Group #13190657 — subgroups

Subgroups — C₇×#13441 #13190657

Order 1792.

378 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂²×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₇×#2151	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#362	C₂	1	2	yes	maximal
4	C₇×#1979	C₂	1	2	yes	maximal
5	C₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₇×Q₃₂	—	2	4	no	—
8	C₂×C₇×Q₃₂	—	2	2	no	—
9	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₇×#108	C₂²	1	2	yes	—
12	C₇×#419	—	2	2	no	—
13	C₂×C₇×#101	—	2	2	no	—
14	C₇×#405	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊C₁₆	—	2	2	no	—
16	C₇×#362	—	2	4	no	—
17	#13441	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₇×Q₈	—	2	2	no	—
19	C₂×C₇×Q₁₆	—	2	2	no	—
20	C₂×C₇×Q₁₆	—	2	2	no	—
21	C₂×C₇×Q₁₆	—	2	2	no	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	D₄	1	4	yes	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×#117	D₄	1	1	yes	characteristic
29	C₇×Q₃₂	—	2	8	no	—
30	C₇×M₃₂	—	2	2	no	—
31	C₂×C₇×C₁₆	—	2	2	no	—
32	C₂×C₇×C₁₆	—	2	1	no	—
33	C₂×C₇×C₁₆	D₄	1	2	yes	—
34	C₇×#108	—	2	2	no	—
35	C₇×#108	—	2	2	no	—
36	C₇×#103	—	4	4	no	—
37	C₇×#101	—	4	4	no	—
38	C₇×#97	—	2	1	no	—
39	C₄×C₇×C₈	—	2	1	no	—
40	C₂²×Q₃₂	C₂×C₇	1	1	yes	characteristic
41	#2151	C₂×C₇	1	1	yes	characteristic
42	C₂×#362	C₂×C₇	1	2	yes	—
43	#1979	C₂×C₇	1	2	yes	—
44	#1381	C₂×C₇	1	1	yes	characteristic
45	C₂×C₇×Q₈	—	2	4	no	—
46	C₂×C₇×Q₈	—	4	2	no	—
47	C₂×C₇×Q₈	—	4	2	no	—
48	C₂×C₇×Q₈	—	4	2	no	—
49	C₂×C₇×Q₈	—	4	2	no	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂²×C₄×C₇	—	4	2	no	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₇×Q₁₆	—	2	4	no	—
54	C₇×Q₁₆	—	2	4	no	—
55	C₇×Q₁₆	—	4	2	no	—
56	C₇×Q₁₆	—	4	2	no	—
57	C₇×M₁₆	—	2	2	no	—
58	C₇×M₁₆	—	4	2	no	—
59	C₇×M₁₆	—	4	2	no	—
60	C₂×C₇×C₈	—	2	1	no	—
61	C₂×C₇×C₈	—	4	2	no	—
62	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₇×C₈	#41	1	1	yes	characteristic
65	C₂×C₇×C₈	#41	1	2	yes	—
66	C₇×C₄⋊C₄	D₈	1	2	yes	—
67	C₇×C₂²⋊C₄	—	2	1	no	—
68	C₄²×C₇	—	2	1	no	—
69	C₇×C₁₆	—	2	2	no	—
70	C₇×C₁₆	—	4	1	no	—
71	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
72	C₂×Q₃₂	—	2	4	no	—
73	C₂×Q₃₂	—	2	2	no	—
74	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
75	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
76	C₂×#108	C₂²×C₇	1	2	yes	—
77	#419	—	2	2	no	—
78	C₂×#101	—	2	2	no	—
79	#405	C₂²×C₇	1	1	yes	characteristic
80	C₄⋊C₁₆	—	2	2	no	—
81	#362	—	2	4	no	—
82	C₂³×C₇	#135	1	1	yes	characteristic
83	C₇×Q₈	—	4	4	no	—
84	C₇×Q₈	—	4	4	no	—
85	C₇×Q₈	—	8	2	no	—
