Group #13258576 — subgroups

Subgroups — #13258576

Order 1792.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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	#11728526	C₂	1	2	yes	maximal
2	#11727194	C₂	1	1	yes	maximal, characteristic
3	#11727191	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1274	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725551	C₂	1	2	yes	maximal, complemented
6	#90168	C₂²	1	1	yes	characteristic
7	C₇×#431	C₂²	1	2	yes	—
8	C₇×#376	C₂²	1	1	yes	characteristic
9	C₂×#6379	—	2	2	no	—
10	#89696	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	#89319	—	2	2	no	—
13	#89281	—	2	2	no	—
14	#89280	—	2	2	no	—
15	#10629	—	7	1	no	maximal, Sylow
16	C₂×#1157	—	2	1	no	—
17	C₂×C₄²×C₇	D₄	1	2	yes	—
18	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
19	#6493	—	2	2	no	—
20	C₂×#1136	—	2	1	no	—
21	C₂²×C₇⋊C₈	—	2	2	no	—
22	#6438	—	2	1	no	—
23	#6437	D₄	1	1	yes	characteristic
24	C₄×D₂₈	—	2	2	no	—
25	#6432	D₄	1	1	yes	characteristic
26	C₇×C₄⋊C₈	—	2	2	no	—
27	C₇×C₂²⋊C₈	—	2	2	no	—
28	C₄×C₇×C₈	—	2	2	no	—
29	C₇×#95	—	2	2	no	—
30	#6379	—	4	4	no	—
31	C₄⋊(C₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	#1963	—	7	1	no	—
35	#1832	—	7	1	no	—
36	#1540	—	7	2	no	—
37	#1301	—	7	2	no	—
38	#1274	D₇	1	1	yes	characteristic, complemented
39	C₂²×C₄×C₇	—	2	2	no	—
40	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	#41	1	2	yes	—
42	C₂×C₂²⋊D₇	—	4	1	no	—
43	#1157	—	4	2	no	—
44	#1157	—	4	2	no	—
45	C₂×D₂₈	—	2	1	no	—
46	C₂×D₂₈	—	2	2	no	—
47	C₂×C₄×D₇	—	4	1	no	—
48	C₂×Dic₁₄	—	2	1	no	—
49	C₂×C₇×C₈	—	4	2	no	—
50	C₄²×C₇	—	2	2	no	—
51	C₄²×C₇	—	2	2	no	—
52	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₇	#41	1	2	yes	—
54	C₄²×C₇	#41	1	1	yes	characteristic
55	(C₂×D₇)⋊C₄	—	4	2	no	—
56	C₄⋊Dic₇	—	2	2	no	—
57	#1136	—	4	2	no	—
58	C₂×C₇⋊C₈	—	4	2	no	—
59	C₂×C₇⋊C₈	—	4	1	no	—
60	C₂×C₇⋊C₈	—	4	2	no	—
61	C₂×C₇⋊C₈	—	4	2	no	—
62	#532	—	7	1	no	—
63	#431	C₂×D₇	1	2	yes	—
64	#423	—	7	1	no	—
65	C₂×C₄⋊C₈	—	7	2	no	—
66	C₂×#104	—	14	2	no	—
67	#376	C₂×D₇	1	1	yes	characteristic
68	#336	—	14	2	no	—
69	#332	—	14	2	no	—
70	#331	—	14	2	no	—
71	C₂³×C₇	#123	1	1	yes	characteristic
72	C₂²×D₇	—	4	1	no	—
73	C₂×C₄×C₇	—	4	2	no	—
74	C₂×C₄×C₇	—	4	4	no	—
75	C₂×C₄×C₇	—	4	2	no	—
76	C₂×C₄×C₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₇	#123	1	2	yes	—
78	C₂×C₄×C₇	#124	1	1	yes	characteristic
79	C₂×C₄×C₇	#126	1	2	yes	—
80	C₂×C₄×C₇	#135	1	2	yes	—
81	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
82	C₂²⋊D₇	—	8	2	no	—
83	C₂×Dic₇	—	4	1	no	—
84	D₂₈	—	4	2	no	—
85	D₂₈	—	8	2	no	—
86	C₄×D₇	—	8	1	no	—
