Group #13142942 — subgroups

Subgroups — C₃×#10588746 #13142942

Order 1728.

326 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 106 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×C₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×#63431	C₂	1	1	yes	maximal, characteristic
3	C₃×#63368	C₂	1	4	yes	maximal
4	C₃×#63362	C₂	1	1	yes	maximal, characteristic
5	C₃×#5069	—	3	1	no	maximal
6	#10588746	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
8	C₃×C₉×C₄⋊C₄	—	2	2	no	—
9	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊Dic₉	—	2	2	no	—
12	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
13	C₃²×#125	D₃	1	1	yes	characteristic
14	C₃×#869	—	3	1	no	—
15	C₃×#810	—	3	4	no	—
16	C₃×#803	—	3	1	no	—
17	C₉×#125	—	2	1	no	—
18	C₉×#125	C₂×C₃	1	1	yes	characteristic
19	#63431	C₂×C₃	1	1	yes	characteristic
20	#63368	C₂×C₃	1	4	yes	—
21	#63362	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	—	2	2	no	—
23	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
24	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂×C₃×Dic₉	—	2	2	no	—
26	C₃×Dic₁₈	—	2	4	no	—
27	C₃×C₉⋊C₈	—	2	2	no	—
28	C₃×#491	—	9	1	no	—
29	#5069	—	3	1	no	—
30	C₂×C₃×Dic₆	—	3	2	no	—
31	C₃²×C₄⋊C₄	—	2	2	no	—
32	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
33	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
34	C₃×C₄⋊Dic₃	—	6	2	no	—
35	C₂×C₃×C₃⋊C₈	—	3	2	no	—
36	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
37	C₉×C₄⋊C₄	—	2	4	no	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	—
41	C₄²×C₉	—	2	1	no	—
42	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
43	C₄⋊Dic₉	—	2	2	no	—
44	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
45	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
46	C₃×C₄×C₉	—	2	2	no	—
47	C₃×C₄×C₉	—	4	2	no	—
48	C₃×C₄×C₉	#41	1	2	yes	—
49	C₃×Dic₉	—	4	2	no	—
50	C₃×C₄⋊Q₈	—	9	1	no	—
51	C₃×#125	—	2	1	no	—
52	C₃×#125	D₉	1	1	yes	characteristic
53	C₃×#125	C₃×D₃	1	1	yes	characteristic
54	#869	—	3	1	no	—
55	C₃×#103	—	9	4	no	—
56	C₃×#97	—	9	1	no	—
57	#810	—	3	4	no	—
58	#803	—	3	1	no	—
59	C₂×C₃²×C₄	—	2	2	no	—
60	C₂×C₃²×C₄	#67	1	2	yes	—
61	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
62	C₂×C₃×Dic₃	—	6	2	no	—
63	C₃×Dic₆	—	6	4	no	—
64	C₃×C₃⋊C₈	—	6	2	no	—
65	C₂×C₄×C₉	—	2	4	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
70	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
71	C₂×Dic₉	—	2	2	no	—
72	Dic₁₈	—	2	4	no	—
73	C₉⋊C₈	—	2	2	no	—
74	#491	—	9	1	no	Sylow
75	C₂×C₃×C₉	#124	1	1	yes	characteristic
76	C₂×C₃×C₉	#137	1	2	yes	—
77	C₂×C₃×Q₈	—	9	2	no	—
78	C₂×Dic₆	—	3	2	no	—
79	C₂×C₃×C₈	—	9	2	no	—
