Group #14554401 — subgroups

Subgroups — #14554401

Order 2187.

1466 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 314 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	#10624884	C₃	1	2	yes	maximal, complemented
3	C₃×#6900	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624781	C₃	1	6	yes	maximal, complemented
5	#10624780	C₃	1	18	yes	maximal, complemented
6	#10624779	C₃	1	3	yes	maximal
7	#10624700	C₃	1	6	yes	maximal, complemented
8	#10624697	C₃	1	3	yes	maximal, complemented
9	3₋¹⁺⁴	—	3	27	no	—
10	C₃×#737	—	3	6	no	—
11	C₃×#737	C₃²	1	6	yes	—
12	C₃×#737	C₃²	1	3	yes	—
13	C₃²×3₋¹⁺²	—	3	3	no	—
14	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
15	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
16	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
18	#6900	—	3	3	no	—
19	#6900	—	3	6	no	—
20	#6900	—	3	6	no	—
21	#6899	—	3	6	no	—
22	#6899	—	3	6	no	—
23	#6895	C₃²	1	12	yes	complemented
24	#6891	C₃²	1	6	yes	—
25	#6887	C₃²	1	6	yes	complemented
26	C₃×C₉⋊C₉	C₃²	1	6	yes	complemented
27	C₃×C₉⋊C₉	C₃²	1	2	yes	complemented
28	C₃×C₉²	C₃²	1	1	yes	characteristic, complemented
29	3₋¹⁺²⋊C₉	C₃²	1	18	yes	complemented
30	3₋¹⁺²⋊C₉	C₃²	1	36	yes	complemented
31	H(3,𝔽₃)⋊C₉	C₃²	1	9	yes	—
32	H(3,𝔽₃)⋊C₉	C₃²	1	18	yes	—
33	C₃⁴	—	3	1	no	—
34	C₃⁴	C₃³	1	1	yes	characteristic
35	#737	—	3	162	no	—
36	#737	—	3	36	no	—
37	#737	—	3	18	no	—
38	#737	—	3	18	no	—
39	#737	3₋¹⁺²	1	27	yes	complemented
40	C₃×3₋¹⁺²	—	3	81	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	3	2	no	—
43	C₃×3₋¹⁺²	—	3	18	no	—
44	C₃×3₋¹⁺²	—	3	3	no	—
45	C₃×3₋¹⁺²	—	3	6	no	—
46	C₃×3₋¹⁺²	—	3	6	no	—
47	C₃×3₋¹⁺²	—	3	12	no	—
48	C₃×3₋¹⁺²	—	3	6	no	—
49	C₃×3₋¹⁺²	—	3	9	no	—
50	C₃×3₋¹⁺²	—	9	9	no	—
51	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
52	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
53	C₃×3₋¹⁺²	C₃³	1	6	yes	—
54	C₃×H(3,𝔽₃)	—	3	2	no	—
55	C₃×H(3,𝔽₃)	—	3	9	no	—
56	C₃×H(3,𝔽₃)	C₃³	1	3	yes	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	3	6	no	—
59	C₃²×C₉	—	3	3	no	—
60	C₃²×C₉	C₃³	1	3	yes	—
61	C₃²×C₉	C₃³	1	2	yes	—
62	C₃²×C₉	C₃³	1	6	yes	—
63	C₃²×C₉	C₃³	1	1	yes	characteristic
64	C₉⋊C₉	—	3	18	no	—
65	C₉⋊C₉	—	3	6	no	—
66	C₉⋊C₉	—	3	12	no	—
67	C₉⋊C₉	—	3	36	no	—
68	C₃²⋊C₉	—	3	6	no	—
69	C₃²⋊C₉	—	3	6	no	—
70	C₃²⋊C₉	C₃³	1	18	yes	—
71	C₉²	—	3	3	no	—
72	C₉²	—	3	6	no	—
73	C₃³	—	3	9	no	—
74	C₃³	—	3	3	no	—
75	C₃³	—	3	3	no	—
76	C₃³	—	9	3	no	—
77	C₃³	—	27	3	no	—
78	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
80	C₃³	C₃⁴	1	1	yes	characteristic
81	3₋¹⁺²	—	3	18	no	—
82	3₋¹⁺²	—	3	18	no	—
83	3₋¹⁺²	—	3	108	no	—
84	3₋¹⁺²	—	3	12	no	—
85	3₋¹⁺²	—	3	9	no	—
86	3₋¹⁺²	—	3	54	no	—
87	3₋¹⁺²	—	3	18	no	—
88	3₋¹⁺²	—	9	18	no	—
89	3₋¹⁺²	—	9	9	no	—
90	3₋¹⁺²	—	27	6	no	—
91	3₋¹⁺²	C₃×3₋¹⁺²	1	18	yes	—
92	H(3,𝔽₃)	—	3	18	no	—
93	H(3,𝔽₃)	—	3	6	no	—
94	H(3,𝔽₃)	C₃×3₋¹⁺²	1	9	yes	—
95	C₃×C₉	—	3	18	no	—
96	C₃×C₉	—	3	54	no	—
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	3	6	no	—
99	C₃×C₉	—	3	6	no	—
100	C₃×C₉	—	3	27	no	—
101	C₃×C₉	—	3	9	no	—
102	C₃×C₉	—	3	18	no	—
103	C₃×C₉	—	3	6	no	—
104	C₃×C₉	—	9	6	no	—
105	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
106	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
107	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
108	C₃²	—	3	3	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	9	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	6	no	—
113	C₃²	—	3	3	no	—
114	C₃²	—	9	9	no	—
115	C₃²	—	27	3	no	—
116	C₃²	—	27	9	no	—
117	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
118	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
119	C₉	—	3	18	no	—
120	C₉	—	3	6	no	—
121	C₉	—	3	9	no	—
122	C₉	—	9	6	no	—
123	C₉	—	9	18	no	—
124	C₉	#6891	1	9	yes	—
125	C₃	—	3	3	no	—
126	C₃	—	9	2	no	—
127	C₃	—	9	1	no	—
128	C₃	—	9	9	no	—
129	C₃	C₃×#6891	1	1	yes	characteristic
130	C₃	#10624934	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic

