Group #13253918 — subgroups

Subgroups — #13253918

Order 1792.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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×#2859	C₂	1	1	yes	maximal, characteristic
2	#11727146	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11725440	C₂	1	4	yes	maximal
5	C₂×C₇×#125	—	2	1	no	—
6	C₇×#389	—	2	2	no	—
7	C₇×#389	C₄	1	4	yes	—
8	C₇×#384	C₂²	1	2	yes	—
9	C₇×#382	—	2	1	no	—
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄×C₇⋊C₈	—	2	1	no	—
14	#89319	—	2	1	no	—
15	#89319	—	2	2	no	—
16	#9897	—	7	1	no	maximal, Sylow
17	C₇×#125	—	4	4	no	—
18	C₂×C₇×C₄⋊C₄	—	2	4	no	—
19	C₂×C₇×C₄⋊C₄	—	2	2	no	—
20	C₂×C₇×C₄⋊C₄	—	2	2	no	—
21	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇	D₄	1	2	yes	—
24	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	—	2	2	no	—
28	C₂²×C₇⋊C₈	—	2	2	no	—
29	C₇×#95	—	2	4	no	—
30	C₄⋊(C₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇⋊C₈)	—	2	4	no	—
32	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
33	C₄×C₇⋊C₈	—	4	2	no	—
34	#2859	D₇	1	1	yes	characteristic
35	#1831	—	7	1	no	—
36	C₄×C₄⋊C₈	—	7	1	no	—
37	#1296	—	7	4	no	—
38	C₂²×C₄×C₇	—	2	4	no	—
39	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₇×C₄⋊C₄	—	4	4	no	—
45	C₇×C₄⋊C₄	—	4	8	no	—
46	C₇×C₄⋊C₄	—	4	4	no	—
47	C₇×C₄⋊C₄	—	4	4	no	—
48	C₄²×C₇	—	2	4	no	—
49	C₄²×C₇	—	2	2	no	—
50	C₄²×C₇	—	4	2	no	—
51	C₄²×C₇	SD₁₆	1	2	yes	—
52	C₄²×C₇	Q₁₆	1	2	yes	—
53	C₄²×C₇	C₂×Q₈	1	2	yes	—
54	C₄²×C₇	#41	1	2	yes	—
55	C₂×C₇⋊C₈	—	2	4	no	—
56	C₂×C₇⋊C₈	—	4	2	no	—
57	C₂×C₇⋊C₈	—	4	2	no	—
58	C₂×C₇⋊C₈	—	4	2	no	—
59	C₂×C₇⋊C₈	—	4	4	no	—
60	C₂×#125	—	2	1	no	—
61	C₂×C₄⋊C₈	—	7	2	no	—
62	C₂×C₄⋊C₈	—	7	2	no	—
63	C₂×C₄×C₈	—	14	1	no	—
64	#389	—	2	2	no	—
65	#389	Dic₇	1	4	yes	—
66	#384	C₂×D₇	1	2	yes	—
67	#382	—	2	1	no	—
68	C₄³	C₂×D₇	1	1	yes	characteristic
69	#336	—	14	2	no	—
70	#336	—	14	1	no	—
71	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	4	4	no	—
79	C₂×C₄×C₇	—	4	4	no	—
80	C₂×C₄×C₇	—	4	4	no	—
81	C₂×C₄×C₇	#103	1	4	yes	—
82	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
83	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₇	#124	1	1	yes	characteristic
85	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇	#135	1	4	yes	—
89	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
90	C₇⋊C₈	—	4	4	no	—
91	C₇⋊C₈	—	8	2	no	—
92	C₂²×C₈	—	14	2	no	—
93	C₂²×C₈	—	14	2	no	—
94	#125	—	4	4	no	—
95	C₂×C₄⋊C₄	—	2	4	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
99	C₂×C₄²	C₂×Dic₇	1	2	yes	—
100	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
102	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
104	C₄⋊C₈	—	7	4	no	—
105	C₄⋊C₈	—	14	2	no	—
106	C₄⋊C₈	—	14	4	no	—
107	C₄×C₈	—	28	2	no	—
108	#95	—	2	4	no	—
109	C₂²×C₇	#391	1	1	yes	characteristic
110	C₂²×C₇	#413	1	2	yes	—
111	C₂²×C₇	C₂×#103	1	1	yes	characteristic
112	C₂²×C₇	#416	1	2	yes	—
113	C₂²×C₇	#418	1	1	yes	characteristic
114	C₄×C₇	—	2	4	no	—
115	C₄×C₇	—	2	2	no	—
116	C₄×C₇	—	4	2	no	—
117	C₄×C₇	—	8	4	no	—
118	C₄×C₇	#476	1	2	yes	—
119	C₄×C₇	#477	1	2	yes	—
120	C₄×C₇	#488	1	2	yes	—
121	C₄×C₇	#499	1	2	yes	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
124	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
125	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
128	C₂×C₈	—	14	4	no	—
129	C₂×C₈	—	28	2	no	—
130	C₂×C₈	—	28	2	no	—
131	C₂×C₈	—	28	2	no	—
132	C₂×C₈	—	28	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	8	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	4	2	no	—
140	C₄²	#1143	1	2	yes	—
141	C₄²	#1144	1	2	yes	—
142	C₄²	Q₈×D₇	1	2	yes	—
143	C₄²	#1161	1	2	yes	—
144	C₂×C₇	#1542	1	2	yes	—
145	C₂×C₇	#1553	1	2	yes	—
146	C₂×C₇	#1987	1	1	yes	characteristic
147	C₂×C₇	#1991	1	1	yes	characteristic
148	C₂×C₇	#1998	1	1	yes	characteristic
149	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	#6409	1	4	yes	—
160	C₂×C₄	#6495	1	2	yes	—
161	C₂×C₄	C₂×#1143	1	1	yes	characteristic
162	C₂×C₄	C₂×#1144	1	1	yes	characteristic
163	C₂×C₄	#6507	1	1	yes	characteristic
164	C₂×C₄	Q₈×Dic₇	1	2	yes	—
165	C₂×C₄	#6509	1	2	yes	—
166	C₂×C₄	#6510	1	1	yes	characteristic
167	C₂×C₄	#6513	1	4	yes	—
168	C₈	—	28	4	no	—
169	C₈	—	56	2	no	—
170	C₇	#9897	1	1	yes	characteristic, complemented, Sylow
171	C₂²	#89987	1	1	yes	characteristic
172	C₂²	C₂×#6409	1	1	yes	characteristic
173	C₂²	#90002	1	1	yes	characteristic
174	C₂²	#90005	1	2	yes	—
175	C₂²	#90009	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	#89836	1	2	yes	—
181	C₄	#89840	1	2	yes	—
182	C₄	#89856	1	2	yes	—
183	C₄	#89863	1	2	yes	—
184	C₂	#11729494	1	1	yes	characteristic
185	C₂	#11729496	1	1	yes	characteristic
186	C₂	#11729533	1	1	yes	characteristic
187	C₂	#11729562	1	2	yes	—
188	C₂	#11729582	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2859

