Group #13271935 — subgroups

Subgroups — #13271935

Order 1792.

310 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₇×#1844	C₂	1	1	yes	maximal, characteristic
2	C₂×#89282	C₂	1	2	yes	maximal
3	#11727533	C₂	1	1	yes	maximal, characteristic
4	C₂×#89258	C₂	1	1	yes	maximal, characteristic
5	#11725527	C₂	1	2	yes	maximal
6	C₂×C₇×#106	—	2	1	no	—
7	C₂×C₇×#106	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#384	—	2	1	no	—
10	C₂×#6393	C₂²	1	1	yes	characteristic
11	#89878	C₂²	1	1	yes	characteristic
12	#89873	—	2	1	no	—
13	C₂×#6386	C₂²	1	2	yes	—
14	#89318	—	2	1	no	—
15	#89282	—	2	4	no	—
16	#89258	C₄	1	4	yes	complemented
17	#13186	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₄	—	2	2	no	—
21	C₂×C₇×C₄⋊C₄	—	4	2	no	—
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	#6489	—	2	1	no	—
24	C₂×#1136	—	2	1	no	—
25	C₇×#106	—	2	2	no	—
26	C₇×#106	—	2	2	no	—
27	C₇×#106	—	4	2	no	—
28	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₇×C₈	Q₈	1	2	yes	—
30	#6393	—	2	1	no	—
31	#6393	D₄	1	2	yes	—
32	#6392	Q₈	1	2	yes	—
33	#6391	—	2	1	no	—
34	#6389	—	4	1	no	—
35	C₈×Dic₇	—	4	1	no	—
36	#6386	—	2	2	no	—
37	#6386	C₂×C₄	1	4	yes	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×#365	—	7	2	no	—
40	C₂×#364	—	7	1	no	—
41	#1844	D₇	1	1	yes	characteristic
42	#1772	—	7	1	no	—
43	#1386	—	7	2	no	—
44	C₂²×C₄×C₇	—	2	1	no	—
45	C₂²×C₄×C₇	—	4	2	no	—
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₇×C₈	SD₁₆	1	2	yes	—
49	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
50	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₇×C₈	#41	1	1	yes	characteristic
56	C₂×C₇×C₈	#41	1	1	yes	characteristic
57	C₇×C₄⋊C₄	—	4	2	no	—
58	C₇×C₄⋊C₄	—	4	2	no	—
59	C₇×C₄⋊C₄	—	8	4	no	—
60	C₄²×C₇	—	2	1	no	—
61	C₄²×C₇	C₄⋊C₄	1	2	yes	—
62	C₂²⋊Dic₇	—	4	1	no	—
63	C₄⋊Dic₇	—	2	2	no	—
64	C₄×Dic₇	—	4	1	no	—
65	#1136	—	2	2	no	—
66	#1136	—	4	1	no	—
67	C₂×C₇⋊C₈	—	4	1	no	—
68	C₇⋊C₁₆	—	2	4	no	—
69	C₂×M₃₂	—	7	2	no	—
70	C₂×#108	—	7	1	no	—
71	#427	—	7	1	no	—
72	C₂×#106	—	2	1	no	—
73	C₂×#106	C₂×D₇	1	2	yes	—
74	#405	—	14	1	no	—
75	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
76	#384	—	2	1	no	—
77	#365	—	14	4	no	—
78	#364	—	7	4	no	—
79	#339	—	14	1	no	—
80	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	8	4	no	—
84	C₂×C₄×C₇	—	8	2	no	—
85	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇	#121	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇	#137	1	1	yes	characteristic
91	C₂×Dic₇	—	4	2	no	—
92	C₇×C₈	#106	1	2	yes	—
93	C₇×C₈	#106	1	2	yes	—
94	C₇×C₈	C₄×D₄	1	1	yes	characteristic
95	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
96	C₇×C₈	#125	1	1	yes	characteristic
97	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
98	C₇⋊C₈	—	4	2	no	—
99	C₂×M₁₆	—	14	1	no	—
100	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
