Group #13265397 — subgroups

Subgroups — #13265397

Order 1792.

250 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 103 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	#11727029	C₂	1	2	yes	maximal
2	C₈²×D₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#1367	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725361	C₂	1	2	yes	maximal, complemented
5	#11725245	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₇	C₄	1	1	yes	characteristic
7	C₄×C₈×D₇	C₄	1	1	yes	characteristic
8	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
9	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₇×C₈²	C₂²	1	1	yes	characteristic
11	(C₂×D₇)⋊C₁₆	—	2	2	no	—
12	#89297	—	2	2	no	—
13	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
14	C₈×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₇⋊C₈	C₄	1	1	yes	characteristic
16	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	#12013	—	7	1	no	maximal, Sylow
18	C₂×C₈×D₇	—	2	1	no	—
19	C₂×C₈×D₇	C₈	1	1	yes	characteristic
20	C₂×C₈×D₇	C₈	1	1	yes	characteristic
21	C₂×C₈×D₇	D₄	1	1	yes	characteristic
22	C₂×C₈×D₇	Q₈	1	1	yes	characteristic
23	C₄²×D₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₇×C₁₆	—	2	2	no	—
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₈×Dic₇	—	2	1	no	—
29	C₈×Dic₇	C₈	1	1	yes	characteristic
30	C₈×Dic₇	C₈	1	1	yes	characteristic
31	C₈×Dic₇	D₄	1	1	yes	characteristic
32	C₈×Dic₇	Q₈	1	1	yes	characteristic
33	C₂×C₇⋊C₁₆	—	2	2	no	—
34	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	#2163	—	7	2	no	—
38	C₂×C₈²	—	7	1	no	—
39	#1370	—	7	2	no	—
40	#1367	—	7	1	no	—
41	#1367	D₇	1	1	yes	characteristic, complemented
42	C₂×C₄×D₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×D₇	M₁₆	1	1	yes	characteristic
45	C₂×C₇×C₈	—	2	1	no	—
46	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
47	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₄×Dic₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
53	C₄×Dic₇	M₁₆	1	1	yes	characteristic
54	C₂×C₇⋊C₈	—	2	1	no	—
55	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
59	C₈×D₇	—	2	4	no	—
60	C₈×D₇	—	2	2	no	—
61	C₈×D₇	—	2	2	no	—
62	C₈×D₇	—	2	2	no	—
63	C₈×D₇	—	2	2	no	—
64	C₇×C₁₆	—	4	2	no	—
65	C₇⋊C₁₆	—	4	2	no	—
66	C₂×C₄×C₈	—	7	1	no	—
67	C₂×C₄×C₈	—	7	1	no	—
68	C₂×C₄×C₈	—	7	1	no	—
69	C₄⋊C₁₆	—	7	2	no	—
70	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
71	C₂²⋊C₁₆	—	14	2	no	—
72	#346	—	14	2	no	—
73	C₈²	—	7	1	no	—
74	C₈²	—	7	1	no	—
75	C₈²	—	7	1	no	—
76	C₈²	C₂×D₇	1	1	yes	characteristic
77	C₂²×D₇	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₇	C₄⋊C₈	1	1	yes	characteristic
82	C₂×Dic₇	C₄⋊C₈	1	1	yes	characteristic
83	C₂×Dic₇	C₄⋊C₈	1	1	yes	characteristic
84	C₄×D₇	—	2	2	no	—
85	C₄×D₇	—	2	1	no	—
86	C₄×D₇	—	2	1	no	—
87	C₄×D₇	—	2	1	no	—
88	C₄×D₇	—	2	1	no	—
89	C₇×C₈	—	2	2	no	—
90	C₇×C₈	—	2	1	no	—
91	C₇×C₈	—	2	1	no	—
