Group #39917705 — subgroups

Subgroups — C₁₀₃×#12430 #39917705

Order 26368.

268 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₁₀₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₃×#1667	C₂	1	2	yes	maximal, complemented
4	C₁₀₃×#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆×C₁₀₃	—	2	1	no	—
7	C₂×D₁₆×C₁₀₃	—	2	1	no	—
8	C₂²×C₁₆×C₁₀₃	—	2	1	no	—
9	#493×C₁₀₃	C₂²	1	2	yes	—
10	#459×C₁₀₃	—	2	2	no	—
11	C₈×D₄×C₁₀₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂²⋊C₁₆×C₁₀₃	—	2	1	no	—
14	C₄×C₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
15	#335×C₁₀₃	C₂²	1	2	yes	—
16	#331×C₁₀₃	—	2	2	no	—
17	C₂×D₈×C₁₀₃	—	2	2	no	—
18	C₂×D₈×C₁₀₃	—	2	2	no	—
19	C₂×D₈×C₁₀₃	—	4	2	no	—
20	C₂²×C₈×C₁₀₃	—	2	1	no	—
21	#127×C₁₀₃	—	2	2	no	—
22	C₄×D₄×C₁₀₃	D₄	1	1	yes	characteristic
23	D₁₆×C₁₀₃	—	4	2	no	—
24	D₁₆×C₁₀₃	—	4	2	no	—
25	C₂×C₁₆×C₁₀₃	—	2	1	no	—
26	C₂×C₁₆×C₁₀₃	—	4	2	no	—
27	C₂×C₁₆×C₁₀₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₀₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₁₀₃	—	2	2	no	—
30	C₄⋊C₈×C₁₀₃	D₄	1	1	yes	characteristic
31	#102×C₁₀₃	—	4	2	no	—
32	C₂²⋊C₈×C₁₀₃	—	2	1	no	—
33	C₄×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
34	C₂×D₄×C₁₀₃	—	4	2	no	—
35	C₂×D₄×C₁₀₃	—	4	2	no	—
36	C₂×D₄×C₁₀₃	—	4	2	no	—
37	C₂×D₄×C₁₀₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₃	—	2	1	no	—
39	D₈×C₁₀₃	—	4	2	no	—
40	D₈×C₁₀₃	—	4	2	no	—
41	D₈×C₁₀₃	—	8	2	no	—
42	D₈×C₁₀₃	—	8	4	no	—
43	C₂×C₈×C₁₀₃	—	2	1	no	—
44	C₂×C₈×C₁₀₃	—	4	1	no	—
45	C₂×C₈×C₁₀₃	—	4	1	no	—
46	C₂×C₈×C₁₀₃	—	4	2	no	—
47	C₂×C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₀₃	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₁₀₃	D₈	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₀₃	—	2	1	no	—
51	C₄²×C₁₀₃	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₁₀₃	—	2	1	no	—
53	C₁₆×C₁₀₃	—	4	1	no	—
54	C₁₆×C₁₀₃	D₈	1	2	yes	complemented
55	C₂³×C₁₀₃	—	2	1	no	—
56	C₂³×C₁₀₃	—	8	2	no	—
57	D₄×C₁₀₃	—	2	1	no	—
58	D₄×C₁₀₃	—	8	2	no	—
59	D₄×C₁₀₃	—	8	2	no	—
60	D₄×C₁₀₃	—	8	2	no	—
61	D₄×C₁₀₃	—	8	2	no	—
62	D₄×C₁₀₃	—	16	2	no	—
63	D₄×C₁₀₃	D₁₆	1	2	yes	complemented
64	C₂×C₄×C₁₀₃	—	2	1	no	—
65	C₂×C₄×C₁₀₃	—	4	1	no	—
66	C₂×C₄×C₁₀₃	—	4	1	no	—
67	C₂×C₄×C₁₀₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₀₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₀₃	#135	1	1	yes	characteristic
70	C₈×C₁₀₃	—	2	1	no	—
71	C₈×C₁₀₃	—	4	1	no	—
72	C₈×C₁₀₃	—	8	2	no	—
73	C₈×C₁₀₃	C₂×D₈	1	1	yes	characteristic
74	C₈×C₁₀₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂²×C₁₀₃	—	2	2	no	—
76	C₂²×C₁₀₃	—	4	1	no	—
77	C₂²×C₁₀₃	—	4	1	no	—
78	C₂²×C₁₀₃	—	16	2	no	—
79	C₂²×C₁₀₃	—	16	2	no	—
80	C₂²×C₁₀₃	—	16	2	no	—
81	C₂²×C₁₀₃	#466	1	1	yes	characteristic
82	C₄×C₁₀₃	—	2	1	no	—
83	C₄×C₁₀₃	—	4	1	no	—
84	C₄×C₁₀₃	#459	1	1	yes	characteristic
85	C₄×C₁₀₃	#472	1	1	yes	characteristic
86	C₄×C₁₀₃	C₂×D₁₆	1	1	yes	characteristic
87	C₄×C₁₀₃	#508	1	1	yes	characteristic