87	C₄×D₇	—	8	1	no	—
88	Dic₁₄	—	4	2	no	—
89	C₇×C₈	—	8	2	no	—
90	C₇⋊C₈	—	8	2	no	—
91	C₇⋊C₈	—	8	1	no	—
92	C₂×#41	—	14	1	no	—
93	C₂×M₁₆	—	14	1	no	—
94	C₂²×C₈	—	14	2	no	—
95	C₄⋊Q₈	—	7	1	no	—
96	#127	—	7	1	no	—
97	#124	—	14	1	no	—
98	#121	—	14	2	no	—
99	C₄×D₄	—	14	2	no	—
100	C₂×C₄²	D₂₈	1	2	yes	—
101	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	14	2	no	—
104	C₄⋊C₈	—	14	2	no	—
105	C₄⋊C₈	—	14	2	no	—
106	#104	—	28	4	no	—
107	C₂²⋊C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	#95	—	2	2	no	—
110	C₂²×C₇	—	2	2	no	—
111	C₂²×C₇	#397	1	1	yes	characteristic
112	C₂²×C₇	#482	1	2	yes	—
113	C₂×D₇	—	8	1	no	—
114	C₂×D₇	—	8	2	no	—
115	C₄×C₇	—	2	2	no	—
116	C₄×C₇	—	2	2	no	—
117	C₄×C₇	—	8	2	no	—
118	C₄×C₇	#463	1	2	yes	—
119	C₄×C₇	#488	1	1	yes	characteristic
120	C₄×C₇	#490	1	1	yes	characteristic
121	Dic₇	—	8	1	no	—
122	#41	—	28	2	no	—
123	#41	—	28	2	no	—
124	C₂×Q₈	—	14	1	no	—
125	C₂×D₄	—	14	2	no	—
126	C₂×D₄	—	14	1	no	—
127	C₂×D₄	—	28	1	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	28	1	no	—
130	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
131	C₂²×C₄	#1159	1	2	yes	—
132	M₁₆	—	28	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	28	1	no	—
135	C₂×C₈	—	28	2	no	—
136	C₂×C₈	—	28	2	no	—
137	C₂×C₈	—	28	2	no	—
138	C₄⋊C₄	—	14	2	no	—
139	C₂²⋊C₄	—	28	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	#1157	1	2	yes	—
143	C₄²	D₄×D₇	1	1	yes	characteristic
144	C₄²	#1161	1	1	yes	characteristic
145	C₂×C₇	—	2	2	no	—
146	C₂×C₇	#2073	1	1	yes	characteristic
147	C₂×C₇	#2080	1	2	yes	—
148	D₇	—	16	1	no	—
149	C₂³	—	28	1	no	—
150	C₂³	#6449	1	1	yes	characteristic
151	Q₈	—	28	2	no	—
152	D₄	—	28	2	no	—
153	D₄	—	56	2	no	—
154	D₄	—	56	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	28	1	no	—
159	C₂×C₄	—	56	1	no	—
160	C₂×C₄	—	56	1	no	—
161	C₂×C₄	#6438	1	1	yes	characteristic
162	C₂×C₄	#6446	1	2	yes	—
163	C₂×C₄	#6458	1	1	yes	characteristic
164	C₂×C₄	#6461	1	2	yes	—
165	C₂×C₄	#6467	1	2	yes	—
166	C₂×C₄	#6471	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	56	2	no	—
169	C₈	—	56	1	no	—
170	C₇	#10629	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	2	2	no	—
172	C₂²	—	56	1	no	—
173	C₂²	—	56	2	no	—
174	C₂²	#89446	1	1	yes	characteristic
175	C₂²	#89509	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	56	1	no	—
180	C₄	#89469	1	1	yes	characteristic
181	C₄	#89485	1	1	yes	characteristic
182	C₄	#89616	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	112	1	no	—
185	C₂	#11726391	1	1	yes	characteristic
186	C₂	#11726454	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11728526

