
126	C₉	—	9	3	no	—
127	C₉	—	9	3	no	—
128	C₉	—	9	3	no	—
129	C₉	—	27	2	no	—
130	C₉	—	27	2	no	—
131	C₉	—	27	2	no	—
132	C₉	—	27	1	no	—
133	C₃	—	3	3	no	—
134	C₃	—	9	2	no	—
135	C₃	—	9	1	no	—
136	C₃	—	9	9	no	—
137	C₃	#10624733	1	1	yes	characteristic
138	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
139	C₃	#10624751	1	1	yes	characteristic
140	C₃	#10624752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624862

C₃

yes

maximal, characteristic

#10624822

C₃

yes

maximal, complemented

#10624782

C₃

yes

maximal, complemented

#10624779

C₃

yes

maximal, characteristic

#10624779

C₃

yes

maximal, characteristic

#10624778

C₃

yes

maximal, characteristic

#10624744

C₃

yes

maximal, complemented

#10624648

C₃

yes

maximal, complemented

C₃×#6860

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#733

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6861

—

#6860

—

C₃⁴

—

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×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

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#733

—

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

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

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

—

—

—

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

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

C₃²⋊H(3,𝔽₃)

yes

characteristic

124

C₃²

C₃×#733

yes

—

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

#10624733

yes

characteristic

138

C₃

UT(4,𝔽₃)

yes

characteristic

139

C₃

#10624751

yes

characteristic

140

C₃

#10624752

yes

characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃²×H(3,𝔽₃)	C₃²
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

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
2. #10624822	C₃	semidirect
3. #10624782	C₃	semidirect
7. #10624744	C₃	semidirect
8. #10624648	C₃	semidirect
9. C₃×#6860	C₃	semidirect
91. H(3,𝔽₃)	#733	semidirect

H ≅ G/N

type

2. #10624822

C₃

semidirect

3. #10624782

C₃

semidirect

7. #10624744

C₃

semidirect

8. #10624648

C₃

semidirect

9. C₃×#6860

C₃

semidirect

91. H(3,𝔽₃)

#733

semidirect

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

subgroup	index
#10624862	3
#10624822	3
#10624782	3
#10624779	3
#10624778	3
#10624744	3
#10624648	3
C₃×#6860	3

subgroup

index

Subgroups — #14548262

Conjugacy classes of subgroups

Complements

Maximal subgroups