Group #13253924 — subgroups

Subgroups — #13253924

Order 1792.

430 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 159 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	#11730752	C₂	1	1	yes	maximal, characteristic
2	C₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11725508	C₂	1	4	yes	maximal
5	C₂×#6433	—	2	1	no	—
6	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₇	C₂²	1	1	yes	characteristic
8	#89708	—	2	1	no	—
9	#89707	C₂²	1	2	yes	—
10	#89700	—	2	2	no	—
11	#89700	C₄	1	4	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄×C₇⋊C₈	—	2	1	no	—
14	C₇×#336	—	2	2	no	—
15	#89319	—	2	1	no	—
16	#9899	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	—	2	2	no	—
18	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇	D₄	1	2	yes	—
22	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇	—	2	2	no	—
24	C₂×C₄⋊Dic₇	—	2	2	no	—
25	C₂×C₄⋊Dic₇	C₂×C₄	1	4	yes	—
26	C₂×Dic₇⋊C₄	—	2	4	no	—
27	C₂²×C₇⋊C₈	—	2	2	no	—
28	#6433	—	4	2	no	—
29	#6433	—	4	2	no	—
30	C₇×C₄⋊C₈	—	2	4	no	—
31	C₇⋊#95	—	2	4	no	—
32	C₄⋊(C₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
34	C₄×C₇⋊C₈	—	4	2	no	—
35	#2859	—	7	1	no	—
36	#1831	D₇	1	1	yes	characteristic
37	C₄×C₄⋊C₈	—	7	1	no	—
38	#1296	—	7	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	C₂²×Dic₇	—	2	4	no	—
45	C₂×C₇×C₈	—	4	2	no	—
46	C₂×C₇×C₈	—	4	4	no	—
47	C₄²×C₇	—	2	4	no	—
48	C₄²×C₇	—	2	2	no	—
49	C₄²×C₇	—	4	2	no	—
50	C₄²×C₇	D₈	1	2	yes	—
51	C₄²×C₇	SD₁₆	1	2	yes	—
52	C₄²×C₇	C₂×Q₈	1	2	yes	—
53	C₄²×C₇	#41	1	2	yes	—
54	C₄⋊Dic₇	—	4	2	no	—
55	C₄⋊Dic₇	—	4	2	no	—
56	C₄⋊Dic₇	—	4	4	no	—
57	C₄⋊Dic₇	—	4	4	no	—
58	Dic₇⋊C₄	—	4	8	no	—
59	C₂×C₇⋊C₈	—	2	4	no	—
60	C₂×C₇⋊C₈	—	4	2	no	—
61	C₂×C₇⋊C₈	—	4	2	no	—
62	C₂×#125	—	14	1	no	—
63	C₂×C₄⋊C₈	—	7	2	no	—
64	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
65	C₂×C₄×C₈	—	14	1	no	—
66	#389	—	7	4	no	—
67	#389	—	14	2	no	—
68	#384	—	7	2	no	—
69	#382	—	14	1	no	—
70	C₄³	C₂×D₇	1	1	yes	characteristic
71	#336	—	2	2	no	—
72	#336	—	14	1	no	—
73	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	#102	1	4	yes	—
81	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
82	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₇	#124	1	1	yes	characteristic
84	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇	#135	1	4	yes	—
88	C₂×C₄×C₇	#137	1	2	yes	—
89	C₂×Dic₇	—	4	4	no	—
90	C₂×Dic₇	—	4	4	no	—
91	C₂×Dic₇	—	4	4	no	—
92	C₇×C₈	—	8	2	no	—
93	C₇⋊C₈	—	4	4	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	—	14	2	no	—
96	#125	—	28	2	no	—
97	#125	—	28	2	no	—
98	C₂×C₄⋊C₄	—	7	4	no	—
