Group #13190673 — subgroups

Subgroups — C₇×#13457 #13190673

Order 1792.

356 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 122 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	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#365	C₂	1	2	yes	maximal
3	C₇×#1945	C₂	1	1	yes	maximal, characteristic
4	C₇×#1386	C₂	1	2	yes	maximal
5	C₇×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₇×M₃₂	C₂²	1	2	yes	—
8	C₇×#498	—	2	4	no	—
9	C₂×C₇×#108	C₂²	1	1	yes	characteristic
10	C₇×#427	C₂²	1	1	yes	characteristic
11	C₂×C₇×#106	—	2	1	no	—
12	C₂×C₇×#106	C₂²	1	2	yes	—
13	C₇×#424	—	2	1	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#365	—	2	4	no	—
16	C₇×#339	—	2	1	no	—
17	#13457	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₇×Q₈	—	4	1	no	—
19	C₂×C₇×M₁₆	—	2	1	no	—
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₇×C₄⋊Q₈	—	2	4	no	—
23	C₇×C₄⋊Q₈	—	4	2	no	—
24	C₇×#117	—	2	1	no	—
25	C₂×C₇×C₄⋊C₄	—	2	2	no	—
26	C₂×C₇×C₄⋊C₄	—	4	1	no	—
27	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
28	C₇×M₃₂	—	2	2	no	—
29	C₇×M₃₂	Q₈	1	4	yes	complemented
30	C₂×C₇×C₁₆	—	2	2	no	—
31	C₇×#108	—	2	2	no	—
32	C₇×#107	Q₈	1	2	yes	complemented
33	C₇×#106	—	2	4	no	—
34	C₇×#106	—	2	2	no	—
35	C₇×#106	—	2	1	no	—
36	C₇×#106	D₄	1	4	yes	—
37	C₇×C₄⋊C₈	—	4	2	no	—
38	C₄×C₇×C₈	—	2	2	no	—
39	C₂×#498	C₂×C₇	1	1	yes	characteristic
40	C₂×#365	C₂×C₇	1	2	yes	—
41	#1945	C₂×C₇	1	1	yes	characteristic
42	#1386	C₂×C₇	1	2	yes	—
43	#1376	C₂×C₇	1	1	yes	characteristic
44	C₂×C₇×Q₈	—	4	4	no	—
45	C₂×C₇×Q₈	—	8	4	no	—
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	—	4	2	no	—
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₇×M₁₆	—	4	1	no	—
50	C₇×M₁₆	—	4	1	no	—
51	C₂×C₇×C₈	—	2	1	no	—
52	C₂×C₇×C₈	—	2	1	no	—
53	C₂×C₇×C₈	—	4	1	no	—
54	C₂×C₇×C₈	SD₁₆	1	2	yes	—
55	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
56	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₇×C₈	#41	1	1	yes	characteristic
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	2	no	—
61	C₇×C₄⋊C₄	—	4	4	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₂²⋊C₄	—	4	1	no	—
64	C₄²×C₇	—	2	1	no	—
65	C₄²×C₇	—	4	1	no	—
66	C₄²×C₇	SD₁₆	1	2	yes	—
67	C₇×C₁₆	—	2	4	no	—
68	C₂×C₄⋊Q₈	—	2	1	no	—
69	C₂×M₃₂	C₂²×C₇	1	2	yes	—
70	#498	—	2	4	no	—
71	C₂×#108	C₂²×C₇	1	1	yes	characteristic
72	#427	C₂²×C₇	1	1	yes	characteristic
73	C₂×#106	—	2	1	no	—
74	C₂×#106	C₂²×C₇	1	2	yes	—
75	#424	—	2	1	no	—
76	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
77	#365	—	2	4	no	—
78	#339	—	2	1	no	—
79	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₇×Q₈	—	8	8	no	—
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	4	4	no	—
84	C₂×C₄×C₇	—	4	2	no	—
85	C₂×C₄×C₇	—	8	4	no	—
86	C₂×C₄×C₇	#120	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇	#137	1	1	yes	characteristic
92	C₇×C₈	—	2	2	no	—
93	C₇×C₈	—	8	1	no	—
94	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
95	C₇×C₈	C₄⋊Q₈	1	2	yes	—
96	C₂²×Q₈	—	4	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
100	C₄⋊Q₈	—	2	4	no	—
101	C₄⋊Q₈	—	4	2	no	—
102	#117	—	2	1	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	1	no	—
105	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₇×Q₈	1	4	yes	complemented
108	C₂×C₁₆	—	2	2	no	—
109	#108	—	2	2	no	—
110	#107	C₇×Q₈	1	2	yes	complemented
111	#106	—	2	4	no	—
112	#106	—	2	2	no	—
113	#106	—	2	1	no	—
114	#106	C₇×D₄	1	4	yes	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₂²×C₇	—	2	2	no	—
118	C₂²×C₇	#450	1	1	yes	characteristic
119	C₂²×C₇	#476	1	1	yes	characteristic
120	C₂²×C₇	#501	1	1	yes	characteristic
121	C₄×C₇	—	2	4	no	—
122	C₄×C₇	—	8	4	no	—
123	C₄×C₇	—	8	2	no	—
124	C₄×C₇	#395	1	1	yes	characteristic
125	C₄×C₇	#448	1	1	yes	characteristic
126	C₄×C₇	#475	1	1	yes	characteristic
127	C₄×C₇	#498	1	1	yes	characteristic
128	C₂×Q₈	—	4	4	no	—
129	C₂×Q₈	—	8	4	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
133	M₁₆	—	4	1	no	—
134	M₁₆	—	4	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	C₇×SD₁₆	1	2	yes	—
139	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
140	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
141	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
142	C₂×C₈	C₇×#41	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₂²⋊C₄	—	4	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	4	1	no	—
150	C₄²	C₇×SD₁₆	1	2	yes	—
151	C₁₆	—	2	4	no	—
152	C₂×C₇	—	2	1	no	—
153	C₂×C₇	—	2	1	no	—
154	C₂×C₇	#2058	1	1	yes	characteristic
155	C₂×C₇	#2230	1	2	yes	—
156	C₂³	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
157	Q₈	—	8	8	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	C₇×#120	1	1	yes	characteristic
164	C₂×C₄	C₇×C₂²⋊Q₈	1	1	yes	characteristic
165	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
167	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
168	C₂×C₄	C₇×#137	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	8	1	no	—
171	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
172	C₈	C₇×C₄⋊Q₈	1	2	yes	—
173	C₇	#13457	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²	—	2	2	no	—
175	C₂²	C₇×#450	1	1	yes	characteristic
176	C₂²	C₇×#476	1	1	yes	characteristic
177	C₂²	C₇×#501	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	8	2	no	—
181	C₄	C₇×#395	1	1	yes	characteristic
182	C₄	C₇×#448	1	1	yes	characteristic
183	C₄	C₇×#475	1	1	yes	characteristic
184	C₄	C₇×#498	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	C₇×#2058	1	1	yes	characteristic
188	C₂	C₇×#2230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#498

