Group #13258848 — subgroups

Subgroups — #13258848

Order 1792.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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×#1956	C₂	1	1	yes	maximal, characteristic
2	#11729541	C₂	1	2	yes	maximal
3	#11727519	C₂	1	1	yes	maximal, characteristic
4	#11727331	C₂	1	2	yes	maximal
5	C₇⋊#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×#103	—	2	2	no	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#384	C₂²	1	1	yes	characteristic
10	C₂×#6392	—	2	2	no	—
11	C₂×#6381	—	2	4	no	—
12	#89707	C₂²	1	2	yes	—
13	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
14	#89324	—	2	4	no	—
15	#10674	—	7	1	no	maximal, Sylow
16	C₂²×C₇×Q₈	—	2	1	no	—
17	C₂²×C₇×C₈	D₄	1	2	yes	—
18	C₇×C₄⋊Q₈	—	2	2	no	—
19	C₇×C₄⋊Q₈	—	2	1	no	—
20	C₇×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₄⋊C₄	—	2	2	no	—
23	C₂×C₇×C₄⋊C₄	—	2	1	no	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	—	2	2	no	—
26	C₂×C₄⋊Dic₇	—	2	2	no	—
27	C₂×C₄⋊Dic₇	—	2	2	no	—
28	C₂²×C₇⋊C₈	D₄	1	4	yes	—
29	C₇×#103	—	4	4	no	—
30	C₄×C₇×C₈	—	2	2	no	—
31	#6392	—	4	4	no	—
32	#6381	—	4	8	no	—
33	C₄×C₇⋊C₈	—	2	2	no	—
34	C₄×C₇⋊C₈	—	2	2	no	—
35	#1990	—	7	2	no	—
36	#1956	D₇	1	1	yes	characteristic
37	#1845	—	7	1	no	—
38	#1845	—	7	2	no	—
39	#1273	—	7	1	no	—
40	C₂×C₇×Q₈	—	2	2	no	—
41	C₂×C₇×Q₈	—	4	1	no	—
42	C₂×C₇×Q₈	—	4	2	no	—
43	C₂×C₇×Q₈	—	4	2	no	—
44	C₂²×C₄×C₇	—	4	1	no	—
45	C₂²×C₄×C₇	—	4	1	no	—
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
48	C₂²×Dic₇	—	4	2	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	—	2	2	no	—
51	C₂×C₇×C₈	D₈	1	4	yes	—
52	C₇×C₄⋊C₄	—	2	4	no	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₇×C₄⋊C₄	—	4	4	no	—
56	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₇	#41	1	1	yes	characteristic
58	C₄²×C₇	#41	1	2	yes	—
59	C₄⋊Dic₇	—	4	4	no	—
60	C₄⋊Dic₇	—	4	4	no	—
61	C₄⋊Dic₇	—	4	4	no	—
62	C₂×C₇⋊C₈	—	2	4	no	—
63	C₂×C₇⋊C₈	—	2	4	no	—
64	C₂×C₇⋊C₈	Q₁₆	1	8	yes	—
65	C₂×C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
66	C₂×#107	—	14	4	no	—
67	C₂×#107	—	14	2	no	—
68	C₂×#103	—	2	2	no	—
69	C₂×C₄×C₈	—	7	2	no	—
70	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
71	#384	—	7	2	no	—
72	#384	C₂×D₇	1	1	yes	characteristic
73	#340	—	14	4	no	—
74	C₂³×C₇	#127	1	1	yes	characteristic
75	C₇×Q₈	—	4	4	no	—
76	C₇×Q₈	—	8	2	no	—
77	C₂×C₄×C₇	—	4	2	no	—
78	C₂×C₄×C₇	—	8	1	no	—
79	C₂×C₄×C₇	—	8	1	no	—
80	C₂×C₄×C₇	—	8	1	no	—
81	C₂×C₄×C₇	—	8	2	no	—
82	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇	#123	1	2	yes	—
85	C₂×C₄×C₇	#123	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
88	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
89	C₂×C₄×C₇	#137	1	2	yes	—
90	C₂×Dic₇	—	8	2	no	—
91	C₂×Dic₇	—	8	2	no	—
