Group #13273688 — subgroups

Subgroups — #13273688

Order 1792.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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	#11729185	C₂	1	1	yes	maximal, characteristic
2	#11729183	C₂	1	1	yes	maximal, characteristic, complemented
3	#11728465	C₂	1	2	yes	maximal
4	C₇×#1385	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725540	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₇	C₂²	1	1	yes	characteristic
7	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₇×#108	C₂²	1	2	yes	—
9	#89898	—	2	2	no	—
10	C₂×#6393	C₂²	1	2	yes	—
11	C₂×#6386	C₂²	1	1	yes	characteristic
12	#89353	—	2	2	no	—
13	(C₂×D₇)⋊C₁₆	—	2	2	no	—
14	#89301	—	2	2	no	—
15	#13498	—	7	1	no	maximal, Sylow
16	C₂²×C₄×D₇	D₄	1	1	yes	characteristic
17	C₂×C₇×M₁₆	—	2	2	no	—
18	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
19	C₂×#1136	—	2	2	no	—
20	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
21	C₂×C₈×D₇	—	2	1	no	—
22	C₂×C₈×D₇	—	2	2	no	—
23	C₂×C₈×D₇	—	2	2	no	—
24	C₂×C₈×D₇	—	2	1	no	—
25	C₇×M₃₂	—	2	2	no	—
26	C₂×C₇×C₁₆	D₄	1	2	yes	—
27	C₇×#108	—	2	2	no	—
28	C₇×#108	—	2	2	no	—
29	(C₂×D₇)⋊C₈	—	4	4	no	—
30	#6393	—	2	2	no	—
31	#6393	—	2	2	no	—
32	#6386	—	2	2	no	—
33	C₂×C₇⋊C₁₆	—	2	1	no	—
34	#2161	—	7	1	no	—
35	#2112	—	7	1	no	—
36	#1810	—	7	2	no	—
37	#1388	—	7	2	no	—
38	#1385	D₇	1	1	yes	characteristic, complemented
39	C₂³×D₇	Q₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₇	Q₁₆	1	1	yes	characteristic
42	C₂×C₄×D₇	—	2	1	no	—
43	C₂×C₄×D₇	—	2	2	no	—
44	C₂×C₄×D₇	—	2	1	no	—
45	C₂×C₄×D₇	—	2	2	no	—
46	C₇×M₁₆	—	4	2	no	—
47	C₇×M₁₆	—	4	2	no	—
48	C₂×C₇×C₈	—	4	2	no	—
49	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₇×C₈	#41	1	2	yes	—
51	C₂×C₇×C₈	#41	1	2	yes	—
52	C₂×C₇×C₈	#41	1	1	yes	characteristic
53	#1136	—	4	2	no	—
54	#1136	—	4	2	no	—
55	C₂×C₇⋊C₈	—	2	1	no	—
56	C₂×C₇⋊C₈	—	2	2	no	—
57	C₂×C₇⋊C₈	—	4	2	no	—
58	C₈×D₇	—	4	1	no	—
59	C₈×D₇	—	4	1	no	—
60	C₈×D₇	—	4	2	no	—
61	C₇×C₁₆	—	2	2	no	—
62	C₇⋊C₁₆	—	4	1	no	—
63	C₂³×C₈	—	7	1	no	—
64	C₂×M₃₂	—	7	1	no	—
65	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
66	C₂×#108	—	7	2	no	—
67	C₂×#108	C₂×D₇	1	2	yes	—
68	#407	—	14	2	no	—
69	#368	—	14	2	no	—
70	C₂²⋊C₁₆	—	14	2	no	—
71	#341	—	14	2	no	—
72	C₂³×C₇	C₂×Q₁₆	1	1	yes	characteristic
73	C₂²×D₇	—	2	1	no	—
74	C₂²×D₇	—	2	2	no	—
75	C₂²×D₇	—	4	2	no	—
76	C₂×C₄×C₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₇	#123	1	2	yes	—
78	C₂×C₄×C₇	#135	1	1	yes	characteristic
79	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×Dic₇	—	2	1	no	—
81	C₂×Dic₇	—	2	2	no	—
82	C₄×D₇	—	4	1	no	—
83	C₄×D₇	—	4	1	no	—
84	C₄×D₇	—	4	2	no	—
85	C₇×C₈	—	8	2	no	—
