Group #13246816 — subgroups

Subgroups — #13246816

Order 1792.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	#11740773	C₂	1	1	yes	maximal, characteristic
2	C₇×#1561	C₂	1	1	yes	maximal, characteristic
3	#11729588	C₂	1	1	yes	maximal, characteristic
4	#11728244	C₂	1	2	yes	maximal
5	C₇⋊#1309	C₂	1	2	yes	maximal
6	C₇×#533	C₂²	1	1	yes	characteristic
7	C₇×#423	C₂²	1	2	yes	—
8	#89796	—	2	2	no	—
9	Q₈×C₇⋊C₈	—	2	2	no	—
10	#89771	C₄	1	2	yes	—
11	#89769	C₄	1	2	yes	—
12	#89696	C₂²	1	2	yes	—
13	#89696	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₇×#332	—	2	2	no	—
16	#89342	—	2	2	no	—
17	#89275	—	2	2	no	—
18	#89275	—	2	2	no	—
19	#8637	—	7	1	no	maximal, Sylow
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₇×C₄⋊Q₈	D₄	1	2	yes	—
22	C₇×#125	Q₈	1	2	yes	—
23	C₇×C₂²⋊Q₈	—	2	2	no	—
24	C₄×C₇×Q₈	—	2	2	no	—
25	C₇×#117	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
27	C₂×#1136	—	2	2	no	—
28	C₂×#1136	C₂×C₄	1	2	yes	—
29	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₇×C₄⋊C₈	—	2	2	no	—
32	C₇×C₄⋊C₈	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	#6377	—	2	2	no	—
38	C₄×C₇⋊C₈	—	2	2	no	—
39	C₄×C₇⋊C₈	D₄	1	2	yes	—
40	C₄×C₇⋊C₈	D₄	1	2	yes	—
41	#2960	—	7	1	no	—
42	#1566	—	7	2	no	—
43	#1561	—	7	1	no	—
44	#1561	D₇	1	1	yes	characteristic
45	#1309	—	7	2	no	—
46	C₂×C₇×Q₈	—	2	2	no	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	#41	1	2	yes	—
49	C₇×M₁₆	—	8	2	no	—
50	C₂×C₇×C₈	—	4	2	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	—	2	4	no	—
53	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
54	C₇×C₂²⋊C₄	—	2	2	no	—
55	C₄²×C₇	—	2	2	no	—
56	C₄²×C₇	C₂×D₄	1	2	yes	—
57	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
59	#1136	—	4	2	no	—
60	#1136	—	8	2	no	—
61	C₂×C₇⋊C₈	—	2	2	no	—
62	C₂×C₇⋊C₈	—	2	2	no	—
63	C₂×C₇⋊C₈	—	4	1	no	—
64	C₂×C₇⋊C₈	—	4	2	no	—
65	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₇⋊C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
68	#533	C₂×D₇	1	1	yes	characteristic
69	#446	—	14	2	no	—
70	C₈×Q₈	—	14	2	no	—
71	#424	—	7	2	no	—
72	#423	—	7	1	no	—
73	#423	—	7	2	no	—
74	#423	C₂×D₇	1	2	yes	—
75	#405	—	7	2	no	—
76	C₂×C₄×C₈	—	7	1	no	—
77	#334	—	14	2	no	—
78	#334	—	14	2	no	—
79	#332	—	2	2	no	—
80	#332	—	14	2	no	—
81	C₂³×C₇	#117	1	1	yes	characteristic
82	C₇×Q₈	—	4	2	no	—
83	C₇×Q₈	—	4	2	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	4	no	—
89	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	#121	1	2	yes	—
92	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
93	C₇×C₈	—	8	2	no	—
94	C₇⋊C₈	—	2	1	no	—
95	C₇⋊C₈	—	2	2	no	—
96	C₇⋊C₈	—	4	2	no	—
97	C₇⋊C₈	—	8	2	no	—
98	C₇⋊C₈	#101	1	2	yes	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	7	2	no	—
101	C₂×M₁₆	—	14	2	no	—
102	C₂²×C₈	—	7	1	no	—
103	C₂²×C₈	—	7	1	no	—
104	C₄⋊Q₈	C₂²⋊D₇	1	2	yes	—
105	#125	Dic₁₄	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	#117	C₄×D₇	1	2	yes	—
109	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	14	4	no	—
113	C₄⋊C₈	—	14	2	no	—
114	C₄⋊C₈	—	14	2	no	—
115	C₄⋊C₈	—	14	2	no	—
116	#97	—	14	2	no	—
117	C₄×C₈	—	7	2	no	—
118	C₄×C₈	—	7	2	no	—
119	C₄×C₈	—	14	2	no	—
120	C₂²×C₇	—	2	1	no	—
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	#380	1	1	yes	characteristic
123	C₄×C₇	—	2	1	no	—
124	C₄×C₇	—	2	2	no	—
125	C₄×C₇	—	4	4	no	—
126	C₄×C₇	C₂×#101	1	1	yes	characteristic
127	C₄×C₇	#413	1	1	yes	characteristic
128	C₄×C₇	#471	1	2	yes	—
129	C₄×C₇	#471	1	2	yes	—
130	C₂×Q₈	—	2	2	no	—
131	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
132	C₂²×C₄	#1159	1	2	yes	—
133	M₁₆	—	8	2	no	—
134	M₁₆	—	28	2	no	—
135	M₁₆	—	56	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	7	1	no	—
138	C₂×C₈	—	7	1	no	—
139	C₂×C₈	—	7	2	no	—
140	C₂×C₈	—	14	2	no	—
141	C₂×C₈	—	14	2	no	—
142	C₂×C₈	—	28	1	no	—
143	C₂×C₈	—	28	2	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	Dic₇⋊C₄	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
150	C₄²	D₄×D₇	1	2	yes	—
151	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
152	C₂×C₇	—	4	1	no	—
153	C₂×C₇	#1801	1	1	yes	characteristic
154	C₂×C₇	#1815	1	2	yes	—
155	C₂³	#6440	1	1	yes	characteristic
156	Q₈	—	4	2	no	—
157	Q₈	—	4	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	#6441	1	2	yes	—
164	C₂×C₄	D₇×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
166	C₂×C₄	#6502	1	2	yes	—
167	C₈	—	7	2	no	—
168	C₈	—	8	2	no	—
169	C₈	—	14	1	no	—
170	C₈	—	14	2	no	—
171	C₈	—	28	2	no	—
172	C₈	—	56	2	no	—
173	C₇	#8637	1	1	yes	characteristic, complemented, Sylow
174	C₂²	—	2	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#89714	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	1	no	—
179	C₄	—	4	4	no	—
180	C₄	D₇×#101	1	1	yes	characteristic
181	C₄	#89523	1	1	yes	characteristic
182	C₄	#89972	1	2	yes	—
183	C₄	#89975	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#11727851	1	1	yes	characteristic
186	C₂	#11728053	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11740773

