Group #14552210 — subgroups

Subgroups — #14552210

Order 2187.

1853 conjugacy classes of subgroups; only the 350 normal subgroups are shown.

normal subgroups only

This group has 1853 conjugacy classes of subgroups — too many to draw. Only its 350 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×#6899	C₃	1	3	yes	maximal, complemented
3	#10624570	C₃	1	6	yes	maximal, complemented
4	#10624565	C₃	1	12	yes	maximal, complemented
5	#10624560	C₃	1	6	yes	maximal, complemented
6	#10624556	C₃	1	6	yes	maximal, complemented
7	#10624548	C₃	1	2	yes	maximal, complemented
8	#10624547	C₃	1	3	yes	maximal
9	C₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃⁵	C₃²	1	1	yes	characteristic
11	C₃²×3₋¹⁺²	—	3	9	no	—
12	C₃²×3₋¹⁺²	—	3	18	no	—
13	C₃²×3₋¹⁺²	—	3	9	no	—
14	C₃²×3₋¹⁺²	C₃²	1	4	yes	—
15	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
16	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
18	C₃³×C₉	C₃²	1	2	yes	—
19	C₃³×C₉	C₃²	1	1	yes	characteristic
20	#6899	—	3	27	no	—
21	C₃×C₉⋊C₉	C₃²	1	6	yes	complemented
22	C₃×C₉⋊C₉	C₃²	1	18	yes	complemented
23	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	18	yes	—
27	C₃×C₉²	C₃²	1	3	yes	complemented
28	#6855	C₃²	1	6	yes	complemented
29	#6855	C₃²	1	12	yes	complemented
30	#6855	C₃²	1	36	yes	complemented
31	C₃⁴	—	3	3	no	—
32	C₃⁴	—	3	6	no	—
33	C₃⁴	—	3	3	no	—
34	C₃⁴	—	3	9	no	—
35	C₃⁴	—	9	6	no	—
36	C₃⁴	C₃³	1	1	yes	characteristic
37	C₃⁴	C₃³	1	2	yes	—
38	C₃⁴	C₃³	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	3	243	no	—
40	C₃×3₋¹⁺²	—	3	6	no	—
41	C₃×3₋¹⁺²	—	3	12	no	—
42	C₃×3₋¹⁺²	—	3	12	no	—
43	C₃×3₋¹⁺²	—	3	6	no	—
44	C₃×3₋¹⁺²	—	3	12	no	—
45	C₃×3₋¹⁺²	—	3	12	no	—
46	C₃×3₋¹⁺²	—	3	12	no	—
47	C₃×3₋¹⁺²	—	3	6	no	—
48	C₃×3₋¹⁺²	—	3	6	no	—
49	C₃×3₋¹⁺²	—	3	6	no	—
50	C₃×3₋¹⁺²	—	3	3	no	—
51	C₃×3₋¹⁺²	—	3	6	no	—
52	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
53	C₃×3₋¹⁺²	3₋¹⁺²	1	18	yes	—
54	C₃²×C₉	—	3	6	no	—
55	C₃²×C₉	—	3	6	no	—
56	C₃²×C₉	—	3	3	no	—
57	C₃²×C₉	—	3	12	no	—
58	C₃²×C₉	—	3	6	no	—
59	C₃²×C₉	3₋¹⁺²	1	9	yes	—
60	C₃²×C₉	C₃³	1	9	yes	—
61	C₃²×C₉	C₃³	1	18	yes	—
62	C₃²×C₉	C₃³	1	2	yes	—
63	C₃²×C₉	C₃³	1	4	yes	—
64	C₃²×C₉	C₃³	1	2	yes	—
65	C₃²×C₉	C₃³	1	1	yes	characteristic
66	C₉⋊C₉	—	3	54	no	—
67	C₉⋊C₉	—	3	18	no	—
68	C₃²⋊C₉	—	3	27	no	—
69	C₃²⋊C₉	—	3	9	no	—
70	C₃²⋊C₉	—	3	18	no	—
71	C₃²⋊C₉	—	9	18	no	—
72	C₉²	—	3	9	no	—
73	C₃³	—	3	1	no	—
74	C₃³	—	3	2	no	—
75	C₃³	—	3	6	no	—
76	C₃³	—	3	27	no	—
77	C₃³	—	3	2	no	—
78	C₃³	—	3	1	no	—
79	C₃³	—	3	2	no	—
80	C₃³	—	3	2	no	—
81	C₃³	—	3	2	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	3	6	no	—
84	C₃³	—	3	6	no	—
85	C₃³	—	3	6	no	—
86	C₃³	—	3	6	no	—
87	C₃³	—	9	27	no	—
88	C₃³	—	9	54	no	—
89	C₃³	—	9	18	no	—
90	C₃³	—	9	9	no	—
91	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
92	C₃³	C₃×3₋¹⁺²	1	6	yes	—
93	C₃³	C₃×3₋¹⁺²	1	3	yes	—
94	C₃³	#737	1	9	yes	—
95	C₃³	C₃⁴	1	1	yes	characteristic
96	3₋¹⁺²	—	3	108	no	—
97	3₋¹⁺²	—	3	54	no	—
98	3₋¹⁺²	—	3	27	no	—
99	3₋¹⁺²	—	3	54	no	—
100	C₃×C₉	—	3	54	no	—
101	C₃×C₉	—	3	4	no	—
102	C₃×C₉	—	3	2	no	—
103	C₃×C₉	—	3	2	no	—
104	C₃×C₉	—	3	4	no	—
105	C₃×C₉	—	3	4	no	—
106	C₃×C₉	—	3	4	no	—
107	C₃×C₉	—	3	4	no	—
108	C₃×C₉	—	3	27	no	—
109	C₃×C₉	—	3	9	no	—
110	C₃×C₉	—	3	18	no	—
111	C₃×C₉	—	3	9	no	—
112	C₃×C₉	—	3	18	no	—
113	C₃×C₉	—	3	18	no	—
114	C₃×C₉	—	3	36	no	—
115	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
116	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
117	C₃×C₉	C₃×3₋¹⁺²	1	12	yes	—
118	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
119	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
120	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
121	C₃²	—	3	3	no	—
122	C₃²	—	3	6	no	—
123	C₃²	—	3	9	no	—
124	C₃²	—	3	18	no	—
125	C₃²	—	3	3	no	—
126	C₃²	—	3	6	no	—
127	C₃²	—	3	9	no	—
128	C₃²	—	3	18	no	—
129	C₃²	—	9	81	no	—
130	C₃²	—	9	1	no	—
131	C₃²	—	9	2	no	—
132	C₃²	—	9	2	no	—
133	C₃²	—	9	2	no	—
134	C₃²	—	9	2	no	—
135	C₃²	—	9	6	no	—
136	C₃²	—	9	6	no	—
137	C₃²	—	9	6	no	—
138	C₃²	#6899	1	9	yes	complemented
139	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
140	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
141	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
142	C₃²	C₃×#737	1	3	yes	—
143	C₃²	3₋¹⁺⁴	1	6	yes	—
144	C₉	—	3	6	no	—
145	C₉	—	3	12	no	—
146	C₉	—	3	6	no	—
147	C₉	—	9	9	no	—
148	C₉	—	9	18	no	—
149	C₉	#6891	1	9	yes	—
150	C₃	—	3	9	no	—
151	C₃	—	9	3	no	—
152	C₃	—	9	6	no	—
153	C₃	C₃×#6891	1	1	yes	characteristic
154	C₃	C₃×#6899	1	1	yes	characteristic
155	C₃	#10624884	1	2	yes	—
156	C₃	#10624896	1	3	yes	—
157	C₃	#10624899	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×#6899