C₂

yes

maximal, characteristic

C₂×C₇×#365

C₂

yes

maximal

C₇×#1945

C₂

yes

maximal, characteristic

C₇×#1386

C₂

yes

maximal

C₇×#1376

C₂

yes

maximal, characteristic

C₂×C₇×C₄⋊Q₈

—

C₂×C₇×M₃₂

C₂²

yes

—

C₇×#498

—

C₂×C₇×#108

C₂²

yes

characteristic

C₇×#427

C₂²

yes

characteristic

C₂×C₇×#106

—

C₂×C₇×#106

C₂²

yes

—

C₇×#424

—

C₂×C₄×C₇×C₈

C₂²

yes

—

—

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×Q₈

—

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

D₄

yes

characteristic

C₂²×C₇×C₈

C₂³

yes

characteristic

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×#117

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₇×M₃₂

—

C₇×M₃₂

Q₈

yes

complemented

C₂×C₇×C₁₆

—

C₇×#108

—

C₇×#107

Q₈

yes

—

—

—

D₄

yes

—

C₇×C₄⋊C₈

—

C₄×C₇×C₈

—

C₂×#498

C₂×C₇

yes

characteristic

C₂×#365

C₂×C₇

yes

—

#1945

C₂×C₇

yes

characteristic

#1386

C₂×C₇

yes

—

#1376

C₂×C₇

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

SD₁₆

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₂×C₇×C₈

C₂×Q₈

yes

—

C₂×C₇×C₈

C₂×Q₈

yes

characteristic

C₂×C₇×C₈

#41

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

SD₁₆

yes

—

C₇×C₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂²×C₇

yes

—

#498

—

C₂×#108

C₂²×C₇

yes

characteristic

#427

C₂²×C₇

yes

characteristic

C₂×#106

—

C₂×#106

C₂²×C₇

yes

—

#424

—

C₂×C₄×C₈

C₂²×C₇

yes

characteristic

#365

—

#339

—

C₂³×C₇

AGL(1,ℤ/8)