88	#12430	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₁₀₃	—	4	2	no	—
90	C₂×C₁₀₃	—	32	2	no	—
91	C₂×C₁₀₃	#1667	1	1	yes	characteristic
92	C₂×C₁₀₃	#2215	1	1	yes	characteristic
93	C₂×C₁₀₃	#2221	1	1	yes	characteristic
94	#2246	C₂×C₁₀₃	1	1	yes	characteristic, complemented
95	D₄×C₁₆	C₂×C₁₀₃	1	1	yes	characteristic, complemented
96	#1667	C₂×C₁₀₃	1	2	yes	complemented
97	#1371	C₂×C₁₀₃	1	1	yes	characteristic, complemented
98	#1361	C₂×C₁₀₃	1	2	yes	complemented
99	C₁₀₃	#12430	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×D₁₆	—	2	1	no	—
101	C₂×D₁₆	—	2	1	no	—
102	C₂²×C₁₆	—	2	1	no	—
103	#493	C₂²×C₁₀₃	1	2	yes	—
104	#459	—	2	2	no	—
105	C₈×D₄	C₂²×C₁₀₃	1	1	yes	characteristic
106	C₄⋊C₁₆	C₂²×C₁₀₃	1	1	yes	characteristic
107	C₂²⋊C₁₆	—	2	1	no	—
108	C₄×C₁₆	C₂²×C₁₀₃	1	1	yes	characteristic
109	#335	C₂²×C₁₀₃	1	2	yes	—
110	#331	—	2	2	no	—
111	C₂×D₈	—	2	2	no	—
112	C₂×D₈	—	2	2	no	—
113	C₂×D₈	—	4	2	no	—
114	C₂²×C₈	—	2	1	no	—
115	#127	—	2	2	no	—
116	C₄×D₄	D₄×C₁₀₃	1	1	yes	characteristic
117	D₁₆	—	4	2	no	—
118	D₁₆	—	4	2	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	4	2	no	—
121	C₂×C₁₆	D₄×C₁₀₃	1	1	yes	characteristic
122	C₂×C₁₆	D₄×C₁₀₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	D₄×C₁₀₃	1	1	yes	characteristic
125	#102	—	4	2	no	—
126	C₂²⋊C₈	—	2	1	no	—
127	C₄×C₈	C₂³×C₁₀₃	1	1	yes	characteristic
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂×D₄	—	4	2	no	—
131	C₂×D₄	D₈×C₁₀₃	1	1	yes	characteristic
132	C₂²×C₄	—	2	1	no	—
133	D₈	—	4	2	no	—
134	D₈	—	4	2	no	—
135	D₈	—	8	2	no	—
136	D₈	—	8	4	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	C₂×D₄×C₁₀₃	1	1	yes	characteristic
142	C₂×C₈	#41×C₁₀₃	1	1	yes	characteristic
143	C₄⋊C₄	D₈×C₁₀₃	1	1	yes	characteristic
144	C₂²⋊C₄	—	2	1	no	—
145	C₄²	C₂×D₄×C₁₀₃	1	1	yes	characteristic
146	C₁₆	—	2	1	no	—
147	C₁₆	—	4	1	no	—
148	C₁₆	D₈×C₁₀₃	1	2	yes	complemented
149	C₂³	—	2	1	no	—
150	C₂³	—	8	2	no	—
151	D₄	—	2	1	no	—
152	D₄	—	8	2	no	—
153	D₄	—	8	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	2	no	—
156	D₄	—	16	2	no	—
157	D₄	D₁₆×C₁₀₃	1	2	yes	complemented
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	#121×C₁₀₃	1	1	yes	characteristic
162	C₂×C₄	C₂×D₈×C₁₀₃	1	1	yes	characteristic
163	C₂×C₄	#135×C₁₀₃	1	1	yes	characteristic
164	C₈	—	2	1	no	—
165	C₈	—	4	1	no	—
166	C₈	—	8	2	no	—
167	C₈	C₂×D₈×C₁₀₃	1	1	yes	characteristic
168	C₈	AGL(1,ℤ/8)×C₁₀₃	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	16	2	no	—
173	C₂²	—	16	2	no	—
174	C₂²	—	16	2	no	—
175	C₂²	#466×C₁₀₃	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	4	1	no	—
178	C₄	#459×C₁₀₃	1	1	yes	characteristic
179	C₄	#472×C₁₀₃	1	1	yes	characteristic
180	C₄	C₂×D₁₆×C₁₀₃	1	1	yes	characteristic
181	C₄	#508×C₁₀₃	1	1	yes	characteristic
182	C₂	—	4	2	no	—
183	C₂	—	32	2	no	—
184	C₂	C₁₀₃×#1667	1	1	yes	characteristic
185	C₂	C₁₀₃×#2215	1	1	yes	characteristic
186	C₂	C₁₀₃×#2221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₃×#2246