86	C₇×C₈	#123	1	1	yes	characteristic
87	C₇×C₈	#124	1	1	yes	characteristic
88	C₇×C₈	#126	1	2	yes	—
89	C₇⋊C₈	—	4	1	no	—
90	C₇⋊C₈	—	8	2	no	—
91	C₂³×C₄	—	7	1	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂×M₁₆	—	14	2	no	—
94	C₂²×C₈	—	7	1	no	—
95	C₂²×C₈	—	14	1	no	—
96	C₂²×C₈	—	14	2	no	—
97	C₂²×C₈	—	14	2	no	—
98	C₂²×C₈	—	14	1	no	—
99	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	14	2	no	—
102	C₂×C₁₆	—	14	1	no	—
103	C₂×C₁₆	D₂₈	1	2	yes	—
104	#108	—	2	2	no	—
105	#108	—	2	2	no	—
106	#108	—	14	2	no	—
107	#108	—	14	2	no	—
108	C₂²⋊C₈	—	28	4	no	—
109	C₂²×C₇	—	4	1	no	—
110	C₂²×C₇	#467	1	1	yes	characteristic
111	C₂²×C₇	#484	1	2	yes	—
112	C₂×D₇	—	2	2	no	—
113	C₂×D₇	—	2	4	no	—
114	C₂×D₇	—	4	1	no	—
115	C₂×D₇	—	4	2	no	—
116	C₂×D₇	—	4	1	no	—
117	C₄×C₇	#397	1	1	yes	characteristic
118	C₄×C₇	#469	1	1	yes	characteristic
119	C₄×C₇	#482	1	2	yes	—
120	Dic₇	—	4	1	no	—
121	C₂⁴	—	7	1	no	—
122	C₂²×C₄	—	7	1	no	—
123	C₂²×C₄	—	14	2	no	—
124	C₂²×C₄	—	14	1	no	—
125	C₂²×C₄	—	14	1	no	—
126	C₂²×C₄	—	14	2	no	—
127	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
128	M₁₆	—	4	2	no	—
129	M₁₆	—	4	2	no	—
130	M₁₆	—	28	2	no	—
131	M₁₆	—	28	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	14	2	no	—
134	C₂×C₈	—	14	1	no	—
135	C₂×C₈	—	28	1	no	—
136	C₂×C₈	—	28	1	no	—
137	C₂×C₈	—	28	2	no	—
138	C₂×C₈	—	28	2	no	—
139	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
140	C₂×C₈	#1157	1	2	yes	—
141	C₂×C₈	#1159	1	2	yes	—
142	C₂×C₈	#1161	1	1	yes	characteristic
143	C₁₆	—	2	2	no	—
144	C₁₆	—	28	1	no	—
145	C₂×C₇	—	2	1	no	—
146	C₂×C₇	—	2	2	no	—
147	C₂×C₇	#2076	1	1	yes	characteristic
148	D₇	—	4	2	no	—
149	C₂³	—	14	1	no	—
150	C₂³	—	14	2	no	—
151	C₂³	—	28	2	no	—
152	C₂³	D₇×Q₁₆	1	1	yes	characteristic
153	C₂×C₄	—	14	1	no	—
154	C₂×C₄	—	14	2	no	—
155	C₂×C₄	—	28	1	no	—
156	C₂×C₄	—	28	1	no	—
157	C₂×C₄	—	28	2	no	—
158	C₂×C₄	#6446	1	2	yes	—
159	C₂×C₄	#6458	1	1	yes	characteristic
160	C₂×C₄	#6475	1	1	yes	characteristic
161	C₂×C₄	#6477	1	2	yes	—
162	C₈	—	8	2	no	—
163	C₈	—	28	1	no	—
164	C₈	—	56	2	no	—
165	C₈	#6438	1	1	yes	characteristic
166	C₈	#6449	1	1	yes	characteristic
167	C₈	#6461	1	2	yes	—
168	C₇	#13498	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	4	1	no	—
170	C₂²	—	14	2	no	—
171	C₂²	—	14	4	no	—
172	C₂²	—	28	1	no	—
173	C₂²	—	28	2	no	—
174	C₂²	—	28	1	no	—
175	C₂²	#89581	1	2	yes	—
176	C₂²	#89659	1	1	yes	characteristic
177	C₄	—	28	1	no	—
178	C₄	#89446	1	1	yes	characteristic
179	C₄	#89556	1	2	yes	—
180	C₄	#89637	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	2	no	—
183	C₂	—	28	2	no	—
184	C₂	#11726585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11729185

