Group #20251974 — subgroups

Subgroups — C₂×#15689166 #20251974

Order 6016.

536 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 269 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	C₂²×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689166	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13093957	C₂	1	2	yes	maximal, complemented
5	C₂×#13093955	C₂	1	2	yes	maximal, complemented
6	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₄₇	—	2	1	no	—
8	C₂×#10625329	C₄	1	8	yes	complemented
9	C₂×#10625329	C₂²	1	4	yes	—
10	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	—	2	1	no	—
12	C₂×C₄⋊Dic₄₇	—	2	2	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
15	#13093957	C₂²	1	8	yes	complemented
16	#13093955	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	#117×C₄₇	—	2	2	no	—
19	#117×C₄₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
23	C₂×C₄²×C₄₇	—	2	1	no	—
24	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	D₄	1	2	yes	—
30	C₂²×C₄×C₄₇	Q₈	1	4	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
33	C₂²×Dic₄₇	—	2	4	no	—
34	C₂²×Dic₄₇	—	2	2	no	—
35	C₂²×Dic₄₇	—	4	4	no	—
36	C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
39	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
40	C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₄²×C₄₇	—	2	4	no	—
42	C₄⋊Dic₄₇	—	2	4	no	—
43	C₄⋊Dic₄₇	—	2	2	no	—
44	C₄⋊Dic₄₇	C₂³	1	4	yes	—
45	#10625329	C₂×C₄	1	16	yes	complemented
46	C₂×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
47	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	C₄⋊C₄	1	4	yes	—
51	C₂³×C₄₇	C₂×D₄	1	2	yes	—
52	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	4	2	no	—
56	C₂×C₄×C₄₇	—	4	2	no	—
57	C₂×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
58	C₂×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	—	4	8	no	—
68	C₂×Dic₄₇	—	4	4	no	—
69	C₄₇⋊C₈	C₂²×C₄	1	8	yes	complemented
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	—	2	8	no	—
73	C₂²×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂²×C₄₇	#137	1	2	yes	—
77	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₄₇	—	4	2	no	—
79	C₄×C₄₇	—	4	2	no	—
80	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
82	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
84	Dic₄₇	—	4	4	no	—
85	C₂×#428	—	47	1	no	maximal, Sylow
86	C₂×C₄₇	—	2	4	no	—
87	C₂×C₄₇	#428	1	4	yes	complemented
88	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
91	C₂²×M₁₆	—	47	1	no	—
92	C₂×#117	D₄₇	1	1	yes	characteristic
93	C₂²×C₄⋊C₄	—	47	1	no	—
94	#428	—	47	8	no	—
95	C₂×#107	—	47	2	no	—
96	C₂×#106	—	47	2	no	—
97	C₄₇	C₂×#428	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	—	94	1	no	—
99	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
100	C₂×M₁₆	—	47	8	no	—
101	C₂×M₁₆	—	47	4	no	—
102	C₂²×C₈	—	47	2	no	—
103	#117	—	2	2	no	—
104	#117	C₂×D₄₇	1	4	yes	—
105	C₂×C₄⋊C₄	—	47	2	no	—
106	C₂×C₄⋊C₄	—	47	4	no	—
107	C₂×C₄⋊C₄	—	94	2	no	—
108	C₂×C₄⋊C₄	—	94	1	no	—
109	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	2	1	no	—
113	#107	—	47	8	no	—
114	#106	—	47	8	no	—
115	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	94	4	no	—
119	C₂²×C₄	—	94	2	no	—
120	C₂²×C₄	—	188	4	no	—
121	C₂²×C₄	Dic₉₄	1	4	yes	—
122	C₂²×C₄	C₄×D₄₇	1	4	yes	—
123	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
126	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
127	M₁₆	—	47	16	no	—
128	C₂×C₈	—	47	8	no	—
129	C₂×C₈	—	47	4	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	47	4	no	—
133	C₄⋊C₄	—	94	4	no	—
134	C₄⋊C₄	—	94	2	no	—
135	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
136	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
137	C₂²⋊C₄	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	Dic₄₇⋊C₄	1	4	yes	—
142	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	94	4	no	—
149	C₂×C₄	—	188	8	no	—
150	C₂×C₄	—	188	4	no	—
151	C₂×C₄	Dic₄₇⋊C₄	1	8	yes	—
152	C₂×C₄	Dic₄₇⋊C₄	1	4	yes	—
153	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
154	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
156	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
157	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
160	C₈	—	47	8	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×Dic₄₇⋊C₄	1	4	yes	—
165	C₂²	C₂×Dic₄₇⋊C₄	1	2	yes	—
166	C₂²	#13094049	1	2	yes	—
167	C₂²	#13094054	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	188	4	no	—
172	C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
173	C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
174	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
175	C₄	C₂²×Dic₉₄	1	1	yes	characteristic
176	C₂	—	2	4	no	—
177	C₂	#15689166	1	4	yes	direct product factor
178	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
179	C₂	C₂×#13094049	1	1	yes	characteristic
180	C₂	C₂×#13094054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625329