C₃

yes

maximal, complemented

#10624570

C₃

yes

maximal, complemented

#10624565

C₃

yes

maximal, complemented

#10624560

C₃

yes

maximal, complemented

#10624556

C₃

yes

maximal, complemented

#10624548

C₃

yes

maximal, complemented

#10624547

C₃

yes

maximal

C₉×C₃²⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

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

#6899

—

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

#6855

C₃²

yes

complemented

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

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

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

3₋¹⁺²

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

—

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

—

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

—

108

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

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

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

116

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

117

C₃×C₉

C₃×3₋¹⁺²

yes

—

118

C₃×C₉

C₃×3₋¹⁺²

yes

—

119

C₃×C₉

C₃×3₋¹⁺²

yes

—

120

C₃×C₉

C₃×3₋¹⁺²

yes

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

#6899

yes

complemented

139

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

140

C₃²

C₃²×3₋¹⁺²

yes

characteristic

141

C₃²

C₃²×3₋¹⁺²

yes

—

142

C₃²

C₃×#737

yes

—

143

C₃²

3₋¹⁺⁴

yes

—

144

C₉

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₉

—

149

C₉

#6891

yes

—

150

C₃

—

151

C₃

—

152

C₃

—

153

C₃

C₃×#6891

yes

characteristic

154

C₃

C₃×#6899

yes

characteristic

155

C₃

#10624884

yes

—

156

C₃

#10624896

yes

—

157

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×#6899	C₃	semidirect
3. #10624570	C₃	semidirect
4. #10624565	C₃	semidirect
5. #10624560	C₃	semidirect
6. #10624556	C₃	semidirect
7. #10624548	C₃	semidirect
9. C₉×C₃²⋊C₉	C₃	semidirect
21. C₃×C₉⋊C₉	C₃²	semidirect
22. C₃×C₉⋊C₉	C₃²	semidirect
27. C₃×C₉²	C₃²	semidirect
28. #6855	C₃²	semidirect
29. #6855	C₃²	semidirect
30. #6855	C₃²	semidirect
138. C₃²	#6899	semidirect

H ≅ G/N

type

2. C₃×#6899

C₃

semidirect

3. #10624570

C₃

semidirect

4. #10624565

C₃

semidirect

5. #10624560

C₃

semidirect

6. #10624556

C₃

semidirect

7. #10624548

C₃

semidirect

9. C₉×C₃²⋊C₉

C₃

semidirect

21. C₃×C₉⋊C₉

C₃²

semidirect

22. C₃×C₉⋊C₉

C₃²

semidirect

27. C₃×C₉²

C₃²

semidirect

28. #6855

C₃²

semidirect

29. #6855

C₃²

semidirect

30. #6855

C₃²

semidirect

138. C₃²

#6899

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×#6899	3
#10624570	3
#10624565	3
#10624560	3
#10624556	3
#10624548	3
#10624547	3
C₉×C₃²⋊C₉	3

subgroup

index

C₃³×3₋¹⁺²

C₉×C₃²⋊C₉

Subgroups — #14552210

Conjugacy classes of subgroups

Complements

Maximal subgroups