Group #14552202 — subgroups

Subgroups — #14552202

Order 2187.

1952 conjugacy classes of subgroups; only the 368 normal subgroups are shown.

normal subgroups only

This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 368 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	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×#6894	C₃	1	2	yes	maximal, complemented
3	C₃×#6891	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624565	C₃	1	18	yes	maximal, complemented
6	#10624560	C₃	1	9	yes	maximal, complemented
7	#10624558	C₃	1	4	yes	maximal, complemented
8	#10624558	C₃	1	2	yes	maximal, complemented
9	#10624547	C₃	1	2	yes	maximal
10	C₃⁵	C₃²	1	1	yes	characteristic
11	C₃²×3₋¹⁺²	—	3	27	no	—
12	C₃²×3₋¹⁺²	—	3	9	no	—
13	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
15	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
17	C₃³×C₉	C₃²	1	2	yes	—
18	C₃³×C₉	C₃²	1	1	yes	characteristic
19	#6894	—	3	18	no	—
20	#6891	—	3	9	no	—
21	C₉×3₋¹⁺²	—	3	9	no	—
22	C₃×C₉⋊C₉	C₃²	1	6	yes	complemented
23	C₃×C₉⋊C₉	C₃²	1	6	yes	complemented
24	C₃×C₉⋊C₉	C₃²	1	12	yes	complemented
25	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	18	yes	—
29	C₃×C₉²	C₃²	1	3	yes	complemented
30	#6855	C₃²	1	36	yes	complemented
31	#6855	C₃²	1	18	yes	complemented
32	C₃⁴	—	3	3	no	—
33	C₃⁴	—	3	9	no	—
34	C₃⁴	—	3	3	no	—
35	C₃⁴	—	3	6	no	—
36	C₃⁴	—	9	6	no	—
37	C₃⁴	C₃³	1	3	yes	—
38	C₃⁴	C₃³	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	3	243	no	—
40	C₃×3₋¹⁺²	—	3	3	no	—
41	C₃×3₋¹⁺²	—	3	18	no	—
42	C₃×3₋¹⁺²	—	3	6	no	—
43	C₃×3₋¹⁺²	—	3	6	no	—
44	C₃×3₋¹⁺²	—	3	6	no	—
45	C₃×3₋¹⁺²	—	3	12	no	—
46	C₃×3₋¹⁺²	—	3	18	no	—
47	C₃×3₋¹⁺²	—	3	9	no	—
48	C₃×3₋¹⁺²	—	3	18	no	—
49	C₃×3₋¹⁺²	—	3	18	no	—
50	C₃×3₋¹⁺²	—	3	9	no	—
51	C₃×3₋¹⁺²	—	3	9	no	—
52	C₃×3₋¹⁺²	3₋¹⁺²	1	27	yes	complemented
53	C₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	complemented
54	C₃²×C₉	—	3	6	no	—
55	C₃²×C₉	—	3	18	no	—
56	C₃²×C₉	—	3	9	no	—
57	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
58	C₃²×C₉	C₃³	1	6	yes	—
59	C₃²×C₉	C₃³	1	6	yes	—
60	C₃²×C₉	C₃³	1	12	yes	—
61	C₃²×C₉	C₃³	1	2	yes	—
62	C₃²×C₉	C₃³	1	6	yes	—
63	C₃²×C₉	C₃³	1	3	yes	—
64	C₃²×C₉	C₃³	1	1	yes	characteristic
65	C₉⋊C₉	—	3	18	no	—
66	C₉⋊C₉	—	3	36	no	—
67	C₉⋊C₉	—	3	18	no	—
68	C₃²⋊C₉	—	3	6	no	—
69	C₃²⋊C₉	—	3	12	no	—
70	C₃²⋊C₉	—	3	27	no	—
71	C₃²⋊C₉	—	9	18	no	—
72	C₉²	—	3	9	no	—
73	C₃³	—	3	3	no	—
74	C₃³	—	3	6	no	—
75	C₃³	—	3	27	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	3	3	no	—
78	C₃³	—	3	3	no	—
79	C₃³	—	3	3	no	—
80	C₃³	—	3	6	no	—
81	C₃³	—	3	6	no	—
82	C₃³	—	3	6	no	—
83	C₃³	—	3	6	no	—
84	C₃³	—	9	27	no	—
85	C₃³	—	9	54	no	—
86	C₃³	—	9	18	no	—
87	C₃³	—	9	9	no	—
88	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
89	C₃³	C₃×3₋¹⁺²	1	9	yes	—
90	C₃³	C₃×3₋¹⁺²	1	3	yes	—
91	C₃³	#737	1	6	yes	—
92	C₃³	C₃⁴	1	1	yes	characteristic
93	3₋¹⁺²	—	3	162	no	—
94	3₋¹⁺²	—	3	27	no	—
95	3₋¹⁺²	—	3	81	no	—
96	3₋¹⁺²	—	9	18	no	—
97	C₃×C₉	—	3	54	no	—
98	C₃×C₉	—	3	3	no	—
99	C₃×C₉	—	3	6	no	—
100	C₃×C₉	—	3	6	no	—
101	C₃×C₉	—	3	6	no	—
102	C₃×C₉	—	3	6	no	—
103	C₃×C₉	—	3	6	no	—
104	C₃×C₉	—	3	27	no	—
105	C₃×C₉	—	3	6	no	—
106	C₃×C₉	—	3	12	no	—
107	C₃×C₉	—	3	6	no	—
108	C₃×C₉	—	3	6	no	—
109	C₃×C₉	—	3	12	no	—
110	C₃×C₉	—	3	18	no	—
111	C₃×C₉	—	3	36	no	—
112	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
113	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
114	C₃×C₉	C₃×3₋¹⁺²	1	18	yes	—
115	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
116	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	complemented
117	C₃²	—	3	3	no	—
118	C₃²	—	3	6	no	—
119	C₃²	—	3	9	no	—
120	C₃²	—	3	18	no	—
121	C₃²	—	3	9	no	—
122	C₃²	—	3	9	no	—
123	C₃²	—	3	18	no	—
124	C₃²	—	9	81	no	—
125	C₃²	—	9	3	no	—
126	C₃²	—	9	6	no	—
127	C₃²	—	9	3	no	—
128	C₃²	—	9	3	no	—
129	C₃²	—	9	6	no	—
130	C₃²	—	9	6	no	—
131	C₃²	C₉⋊H(3,𝔽₃)	1	9	yes	—
132	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
133	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
134	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
135	C₃²	C₃×#737	1	2	yes	—
136	C₃²	3₋¹⁺⁴	1	6	yes	—
137	C₉	—	3	6	no	—
138	C₉	—	3	18	no	—
139	C₉	—	3	9	no	—
140	C₉	—	9	6	no	—
141	C₉	—	9	6	no	—
142	C₉	—	9	12	no	—
143	C₉	#6891	1	9	yes	complemented
144	C₃	—	3	9	no	—
145	C₃	—	9	9	no	—
146	C₃	C₃×#6891	1	1	yes	characteristic
147	C₃	C₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
148	C₃	#10624882	1	2	yes	—
149	C₃	3₋¹⁺²²	1	3	yes	—
150	C₃	#10624899	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×#6894