102	#117	—	14	1	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	2	no	—
105	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
106	M₃₂	—	7	4	no	—
107	M₃₂	—	14	2	no	—
108	C₂×C₁₆	—	14	2	no	—
109	#108	—	7	2	no	—
110	#108	—	14	1	no	—
111	#107	—	7	2	no	—
112	#106	—	2	2	no	—
113	#106	—	2	2	no	—
114	#106	—	4	2	no	—
115	#106	—	14	1	no	—
116	#97	—	28	1	no	—
117	C₄×C₈	—	28	1	no	—
118	C₄×C₈	Dic₁₄	1	2	yes	—
119	C₄×C₈	C₄×D₇	1	2	yes	—
120	C₂²×C₇	—	2	2	no	—
121	C₂²×C₇	#428	1	1	yes	characteristic
122	C₂²×C₇	#460	1	1	yes	characteristic
123	C₂²×C₇	#476	1	1	yes	characteristic
124	C₄×C₇	—	2	4	no	—
125	C₄×C₇	—	16	2	no	—
126	C₄×C₇	#389	1	1	yes	characteristic
127	C₄×C₇	C₂×#106	1	1	yes	characteristic
128	C₄×C₇	#461	1	1	yes	characteristic
129	C₄×C₇	#475	1	1	yes	characteristic
130	Dic₇	—	8	2	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
134	M₁₆	—	14	2	no	—
135	M₁₆	—	28	1	no	—
136	C₂×C₈	—	28	1	no	—
137	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
138	C₂×C₈	#1142	1	2	yes	—
139	C₂×C₈	#1143	1	1	yes	characteristic
140	C₂×C₈	#1143	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
142	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
143	C₂×C₈	D₄×D₇	1	1	yes	characteristic
144	C₂×C₈	#1159	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
146	C₂×C₈	#1161	1	1	yes	characteristic
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	8	4	no	—
150	C₄⋊C₄	—	14	2	no	—
151	C₂²⋊C₄	—	28	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	28	1	no	—
154	C₄²	Dic₇⋊C₄	1	2	yes	—
155	C₁₆	—	14	4	no	—
156	C₂×C₇	—	2	1	no	—
157	C₂×C₇	—	2	1	no	—
158	C₂×C₇	#1948	1	1	yes	characteristic
159	C₂×C₇	#2230	1	2	yes	—
160	C₂³	#6495	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	28	2	no	—
166	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₂×#1142	1	1	yes	characteristic
168	C₂×C₄	#6501	1	1	yes	characteristic
169	C₂×C₄	C₂×#1143	1	1	yes	characteristic
170	C₂×C₄	#6505	1	1	yes	characteristic
171	C₂×C₄	#6509	1	1	yes	characteristic
172	C₈	—	28	2	no	—
173	C₈	#6382	1	2	yes	—
174	C₈	#6382	1	2	yes	—
175	C₈	#6450	1	1	yes	characteristic
176	C₈	#6451	1	1	yes	characteristic
177	C₈	#6453	1	1	yes	characteristic
178	C₈	#6456	1	1	yes	characteristic
179	C₇	#13186	1	1	yes	characteristic, complemented, Sylow
180	C₂²	—	2	2	no	—
181	C₂²	#89736	1	1	yes	characteristic
182	C₂²	#89847	1	1	yes	characteristic
183	C₂²	#89856	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	16	2	no	—
186	C₄	—	56	2	no	—
187	C₄	C₂×#6382	1	1	yes	characteristic
188	C₄	#89745	1	1	yes	characteristic
189	C₄	#89855	1	1	yes	characteristic
190	C₄	#89866	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#11727322	1	1	yes	characteristic
194	C₂	#11729104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1844