92	C₂×Dic₇	—	8	2	no	—
93	C₇×C₈	—	2	4	no	—
94	C₇⋊C₈	—	2	8	no	—
95	C₂²×Q₈	—	2	1	no	—
96	C₂²×C₈	—	7	4	no	—
97	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
98	C₄⋊Q₈	—	2	1	no	—
99	C₄⋊Q₈	—	2	2	no	—
100	C₄⋊Q₈	Dic₁₄	1	2	yes	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	2	1	no	—
104	C₂×C₄⋊C₄	—	14	2	no	—
105	C₂×C₄⋊C₄	—	14	2	no	—
106	C₂×C₄⋊C₄	—	14	2	no	—
107	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
108	#107	—	28	8	no	—
109	#107	—	28	4	no	—
110	#103	—	4	4	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	14	2	no	—
113	C₄×C₈	—	14	2	no	—
114	C₂²×C₇	#399	1	1	yes	characteristic
115	C₂²×C₇	#493	1	1	yes	characteristic
116	C₂²×C₇	#494	1	2	yes	—
117	C₂²×C₇	#496	1	2	yes	—
118	C₂²×C₇	#497	1	1	yes	characteristic
119	C₄×C₇	—	8	2	no	—
120	C₄×C₇	—	16	1	no	—
121	C₄×C₇	#474	1	2	yes	—
122	C₄×C₇	#477	1	4	yes	—
123	C₄×C₇	#480	1	2	yes	—
124	C₄×C₇	#484	1	4	yes	—
125	Dic₇	—	16	2	no	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	4	1	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂²×C₄	—	4	1	no	—
131	C₂²×C₄	—	4	1	no	—
132	C₂²×C₄	—	28	2	no	—
133	C₂²×C₄	D₄×D₇	1	2	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	7	8	no	—
138	C₂×C₈	—	14	4	no	—
139	C₂×C₈	—	14	4	no	—
140	C₂×C₈	#1141	1	4	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	28	4	no	—
146	C₄⋊C₄	—	28	4	no	—
147	C₄⋊C₄	—	28	4	no	—
148	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
149	C₄²	#1157	1	1	yes	characteristic
150	C₄²	#1159	1	2	yes	—
151	C₂×C₇	#1713	1	1	yes	characteristic
152	C₂×C₇	#1721	1	1	yes	characteristic
153	C₂×C₇	#1723	1	2	yes	—
154	C₂×C₇	#2093	1	1	yes	characteristic
155	C₂×C₇	#2095	1	2	yes	—
156	C₂³	#6503	1	1	yes	characteristic
157	Q₈	—	4	4	no	—
158	Q₈	—	8	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	56	2	no	—
165	C₂×C₄	—	56	2	no	—
166	C₂×C₄	—	56	2	no	—
167	C₂×C₄	#6441	1	2	yes	—
168	C₂×C₄	#6446	1	2	yes	—
169	C₂×C₄	#6477	1	4	yes	—
170	C₂×C₄	D₇×Q₁₆	1	4	yes	—
171	C₂×C₄	#6487	1	1	yes	characteristic
172	C₂×C₄	C₂×#1141	1	2	yes	—
173	C₂×C₄	#6498	1	1	yes	characteristic
174	C₂×C₄	#6514	1	2	yes	—
175	C₈	—	2	4	no	—
176	C₈	—	14	8	no	—
177	C₇	#10674	1	1	yes	characteristic, complemented, Sylow
178	C₂²	#89721	1	1	yes	characteristic
179	C₂²	#89832	1	1	yes	characteristic
180	C₂²	#89845	1	1	yes	characteristic
181	C₂²	#89948	1	2	yes	—
182	C₂²	#89953	1	2	yes	—
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	112	2	no	—
186	C₄	#89566	1	4	yes	—
187	C₄	#89581	1	4	yes	—
188	C₄	#89779	1	2	yes	—
189	C₄	#89805	1	2	yes	—
190	C₂	#11727409	1	1	yes	characteristic
191	C₂	#11728003	1	2	yes	—
192	C₂	#11728981	1	2	yes	—
193	C₂	#11729016	1	1	yes	characteristic
194	C₂	#11729052	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1956