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15689166

C₂

yes

maximal, direct product factor

C₂×#13093957

C₂

yes

maximal, complemented

C₂×#13093955

C₂

yes

maximal, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

—

C₂×#10625329

C₄

yes

complemented

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093957

C₂²

yes

complemented

#13093955

C₂²

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

#10625329

C₂×C₄

yes

complemented

C₂×C₄₇⋊C₈

C₂×C₄

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

C₂²×C₄

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²×C₄₇

#137

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

Dic₄₇

—

C₂×#428

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#428

yes

complemented

C₂×C₄₇

C₂²×C₄⋊C₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×M₁₆

—

C₂×#117

D₄₇

yes

characteristic

C₂²×C₄⋊C₄

—

#428

—

C₂×#107

—

C₂×#106

—

C₄₇

C₂×#428

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

#117

—

104

#117

C₂×D₄₇

yes

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

110

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

—

113

#107

—

114

#106

—

115

C₂⁴

C₂²⋊D₄₇

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

188

—

121

C₂²×C₄

Dic₉₄

yes

—

122

C₂²×C₄

C₄×D₄₇

yes

—

123

C₂²×C₄

C₂²⋊D₄₇

yes

—

124

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

125

C₂²×C₄

C₂²×D₄₇

yes

—

126

C₂²×C₄

C₂²×D₄₇

yes

characteristic

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

C₂²×D₄₇

yes

—

136

C₄⋊C₄

C₂²×D₄₇

yes

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₂³

—

140

C₂³

—

141

C₂³

Dic₄₇⋊C₄

yes

—

142

C₂³

C₂×C₂²⋊D₄₇

yes

—

143

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

188

—

150

C₂×C₄

—

188

—

151

C₂×C₄

Dic₄₇⋊C₄

yes

—

152

C₂×C₄

Dic₄₇⋊C₄

yes

—

153

C₂×C₄

C₂×Dic₉₄

yes

—

154

C₂×C₄

C₂×Dic₉₄

yes

—

155

C₂×C₄

C₂×C₄×D₄₇

yes

—

156

C₂×C₄

C₂×C₄×D₄₇

yes

—

157

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

158

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

159

C₂×C₄

C₂³×D₄₇

yes

characteristic

160

C₈

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂×Dic₄₇⋊C₄

yes

—

165

C₂²

C₂×Dic₄₇⋊C₄

yes

—

166

C₂²

#13094049

yes

—

167

C₂²

#13094054

yes

—

168

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

188

—

172

C₄

C₂×Dic₄₇⋊C₄

yes

—

173

C₄

C₂×Dic₄₇⋊C₄

yes

—

174

C₄

C₂²×C₄×D₄₇

yes

characteristic

175

C₄

C₂²×Dic₉₄

yes

characteristic

176

C₂

—

177

C₂

#15689166

yes

direct product factor

178

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

179

C₂

C₂×#13094049

yes

characteristic

180

C₂

C₂×#13094054

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	C₂×#428
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#117×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
3. #15689166	C₂	direct (N × H)
4. C₂×#13093957	C₂	semidirect
5. C₂×#13093955	C₂	semidirect
8. C₂×#10625329	C₄	semidirect
15. #13093957	C₂²	semidirect
16. #13093955	C₂²	semidirect
45. #10625329	C₂×C₄	semidirect
46. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
69. C₄₇⋊C₈	C₂²×C₄	semidirect
87. C₂×C₄₇	#428	semidirect
97. C₄₇	C₂×#428	semidirect
177. C₂	#15689166	direct (N × H)

H ≅ G/N

type

3. #15689166

C₂

direct (N × H)

4. C₂×#13093957

C₂

semidirect

5. C₂×#13093955

C₂

semidirect

8. C₂×#10625329

C₄

semidirect

15. #13093957

C₂²

semidirect

16. #13093955

C₂²

semidirect

45. #10625329

C₂×C₄

semidirect

46. C₂×C₄₇⋊C₈

C₂×C₄

semidirect

69. C₄₇⋊C₈

C₂²×C₄

semidirect

87. C₂×C₄₇

#428

semidirect

97. C₄₇

C₂×#428

semidirect

177. C₂

#15689166

direct (N × H)

Subgroups — C₂×#15689166 #20251974

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689166 #20251974

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689166 #20251974