Group #14548578 — subgroups

Subgroups — #14548578

Order 2187.

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

normal subgroups only

This group has 326 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	#10624913	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624809	C₃	1	2	yes	maximal, complemented
3	#10624773	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624741	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624661	C₃	1	2	yes	maximal, complemented
6	#10624639	C₃	1	2	yes	maximal, complemented
7	#10624636	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624617	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624596	C₃	1	2	yes	maximal, complemented
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃×#731	—	3	3	no	—
12	#6898	—	3	6	no	—
13	#6896	—	3	3	no	—
14	#6896	—	3	3	no	—
15	#6891	C₃²	1	2	yes	—
16	#6891	C₃²	1	1	yes	characteristic
17	#6891	C₃²	1	1	yes	characteristic
18	#6891	C₃²	1	1	yes	characteristic
19	#6891	C₃²	1	2	yes	—
20	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	#6867	—	3	6	no	—
24	#6866	—	3	3	no	—
25	#6861	—	3	3	no	—
26	#6859	—	3	6	no	—
27	#6857	—	3	6	no	—
28	#6856	—	3	6	no	—
29	#6856	—	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	1	no	—
34	C₃×3₋¹⁺²	—	3	1	no	—
35	C₃×3₋¹⁺²	—	3	2	no	—
36	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
37	C₃×H(3,𝔽₃)	—	3	1	no	—
38	C₃×H(3,𝔽₃)	—	3	2	no	—
39	C₃×H(3,𝔽₃)	—	3	2	no	—
40	C₃×H(3,𝔽₃)	—	3	2	no	—
41	C₃×H(3,𝔽₃)	—	3	2	no	—
42	C₃×H(3,𝔽₃)	—	3	2	no	—
43	C₃×H(3,𝔽₃)	—	9	3	no	—
44	C₃²×C₉	—	3	2	no	—
45	C₃²×C₉	—	3	1	no	—
46	#731	—	9	9	no	—
47	C₉⋊C₉	—	3	24	no	—
48	C₃²⋊C₉	—	3	1	no	—
49	C₃²⋊C₉	—	3	2	no	—
50	C₃²⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	3	2	no	—
52	C₃²⋊C₉	—	3	2	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	1	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	1	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
64	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
65	C₉²	—	3	3	no	—
66	C₃³	—	3	3	no	—
67	C₃³	—	9	1	no	—
68	C₃³	—	9	1	no	—
69	C₃³	—	9	2	no	—
70	C₃³	—	9	2	no	—
71	C₃³	—	9	1	no	—
72	C₃³	—	9	2	no	—
73	C₃³	—	27	1	no	—
74	C₃³	—	27	2	no	—
75	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
76	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	3₋¹⁺²	—	9	3	no	—
79	3₋¹⁺²	—	9	3	no	—
80	3₋¹⁺²	—	27	1	no	—
81	3₋¹⁺²	—	27	2	no	—
82	3₋¹⁺²	—	27	2	no	—
83	H(3,𝔽₃)	—	3	3	no	—
84	H(3,𝔽₃)	—	9	6	no	—
85	H(3,𝔽₃)	—	9	2	no	—
86	H(3,𝔽₃)	—	27	2	no	—
87	H(3,𝔽₃)	—	27	2	no	—
88	H(3,𝔽₃)	—	27	2	no	—
89	H(3,𝔽₃)	—	27	2	no	—
90	C₃×C₉	—	3	3	no	—
91	C₃×C₉	—	3	3	no	—
92	C₃×C₉	—	3	6	no	—
93	C₃×C₉	—	9	3	no	—
94	C₃×C₉	—	9	6	no	—
95	C₃×C₉	—	9	1	no	—
96	C₃×C₉	—	9	2	no	—
97	C₃×C₉	—	9	1	no	—
98	C₃×C₉	—	9	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	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	3	2	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	9	3	no	—
110	C₃²	—	27	3	no	—
111	C₃²	—	27	3	no	—
112	C₃²	—	27	6	no	—
113	C₃²	—	27	1	no	—
114	C₃²	—	27	2	no	—
115	C₃²	—	27	2	no	—
116	C₃²	—	27	2	no	—
117	C₃²	—	27	2	no	—
118	C₃²	—	27	2	no	—
119	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
120	C₉	—	9	3	no	—
121	C₉	—	9	3	no	—
122	C₉	—	9	6	no	—
123	C₉	—	27	1	no	—
124	C₉	—	27	1	no	—
125	C₉	—	27	2	no	—
126	C₉	—	27	2	no	—
127	C₃	—	9	1	no	—
128	C₃	—	9	2	no	—
129	C₃	—	9	1	no	—
130	C₃	—	27	3	no	—
131	C₃	—	81	2	no	—
132	C₃	#10624749	1	1	yes	characteristic
133	C₃	#10624753	1	1	yes	characteristic
134	C₃	#10624763	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624913

C₃

yes

maximal, characteristic, complemented

#10624809

C₃

yes

maximal, complemented

#10624773

C₃

yes

maximal, characteristic, complemented

#10624741

C₃

yes

maximal, characteristic, complemented

#10624661

C₃

yes

maximal, complemented

#10624639

C₃

yes

maximal, complemented

#10624636

C₃

yes

maximal, characteristic, complemented

#10624617

C₃

yes

maximal, characteristic, complemented

#10624596

C₃

yes

maximal, complemented

C₃²×H(3,𝔽₃)

C₃²

yes

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

—

—

—

—

—

—

—

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

H(3,𝔽₃)

yes

—

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

—

#731

—

C₉⋊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

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

C₃²⋊H(3,𝔽₃)

yes

characteristic

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

—

132

C₃

#10624749

yes

characteristic

133

C₃

#10624753

yes

characteristic

134

C₃

#10624763

yes

—

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

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. #10624913	C₃	semidirect
2. #10624809	C₃	semidirect
3. #10624773	C₃	semidirect
4. #10624741	C₃	semidirect
5. #10624661	C₃	semidirect
6. #10624639	C₃	semidirect
7. #10624636	C₃	semidirect
8. #10624617	C₃	semidirect
9. #10624596	C₃	semidirect

H ≅ G/N

type

1. #10624913

C₃

semidirect

2. #10624809

C₃

semidirect

3. #10624773

C₃

semidirect

4. #10624741

C₃

semidirect

5. #10624661

C₃

semidirect

6. #10624639

C₃

semidirect

7. #10624636

C₃

semidirect

8. #10624617

C₃

semidirect

9. #10624596

C₃

semidirect

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

subgroup	index
#10624913	3
#10624809	3
#10624773	3
#10624741	3
#10624661	3
#10624639	3
#10624636	3
#10624617	3
#10624596	3

subgroup

index

Subgroups — #14548578

Conjugacy classes of subgroups

Complements

Maximal subgroups