99	C₂×C₄⋊C₄	—	14	4	no	—
100	C₂×C₄⋊C₄	—	14	2	no	—
101	C₂×C₄⋊C₄	—	14	2	no	—
102	C₂×C₄²	C₄×D₇	1	2	yes	—
103	C₂×C₄²	D₂₈	1	1	yes	characteristic
104	C₂×C₄²	D₂₈	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
106	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
107	C₄⋊C₈	—	2	4	no	—
108	C₄⋊C₈	—	7	4	no	—
109	C₄⋊C₈	—	14	2	no	—
110	C₄×C₈	—	28	2	no	—
111	#95	—	14	4	no	—
112	C₂²×C₇	#391	1	1	yes	characteristic
113	C₂²×C₇	#413	1	2	yes	—
114	C₂²×C₇	C₂×#102	1	1	yes	characteristic
115	C₂²×C₇	#416	1	2	yes	—
116	C₂²×C₇	#419	1	1	yes	characteristic
117	C₄×C₇	—	2	4	no	—
118	C₄×C₇	—	2	2	no	—
119	C₄×C₇	—	4	2	no	—
120	C₄×C₇	#474	1	2	yes	—
121	C₄×C₇	#475	1	2	yes	—
122	C₄×C₇	#488	1	2	yes	—
123	C₄×C₇	#499	1	2	yes	—
124	Dic₇	—	8	4	no	—
125	C₂²×C₄	—	14	4	no	—
126	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
127	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
128	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	14	4	no	—
134	C₂×C₈	—	28	2	no	—
135	C₂×C₈	—	28	2	no	—
136	C₄⋊C₄	—	28	4	no	—
137	C₄⋊C₄	—	28	2	no	—
138	C₄⋊C₄	—	28	2	no	—
139	C₄⋊C₄	—	28	4	no	—
140	C₄⋊C₄	—	28	8	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	4	2	no	—
144	C₄²	#1132	1	2	yes	—
145	C₄²	D₅₆	1	2	yes	—
146	C₄²	#1159	1	2	yes	—
147	C₄²	Q₈×D₇	1	2	yes	—
148	C₂×C₇	#1540	1	2	yes	—
149	C₂×C₇	#1554	1	2	yes	—
150	C₂×C₇	#1988	1	1	yes	characteristic
151	C₂×C₇	#1991	1	1	yes	characteristic
152	C₂×C₇	#1998	1	1	yes	characteristic
153	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	28	4	no	—
161	C₂×C₄	—	28	4	no	—
162	C₂×C₄	—	28	4	no	—
163	C₂×C₄	#6395	1	4	yes	—
164	C₂×C₄	#6450	1	2	yes	—
165	C₂×C₄	#6460	1	2	yes	—
166	C₂×C₄	C₂×#1132	1	1	yes	characteristic
167	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
168	C₂×C₄	#6472	1	2	yes	—
169	C₂×C₄	#6499	1	1	yes	characteristic
170	C₂×C₄	#6507	1	1	yes	characteristic
171	C₂×C₄	#6513	1	4	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	28	4	no	—
174	C₇	#9899	1	1	yes	characteristic, complemented, Sylow
175	C₂²	#89746	1	1	yes	characteristic
176	C₂²	#89774	1	2	yes	—
177	C₂²	C₂×#6395	1	1	yes	characteristic
178	C₂²	#89892	1	1	yes	characteristic
179	C₂²	#89904	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	56	4	no	—
184	C₄	#89618	1	2	yes	—
185	C₄	#89623	1	2	yes	—
186	C₄	#89828	1	2	yes	—
187	C₄	#89836	1	2	yes	—
188	C₂	#11727308	1	1	yes	characteristic
189	C₂	#11728214	1	1	yes	characteristic
190	C₂	#11728279	1	1	yes	characteristic
191	C₂	#11728526	1	2	yes	—
192	C₂	#11728575	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11730752