86	C₇×Q₈	—	8	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	—	8	2	no	—
90	C₂×C₄×C₇	—	8	2	no	—
91	C₂×C₄×C₇	#121	1	2	yes	—
92	C₂×C₄×C₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
95	C₇×C₈	—	4	1	no	—
96	C₇×C₈	—	8	2	no	—
97	C₇×C₈	#120	1	1	yes	characteristic
98	C₇×C₈	#123	1	2	yes	—
99	C₇×C₈	#124	1	1	yes	characteristic
100	C₂²×Q₈	—	2	2	no	—
101	C₂×Q₁₆	—	2	2	no	—
102	C₂×Q₁₆	—	2	2	no	—
103	C₂×Q₁₆	—	2	2	no	—
104	C₂×Q₁₆	—	2	2	no	—
105	C₂×Q₁₆	—	2	2	no	—
106	C₂×Q₁₆	C₇×D₄	1	4	yes	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
110	#117	C₇×D₄	1	1	yes	characteristic
111	Q₃₂	—	2	8	no	—
112	M₃₂	—	2	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	C₇×D₄	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	—	2	2	no	—
118	#103	—	4	4	no	—
119	#101	—	4	4	no	—
120	#97	—	2	1	no	—
121	C₄×C₈	—	2	1	no	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	#463	1	2	yes	—
124	C₂²×C₇	#466	1	1	yes	characteristic
125	C₄×C₇	—	4	2	no	—
126	C₄×C₇	—	8	4	no	—
127	C₄×C₇	#394	1	1	yes	characteristic
128	C₄×C₇	#459	1	2	yes	—
129	C₄×C₇	#469	1	1	yes	characteristic
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	4	2	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
138	Q₁₆	—	2	4	no	—
139	Q₁₆	—	2	4	no	—
140	Q₁₆	—	4	2	no	—
141	Q₁₆	—	4	2	no	—
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
148	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
149	C₂×C₈	C₇×#41	1	1	yes	characteristic
150	C₂×C₈	C₇×#41	1	2	yes	—
151	C₄⋊C₄	C₇×D₈	1	2	yes	—
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	—	4	1	no	—
156	C₂×C₇	—	2	2	no	—
157	C₂×C₇	#2054	1	1	yes	characteristic
158	C₂×C₇	#2222	1	2	yes	—
159	C₂³	C₇×#135	1	1	yes	characteristic
160	Q₈	—	4	4	no	—
161	Q₈	—	4	4	no	—
162	Q₈	—	8	2	no	—
163	Q₈	—	8	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₇×#121	1	1	yes	characteristic
169	C₂×C₄	C₇×#121	1	2	yes	—
170	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
171	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
172	C₈	—	4	1	no	—
173	C₈	—	8	2	no	—
174	C₈	C₇×#120	1	1	yes	characteristic
175	C₈	C₇×#123	1	2	yes	—
176	C₈	C₇×#124	1	1	yes	characteristic
177	C₇	#13441	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²	—	2	2	no	—
179	C₂²	C₇×#463	1	2	yes	—
180	C₂²	C₇×#466	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	8	4	no	—
183	C₄	C₇×#394	1	1	yes	characteristic
184	C₄	C₇×#459	1	2	yes	—
185	C₄	C₇×#469	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	C₇×#2054	1	1	yes	characteristic
188	C₂	C₇×#2222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇×Q₃₂