80	C₃×C₄⋊C₄	—	2	4	no	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	—	18	2	no	—
85	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
86	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
89	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
90	C₄⋊Dic₃	—	6	2	no	—
91	C₂×C₃⋊C₈	—	3	2	no	—
92	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
93	C₃²×C₄	—	2	2	no	—
94	C₃²×C₄	—	4	2	no	—
95	C₃²×C₄	#239	1	2	yes	—
96	C₃×Dic₃	—	12	2	no	—
97	C₂²×C₉	—	2	1	no	—
98	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	2	2	no	—
101	C₄×C₉	—	4	2	no	—
102	C₄×C₉	—	4	4	no	—
103	C₄×C₉	—	4	2	no	—
104	C₄×C₉	C₃×#41	1	2	yes	—
105	Dic₉	—	4	2	no	—
106	C₄⋊Q₈	—	9	1	no	—
107	#125	C₃×D₉	1	1	yes	characteristic
108	#103	—	9	4	no	—
109	#97	—	9	1	no	—
110	C₃×C₉	#491	1	1	yes	characteristic, complemented, Sylow
111	C₃×Q₈	—	18	4	no	—
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	18	2	no	—
118	C₂×C₃×C₄	#611	1	2	yes	—
119	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
120	C₂×C₃×C₄	C₃×#67	1	2	yes	—
121	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
122	C₂×Dic₃	—	6	2	no	—
123	Dic₆	—	6	4	no	—
124	C₃×C₈	—	18	2	no	—
125	C₃⋊C₈	—	6	2	no	—
126	C₂×C₃²	#947	1	1	yes	characteristic
127	C₂×C₃²	#951	1	2	yes	—
128	C₂×C₉	—	2	1	no	—
129	C₂×C₉	—	2	2	no	—
130	C₂×C₉	C₃×#124	1	1	yes	characteristic
131	C₂×C₉	C₃×#137	1	2	yes	—
132	C₂×Q₈	—	9	2	no	—
133	C₂×C₈	—	9	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	18	2	no	—
136	C₄⋊C₄	C₂×C₃×D₉	1	2	yes	—
137	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
138	C₂²×C₃	—	2	1	no	—
139	C₂²×C₃	C₂×#611	1	1	yes	characteristic
140	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	—	4	2	no	—
145	C₃×C₄	—	4	4	no	—
146	C₃×C₄	—	4	2	no	—
147	C₃×C₄	—	4	2	no	—
148	C₃×C₄	—	36	2	no	—
149	C₃×C₄	#3701	1	2	yes	—
150	C₃×C₄	C₃×#239	1	2	yes	—
151	Dic₃	—	12	2	no	—
152	C₃²	#5069	1	1	yes	characteristic
153	C₉	—	2	1	no	—
154	C₉	C₃×#491	1	1	yes	characteristic, complemented
155	Q₈	—	18	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	18	2	no	—
158	C₂×C₄	C₃×#611	1	2	yes	—
159	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
160	C₈	—	18	2	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	2	no	—
163	C₂×C₃	#63509	1	1	yes	characteristic
164	C₂×C₃	#63511	1	2	yes	—
165	C₂×C₃	C₃×#947	1	1	yes	characteristic
166	C₂×C₃	C₃×#951	1	2	yes	—
167	C₂²	C₂×C₃×#611	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	36	2	no	—
171	C₄	C₃×#3701	1	2	yes	—
172	C₃	—	2	1	no	—
173	C₃	#10588746	1	1	yes	characteristic, direct product factor
174	C₃	C₃×#5069	1	1	yes	characteristic
175	C₂	C₃×#63509	1	1	yes	characteristic
176	C₂	C₃×#63511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#125