C₂

yes

maximal, characteristic

#11729183

C₂

yes

maximal, characteristic, complemented

#11728465

C₂

yes

maximal

C₇×#1385

C₂

yes

maximal, characteristic, complemented

#11725540

C₂

yes

maximal, complemented

C₂²×C₈×D₇

C₂²

yes

characteristic

C₂×C₇×M₃₂

C₂²

yes

characteristic

C₂×C₇×#108

C₂²

yes

—

#89898

—

C₂×#6393

C₂²

yes

—

C₂×#6386

C₂²

yes

characteristic

#89353

—

(C₂×D₇)⋊C₁₆

—

#89301

—

#13498

—

maximal, Sylow

C₂²×C₄×D₇

D₄

yes

characteristic

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

C₂³

yes

characteristic

C₂×#1136

—

C₂²×C₇⋊C₈

D₄

yes

characteristic

C₂×C₈×D₇

—

C₂×C₈×D₇

—

C₂×C₈×D₇

—

C₂×C₈×D₇

—

C₇×M₃₂

—

C₂×C₇×C₁₆

D₄

yes

—

C₇×#108

—

C₇×#108

—

(C₂×D₇)⋊C₈

—

#6393

—

#6393

—

#6386

—

C₂×C₇⋊C₁₆

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂³×D₇

Q₁₆

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×Dic₇

Q₁₆

yes

characteristic

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₂×C₄×D₇

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

#41

yes

—

C₂×C₇×C₈

#41

yes

—

C₂×C₇×C₈

yes

—

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₈×D₇

—

C₈×D₇

—

C₈×D₇

—

C₇×C₁₆

—

C₇⋊C₁₆

—

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₇

yes

—

#407

—

#368

—

C₂²⋊C₁₆

—

#341

—

C₂³×C₇

C₂×Q₁₆

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂×C₄×C₇

#121

yes

characteristic

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

#135

yes

characteristic

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₇×C₈

—

C₇×C₈

yes

C₇×C₈

yes

C₇×C₈

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

D₂₈

yes

—

104

#108

—

105

#108

—

106

#108

—

107

#108

—

108

C₂²⋊C₈

—

109

C₂²×C₇

—

110

C₂²×C₇

#467

yes

characteristic

111

C₂²×C₇

#484

yes

—

112

C₂×D₇

—

113

C₂×D₇

—

114

C₂×D₇

—

115

C₂×D₇

—

116

C₂×D₇

—

117

C₄×C₇

#397

yes

characteristic

118

C₄×C₇

#469

yes

characteristic

119

C₄×C₇

#482

yes

—

120

Dic₇

—

121

C₂⁴

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

D₄×D₇

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₂×D₂₈

yes

characteristic

140

C₂×C₈

#1157

yes

—

141

C₂×C₈

#1159

yes

—

142

C₂×C₈

#1161

yes

characteristic

143

C₁₆

—

144

C₁₆

—

145

C₂×C₇

—

146

C₂×C₇

—

147

C₂×C₇

#2076

yes

characteristic

148

D₇

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

D₇×Q₁₆

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#6446

yes

—

159

C₂×C₄

#6458

yes

characteristic

160

C₂×C₄

#6475

yes

characteristic

161

C₂×C₄

#6477

yes

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

#6438

yes

characteristic

166

C₈

#6449

yes

characteristic

167

C₈

#6461

yes

—

168

C₇

#13498

yes

characteristic, complemented, Sylow

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#89581

yes

—

176

C₂²

#89659

yes

characteristic

177

C₄

—

178

C₄

#89446

yes

characteristic

179

C₄

#89556

yes

—

180

C₄

#89637

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#11726585

yes

characteristic

	subgroup	G/H
center	C₄	#89637
commutator, cosocle	C₂²×C₇×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1385	C₂
socle	C₂×C₇	#2076
prime radical	C₂³×D₇	Q₁₆
hypercenter	#1385	D₇
hypocenter	C₇	#13498
LCS	G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#89637

commutator, cosocle

C₂²×C₇×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₇

C₂

C₂×C₇

C₂³×D₇

Q₁₆

D₇

C₇

G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. #11729183	C₂	semidirect
4. C₇×#1385	C₂	semidirect
5. #11725540	C₂	semidirect
38. #1385	D₇	semidirect
168. C₇	#13498	semidirect

H ≅ G/N

type

2. #11729183

C₂

semidirect

4. C₇×#1385

C₂

semidirect

5. #11725540

C₂

semidirect

38. #1385

D₇

semidirect

168. C₇

#13498

semidirect

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

subgroup	index
#11729185	2
#11729183	2
#11728465	2
C₇×#1385	2
#11725540	2
#13498	7

subgroup

index

Subgroups — #13273688

Conjugacy classes of subgroups

Complements

Maximal subgroups