Group #13253946 — subgroups

Subgroups — #13253946

Order 1792.

438 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 159 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	#11730752	C₂	1	1	yes	maximal, characteristic
2	C₇×#1842	C₂	1	1	yes	maximal, characteristic
3	#11727145	C₂	1	1	yes	maximal, characteristic
4	#11725508	C₂	1	4	yes	maximal
5	C₂×#6433	—	2	1	no	—
6	C₂×C₇×C₄⋊C₈	—	2	2	no	—
7	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₇	C₂²	1	1	yes	characteristic
9	#89708	—	2	1	no	—
10	#89707	C₂²	1	2	yes	—
11	#89700	—	2	2	no	—
12	#89700	C₄	1	4	yes	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×#6377	—	2	1	no	—
15	#89319	—	2	1	no	—
16	#9904	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	—	2	2	no	—
18	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇	D₄	1	2	yes	—
22	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇	—	2	2	no	—
24	C₂×C₄⋊Dic₇	—	2	2	no	—
25	C₂×C₄⋊Dic₇	C₂×C₄	1	4	yes	—
26	C₂×Dic₇⋊C₄	—	2	4	no	—
27	C₂²×C₇⋊C₈	—	2	2	no	—
28	#6433	—	4	2	no	—
29	#6433	—	4	2	no	—
30	C₇×C₄⋊C₈	—	2	4	no	—
31	C₇×C₄⋊C₈	—	4	4	no	—
32	C₇⋊#95	—	2	4	no	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
35	#6377	—	4	2	no	—
36	#2859	—	7	1	no	—
37	#1842	D₇	1	1	yes	characteristic
38	#1767	—	7	1	no	—
39	#1296	—	7	4	no	—
40	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₇	—	2	4	no	—
46	C₂×C₇×C₈	—	4	2	no	—
47	C₂×C₇×C₈	—	4	4	no	—
48	C₄²×C₇	—	2	2	no	—
49	C₄²×C₇	—	2	2	no	—
50	C₄²×C₇	—	2	2	no	—
51	C₄²×C₇	—	4	2	no	—
52	C₄²×C₇	D₈	1	2	yes	—
53	C₄²×C₇	SD₁₆	1	2	yes	—
54	C₄²×C₇	C₂×Q₈	1	2	yes	—
55	C₄²×C₇	#41	1	2	yes	—
56	C₄⋊Dic₇	—	4	2	no	—
57	C₄⋊Dic₇	—	4	2	no	—
58	C₄⋊Dic₇	—	4	2	no	—
59	C₄⋊Dic₇	—	4	2	no	—
60	C₄⋊Dic₇	—	4	4	no	—
61	Dic₇⋊C₄	—	4	8	no	—
62	C₂×C₇⋊C₈	—	2	4	no	—
63	C₂×C₇⋊C₈	—	4	2	no	—
64	C₂×C₇⋊C₈	—	4	2	no	—
65	C₂×#125	—	14	1	no	—
66	C₂×C₄⋊C₈	—	2	2	no	—
67	C₂×C₄⋊C₈	—	7	2	no	—
68	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
69	C₂×#97	—	14	1	no	—
70	#389	—	7	4	no	—
71	#389	—	14	2	no	—
72	#384	—	7	2	no	—
73	#382	—	14	1	no	—
74	C₄³	C₂×D₇	1	1	yes	characteristic
75	#336	—	14	1	no	—
76	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	#102	1	4	yes	—
84	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₇	#137	1	2	yes	—
92	C₂×C₄×C₇	#137	1	2	yes	—
93	C₂×Dic₇	—	4	4	no	—
94	C₂×Dic₇	—	4	4	no	—
95	C₂×Dic₇	—	4	4	no	—
96	C₇×C₈	—	8	2	no	—
97	C₇⋊C₈	—	4	4	no	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	14	2	no	—
100	#125	—	28	2	no	—
101	#125	—	28	2	no	—
102	C₂×C₄⋊C₄	—	7	4	no	—
103	C₂×C₄⋊C₄	—	14	4	no	—
104	C₂×C₄⋊C₄	—	14	2	no	—