C₂

yes

maximal, characteristic

C₃×#63431

C₂

yes

maximal, characteristic

C₃×#63368

C₂

yes

maximal

C₃×#63362

C₂

yes

maximal, characteristic

C₃×#5069

—

maximal

#10588746

C₃

yes

maximal, characteristic, direct product factor

C₂×C₃×Dic₁₈

C₂²

yes

—

C₃×C₉×C₄⋊C₄

—

C₃×C₉×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₃×C₄⋊Dic₉

—

C₂×C₃×C₉⋊C₈

C₂²

yes

—

C₃²×#125

D₃

yes

—

—

—

—

C₂×C₃

yes

characteristic

#63431

C₂×C₃

yes

characteristic

#63368

C₂×C₃

yes

—

#63362

C₂×C₃

yes

characteristic

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃×Dic₉

—

C₃×Dic₁₈

—

C₃×C₉⋊C₈

—

C₃×#491

—

#5069

—

C₂×C₃×Dic₆

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₂×Dic₁₈

C₂²×C₃

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₄⋊Dic₉

—

C₂×C₉⋊C₈

C₂²×C₃

yes

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

#41

yes

—

C₃×Dic₉

—

C₃×C₄⋊Q₈

—

C₃×#125

—

C₃×#125

D₉

yes

characteristic

C₃×#125

C₃×D₃

yes

characteristic

#869

—

C₃×#103

—

C₃×#97

—

#810

—

#803

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×C₃⋊C₈

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₂×Dic₉

—

Dic₁₈

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×D₉

yes

—

C₃×C₄⋊C₄

C₂×C₃×D₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#239

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

C₄×C₉

—

100

C₄×C₉

—

101

C₄×C₉

—

102

C₄×C₉

—

103

C₄×C₉

—

104

C₄×C₉

C₃×#41

yes

—

105

Dic₉

—

106

C₄⋊Q₈

—

107

#125

C₃×D₉

yes

characteristic

108

#103

—

109

#97

—

110

C₃×C₉

#491

yes

characteristic, complemented, Sylow

111

C₃×Q₈

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

#611

yes

—

119

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

120

C₂×C₃×C₄

C₃×#67

yes

—

121

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

122

C₂×Dic₃

—

123

Dic₆

—

124

C₃×C₈

—

125

C₃⋊C₈

—

126

C₂×C₃²

#947

yes

characteristic

127

C₂×C₃²

#951

yes

—

128

C₂×C₉

—

129

C₂×C₉

—

130

C₂×C₉

C₃×#124

yes

characteristic

131

C₂×C₉

C₃×#137

yes

—

132

C₂×Q₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

C₂×C₃×D₉

yes

—

137

C₄²

C₂×C₃×D₉

yes

characteristic

138

C₂²×C₃

—

139

C₂²×C₃

C₂×#611

yes

characteristic

140

C₂²×C₃

C₂×C₃×#67

yes

characteristic

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

—

149

C₃×C₄

#3701

yes

—

150

C₃×C₄

C₃×#239

yes

—

151

Dic₃

—

152

C₃²

#5069

yes

characteristic

153

C₉

—

154

C₉

C₃×#491

yes

characteristic, complemented

155

Q₈

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₃×#611

yes

—

159

C₂×C₄

C₂²×C₃×D₉

yes

characteristic

160

C₈

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

#63509

yes

characteristic

164

C₂×C₃

#63511

yes

—

165

C₂×C₃

C₃×#947

yes

characteristic

166

C₂×C₃

C₃×#951

yes

—

167

C₂²

C₂×C₃×#611

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

C₃×#3701

yes

—

172

C₃

—

173

C₃

#10588746

yes

characteristic, direct product factor

174

C₃

C₃×#5069

yes

characteristic

175

C₂

C₃×#63509

yes

characteristic

176

C₂

C₃×#63511

yes

—

	subgroup	G/H
center	C₂²×C₃	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle, prime radical	C₂²×C₃²	C₂×#67
hypercenter	C₃×#125	D₉
hypocenter	C₉	C₃×#491
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#125

subgroup

G/H

center

C₂²×C₃

C₂×#611

commutator, cosocle

C₂×C₄×C₉

C₂³×C₃

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₃×C₉×#125

C₂

socle, prime radical

C₂²×C₃²

C₂×#67

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#125

N	H ≅ G/N	type
6. #10588746	C₃	direct (N × H)
110. C₃×C₉	#491	semidirect
154. C₉	C₃×#491	semidirect
173. C₃	#10588746	direct (N × H)

H ≅ G/N

type

6. #10588746

C₃

direct (N × H)

110. C₃×C₉

#491

semidirect

154. C₉

C₃×#491

semidirect

173. C₃

#10588746

direct (N × H)

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

subgroup	index
C₃×C₉×#125	2
C₃×#63431	2
C₃×#63368	2
C₃×#63362	2
C₃×#5069	3
#10588746	3

subgroup

index

C₃×C₉×#125

Subgroups — C₃×#10588746 #13142942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#10588746 #13142942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588746 #13142942