Group #14547607 — subgroups

Subgroups — #14547607

Order 2187.

1118 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1118 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	C₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic
2	#10624823	C₃	1	3	yes	maximal, complemented
3	#10624771	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624738	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624611	C₃	1	3	yes	maximal, complemented
6	#10624602	C₃	1	3	yes	maximal
7	#10624570	C₃	1	1	yes	maximal, characteristic, complemented
8	3₋¹⁺⁴	—	3	27	no	—
9	C₃×#737	—	3	3	no	—
10	C₃×#737	—	3	3	no	—
11	C₃×#737	—	3	3	no	—
12	C₃²×3₋¹⁺²	—	3	3	no	—
13	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
14	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃×#733	—	3	3	no	—
18	#6891	C₃²	1	3	yes	—
19	#6891	C₃²	1	3	yes	—
20	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
21	3₋¹⁺²⋊C₉	—	3	9	no	—
22	#6867	—	3	3	no	—
23	#6859	—	3	9	no	—
24	#6858	—	3	9	no	—
25	#6855	—	3	3	no	—
26	C₃⁴	—	3	1	no	—
27	C₃⁴	C₃³	1	1	yes	characteristic
28	#737	—	3	81	no	—
29	#737	—	3	54	no	—
30	#737	—	3	81	no	—
31	#737	—	3	27	no	—
32	C₃×3₋¹⁺²	—	3	18	no	—
33	C₃×3₋¹⁺²	—	3	81	no	—
34	C₃×3₋¹⁺²	—	3	1	no	—
35	C₃×3₋¹⁺²	—	3	1	no	—
36	C₃×3₋¹⁺²	—	3	1	no	—
37	C₃×3₋¹⁺²	—	3	1	no	—
38	C₃×3₋¹⁺²	—	3	1	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	1	no	—
41	C₃×3₋¹⁺²	—	3	1	no	—
42	C₃×3₋¹⁺²	—	3	9	no	—
43	C₃×3₋¹⁺²	—	3	9	no	—
44	C₃×3₋¹⁺²	—	3	9	no	—
45	C₃×3₋¹⁺²	—	3	3	no	—
46	C₃×3₋¹⁺²	—	3	3	no	—
47	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
48	C₃×H(3,𝔽₃)	—	3	1	no	—
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃×H(3,𝔽₃)	—	3	9	no	—
51	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
52	C₃²×C₉	—	3	9	no	—
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	3	3	no	—
55	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
56	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
57	#733	—	9	9	no	—
58	C₃²⋊C₉	—	3	3	no	—
59	C₃²⋊C₉	—	3	3	no	—
60	C₃²⋊C₉	—	3	3	no	—
61	C₃²⋊C₉	—	3	3	no	—
62	C₃²⋊C₉	—	3	3	no	—
63	C₃²⋊C₉	—	3	3	no	—
64	C₃²⋊C₉	—	3	3	no	—
65	C₃²⋊C₉	—	3	3	no	—
66	C₃²⋊C₉	—	3	3	no	—
67	C₃³	—	3	9	no	—
68	C₃³	—	3	3	no	—
69	C₃³	—	3	3	no	—
70	C₃³	—	9	3	no	—
71	C₃³	—	27	3	no	—
72	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
73	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
75	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
76	3₋¹⁺²	—	3	9	no	—
77	3₋¹⁺²	—	3	9	no	—
78	3₋¹⁺²	—	3	9	no	—
79	3₋¹⁺²	—	3	162	no	—
80	3₋¹⁺²	—	3	9	no	—
81	3₋¹⁺²	—	3	9	no	—
82	3₋¹⁺²	—	3	9	no	—
83	3₋¹⁺²	—	3	9	no	—
84	3₋¹⁺²	—	3	9	no	—
85	3₋¹⁺²	—	3	9	no	—
86	3₋¹⁺²	—	9	9	no	—
87	3₋¹⁺²	—	9	9	no	—
88	H(3,𝔽₃)	—	3	9	no	—
89	H(3,𝔽₃)	—	3	9	no	—
90	H(3,𝔽₃)	—	3	9	no	—
91	C₃×C₉	—	3	9	no	—
92	C₃×C₉	—	3	81	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	3	3	no	—
95	C₃×C₉	—	3	3	no	—
96	C₃×C₉	—	3	6	no	—
97	C₃×C₉	—	3	9	no	—
98	C₃×C₉	—	9	9	no	—
99	C₃×C₉	—	9	3	no	—
100	C₃×C₉	—	9	3	no	—
101	C₃×C₉	—	9	3	no	—
102	C₃×C₉	AΣL(1,𝔽₂₇)	1	9	yes	—
103	C₃²	—	3	3	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	9	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	3	no	—
109	C₃²	—	3	3	no	—
110	C₃²	—	3	3	no	—
111	C₃²	—	9	9	no	—
112	C₃²	—	27	3	no	—
113	C₃²	—	27	9	no	—
114	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
115	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
116	C₉	—	3	9	no	—
117	C₉	—	3	9	no	—
118	C₉	—	3	9	no	—
119	C₉	—	27	3	no	—
120	C₉	—	27	3	no	—
121	C₉	—	27	3	no	—
122	C₃	—	3	3	no	—
123	C₃	—	9	1	no	—
124	C₃	—	9	1	no	—
125	C₃	—	9	1	no	—
126	C₃	—	9	9	no	—
127	C₃	#10624730	1	1	yes	characteristic
128	C₃	#10624751	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic

#10624823

C₃

yes

maximal, complemented

#10624771

C₃

yes

maximal, characteristic, complemented

#10624738

C₃

yes

maximal, characteristic, complemented

#10624611

C₃

yes

maximal, complemented

#10624602

C₃

yes

maximal

#10624570

C₃

yes

maximal, characteristic, complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

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

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

—

—

—

—

C₃⁴

—

C₃⁴

C₃³

yes

—

—

—

—

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

—

C₃×3₋¹⁺²

—

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)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

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

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

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

—

162

—

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)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

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

AΣL(1,𝔽₂₇)

yes

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

115

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₃

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

#10624730

yes

characteristic

128

C₃

#10624751

yes

—

	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. #10624823	C₃	semidirect
3. #10624771	C₃	semidirect
4. #10624738	C₃	semidirect
5. #10624611	C₃	semidirect
7. #10624570	C₃	semidirect

H ≅ G/N

type

2. #10624823

C₃

semidirect

3. #10624771

C₃

semidirect

4. #10624738

C₃

semidirect

5. #10624611

C₃

semidirect

7. #10624570

C₃

semidirect

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

subgroup	index
C₃×3₋¹⁺⁴	3
#10624823	3
#10624771	3
#10624738	3
#10624611	3
#10624602	3
#10624570	3

subgroup

index

C₃×3₋¹⁺⁴

Subgroups — #14547607

Conjugacy classes of subgroups

Complements

Maximal subgroups