105	C₂×C₄⋊C₄	—	14	2	no	—
106	C₂×C₄²	C₄×D₇	1	2	yes	—
107	C₂×C₄²	D₂₈	1	1	yes	characteristic
108	C₂×C₄²	D₂₈	1	1	yes	characteristic
109	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
110	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	4	4	no	—
113	C₄⋊C₈	—	7	4	no	—
114	C₄⋊C₈	—	14	2	no	—
115	#97	—	28	2	no	—
116	#95	—	14	4	no	—
117	C₂²×C₇	#391	1	1	yes	characteristic
118	C₂²×C₇	#413	1	2	yes	—
119	C₂²×C₇	C₂×#102	1	1	yes	characteristic
120	C₂²×C₇	#417	1	2	yes	—
121	C₂²×C₇	#419	1	1	yes	characteristic
122	C₄×C₇	—	2	4	no	—
123	C₄×C₇	—	2	2	no	—
124	C₄×C₇	—	4	2	no	—
125	C₄×C₇	#474	1	2	yes	—
126	C₄×C₇	#475	1	2	yes	—
127	C₄×C₇	#490	1	1	yes	characteristic
128	C₄×C₇	#491	1	1	yes	characteristic
129	C₄×C₇	#501	1	2	yes	—
130	Dic₇	—	8	4	no	—
131	C₂²×C₄	—	14	4	no	—
132	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
133	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	14	4	no	—
140	C₂×C₈	—	28	2	no	—
141	C₂×C₈	—	28	2	no	—
142	C₄⋊C₄	—	28	4	no	—
143	C₄⋊C₄	—	28	2	no	—
144	C₄⋊C₄	—	28	2	no	—
145	C₄⋊C₄	—	28	2	no	—
146	C₄⋊C₄	—	28	2	no	—
147	C₄⋊C₄	—	28	8	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	4	2	no	—
152	C₄²	#1132	1	2	yes	—
153	C₄²	D₅₆	1	2	yes	—
154	C₄²	#1159	1	2	yes	—
155	C₄²	Q₈×D₇	1	2	yes	—
156	C₂×C₇	#1547	1	2	yes	—
157	C₂×C₇	#1559	1	2	yes	—
158	C₂×C₇	#1988	1	1	yes	characteristic
159	C₂×C₇	#1992	1	1	yes	characteristic
160	C₂×C₇	#1998	1	1	yes	characteristic
161	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	28	4	no	—
169	C₂×C₄	—	28	4	no	—
170	C₂×C₄	—	28	4	no	—
171	C₂×C₄	#6395	1	4	yes	—
172	C₂×C₄	#6450	1	2	yes	—
173	C₂×C₄	#6460	1	2	yes	—
174	C₂×C₄	C₂×#1132	1	1	yes	characteristic
175	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
176	C₂×C₄	#6472	1	2	yes	—
177	C₂×C₄	#6495	1	2	yes	—
178	C₂×C₄	#6499	1	1	yes	characteristic
179	C₂×C₄	#6505	1	2	yes	—
180	C₂×C₄	#6507	1	1	yes	characteristic
181	C₈	—	8	2	no	—
182	C₈	—	28	4	no	—
183	C₇	#9904	1	1	yes	characteristic, complemented, Sylow
184	C₂²	#89738	1	2	yes	—
185	C₂²	#89746	1	1	yes	characteristic
186	C₂²	C₂×#6395	1	1	yes	characteristic
187	C₂²	#89892	1	1	yes	characteristic
188	C₂²	#89904	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	56	4	no	—
193	C₄	#89618	1	2	yes	—
194	C₄	#89623	1	2	yes	—
195	C₄	#89843	1	1	yes	characteristic
196	C₄	#89852	1	2	yes	—
197	C₄	#89854	1	1	yes	characteristic
198	C₂	#11727337	1	1	yes	characteristic
199	C₂	#11728186	1	2	yes	—
200	C₂	#11728214	1	1	yes	characteristic
201	C₂	#11728264	1	2	yes	—
202	C₂	#11728279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11730752