C₂

yes

maximal, characteristic

C₇×#2151

C₂

yes

maximal, characteristic

C₂×C₇×#362

C₂

yes

maximal

C₇×#1979

C₂

yes

maximal

C₇×#1381

C₂

yes

maximal, characteristic

C₂²×C₇×Q₁₆

C₂²

yes

—

C₂×C₇×Q₃₂

—

C₂×C₇×Q₃₂

—

C₂×C₇×M₃₂

C₂²

yes

characteristic

C₂²×C₇×C₁₆

C₂²

yes

characteristic

C₂×C₇×#108

C₂²

yes

—

C₇×#419

—

C₂×C₇×#101

—

C₇×#405

C₂²

yes

characteristic

C₇×C₄⋊C₁₆

—

C₇×#362

—

#13441

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×Q₈

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

D₄

yes

—

C₂×C₇×M₁₆

—

C₂×C₇×M₁₆

D₄

yes

characteristic

C₂²×C₇×C₈

C₂³

yes

characteristic

C₇×#117

D₄

yes

characteristic

C₇×Q₃₂

—

C₇×M₃₂

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

D₄

yes

—

—

—

—

—

C₇×#97

—

C₄×C₇×C₈

—

C₂²×Q₃₂

C₂×C₇

yes

characteristic

#2151

C₂×C₇

yes

characteristic

C₂×#362

C₂×C₇

yes

—

#1979

C₂×C₇

yes

—

#1381

C₂×C₇

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₇×Q₁₆

—

C₇×Q₁₆

—

C₇×Q₁₆

—

C₇×Q₁₆

—

C₇×M₁₆

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₂×C₇×C₈

C₂×D₄

yes

C₂×C₇×C₈

yes

C₂×C₇×C₈

yes

—

C₇×C₄⋊C₄

D₈

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₇×C₁₆

—

C₇×C₁₆

—

C₂²×Q₁₆

C₂²×C₇

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

C₂²×C₇

yes

characteristic

C₂²×C₁₆

C₂²×C₇

yes

characteristic

C₂×#108

C₂²×C₇

yes

—

#419

—

C₂×#101

—

#405

C₂²×C₇

yes

characteristic

C₄⋊C₁₆

—

#362

—

C₂³×C₇

#135

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

#120

yes

characteristic

C₇×C₈

#123

yes

—

C₇×C₈

#124

yes

characteristic

100

C₂²×Q₈

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

C₇×D₄

yes

—

107

C₂×M₁₆

—

108

C₂×M₁₆

C₇×D₄

yes

characteristic

109

C₂²×C₈

C₂³×C₇

yes

characteristic

110

#117

C₇×D₄

yes

characteristic

111

Q₃₂

—

112

M₃₂

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₂×C₁₆

C₇×D₄

yes

—

116

#108

—

117

#108

—

118

#103

—

119

#101

—

120

#97

—

121

C₄×C₈

—

122

C₂²×C₇

—

123

C₂²×C₇

#463

yes

—

124

C₂²×C₇

#466

yes

characteristic

125

C₄×C₇

—

126

C₄×C₇

—

127

C₄×C₇

#394

yes

characteristic

128

C₄×C₇

#459

yes

—

129

C₄×C₇

#469

yes

characteristic

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

138

Q₁₆

—

139

Q₁₆

—

140

Q₁₆

—

141

Q₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₂×C₇×D₄

yes

—

148

C₂×C₈

C₂×C₇×D₄

yes

characteristic

149

C₂×C₈

C₇×#41

yes

characteristic

150

C₂×C₈

C₇×#41

yes

—

151

C₄⋊C₄

C₇×D₈

yes

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₁₆

—

155

C₁₆

—

156

C₂×C₇

—

157

C₂×C₇

#2054

yes

characteristic

158

C₂×C₇

#2222

yes

—

159

C₂³

C₇×#135

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

C₇×#121

yes

characteristic

169

C₂×C₄

C₇×#121

yes

—

170

C₂×C₄

C₂×C₇×D₈

yes

characteristic

171

C₂×C₄

C₇×AGL(1,ℤ/8)

yes

—

172

C₈

—

173

C₈

—

174

C₈

C₇×#120

yes

characteristic

175

C₈

C₇×#123

yes

—

176

C₈

C₇×#124

yes

characteristic

177

C₇

#13441

yes

characteristic, direct product factor, Sylow

178

C₂²

—

179

C₂²

C₇×#463

yes

—

180

C₂²

C₇×#466

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

C₇×#394

yes

characteristic

184

C₄

C₇×#459

yes

—

185

C₄

C₇×#469

yes

characteristic

186

C₂

—

187

C₂

C₇×#2054

yes

characteristic

188

C₂

C₇×#2222

yes

—

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#405 ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₇

prime radical

C₂³×C₇

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#405 ◁ G

N	H ≅ G/N	type
17. #13441	C₇	direct (N × H)
177. C₇	#13441	direct (N × H)

H ≅ G/N

type

17. #13441

C₇

direct (N × H)

177. C₇

#13441

direct (N × H)

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

subgroup	index
C₂²×C₇×Q₃₂	2
C₇×#2151	2
C₂×C₇×#362	2
C₇×#1979	2
C₇×#1381	2
#13441	7

subgroup

index

C₂²×C₇×Q₃₂

C₂×C₇×#362

Subgroups — C₇×#13441 #13190657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#13441 #13190657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13441 #13190657