C₂

yes

maximal

#11727194

C₂

yes

maximal, characteristic

#11727191

C₂

yes

maximal, characteristic, complemented

C₇×#1274

C₂

yes

maximal, characteristic, complemented

#11725551

C₂

yes

maximal, complemented

#90168

C₂²

yes

characteristic

C₇×#431

C₂²

yes

—

C₇×#376

C₂²

yes

characteristic

C₂×#6379

—

#89696

C₂²

yes

characteristic

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

—

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

#6493

—

C₂×#1136

—

C₂²×C₇⋊C₈

—

#6438

—

#6437

D₄

yes

characteristic

C₄×D₂₈

—

#6432

D₄

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

—

C₇×#95

—

#6379

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₂²⋊D₇

—

#1157

—

#1157

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×C₄×D₇

—

C₂×Dic₁₄

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

#41

yes

—

C₄²×C₇

#41

yes

characteristic

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

#532

—

#431

C₂×D₇

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×#104

—

#376

C₂×D₇

yes

—

—

—

C₂³×C₇

#123

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

#126

yes

—

C₂×C₄×C₇

#135

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₇

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

C₄×D₇

—

C₄×D₇

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

100

C₂×C₄²

D₂₈

yes

—

101

C₂×C₄²

C₂²×D₇

yes

characteristic

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#104

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

#95

—

110

C₂²×C₇

—

111

C₂²×C₇

#397

yes

characteristic

112

C₂²×C₇

#482

yes

—

113

C₂×D₇

—

114

C₂×D₇

—

115

C₄×C₇

—

116

C₄×C₇

—

117

C₄×C₇

—

118

C₄×C₇

#463

yes

—

119

C₄×C₇

#488

yes

characteristic

120

C₄×C₇

#490

yes

characteristic

121

Dic₇

—

122

#41

—

123

#41

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×D₂₈

yes

characteristic

131

C₂²×C₄

#1159

yes

—

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

#1157

yes

—

143

C₄²

D₄×D₇

yes

characteristic

144

C₄²

#1161

yes

characteristic

145

C₂×C₇

—

146

C₂×C₇

#2073

yes

characteristic

147

C₂×C₇

#2080

yes

—

148

D₇

—

149

C₂³

—

150

C₂³

#6449

yes

characteristic

151

Q₈

—

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#6438

yes

characteristic

162

C₂×C₄

#6446

yes

—

163

C₂×C₄

#6458

yes

characteristic

164

C₂×C₄

#6461

yes

—

165

C₂×C₄

#6467

yes

—

166

C₂×C₄

#6471

yes

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₇

#10629

yes

characteristic, complemented, Sylow

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#89446

yes

characteristic

175

C₂²

#89509

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#89469

yes

characteristic

181

C₄

#89485

yes

characteristic

182

C₄

#89616

yes

—

183

C₂

—

184

C₂

—

112

—

185

C₂

#11726391

yes

characteristic

186

C₂

#11726454

yes

—

	subgroup	G/H
center	C₂×C₄	#6438
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1274	C₂
socle	C₂²×C₇	#397
prime radical	#90168	C₂²
hypercenter	#1274	D₇
hypocenter	C₇	#10629
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#6438

commutator, cosocle

C₂×C₄²×C₇

C₂³

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂²

D₇

C₇

G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
3. #11727191	C₂	semidirect
4. C₇×#1274	C₂	semidirect
5. #11725551	C₂	semidirect
38. #1274	D₇	semidirect
170. C₇	#10629	semidirect

H ≅ G/N

type

3. #11727191

C₂

semidirect

4. C₇×#1274

C₂

semidirect

5. #11725551

C₂

semidirect

38. #1274

D₇

semidirect

170. C₇

#10629

semidirect

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

subgroup	index
#11728526	2
#11727194	2
#11727191	2
C₇×#1274	2
#11725551	2
#10629	7

subgroup

index

Subgroups — #13258576

Conjugacy classes of subgroups

Complements

Maximal subgroups