Group #13142992 — subgroups

Subgroups — #10588751⋊C₃ #13142992

Order 1728.

384 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 92 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	3₋¹⁺²×#127	C₂	1	1	yes	maximal, characteristic
2	#63432⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#63392⋊C₃	C₂	1	4	yes	maximal, complemented
4	#63362⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#5071	—	3	1	no	maximal
6	#10588751	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×3₋¹⁺²	—	2	2	no	—
8	C₂×D₄×3₋¹⁺²	C₂²	1	2	yes	—
9	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	2	yes	—
11	(3₋¹⁺²⋊C₄)⋊C₄	—	2	2	no	—
12	C₂×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
13	C₃²×#127	D₃	1	1	yes	characteristic
14	C₃×#870	—	3	1	no	—
15	C₃×#832	—	3	4	no	—
16	C₃×#803	—	3	1	no	—
17	C₉×#127	—	2	1	no	—
18	C₉×#127	C₂×C₃	1	1	yes	characteristic
19	#63432	C₂×C₃	1	1	yes	characteristic, complemented
20	#63392	C₂×C₃	1	4	yes	complemented
21	#63362	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂³×3₋¹⁺²	—	2	2	no	—
23	D₄×3₋¹⁺²	—	2	4	no	—
24	D₄×3₋¹⁺²	—	4	4	no	—
25	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
26	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
27	C₂×3₋¹⁺²⋊C₄	—	2	2	no	—
28	3₋¹⁺²⋊C₈	—	2	2	no	—
29	C₃×#490	—	9	1	no	—
30	#5071	—	3	1	no	—
31	C₂×C₃²×D₄	—	2	2	no	—
32	C₂×C₃²×D₄	C₂×D₃	1	2	yes	—
33	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
34	C₃×C₄⋊Dic₃	—	3	2	no	—
35	C₃×Dic₃⋊C₄	—	6	2	no	—
36	C₂×C₃×C₃⋊C₈	—	3	2	no	—
37	C₂×D₄×C₉	—	2	4	no	—
38	C₂×D₄×C₉	—	2	2	no	—
39	C₂×D₄×C₉	—	2	2	no	—
40	C₂×D₄×C₉	C₂²×C₃	1	2	yes	—
41	C₄²×C₉	—	2	1	no	—
42	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
43	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
44	Dic₉⋊C₄	—	2	2	no	—
45	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
46	C₂²×3₋¹⁺²	—	4	2	no	—
47	C₂²×3₋¹⁺²	—	4	4	no	—
48	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
49	C₄×3₋¹⁺²	—	2	2	no	—
50	C₄×3₋¹⁺²	#41	1	2	yes	—
51	3₋¹⁺²⋊C₄	—	4	2	no	—
52	C₃×#127	—	3	1	no	—
53	C₃×#127	C₃×D₃	1	1	yes	characteristic
54	C₃×#125	—	9	1	no	—
55	#870	—	3	1	no	—
56	C₃×#102	—	9	4	no	—
57	C₃×#97	—	9	1	no	—
58	#832	—	3	4	no	—
59	#803	—	3	1	no	—
60	C₂³×C₃²	—	2	2	no	—
61	C₃²×D₄	—	2	4	no	—
62	C₃²×D₄	—	4	4	no	—
63	C₂×C₃²×C₄	#67	1	2	yes	—
64	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₂×C₃×Dic₃	—	6	2	no	—
66	C₂³×C₉	—	2	4	no	—
67	C₂³×C₉	—	2	2	no	—
68	C₃×C₃⋊C₈	—	6	2	no	—
69	D₄×C₉	—	2	4	no	—
70	D₄×C₉	—	4	4	no	—
71	D₄×C₉	—	4	8	no	—
72	D₄×C₉	—	4	4	no	—
73	C₂×C₄×C₉	—	2	2	no	—
74	C₂×C₄×C₉	—	2	1	no	—
75	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
76	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
77	C₂×Dic₉	—	2	2	no	—
78	C₉⋊C₈	—	2	2	no	—
79	#490	—	9	1	no	Sylow
80	C₂×3₋¹⁺²	—	8	2	no	—
81	C₂×3₋¹⁺²	#124	1	1	yes	characteristic