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#120

yes

characteristic

C₂×C₄×C₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

#137

yes

characteristic

C₇×C₈

—

C₇×C₈

—

C₇×C₈

C₂²⋊Q₈

yes

—

C₇×C₈

C₄⋊Q₈

yes

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₇×D₄

yes

characteristic

C₂²×C₈

C₂³×C₇

yes

characteristic

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

#117

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₇×D₄

yes

characteristic

106

M₃₂

—

107

M₃₂

C₇×Q₈

yes

complemented

108

C₂×C₁₆

—

109

#108

—

110

#107

C₇×Q₈

yes

complemented

111

#106

—

112

#106

—

113

#106

—

114

#106

C₇×D₄

yes

—

115

C₄⋊C₈

—

116

C₄×C₈

—

117

C₂²×C₇

—

118

C₂²×C₇

#450

yes

characteristic

119

C₂²×C₇

#476

yes

characteristic

120

C₂²×C₇

#501

yes

characteristic

121

C₄×C₇

—

122

C₄×C₇

—

123

C₄×C₇

—

124

C₄×C₇

#395

yes

characteristic

125

C₄×C₇

#448

yes

characteristic

126

C₄×C₇

#475

yes

characteristic

127

C₄×C₇

#498

yes

characteristic

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

C₇×SD₁₆

yes

—

139

C₂×C₈

C₂×C₇×D₄

yes

—

140

C₂×C₈

C₂×C₇×Q₈

yes

—

141

C₂×C₈

C₂×C₇×Q₈

yes

characteristic

142

C₂×C₈

C₇×#41

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

C₇×SD₁₆

yes

—

151

C₁₆

—

152

C₂×C₇

—

153

C₂×C₇

—

154

C₂×C₇

#2058

yes

characteristic

155

C₂×C₇

#2230

yes

—

156

C₂³

C₇×AGL(1,ℤ/8)

yes

characteristic

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₇×#120

yes

characteristic

164

C₂×C₄

C₇×C₂²⋊Q₈

yes

characteristic

165

C₂×C₄

C₇×C₄⋊Q₈

yes

characteristic

166

C₂×C₄

C₂×C₇×SD₁₆

yes

characteristic

167

C₂×C₄

C₂×C₇×SD₁₆

yes

characteristic

168

C₂×C₄

C₇×#137

yes

characteristic

169

C₈

—

170

C₈

—

171

C₈

C₇×C₂²⋊Q₈

yes

—

172

C₈

C₇×C₄⋊Q₈

yes

—

173

C₇

#13457

yes

characteristic, direct product factor, Sylow

174

C₂²

—

175

C₂²

C₇×#450

yes

characteristic

176

C₂²

C₇×#476

yes

characteristic

177

C₂²

C₇×#501

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

C₇×#395

yes

characteristic

182

C₄

C₇×#448

yes

characteristic

183

C₄

C₇×#475

yes

characteristic

184

C₄

C₇×#498

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

C₇×#2058

yes

characteristic

188

C₂

C₇×#2230

yes

—

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

subgroup

G/H

center, socle

C₂²×C₇

#476

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₇

prime radical

C₂³×C₇

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

N	H ≅ G/N	type
17. #13457	C₇	direct (N × H)
29. C₇×M₃₂	Q₈	semidirect
32. C₇×#107	Q₈	semidirect
107. M₃₂	C₇×Q₈	semidirect
110. #107	C₇×Q₈	semidirect
173. C₇	#13457	direct (N × H)

H ≅ G/N

type

17. #13457

C₇

direct (N × H)

29. C₇×M₃₂

Q₈

semidirect

32. C₇×#107

Q₈

semidirect

107. M₃₂

C₇×Q₈

semidirect

110. #107

C₇×Q₈

semidirect

173. C₇

#13457

direct (N × H)

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

subgroup	index
C₂×C₇×#498	2
C₂×C₇×#365	2
C₇×#1945	2
C₇×#1386	2
C₇×#1376	2
#13457	7

subgroup

index

C₂×C₇×#498

C₂×C₇×#365

Subgroups — C₇×#13457 #13190673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#13457 #13190673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13457 #13190673