C₂

yes

maximal, characteristic

C₂×#89282

C₂

yes

maximal

#11727533

C₂

yes

maximal, characteristic

C₂×#89258

C₂

yes

maximal, characteristic

#11725527

C₂

yes

maximal

C₂×C₇×#106

—

C₂×C₇×#106

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#384

—

C₂×#6393

C₂²

yes

characteristic

#89878

C₂²

yes

characteristic

#89873

—

C₂×#6386

C₂²

yes

—

#89318

—

#89282

—

#89258

C₄

yes

complemented

#13186

—

maximal, Sylow

C₂²×C₇×C₈

D₄

yes

characteristic

C₂²×C₇×C₈

C₂³

yes

characteristic

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

D₄

yes

—

—

—

—

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

Q₈

yes

—

#6393

—

#6393

D₄

yes

—

#6392

Q₈

yes

—

#6391

—

#6389

—

C₈×Dic₇

—

#6386

—

#6386

C₂×C₄

yes

—

C₂×C₇⋊C₁₆

—

C₂×#365

—

C₂×#364

—

#1844

D₇

yes

characteristic

#1772

—

#1386

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

SD₁₆

yes

—

C₂×C₇×C₈

SD₁₆

yes

characteristic

C₂×C₇×C₈

SD₁₆

yes

characteristic

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×Q₈

yes

characteristic

C₂×C₇×C₈

C₂×Q₈

yes

C₂×C₇×C₈

yes

C₂×C₇×C₈

yes

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₄⋊C₄

yes

—

C₂²⋊Dic₇

—

C₄⋊Dic₇

—

C₄×Dic₇

—

#1136

—

#1136

—

C₂×C₇⋊C₈

—

C₇⋊C₁₆

—

C₂×M₃₂

—

—

—

—

C₂×D₇

yes

—

#405

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

—

—

C₂³×C₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

#137

yes

characteristic

C₂×Dic₇

—

C₇×C₈

#106

yes

—

C₇×C₈

#106

yes

—

C₇×C₈

C₄×D₄

yes

characteristic

C₇×C₈

C₄×Q₈

yes

characteristic

C₇×C₈

#125

yes

characteristic

C₇×C₈

C₄⋊Q₈

yes

characteristic

C₇⋊C₈

—

C₂×M₁₆

—

100

C₂²×C₈

C₂²⋊D₇

yes

characteristic

101

C₂²×C₈

C₂²×D₇

yes

characteristic

102

#117

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₂²⋊D₇

yes

characteristic

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

#108

—

110

#108

—

111

#107

—

112

#106

—

113

#106

—

114

#106

—

115

#106

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

Dic₁₄

yes

—

119

C₄×C₈

C₄×D₇

yes

—

120

C₂²×C₇

—

121

C₂²×C₇

#428

yes

characteristic

122

C₂²×C₇

#460

yes

characteristic

123

C₂²×C₇

#476

yes

characteristic

124

C₄×C₇

—

125

C₄×C₇

—

126

C₄×C₇

#389

yes

characteristic

127

C₄×C₇

C₂×#106

yes

characteristic

128

C₄×C₇

#461

yes

characteristic

129

C₄×C₇

#475

yes

characteristic

130

Dic₇

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

134

M₁₆

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

Dic₇⋊C₄

yes

—

138

C₂×C₈

#1142

yes

—

139

C₂×C₈

#1143

yes

characteristic

140

C₂×C₈

#1143

yes

characteristic

141

C₂×C₈

C₂×Dic₁₄

yes

characteristic

142

C₂×C₈

C₂×C₄×D₇

yes

characteristic

143

C₂×C₈

D₄×D₇

yes

characteristic

144

C₂×C₈

#1159

yes

characteristic

145

C₂×C₈

Q₈×D₇

yes

characteristic

146

C₂×C₈

#1161

yes

characteristic

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

—

154

C₄²

Dic₇⋊C₄

yes

—

155

C₁₆

—

156

C₂×C₇

—

157

C₂×C₇

—

158

C₂×C₇

#1948

yes

characteristic

159

C₂×C₇

#2230

yes

—

160

C₂³

#6495

yes

characteristic

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₂×Dic₇⋊C₄

yes

characteristic

167

C₂×C₄

C₂×#1142

yes

characteristic

168

C₂×C₄

#6501

yes

characteristic

169

C₂×C₄

C₂×#1143

yes

characteristic

170

C₂×C₄

#6505

yes

characteristic

171

C₂×C₄

#6509

yes

characteristic

172

C₈

—

173

C₈

#6382

yes

—

174

C₈

#6382

yes

—

175

C₈

#6450

yes

characteristic

176

C₈

#6451

yes

characteristic

177

C₈

#6453

yes

characteristic

178

C₈

#6456

yes

characteristic

179

C₇

#13186

yes

characteristic, complemented, Sylow

180

C₂²

—

181

C₂²

#89736

yes

characteristic

182

C₂²

#89847

yes

characteristic

183

C₂²

#89856

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₂×#6382

yes

characteristic

188

C₄

#89745

yes

characteristic

189

C₄

#89855

yes

characteristic

190

C₄

#89866

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

#11727322

yes

characteristic

194

C₂

#11729104

yes

—

	subgroup	G/H
center	C₂²	#89856
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1844	C₂
socle	C₂²×C₇	#476
prime radical	C₂³×C₇	AGL(1,ℤ/8)
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1844	D₇
hypocenter	C₇	#13186
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#89856

commutator

C₂×C₇×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂²×C₇×C₈

C₂³

D₇

C₇

G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
16. #89258	C₄	semidirect
179. C₇	#13186	semidirect

H ≅ G/N

type

16. #89258

C₄

semidirect

179. C₇

#13186

semidirect

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

subgroup	index
C₇×#1844	2
C₂×#89282	2
#11727533	2
C₂×#89258	2
#11725527	2
#13186	7

subgroup

index

Subgroups — #13271935

Conjugacy classes of subgroups

Complements

Maximal subgroups