#10624884

C₃

yes

maximal, complemented

C₃×#6900

C₃

yes

maximal, characteristic, complemented

#10624781

C₃

yes

maximal, complemented

#10624780

C₃

yes

maximal, complemented

#10624779

C₃

yes

maximal

#10624700

C₃

yes

maximal, complemented

#10624697

C₃

yes

maximal, complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

C₃²

yes

—

C₃×#737

C₃²

yes

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

—

—

—

—

—

C₃²

yes

complemented

#6891

C₃²

yes

—

#6887

C₃²

yes

complemented

C₃×C₉⋊C₉

C₃²

yes

complemented

C₃×C₉⋊C₉

C₃²

yes

complemented

C₃×C₉²

C₃²

yes

characteristic, complemented

3₋¹⁺²⋊C₉

C₃²

yes

complemented

3₋¹⁺²⋊C₉

C₃²

yes

complemented

H(3,𝔽₃)⋊C₉

C₃²

yes

—

H(3,𝔽₃)⋊C₉

C₃²

yes

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

#737

—

162

—

—

—

—

3₋¹⁺²

yes

complemented

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

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₉²

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

108

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×3₋¹⁺²

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×3₋¹⁺²

yes

—

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

C₃×H(3,𝔽₃)

yes

—

106

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

107

C₃×C₉

C₃×3₋¹⁺²

yes

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

118

C₃²

C₃²×3₋¹⁺²

yes

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

#6891

yes

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

C₃×#6891

yes

characteristic

130

C₃

#10624934

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624884	C₃	semidirect
3. C₃×#6900	C₃	semidirect
4. #10624781	C₃	semidirect
5. #10624780	C₃	semidirect
7. #10624700	C₃	semidirect
8. #10624697	C₃	semidirect
23. #6895	C₃²	semidirect
25. #6887	C₃²	semidirect
26. C₃×C₉⋊C₉	C₃²	semidirect
27. C₃×C₉⋊C₉	C₃²	semidirect
28. C₃×C₉²	C₃²	semidirect
29. 3₋¹⁺²⋊C₉	C₃²	semidirect
30. 3₋¹⁺²⋊C₉	C₃²	semidirect
39. #737	3₋¹⁺²	semidirect

H ≅ G/N

type

2. #10624884

C₃

semidirect

3. C₃×#6900

C₃

semidirect

4. #10624781

C₃

semidirect

5. #10624780

C₃

semidirect

7. #10624700

C₃

semidirect

8. #10624697

C₃

semidirect

23. #6895

C₃²

semidirect

25. #6887

C₃²

semidirect

26. C₃×C₉⋊C₉

C₃²

semidirect

27. C₃×C₉⋊C₉

C₃²

semidirect

28. C₃×C₉²

C₃²

semidirect

29. 3₋¹⁺²⋊C₉

C₃²

semidirect

30. 3₋¹⁺²⋊C₉

C₃²

semidirect

39. #737

3₋¹⁺²

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
#10624884	3
C₃×#6900	3
#10624781	3
#10624780	3
#10624779	3
#10624700	3
#10624697	3

subgroup

index

C₃×3₋¹⁺⁴

Subgroups — #14554401

Conjugacy classes of subgroups

Complements

Maximal subgroups