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₁₀₃×#1667

C₂

yes

maximal, complemented

C₁₀₃×#1371

C₂

yes

maximal, characteristic, complemented

C₁₀₃×#1361

C₂

yes

maximal, complemented

C₂×D₁₆×C₁₀₃

—

C₂×D₁₆×C₁₀₃

—

C₂²×C₁₆×C₁₀₃

—

#493×C₁₀₃

C₂²

yes

—

#459×C₁₀₃

—

C₈×D₄×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₀₃

C₂²

yes

characteristic

C₂²⋊C₁₆×C₁₀₃

—

C₄×C₁₆×C₁₀₃

C₂²

yes

characteristic

#335×C₁₀₃

C₂²

yes

—

#331×C₁₀₃

—

C₂×D₈×C₁₀₃

—

C₂×D₈×C₁₀₃

—

C₂×D₈×C₁₀₃

—

C₂²×C₈×C₁₀₃

—

#127×C₁₀₃

—

C₄×D₄×C₁₀₃

D₄

yes

characteristic

D₁₆×C₁₀₃

—

D₁₆×C₁₀₃

—

C₂×C₁₆×C₁₀₃

—

C₂×C₁₆×C₁₀₃

—

C₂×C₁₆×C₁₀₃

D₄

yes

characteristic

C₂×C₁₆×C₁₀₃

D₄

yes

characteristic

C₄⋊C₈×C₁₀₃

—

C₄⋊C₈×C₁₀₃

D₄

yes

characteristic

#102×C₁₀₃

—

C₂²⋊C₈×C₁₀₃

—

C₄×C₈×C₁₀₃

C₂³

yes

characteristic

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

D₈

yes

characteristic

C₂²×C₄×C₁₀₃

—

D₈×C₁₀₃

—

D₈×C₁₀₃

—

D₈×C₁₀₃

—

D₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₃

#41

yes

characteristic

C₄⋊C₄×C₁₀₃

D₈

yes

characteristic

C₂²⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂×D₄

yes

characteristic

C₁₆×C₁₀₃

—

C₁₆×C₁₀₃

—

C₁₆×C₁₀₃

D₈

yes

complemented

C₂³×C₁₀₃

—

C₂³×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

D₁₆

yes

complemented

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

#121

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₃

#135

yes

characteristic

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₈×C₁₀₃

C₂×D₈

yes

characteristic

C₈×C₁₀₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#466

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

yes

C₄×C₁₀₃

yes

C₄×C₁₀₃

C₂×D₁₆

yes

C₄×C₁₀₃

yes

C₁₀₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

characteristic, complemented

D₄×C₁₆

C₂×C₁₀₃

yes

characteristic, complemented

#1667

C₂×C₁₀₃

yes

complemented

#1371

C₂×C₁₀₃

yes

characteristic, complemented

#1361

C₂×C₁₀₃

yes

complemented

C₁₀₃

#12430

yes

characteristic, direct product factor, Sylow

100

C₂×D₁₆

—

101

C₂×D₁₆

—

102

C₂²×C₁₆

—

103

#493

C₂²×C₁₀₃

yes

—

104

#459

—

105

C₈×D₄

C₂²×C₁₀₃

yes

characteristic

106

C₄⋊C₁₆

C₂²×C₁₀₃

yes

characteristic

107

C₂²⋊C₁₆

—

108

C₄×C₁₆

C₂²×C₁₀₃

yes

characteristic

109

#335

C₂²×C₁₀₃

yes

—

110

#331

—

111

C₂×D₈

—

112

C₂×D₈

—

113

C₂×D₈

—

114

C₂²×C₈

—

115

#127

—

116

C₄×D₄

D₄×C₁₀₃

yes

characteristic

117

D₁₆

—

118

D₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

D₄×C₁₀₃

yes

characteristic

122

C₂×C₁₆

D₄×C₁₀₃

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

D₄×C₁₀₃

yes

characteristic

125

#102

—

126

C₂²⋊C₈

—

127

C₄×C₈

C₂³×C₁₀₃

yes

characteristic

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

D₈×C₁₀₃

yes

characteristic

132