C₂

yes

maximal, characteristic

#11729541

C₂

yes

maximal

#11727519

C₂

yes

maximal, characteristic

#11727331

C₂

yes

maximal

C₇⋊#1273

C₂

yes

maximal, characteristic

C₂×C₇×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₇×#103

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#384

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₇⋊C₈

C₂²

yes

—

#89324

—

#10674

—

maximal, Sylow

C₂²×C₇×Q₈

—

C₂²×C₇×C₈

D₄

yes

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

Q₈

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂²×C₇⋊C₈

D₄

yes

—

C₇×#103

—

C₄×C₇×C₈

—

#6392

—

#6381

—

C₄×C₇⋊C₈

—

C₄×C₇⋊C₈

—

#1990

—

#1956

D₇

yes

—

—

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₈

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×Q₈

yes

C₄²×C₇

yes

C₄²×C₇

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

Q₁₆

yes

—

C₂×C₄⋊Q₈

C₂×D₇

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

#384

—

#384

C₂×D₇

yes

characteristic

#340

—

C₂³×C₇

#127

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

#123

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×C₄×C₇

C₂×Q₁₆

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

Dic₁₄

yes

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₇

yes

characteristic

108

#107

—

109

#107

—

110

#103

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₂²×C₇

#399

yes

characteristic

115

C₂²×C₇

#493

yes

characteristic

116

C₂²×C₇

#494

yes

—

117

C₂²×C₇

#496

yes

—

118

C₂²×C₇

#497

yes

characteristic

119

C₄×C₇

—

120

C₄×C₇

—

121

C₄×C₇

#474

yes

—

122

C₄×C₇

#477

yes

—

123

C₄×C₇

#480

yes

—

124

C₄×C₇

#484

yes

—

125

Dic₇

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

D₄×D₇

yes

—

134

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

#1141

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

C₂×Dic₁₄

yes

characteristic

149

C₄²

#1157

yes

characteristic

150

C₄²

#1159

yes

—

151

C₂×C₇

#1713

yes

characteristic

152

C₂×C₇

#1721

yes

characteristic

153

C₂×C₇

#1723

yes

—

154

C₂×C₇

#2093

yes

characteristic

155

C₂×C₇

#2095

yes

—

156

C₂³

#6503

yes

characteristic

157

Q₈

—

158

Q₈

—

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

#6441

yes

—

168

C₂×C₄

#6446

yes

—

169

C₂×C₄

#6477

yes

—

170

C₂×C₄

D₇×Q₁₆

yes

—

171

C₂×C₄

#6487

yes

characteristic

172

C₂×C₄

C₂×#1141

yes

—

173

C₂×C₄

#6498

yes

characteristic

174

C₂×C₄

#6514

yes

—

175

C₈

—

176

C₈

—

177

C₇

#10674

yes

characteristic, complemented, Sylow

178

C₂²

#89721

yes

characteristic

179

C₂²

#89832

yes

characteristic

180

C₂²

#89845

yes

characteristic

181

C₂²

#89948

yes

—

182

C₂²

#89953

yes

—

183

C₄

—

184

C₄

—

185

C₄

—

112

—

186

C₄

#89566

yes

—

187

C₄

#89581

yes

—

188

C₄

#89779

yes

—

189

C₄

#89805

yes

—

190

C₂

#11727409

yes

characteristic

191

C₂

#11728003

yes

—

192

C₂

#11728981

yes

—

193

C₂

#11729016

yes

characteristic

194

C₂

#11729052

yes

characteristic

	subgroup	G/H
center	C₂³	#6503
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1956	C₂
socle, prime radical	C₂³×C₇	#127
hypercenter	#1956	D₇
hypocenter	C₇	#10674
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#6503

commutator, cosocle

C₂×C₄²×C₇

C₂³

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂³×C₇

D₇

C₇

G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
177. C₇	#10674	semidirect

H ≅ G/N

type

177. C₇

#10674

semidirect

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

subgroup	index
C₇×#1956	2
#11729541	2
#11727519	2
#11727331	2
C₇⋊#1273	2
#10674	7

subgroup

index

Subgroups — #13258848

Conjugacy classes of subgroups

Complements

Maximal subgroups