C₂

yes

maximal, characteristic

C₇×#1842

C₂

yes

maximal, characteristic

#11727145

C₂

yes

maximal, characteristic

#11725508

C₂

yes

maximal

C₂×#6433

—

C₂×C₇×C₄⋊C₈

—

C₂×C₇×C₄⋊C₈

C₂²

yes

—

C₄³×C₇

C₂²

yes

characteristic

#89708

—

#89707

C₂²

yes

—

#89700

—

#89700

C₄

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

C₂²×C₇×C₈

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂×C₄

yes

—

C₂×Dic₇⋊C₄

—

C₂²×C₇⋊C₈

—

#6433

—

#6433

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇⋊#95

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

Q₈

yes

—

#6377

—

#2859

—

#1842

D₇

yes

characteristic

#1767

—

#1296

—

C₂²×C₄×C₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

D₈

yes

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

C₂×Q₈

yes

—

C₄²×C₇

#41

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₇

yes

—

C₂×#97

—

—

—

—

—

C₄³

C₂×D₇

yes

characteristic

#336

—

C₂³×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#102

yes

—

C₂×C₄×C₇

C₄×Q₈

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

100

#125

—

101

#125

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₄×D₇

yes

—

107

C₂×C₄²

D₂₈

yes

characteristic

108

C₂×C₄²

D₂₈

yes

characteristic

109

C₂×C₄²

C₂²⋊D₇

yes

—

110

C₂×C₄²

C₂²×D₇

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

#97

—

116

#95

—

117

C₂²×C₇

#391

yes

characteristic

118

C₂²×C₇

#413

yes

—

119

C₂²×C₇

C₂×#102

yes

characteristic

120

C₂²×C₇

#417

yes

—

121

C₂²×C₇

#419

yes

characteristic

122

C₄×C₇

—

123

C₄×C₇

—

124

C₄×C₇

—

125

C₄×C₇

#474

yes

—

126

C₄×C₇

#475

yes

—

127

C₄×C₇

#490

yes

characteristic

128

C₄×C₇

#491

yes

characteristic

129

C₄×C₇

#501

yes

—

130

Dic₇

—

131

C₂²×C₄

—

132

C₂²×C₄

(C₂×D₇)⋊C₄

yes

—

133

C₂²×C₄

(C₂×D₇)⋊C₄

yes

—

134

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

135

C₂²×C₄

C₂×D₂₈

yes

characteristic

136

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

#1132

yes

—

153

C₄²

D₅₆

yes

—

154

C₄²

#1159

yes

—

155

C₄²

Q₈×D₇

yes

—

156

C₂×C₇

#1547

yes

—

157

C₂×C₇

#1559

yes

—

158

C₂×C₇

#1988

yes

characteristic

159

C₂×C₇

#1992

yes

characteristic

160

C₂×C₇

#1998

yes

characteristic

161

C₂³

C₂×(C₂×D₇)⋊C₄

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#6395

yes

—

172

C₂×C₄

#6450

yes

—

173

C₂×C₄

#6460

yes

—

174

C₂×C₄

C₂×#1132

yes

characteristic

175

C₂×C₄

C₂×D₅₆

yes

characteristic

176

C₂×C₄

#6472

yes

—

177

C₂×C₄

#6495

yes

—

178

C₂×C₄

#6499

yes

characteristic

179

C₂×C₄

#6505

yes

—

180

C₂×C₄

#6507

yes

characteristic

181

C₈

—

182

C₈

—

183

C₇

#9904

yes

characteristic, complemented, Sylow

184

C₂²

#89738

yes

—

185

C₂²

#89746

yes

characteristic

186

C₂²

C₂×#6395

yes

characteristic

187

C₂²

#89892

yes

characteristic

188

C₂²

#89904

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#89618

yes

—

194

C₄

#89623

yes

—

195

C₄

#89843

yes

characteristic

196

C₄

#89852

yes

—

197

C₄

#89854

yes

characteristic

198

C₂

#11727337

yes

characteristic

199

C₂

#11728186

yes

—

200

C₂

#11728214

yes

characteristic

201

C₂

#11728264

yes

—

202

C₂

#11728279

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇)⋊C₄
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1842	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1842	D₇
hypocenter	C₇	#9904
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×(C₂×D₇)⋊C₄

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

Fitting

C₇×#1842

C₂

socle, prime radical

C₂³×C₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
183. C₇	#9904	semidirect

H ≅ G/N

type

183. C₇

#9904

semidirect

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

subgroup	index
#11730752	2
C₇×#1842	2
#11727145	2
#11725508	2
#9904	7

subgroup

index

Subgroups — #13253946

Conjugacy classes of subgroups

Complements

Maximal subgroups