82	C₂×3₋¹⁺²	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₃×D₄	—	2	2	no	—
84	C₂×C₃×D₄	—	3	2	no	—
85	C₂×C₃×D₄	—	6	2	no	—
86	C₂×C₃×D₄	C₂×C₃×D₃	1	2	yes	—
87	C₂×C₃×C₈	—	9	2	no	—
88	C₃×C₄⋊C₄	—	9	2	no	—
89	C₃×C₄⋊C₄	—	18	2	no	—
90	C₃×C₄²	—	3	1	no	—
91	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
92	C₄⋊Dic₃	—	3	2	no	—
93	Dic₃⋊C₄	—	6	2	no	—
94	C₂×C₃⋊C₈	—	3	2	no	—
95	C₂²×C₃²	—	4	4	no	—
96	C₂²×C₃²	—	4	2	no	—
97	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
98	C₃²×C₄	—	2	2	no	—
99	C₃²×C₄	#237	1	2	yes	—
100	C₃×Dic₃	—	12	2	no	—
101	C₂²×C₉	—	2	1	no	—
102	C₂²×C₉	—	4	8	no	—
103	C₂²×C₉	—	4	4	no	—
104	C₂²×C₉	—	4	2	no	—
105	C₂²×C₉	—	4	4	no	—
106	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
107	C₄×C₉	—	2	4	no	—
108	C₄×C₉	—	2	2	no	—
109	C₄×C₉	—	2	2	no	—
110	C₄×C₉	C₃×#41	1	2	yes	—
111	Dic₉	—	4	2	no	—
112	#127	AGL(1,ℤ/9)	1	1	yes	characteristic
113	#125	—	9	1	no	—
114	#102	—	9	4	no	—
115	#97	—	9	1	no	—
116	3₋¹⁺²	#490	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₃	—	2	2	no	—
118	C₂³×C₃	—	6	2	no	—
119	C₃×D₄	—	2	4	no	—
120	C₃×D₄	—	4	4	no	—
121	C₃×D₄	—	6	4	no	—
122	C₃×D₄	—	12	4	no	—
123	C₂×C₃×C₄	—	3	2	no	—
124	C₂×C₃×C₄	—	3	1	no	—
125	C₂×C₃×C₄	—	18	2	no	—
126	C₂×C₃×C₄	C₃×#67	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
128	C₂×Dic₃	—	6	2	no	—
129	C₃×C₈	—	18	2	no	—
130	C₃⋊C₈	—	6	2	no	—
131	C₂×C₃²	—	8	2	no	—
132	C₂×C₃²	#932	1	2	yes	—
133	C₂×C₃²	#936	1	1	yes	characteristic
134	C₂×C₉	—	2	2	no	—
135	C₂×C₉	—	2	1	no	—
136	C₂×C₉	—	8	4	no	—
137	C₂×C₉	—	8	2	no	—
138	C₂×C₉	C₃×#124	1	1	yes	characteristic
139	C₂×C₉	C₃×AGL(1,ℤ/8)	1	2	yes	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
142	C₂×C₈	—	9	2	no	—
143	C₄⋊C₄	—	9	2	no	—
144	C₄⋊C₄	—	18	2	no	—
145	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
146	C₂²×C₃	—	3	1	no	—
147	C₂²×C₃	—	4	4	no	—
148	C₂²×C₃	—	4	2	no	—
149	C₂²×C₃	—	12	2	no	—
150	C₂²×C₃	—	12	4	no	—
151	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
152	C₃×C₄	—	2	2	no	—
153	C₃×C₄	—	3	2	no	—
154	C₃×C₄	—	6	2	no	—
155	C₃×C₄	—	36	2	no	—
156	C₃×C₄	C₃×#237	1	2	yes	—
157	Dic₃	—	12	2	no	—
158	C₃²	#5071	1	1	yes	characteristic
159	C₉	—	2	1	no	—
160	C₉	C₃×#490	1	1	yes	characteristic, complemented
161	C₂³	—	2	2	no	—
162	D₄	—	2	4	no	—
163	D₄	—	4	4	no	—
164	C₂×C₄	—	18	2	no	—
165	C₂×C₄	#611⋊C₃	1	2	yes	—
166	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
167	C₈	—	18	2	no	—
168	C₂×C₃	—	3	2	no	—
169	C₂×C₃	—	3	1	no	—
170	C₂×C₃	—	8	2	no	—
171	C₂×C₃	—	24	2	no	—
172	C₂×C₃	C₃×#932	1	2	yes	—
173	C₂×C₃	C₃×#936	1	1	yes	characteristic
174	C₂²	—	4	4	no	—
175	C₂²	—	4	2	no	—
176	C₂²	C₂×#611⋊C₃	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	36	2	no	—
179	C₄	#3699⋊C₃	1	2	yes	—
180	C₃	—	3	1	no	—
181	C₃	C₃×#5071	1	1	yes	characteristic
182	C₂	—	8	2	no	—
183	C₂	#63495⋊C₃	1	2	yes	—
184	C₂	#63498⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#127

