Group #14549773 — subgroups

Subgroups — #14549773

Order 2187.

377 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₃×#6893	C₃	1	1	yes	maximal, characteristic
2	#10624556	C₃	1	3	yes	maximal, complemented
3	#10624452	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624452	C₃	1	3	yes	maximal, complemented
5	#10624452	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624452	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624449	C₃	1	3	yes	maximal
8	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	#6893	—	3	3	no	—
13	#6893	—	3	3	no	—
14	#6893	—	3	3	no	—
15	C₃×C₉⋊C₉	—	3	3	no	—
16	C₃×C₃²⋊C₉	—	3	3	no	—
17	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
20	#6855	—	3	6	no	—
21	C₃⁴	—	3	1	no	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	3	1	no	—
24	C₃×3₋¹⁺²	—	3	1	no	—
25	C₃×3₋¹⁺²	—	3	1	no	—
26	C₃×3₋¹⁺²	—	3	1	no	—
27	C₃×3₋¹⁺²	—	9	3	no	—
28	C₃×3₋¹⁺²	—	9	3	no	—
29	C₃×3₋¹⁺²	3₋¹⁺²	1	3	yes	—
30	C₃×3₋¹⁺²	3₋¹⁺²	1	3	yes	—
31	C₃×H(3,𝔽₃)	—	3	1	no	—
32	C₃×H(3,𝔽₃)	—	3	1	no	—
33	C₃×H(3,𝔽₃)	—	9	3	no	—
34	C₃×H(3,𝔽₃)	3₋¹⁺²	1	3	yes	—
35	C₃²×C₉	—	3	1	no	—
36	C₃²×C₉	—	3	1	no	—
37	C₃²×C₉	—	3	3	no	—
38	C₃²×C₉	—	3	1	no	—
39	C₃²×C₉	—	3	1	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	—	3	3	no	—
42	C₃²×C₉	—	3	3	no	—
43	C₃²×C₉	—	9	3	no	—
44	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
45	C₉⋊C₉	—	9	9	no	—
46	C₃²⋊C₉	—	3	27	no	—
47	C₃²⋊C₉	—	3	9	no	—
48	C₃²⋊C₉	—	9	3	no	—
49	C₃²⋊C₉	—	9	3	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	3	3	no	—
52	C₃³	—	3	3	no	—
53	C₃³	—	3	3	no	—
54	C₃³	—	3	3	no	—
55	C₃³	—	9	3	no	—
56	C₃³	—	9	3	no	—
57	C₃³	—	9	3	no	—
58	C₃³	—	9	3	no	—
59	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
61	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
62	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
63	3₋¹⁺²	—	9	3	no	—
64	3₋¹⁺²	—	9	3	no	—
65	3₋¹⁺²	—	9	3	no	—
66	3₋¹⁺²	—	9	3	no	—
67	3₋¹⁺²	—	9	3	no	—
68	3₋¹⁺²	—	9	3	no	—
69	H(3,𝔽₃)	—	9	3	no	—
70	H(3,𝔽₃)	—	9	3	no	—
71	H(3,𝔽₃)	—	9	3	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	3	no	—
74	C₃×C₉	—	3	3	no	—
75	C₃×C₉	—	3	9	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	6	no	—
78	C₃×C₉	—	9	3	no	—
79	C₃×C₉	—	9	3	no	—
80	C₃×C₉	—	9	3	no	—
81	C₃×C₉	—	9	3	no	—
82	C₃×C₉	—	9	3	no	—
83	C₃×C₉	—	9	3	no	—
84	C₃×C₉	—	9	3	no	—
85	C₃×C₉	—	9	3	no	—
86	C₃×C₉	—	9	3	no	—
87	C₃×C₉	—	9	3	no	—
88	C₃×C₉	—	9	3	no	—
89	C₃×C₉	—	9	3	no	—
90	C₃×C₉	—	9	3	no	—
91	C₃×C₉	—	9	3	no	—
92	C₃×C₉	—	9	3	no	—
93	C₃×C₉	—	9	3	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	3	1	no	—
96	C₃²	—	3	1	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	3	3	no	—
99	C₃²	—	3	3	no	—
100	C₃²	—	9	3	no	—
101	C₃²	—	9	3	no	—
102	C₃²	—	9	3	no	—
103	C₃²	—	9	3	no	—
104	C₃²	—	9	9	no	—
105	C₃²	—	27	3	no	—
106	C₃²	—	27	3	no	—
107	C₃²	—	27	3	no	—
108	C₃²	#6891	1	1	yes	characteristic
109	C₃²	#6908	1	3	yes	—
110	C₃²	#6909	1	3	yes	—
111	C₃²	#6913	1	3	yes	—
112	C₃²	#6913	1	3	yes	—
113	C₉	—	9	3	no	—
114	C₉	—	9	3	no	—
115	C₉	—	9	3	no	—
116	C₉	—	9	9	no	—
117	C₉	—	27	3	no	—
118	C₉	—	27	3	no	—
119	C₃	—	9	1	no	—
120	C₃	—	9	1	no	—
121	C₃	—	9	1	no	—
122	C₃	—	27	3	no	—
123	C₃	#10624623	1	3	yes	—
124	C₃	#10624651	1	3	yes	—
125	C₃	#10624664	1	3	yes	—
126	C₃	#10624769	1	1	yes	characteristic
127	C₃	#10624780	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6893

C₃

yes

maximal, characteristic

#10624556

C₃

yes

maximal, complemented

#10624452

C₃

yes

maximal, characteristic, complemented

#10624452

C₃

yes

maximal, complemented

#10624452

C₃

yes

maximal, characteristic, complemented

#10624452

C₃

yes

maximal, characteristic, complemented

#10624449

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

#6855

—

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

3₋¹⁺²

yes

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

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

H(3,𝔽₃)

yes

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

#6891

yes

characteristic

109

C₃²

#6908

yes

—

110

C₃²

#6909

yes

—

111

C₃²

#6913

yes

—

112

C₃²

#6913

yes

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

#10624623

yes

—

124

C₃

#10624651

yes

—

125

C₃

#10624664

yes

—

126

C₃

#10624769

yes

characteristic

127

C₃

#10624780

yes

—

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

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
2. #10624556	C₃	semidirect
3. #10624452	C₃	semidirect
4. #10624452	C₃	semidirect
5. #10624452	C₃	semidirect
6. #10624452	C₃	semidirect

H ≅ G/N

type

2. #10624556

C₃

semidirect

3. #10624452

C₃

semidirect

4. #10624452

C₃

semidirect

5. #10624452

C₃

semidirect

6. #10624452

C₃

semidirect

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

subgroup	index
C₃×#6893	3
#10624556	3
#10624452	3
#10624449	3

subgroup

index

Subgroups — #14549773

Conjugacy classes of subgroups

Complements

Maximal subgroups