92	C₇×C₈	M₃₂	1	2	yes	—
93	C₇×C₈	#135	1	2	yes	—
94	C₇⋊C₈	—	2	2	no	—
95	C₇⋊C₈	—	2	1	no	—
96	C₇⋊C₈	—	2	1	no	—
97	C₇⋊C₈	#108	1	2	yes	—
98	C₇⋊C₈	M₃₂	1	2	yes	—
99	C₂²×C₈	—	7	1	no	—
100	C₂²×C₈	—	7	1	no	—
101	C₂²×C₈	—	7	1	no	—
102	C₂²×C₈	—	7	1	no	—
103	C₂²×C₈	—	14	1	no	—
104	C₂×C₄²	—	7	1	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	14	2	no	—
107	C₄×C₈	—	7	1	no	—
108	C₄×C₈	—	7	1	no	—
109	C₄×C₈	—	7	1	no	—
110	C₄×C₈	—	7	1	no	—
111	C₄×C₈	—	7	1	no	—
112	C₄×C₈	—	7	1	no	—
113	C₄×C₈	—	7	1	no	—
114	C₄×C₈	—	14	1	no	—
115	C₄×C₈	C₄×D₇	1	1	yes	characteristic
116	C₄×C₈	C₄×D₇	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
118	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂×D₇	—	2	1	no	—
120	C₂×D₇	—	2	1	no	—
121	C₂×D₇	#364	1	2	yes	—
122	C₄×C₇	—	2	1	no	—
123	C₄×C₇	#427	1	1	yes	characteristic
124	C₄×C₇	C₂×#108	1	1	yes	characteristic
125	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
126	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
127	Dic₇	—	2	1	no	—
128	Dic₇	—	2	1	no	—
129	Dic₇	—	2	1	no	—
130	C₂²×C₄	—	7	1	no	—
131	C₂²×C₄	—	7	1	no	—
132	C₂²×C₄	—	7	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	7	1	no	—
135	C₂×C₈	—	7	1	no	—
136	C₂×C₈	—	7	1	no	—
137	C₂×C₈	—	7	1	no	—
138	C₂×C₈	—	14	2	no	—
139	C₂×C₈	—	14	2	no	—
140	C₂×C₈	—	14	4	no	—
141	C₂×C₈	—	14	1	no	—
142	C₂×C₈	—	14	2	no	—
143	C₂×C₈	—	14	2	no	—
144	C₂×C₈	C₈×D₇	1	1	yes	characteristic
145	C₂×C₈	C₈×D₇	1	1	yes	characteristic
146	C₂×C₈	D₄×D₇	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
148	C₄²	—	7	1	no	—
149	C₄²	—	7	1	no	—
150	C₄²	—	7	1	no	—
151	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
152	C₁₆	—	4	2	no	—
153	C₁₆	—	28	2	no	—
154	C₂×C₇	#1564	1	1	yes	characteristic
155	C₂×C₇	#2150	1	1	yes	characteristic
156	C₂×C₇	C₂×#364	1	1	yes	characteristic
157	D₇	—	2	2	no	—
158	C₂³	—	7	1	no	—
159	C₂×C₄	—	7	1	no	—
160	C₂×C₄	—	7	1	no	—
161	C₂×C₄	—	7	1	no	—
162	C₂×C₄	—	14	2	no	—
163	C₂×C₄	—	14	1	no	—
164	C₂×C₄	—	14	1	no	—
165	C₂×C₄	—	14	1	no	—
166	C₂×C₄	—	14	1	no	—
167	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
169	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	7	2	no	—
174	C₈	—	7	2	no	—
175	C₈	—	14	2	no	—
176	C₈	—	14	1	no	—
177	C₈	—	14	1	no	—
178	C₈	#6372	1	2	yes	—
179	C₈	#6479	1	2	yes	—
180	C₇	#12013	1	1	yes	characteristic, complemented, Sylow
181	C₂²	—	7	2	no	—
182	C₂²	—	14	1	no	—
183	C₂²	—	14	1	no	—
184	C₂²	D₇×C₄⋊C₈	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	14	1	no	—
187	C₄	—	14	1	no	—
188	C₄	—	14	1	no	—
189	C₄	#89632	1	1	yes	characteristic
190	C₄	D₇×#108	1	1	yes	characteristic
191	C₄	C₂×#6372	1	1	yes	characteristic
192	C₄	D₇×M₃₂	1	1	yes	characteristic
193	C₂	—	14	2	no	—
194	C₂	#11726281	1	1	yes	characteristic
195	C₂	#11727038	1	1	yes	characteristic
196	C₂	D₇×#364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11727029