C₂

yes

maximal, characteristic

#11727146

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₇⋊C₈)

C₂

yes

maximal, characteristic

#11725440

C₂

yes

maximal

C₂×C₇×#125

—

C₇×#389

—

C₇×#389

C₄

yes

—

C₇×#384

C₂²

yes

—

C₇×#382

—

C₄³×C₇

C₂²

yes

characteristic

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₇⋊C₈

—

—

—

—

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂²×C₇⋊C₈

—

C₂²×C₇⋊C₈

—

C₇×#95

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

Q₈

yes

—

C₄×C₇⋊C₈

—

#2859

D₇

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

#1296

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

Q₁₆

yes

—

C₄²×C₇

C₂×Q₈

yes

—

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#389

—

#389

Dic₇

yes

—

#384

C₂×D₇

yes

—

#382

—

C₄³

C₂×D₇

yes

characteristic

#336

—

#336

—

C₂³×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#103

yes

—

C₂×C₄×C₇

C₄×Q₈

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇

#135

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₇

yes

—

C₂×C₄²

C₂×Dic₇

yes

—

100

C₂×C₄²

C₂²⋊D₇

yes

characteristic

101

C₂×C₄²

C₂²⋊D₇

yes

—

102

C₂×C₄²

C₂²⋊D₇

yes

characteristic

103

C₂×C₄²

C₂²×D₇

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄×C₈

—

108

#95

—

109

C₂²×C₇

#391

yes

characteristic

110

C₂²×C₇

#413

yes

—

111

C₂²×C₇

C₂×#103

yes

characteristic

112

C₂²×C₇

#416

yes

—

113

C₂²×C₇

#418

yes

characteristic

114

C₄×C₇

—

115

C₄×C₇

—

116

C₄×C₇

—

117

C₄×C₇

—

118

C₄×C₇

#476

yes

—

119

C₄×C₇

#477

yes

—

120

C₄×C₇

#488

yes

—

121

C₄×C₇

#499

yes

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊Dic₇

yes

—

124

C₂²×C₄

C₂²⋊Dic₇

yes

—

125

C₂²×C₄

C₂²×Dic₇

yes

characteristic

126

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

127

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

#1143

yes

—

141

C₄²

#1144

yes

—

142

C₄²

Q₈×D₇

yes

—

143

C₄²

#1161

yes

—

144

C₂×C₇

#1542

yes

—

145

C₂×C₇

#1553

yes

—

146

C₂×C₇

#1987

yes

characteristic

147

C₂×C₇

#1991

yes

characteristic

148

C₂×C₇

#1998

yes

characteristic

149

C₂³

C₂×C₂²⋊Dic₇

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

#6409

yes

—

160

C₂×C₄

#6495

yes

—

161

C₂×C₄

C₂×#1143

yes

characteristic

162

C₂×C₄

C₂×#1144

yes

characteristic

163

C₂×C₄

#6507

yes

characteristic

164

C₂×C₄

Q₈×Dic₇

yes

—

165

C₂×C₄

#6509

yes

—

166

C₂×C₄

#6510

yes

characteristic

167

C₂×C₄

#6513

yes

—

168

C₈

—

169

C₈

—

170

C₇

#9897

yes

characteristic, complemented, Sylow

171

C₂²

#89987

yes

characteristic

172

C₂²

C₂×#6409

yes

characteristic

173

C₂²

#90002

yes

characteristic

174

C₂²

#90005

yes

—

175

C₂²

#90009

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#89836

yes

—

181

C₄

#89840

yes

—

182

C₄

#89856

yes

—

183

C₄

#89863

yes

—

184

C₂

#11729494

yes

characteristic

185

C₂

#11729496

yes

characteristic

186

C₂

#11729533

yes

characteristic

187

C₂

#11729562

yes

—

188

C₂

#11729582

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2859	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2859	D₇
hypocenter	C₇	#9897
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₇

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

Fitting

C₇×#2859

C₂

socle, prime radical

C₂³×C₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
170. C₇	#9897	semidirect

H ≅ G/N

type

170. C₇

#9897

semidirect

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

subgroup	index
C₇×#2859	2
#11727146	2
C₄×C₄⋊(C₇⋊C₈)	2
#11725440	2
#9897	7

subgroup

index

C₄×C₄⋊(C₇⋊C₈)

Subgroups — #13253918

Conjugacy classes of subgroups

Complements

Maximal subgroups