C₂

yes

maximal, characteristic

C₇×#1831

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₇⋊C₈)

C₂

yes

maximal, characteristic

#11725508

C₂

yes

maximal

C₂×#6433

—

C₂×C₇×C₄⋊C₈

C₂²

yes

—

C₄³×C₇

C₂²

yes

characteristic

#89708

—

#89707

C₂²

yes

—

#89700

—

#89700

C₄

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₇⋊C₈

—

—

—

—

C₂²×C₇×C₈

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂×C₄

yes

—

C₂×Dic₇⋊C₄

—

C₂²×C₇⋊C₈

—

#6433

—

#6433

—

C₇×C₄⋊C₈

—

C₇⋊#95

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

Q₈

yes

—

C₄×C₇⋊C₈

—

#2859

—

#1831

D₇

yes

characteristic

C₄×C₄⋊C₈

—

#1296

—

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

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

D₈

yes

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

C₂×Q₈

yes

—

C₄²×C₇

#41

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₇

yes

—

C₂×C₄×C₈

—

—

—

—

—

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

#102

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×D₈

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

#135

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₄×D₇

yes

—

103

C₂×C₄²

D₂₈

yes

characteristic

104

C₂×C₄²

D₂₈

yes

characteristic

105

C₂×C₄²

C₂²⋊D₇

yes

—

106

C₂×C₄²

C₂²×D₇

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄×C₈

—

111

#95

—

112

C₂²×C₇

#391

yes

characteristic

113

C₂²×C₇

#413

yes

—

114

C₂²×C₇

C₂×#102

yes

characteristic

115

C₂²×C₇

#416

yes

—

116

C₂²×C₇

#419

yes

characteristic

117

C₄×C₇

—

118

C₄×C₇

—

119

C₄×C₇

—

120

C₄×C₇

#474

yes

—

121

C₄×C₇

#475

yes

—

122

C₄×C₇

#488

yes

—

123

C₄×C₇

#499

yes

—

124

Dic₇

—

125

C₂²×C₄

—

126

C₂²×C₄

(C₂×D₇)⋊C₄

yes

—

127

C₂²×C₄

(C₂×D₇)⋊C₄

yes

—

128

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

129

C₂²×C₄

C₂×D₂₈

yes

characteristic

130

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

#1132

yes

—

145

C₄²

D₅₆

yes

—

146

C₄²

#1159

yes

—

147

C₄²

Q₈×D₇

yes

—

148

C₂×C₇

#1540

yes

—

149

C₂×C₇

#1554

yes

—

150

C₂×C₇

#1988

yes

characteristic

151

C₂×C₇

#1991

yes

characteristic

152

C₂×C₇

#1998

yes

characteristic

153

C₂³

C₂×(C₂×D₇)⋊C₄

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

#6395

yes

—

164

C₂×C₄

#6450

yes

—

165

C₂×C₄

#6460

yes

—

166

C₂×C₄

C₂×#1132

yes

characteristic

167

C₂×C₄

C₂×D₅₆

yes

characteristic

168

C₂×C₄

#6472

yes

—

169

C₂×C₄

#6499

yes

characteristic

170

C₂×C₄

#6507

yes

characteristic

171

C₂×C₄

#6513

yes

—

172

C₈

—

173

C₈

—

174

C₇

#9899

yes

characteristic, complemented, Sylow

175

C₂²

#89746

yes

characteristic

176

C₂²

#89774

yes

—

177

C₂²

C₂×#6395

yes

characteristic

178

C₂²

#89892

yes

characteristic

179

C₂²

#89904

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#89618

yes

—

185

C₄

#89623

yes

—

186

C₄

#89828

yes

—

187

C₄

#89836

yes

—

188

C₂

#11727308

yes

characteristic

189

C₂

#11728214

yes

characteristic

190

C₂

#11728279

yes

characteristic

191

C₂

#11728526

yes

—

192

C₂

#11728575

yes

—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇)⋊C₄
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1831	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1831	D₇
hypocenter	C₇	#9899
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×(C₂×D₇)⋊C₄

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

Fitting

C₇×#1831

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 ◁ #1831

N	H ≅ G/N	type
174. C₇	#9899	semidirect

H ≅ G/N

type

174. C₇

#9899

semidirect

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

subgroup	index
#11730752	2
C₇×#1831	2
C₄×C₄⋊(C₇⋊C₈)	2
#11725508	2
#9899	7

subgroup

index

C₄×C₄⋊(C₇⋊C₈)

Subgroups — #13253924

Conjugacy classes of subgroups

Complements

Maximal subgroups