C₃

yes

maximal, complemented

C₃×#6891

C₃

yes

maximal, characteristic

C₃×C₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

#10624565

C₃

yes

maximal, complemented

#10624560

C₃

yes

maximal, complemented

#10624558

C₃

yes

maximal, complemented

#10624558

C₃

yes

maximal, complemented

#10624547

C₃

yes

maximal

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

#6894

—

#6891

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

C₃²

yes

complemented

C₃×C₉⋊C₉

C₃²

yes

complemented

C₃×C₉⋊C₉

C₃²

yes

complemented

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₉²

C₃²

yes

complemented

#6855

C₃²

yes

complemented

#6855

C₃²

yes

complemented

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃³

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

243

—

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

3₋¹⁺²

yes

complemented

C₃×3₋¹⁺²

3₋¹⁺²

yes

complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

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

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

113

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

114

C₃×C₉

C₃×3₋¹⁺²

yes

—

115

C₃×C₉

C₃×3₋¹⁺²

yes

—

116

C₃×C₉

C₃×3₋¹⁺²

yes

complemented

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

C₉⋊H(3,𝔽₃)

yes

—

132

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

133

C₃²

C₃²×3₋¹⁺²

yes

—

134

C₃²

C₃²×3₋¹⁺²

yes

characteristic

135

C₃²

C₃×#737

yes

—

136

C₃²

3₋¹⁺⁴

yes

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

—

141

C₉

—

142

C₉

—

143

C₉

#6891

yes

complemented

144

C₃

—

145

C₃

—

146

C₃

C₃×#6891

yes

characteristic

147

C₃

C₃×C₉⋊H(3,𝔽₃)

yes

characteristic

148

C₃

#10624882

yes

—

149

C₃

3₋¹⁺²²

yes

—

150

C₃

#10624899

yes

—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₃³

C₃⁴

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃³ ▷ C₁

UCS

C₁ ◁ C₃³ ◁ G

LECS

G ▷ C₃³ ▷ C₁

N	H ≅ G/N	type
2. C₃×#6894	C₃	semidirect
4. C₃×C₉×3₋¹⁺²	C₃	semidirect
5. #10624565	C₃	semidirect
6. #10624560	C₃	semidirect
7. #10624558	C₃	semidirect
8. #10624558	C₃	semidirect
22. C₃×C₉⋊C₉	C₃²	semidirect
23. C₃×C₉⋊C₉	C₃²	semidirect
24. C₃×C₉⋊C₉	C₃²	semidirect
29. C₃×C₉²	C₃²	semidirect
30. #6855	C₃²	semidirect
31. #6855	C₃²	semidirect
52. C₃×3₋¹⁺²	3₋¹⁺²	semidirect
53. C₃×3₋¹⁺²	3₋¹⁺²	semidirect
116. C₃×C₉	C₃×3₋¹⁺²	semidirect
143. C₉	#6891	semidirect

H ≅ G/N

type

2. C₃×#6894

C₃

semidirect

4. C₃×C₉×3₋¹⁺²

C₃

semidirect

5. #10624565

C₃

semidirect

6. #10624560

C₃

semidirect

7. #10624558

C₃

semidirect

8. #10624558

C₃

semidirect

22. C₃×C₉⋊C₉

C₃²

semidirect

23. C₃×C₉⋊C₉

C₃²

semidirect

24. C₃×C₉⋊C₉

C₃²

semidirect

29. C₃×C₉²

C₃²

semidirect

30. #6855

C₃²

semidirect

31. #6855

C₃²

semidirect

52. C₃×3₋¹⁺²

3₋¹⁺²

semidirect

53. C₃×3₋¹⁺²

3₋¹⁺²

semidirect

116. C₃×C₉

C₃×3₋¹⁺²

semidirect

143. C₉

#6891

semidirect

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃×#6894	3
C₃×#6891	3
C₃×C₉×3₋¹⁺²	3
#10624565	3
#10624560	3
#10624558	3
#10624547	3

subgroup

index

C₃³×3₋¹⁺²

C₃×C₉×3₋¹⁺²

Subgroups — #14552202

Conjugacy classes of subgroups

Complements

Maximal subgroups