C₂

yes

maximal

C₈²×D₇

C₂

yes

maximal, characteristic

C₇×#1367

C₂

yes

maximal, characteristic, complemented

#11725361

C₂

yes

maximal, complemented

#11725245

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₇

C₄

yes

characteristic

C₄×C₈×D₇

C₄

yes

characteristic

C₄×C₈×D₇

C₂²

yes

characteristic

C₇×C₄⋊C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

characteristic

(C₂×D₇)⋊C₁₆

—

#89297

—

C₄⋊(C₇⋊C₁₆)

C₂²

yes

—

C₈×C₇⋊C₈

C₄

yes

characteristic

C₈×C₇⋊C₈

C₄

yes

characteristic

C₈×C₇⋊C₈

C₂²

yes

characteristic

#12013

—

maximal, Sylow

C₂×C₈×D₇

—

C₂×C₈×D₇

C₈

yes

characteristic

C₂×C₈×D₇

C₈

yes

characteristic

C₂×C₈×D₇

D₄

yes

characteristic

C₂×C₈×D₇

Q₈

yes

characteristic

C₄²×D₇

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

—

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

characteristic

C₈×Dic₇

—

C₈×Dic₇

C₈

yes

characteristic

C₈×Dic₇

C₈

yes

characteristic

C₈×Dic₇

D₄

yes

characteristic

C₈×Dic₇

Q₈

yes

characteristic

C₂×C₇⋊C₁₆

—

C₄×C₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇⋊C₈

C₂×C₄

yes

characteristic

#2163

—

C₂×C₈²

—

#1370

—

#1367

—

#1367

D₇

yes

characteristic, complemented

C₂×C₄×D₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇

C₂×C₈

yes

characteristic

C₂×C₄×D₇

M₁₆

yes

characteristic

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×Q₈

yes

characteristic

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄×Dic₇

C₄⋊C₄

yes

characteristic

C₄×Dic₇

C₂×C₈

yes

characteristic

C₄×Dic₇

M₁₆

yes

characteristic

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₂×C₈

yes

characteristic

C₂×C₇⋊C₈

C₂×C₈

yes

characteristic

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₇

yes

—

C₂²⋊C₁₆

—

#346

—

C₈²

—

C₈²

—

C₈²

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₇

C₂×M₁₆

yes

characteristic

C₂×Dic₇

C₄⋊C₈

yes

characteristic

C₂×Dic₇

C₄⋊C₈

yes

characteristic

C₂×Dic₇

C₄⋊C₈

yes

characteristic

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

M₃₂

yes

—

C₇×C₈

#135

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

#108

yes

—

C₇⋊C₈

M₃₂

yes

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

C₄×D₇

yes

characteristic

116

C₄×C₈

C₄×D₇

yes

characteristic

117

C₄×C₈

C₂²×D₇

yes

characteristic

118

C₂²×C₇

C₂×C₄⋊C₈

yes

characteristic

119

C₂×D₇

—

120

C₂×D₇

—

121

C₂×D₇

#364

yes

—

122

C₄×C₇

—

123

C₄×C₇

#427

yes

characteristic

124

C₄×C₇

C₂×#108

yes

characteristic

125

C₄×C₇

C₂×M₃₂

yes

characteristic

126

C₄×C₇

C₂×M₃₂

yes

characteristic

127

Dic₇

—

128

Dic₇

—

129

Dic₇

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

C₈×D₇

yes

characteristic

145

C₂×C₈

C₈×D₇

yes

characteristic

146

C₂×C₈

D₄×D₇

yes

characteristic

147

C₂×C₈

Q₈×D₇

yes

characteristic

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×C₄×D₇

yes

characteristic

152

C₁₆

—

153

C₁₆

—

154

C₂×C₇

#1564

yes

characteristic

155

C₂×C₇

#2150

yes

characteristic

156

C₂×C₇

C₂×#364

yes

characteristic

157

D₇

—

158

C₂³

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

D₇×C₄⋊C₄

yes

characteristic

168

C₂×C₄

C₂×C₈×D₇

yes

characteristic

169

C₂×C₄

D₇×M₁₆

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

#6372

yes

—

179

C₈

#6479

yes

—

180

C₇

#12013

yes

characteristic, complemented, Sylow

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

D₇×C₄⋊C₈

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#89632

yes

characteristic

190

C₄

D₇×#108

yes

characteristic

191

C₄

C₂×#6372

yes

characteristic

192

C₄

D₇×M₃₂

yes

characteristic

193

C₂

—

194

C₂

#11726281

yes

characteristic

195

C₂

#11727038

yes

characteristic

196

C₂

D₇×#364

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1367	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂²×D₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#1367	D₇
hypocenter	C₇	#12013
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₇

commutator

C₂×C₄×C₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇

Fitting

C₇×#1367

C₂

socle

C₂²×C₇

C₂×C₄⋊C₈

prime radical

C₂²×D₇

C₄⋊C₈

cosocle

C₄×C₇×C₈

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
3. C₇×#1367	C₂	semidirect
4. #11725361	C₂	semidirect
5. #11725245	C₂	semidirect
41. #1367	D₇	semidirect
180. C₇	#12013	semidirect

H ≅ G/N

type

3. C₇×#1367

C₂

semidirect

4. #11725361

C₂

semidirect

5. #11725245

C₂

semidirect

41. #1367

D₇

semidirect

180. C₇

#12013

semidirect

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

subgroup	index
#11727029	2
C₈²×D₇	2
C₇×#1367	2
#11725361	2
#11725245	2
#12013	7

subgroup

index

C₈²×D₇

Subgroups — #13265397

Conjugacy classes of subgroups

Complements

Maximal subgroups