C₂

yes

maximal, characteristic

C₇×#1561

C₂

yes

maximal, characteristic

#11729588

C₂

yes

maximal, characteristic

#11728244

C₂

yes

maximal

C₇⋊#1309

C₂

yes

maximal

C₇×#533

C₂²

yes

characteristic

C₇×#423

C₂²

yes

—

#89796

—

Q₈×C₇⋊C₈

—

#89771

C₄

yes

—

#89769

C₄

yes

—

#89696

C₂²

yes

—

#89696

C₂²

yes

characteristic

C₂×C₄×C₇⋊C₈

C₂²

yes

—

—

—

—

—

C₂×C₇×M₁₆

—

C₇×C₄⋊Q₈

D₄

yes

—

C₇×#125

Q₈

yes

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₇×#117

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×#1136

—

C₂×#1136

C₂×C₄

yes

—

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇⋊C₈

C₂×C₄

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

C₄×C₇⋊C₈

—

C₄×C₇⋊C₈

D₄

yes

—

C₄×C₇⋊C₈

D₄

yes

—

—

—

—

D₇

yes

characteristic

#1309

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₄⋊C₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

#1136

—

#1136

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇⋊C₈

C₂²⋊C₄

yes

characteristic

#533

C₂×D₇

yes

characteristic

#446

—

C₈×Q₈

—

—

—

—

C₂×D₇

yes

—

#405

—

C₂×C₄×C₈

—

—

—

—

—

C₂³×C₇

#117

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₇⋊C₈

#101

yes

—

C₂×M₁₆

—

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₄⋊Q₈

C₂²⋊D₇

yes

—

105

#125

Dic₁₄

yes

—

106

C₂²⋊Q₈

—

107

C₄×Q₈

—

108

#117

C₄×D₇

yes

—

109

C₂×C₄²

C₂²×D₇

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂²×C₇

—

121

C₂²×C₇

—

122

C₂²×C₇

#380

yes

characteristic

123

C₄×C₇

—

124

C₄×C₇

—

125

C₄×C₇

—

126

C₄×C₇

C₂×#101

yes

characteristic

127

C₄×C₇

#413

yes

characteristic

128

C₄×C₇

#471

yes

—

129

C₄×C₇

#471

yes

—

130

C₂×Q₈

—

131

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

132

C₂²×C₄

#1159

yes

—

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

Dic₇⋊C₄

yes

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

C₂×Dic₁₄

yes

characteristic

150

C₄²

D₄×D₇

yes

—

151

C₄²

C₂×C₂²⋊D₇

yes

characteristic

152

C₂×C₇

—

153

C₂×C₇

#1801

yes

characteristic

154

C₂×C₇

#1815

yes

—

155

C₂³

#6440

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

#6441

yes

—

164

C₂×C₄

D₇×C₂²⋊C₄

yes

characteristic

165

C₂×C₄

C₂×Dic₇⋊C₄

yes

characteristic

166

C₂×C₄

#6502

yes

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₇

#8637

yes

characteristic, complemented, Sylow

174

C₂²

—

175

C₂²

—

176

C₂²

#89714

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

D₇×#101

yes

characteristic

181

C₄

#89523

yes

characteristic

182

C₄

#89972

yes

—

183

C₄

#89975

yes

—

184

C₂

—

185

C₂

#11727851

yes

characteristic

186

C₂

#11728053

yes

—

	subgroup	G/H
center	C₂²	#89714
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1561	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1561	D₇
hypocenter	C₇	#8637
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

subgroup

G/H

center

C₂²

#89714

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

N	H ≅ G/N	type
173. C₇	#8637	semidirect

H ≅ G/N

type

173. C₇

#8637

semidirect

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

subgroup	index
#11740773	2
C₇×#1561	2
#11729588	2
#11728244	2
C₇⋊#1309	2
#8637	7

subgroup

index

Subgroups — #13246816

Conjugacy classes of subgroups

Complements

Maximal subgroups