C₂²×C₄

—

133

D₈

—

134

D₈

—

135

D₈

—

136

D₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

C₂×D₄×C₁₀₃

yes

characteristic

142

C₂×C₈

#41×C₁₀₃

yes

characteristic

143

C₄⋊C₄

D₈×C₁₀₃

yes

characteristic

144

C₂²⋊C₄

—

145

C₄²

C₂×D₄×C₁₀₃

yes

characteristic

146

C₁₆

—

147

C₁₆

—

148

C₁₆

D₈×C₁₀₃

yes

complemented

149

C₂³

—

150

C₂³

—

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

D₁₆×C₁₀₃

yes

complemented

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#121×C₁₀₃

yes

characteristic

162

C₂×C₄

C₂×D₈×C₁₀₃

yes

characteristic

163

C₂×C₄

#135×C₁₀₃

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

C₂×D₈×C₁₀₃

yes

characteristic

168

C₈

AGL(1,ℤ/8)×C₁₀₃

yes

characteristic

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#466×C₁₀₃

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

#459×C₁₀₃

yes

characteristic

179

C₄

#472×C₁₀₃

yes

characteristic

180

C₄

C₂×D₁₆×C₁₀₃

yes

characteristic

181

C₄

#508×C₁₀₃

yes

characteristic

182

C₂

—

183

C₂

—

184

C₂

C₁₀₃×#1667

yes

characteristic

185

C₂

C₁₀₃×#2215

yes

characteristic

186

C₂

C₁₀₃×#2221

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₃	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₄²×C₁₀₃ ◁ C₈×D₄×C₁₀₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₃

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₃ ◁ C₄²×C₁₀₃ ◁ C₈×D₄×C₁₀₃ ◁ G

N	H ≅ G/N	type
1. C₁₀₃×#2246	C₂	semidirect
2. D₄×C₁₆×C₁₀₃	C₂	semidirect
3. C₁₀₃×#1667	C₂	semidirect
4. C₁₀₃×#1371	C₂	semidirect
5. C₁₀₃×#1361	C₂	semidirect
54. C₁₆×C₁₀₃	D₈	semidirect
63. D₄×C₁₀₃	D₁₆	semidirect
88. #12430	C₁₀₃	direct (N × H)
94. #2246	C₂×C₁₀₃	semidirect
95. D₄×C₁₆	C₂×C₁₀₃	semidirect
96. #1667	C₂×C₁₀₃	semidirect
97. #1371	C₂×C₁₀₃	semidirect
98. #1361	C₂×C₁₀₃	semidirect
99. C₁₀₃	#12430	direct (N × H)
148. C₁₆	D₈×C₁₀₃	semidirect
157. D₄	D₁₆×C₁₀₃	semidirect

H ≅ G/N

type

1. C₁₀₃×#2246

C₂

semidirect

2. D₄×C₁₆×C₁₀₃

C₂

semidirect

3. C₁₀₃×#1667

C₂

semidirect

4. C₁₀₃×#1371

C₂

semidirect

5. C₁₀₃×#1361

C₂

semidirect

54. C₁₆×C₁₀₃

D₈

semidirect

63. D₄×C₁₀₃

D₁₆

semidirect

88. #12430

C₁₀₃

direct (N × H)

94. #2246

C₂×C₁₀₃

semidirect

95. D₄×C₁₆

C₂×C₁₀₃

semidirect

96. #1667

C₂×C₁₀₃

semidirect

97. #1371

C₂×C₁₀₃

semidirect

98. #1361

C₂×C₁₀₃

semidirect

99. C₁₀₃

#12430

direct (N × H)

148. C₁₆

D₈×C₁₀₃

semidirect

157. D₄

D₁₆×C₁₀₃

semidirect

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

subgroup	index
C₁₀₃×#2246	2
D₄×C₁₆×C₁₀₃	2
C₁₀₃×#1667	2
C₁₀₃×#1371	2
C₁₀₃×#1361	2
#12430	103

subgroup

index

C₁₀₃×#2246

D₄×C₁₆×C₁₀₃

C₁₀₃×#1667

C₁₀₃×#1371

C₁₀₃×#1361

#12430

103

Subgroups — C₁₀₃×#12430 #39917705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C103×#12430 #39917705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₃×#12430 #39917705