C₂

yes

maximal, characteristic

#63432⋊C₃

C₂

yes

maximal, characteristic, complemented

#63392⋊C₃

C₂

yes

maximal, complemented

#63362⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×#5071

—

maximal

#10588751

C₃

yes

maximal, characteristic, complemented

C₂×D₄×3₋¹⁺²

—

C₂×D₄×3₋¹⁺²

C₂²

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

C₂²

yes

—

(3₋¹⁺²⋊C₄)⋊C₄

—

C₂×3₋¹⁺²⋊C₈

C₂²

yes

—

C₃²×#127

D₃

yes

—

—

—

—

C₂×C₃

yes

characteristic

#63432

C₂×C₃

yes

characteristic, complemented

#63392

C₂×C₃

yes

complemented

#63362

C₂×C₃

yes

characteristic, complemented

C₂³×3₋¹⁺²

—

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊C₈

—

C₃×#490

—

#5071

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₂×C₃×C₃⋊C₈

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₄⋊Dic₉

C₂²×C₃

yes

—

Dic₉⋊C₄

—

C₂×C₉⋊C₈

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂×D₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

#41

yes

—

3₋¹⁺²⋊C₄

—

C₃×#127

—

C₃×#127

C₃×D₃

yes

—

—

—

C₃×#97

—

#832

—

#803

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

—

C₃×C₃⋊C₈

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₂×Dic₉

—

C₉⋊C₈

—

#490

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#124

yes

characteristic

C₂×3₋¹⁺²

AGL(1,ℤ/8)

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×C₃×D₃

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#237

yes

—

100

C₃×Dic₃

—

101

C₂²×C₉

—

102

C₂²×C₉

—

103

C₂²×C₉

—

104

C₂²×C₉

—

105

C₂²×C₉

—

106

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

107

C₄×C₉

—

108

C₄×C₉

—

109

C₄×C₉

—

110

C₄×C₉

C₃×#41

yes

—

111

Dic₉

—

112

yes

113

—

114

#102

—

115

#97

—

116

3₋¹⁺²

#490

yes

characteristic, complemented, Sylow

117

C₂³×C₃

—

118

C₂³×C₃

—

119

C₃×D₄

—

120

C₃×D₄

—

121

C₃×D₄

—

122

C₃×D₄

—

123

C₂×C₃×C₄

—

124

C₂×C₃×C₄

—

125

C₂×C₃×C₄

—

126

C₂×C₃×C₄

C₃×#67

yes

—

127

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

128

C₂×Dic₃

—

129

C₃×C₈

—

130

C₃⋊C₈

—

131

C₂×C₃²

—

132

C₂×C₃²

#932

yes

—

133

C₂×C₃²

#936

yes

characteristic

134

C₂×C₉

—

135

C₂×C₉

—

136

C₂×C₉

—

137

C₂×C₉

—

138

C₂×C₉

C₃×#124

yes

characteristic

139

C₂×C₉

C₃×AGL(1,ℤ/8)

yes

—

140

C₂×D₄

—

141

C₂×D₄

C₂×AGL(1,ℤ/9)

yes

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

146

C₂²×C₃

—

147

C₂²×C₃

—

148

C₂²×C₃

—

149

C₂²×C₃

—

150

C₂²×C₃

—

151

C₂²×C₃

C₂×C₃×#67

yes

characteristic

152

C₃×C₄

—

153

C₃×C₄

—

154

C₃×C₄

—

155

C₃×C₄

—

156

C₃×C₄

C₃×#237

yes

—

157

Dic₃

—

158

C₃²

#5071

yes

characteristic

159

C₉

—

160

C₉

C₃×#490

yes

characteristic, complemented

161

C₂³

—

162

D₄

—

163

D₄

—

164

C₂×C₄

—

165

C₂×C₄

#611⋊C₃

yes

—

166

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

167

C₈

—

168

C₂×C₃

—

169

C₂×C₃

—

170

C₂×C₃

—

171

C₂×C₃

—

172

C₂×C₃

C₃×#932

yes

—

173

C₂×C₃

C₃×#936

yes

characteristic

174

C₂²

—

175

C₂²

—

176

C₂²

C₂×#611⋊C₃

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

#3699⋊C₃

yes

—

180

C₃

—

181

C₃

C₃×#5071

yes

characteristic

182

C₂

—

183

C₂

#63495⋊C₃

yes

—

184

C₂

#63498⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#611⋊C₃
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#127	C₂
socle	C₂²×C₃	C₂×C₃×#67
prime radical	C₃²×#127	D₃
hypercenter	#127	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#490
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×#611⋊C₃

commutator, cosocle

C₂×C₄×C₉

C₂³×C₃

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

3₋¹⁺²×#127

C₂

socle

C₂²×C₃

C₂×C₃×#67

prime radical

C₃²×#127

D₃

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. #63432⋊C₃	C₂	semidirect
3. #63392⋊C₃	C₂	semidirect
4. #63362⋊C₃	C₂	semidirect
6. #10588751	C₃	semidirect
19. #63432	C₂×C₃	semidirect
20. #63392	C₂×C₃	semidirect
21. #63362	C₂×C₃	semidirect
116. 3₋¹⁺²	#490	semidirect
160. C₉	C₃×#490	semidirect

H ≅ G/N

type

2. #63432⋊C₃

C₂

semidirect

3. #63392⋊C₃

C₂

semidirect

4. #63362⋊C₃

C₂

semidirect

6. #10588751

C₃

semidirect

19. #63432

C₂×C₃

semidirect

20. #63392

C₂×C₃

semidirect

21. #63362

C₂×C₃

semidirect

116. 3₋¹⁺²

#490

semidirect

160. C₉

C₃×#490

semidirect

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

subgroup	index
3₋¹⁺²×#127	2
#63432⋊C₃	2
#63392⋊C₃	2
#63362⋊C₃	2
C₃×#5071	3
#10588751	3

subgroup

index

3₋¹⁺²×#127

Subgroups — #10588751⋊C₃ #13142992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588751⋊C3 #13142992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588751⋊C₃ #13142992