Group #22568792 — subgroups

Subgroups — C₂×#16863436 #22568792

Order 8576.

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×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂×#14524002	C₂	1	1	yes	maximal, characteristic
3	#16863436	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14523976	C₂	1	2	yes	maximal, complemented
5	C₂×#14523974	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#11761612	C₄	1	8	yes	complemented
8	C₂×#11761612	C₂²	1	4	yes	—
9	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
10	#14524002	—	2	2	no	—
11	#14524002	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
15	C₂×C₄×Dic₆₇	—	2	1	no	—
16	#14523976	C₂²	1	8	yes	complemented
17	#14523974	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₆₇	—	2	1	no	—
19	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×Dic₆₇	—	2	1	no	—
25	C₂²×Dic₆₇	—	2	1	no	—
26	#11761612	C₂×C₄	1	16	yes	complemented
27	C₂×C₆₇⋊C₈	C₂×C₄	1	8	yes	complemented
28	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	C₂³	1	2	yes	—
32	C₄⋊Dic₆₇	C₂³	1	2	yes	—
33	C₂²⋊Dic₆₇	—	2	4	no	—
34	C₄×Dic₆₇	—	2	4	no	—
35	C₂⁴×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	—	2	4	no	—
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	—	4	4	no	—
39	C₂²×C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	D₄	1	2	yes	—
42	C₂²×C₄×C₆₇	Q₈	1	4	yes	—
43	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
45	C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
48	C₂³×C₆₇	—	2	2	no	—
49	C₂³×C₆₇	—	2	2	no	—
50	C₂³×C₆₇	C₄⋊C₄	1	4	yes	—
51	C₂³×C₆₇	C₂×D₄	1	2	yes	—
52	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	—	2	4	no	—
54	C₂×C₄×C₆₇	—	4	8	no	—
55	C₂×C₄×C₆₇	—	4	4	no	—
56	C₂×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
57	C₂×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
58	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₆₇	—	2	2	no	—
66	C₂×Dic₆₇	—	2	2	no	—
67	C₂×Dic₆₇	—	4	2	no	—
68	C₂×Dic₆₇	—	4	2	no	—
69	C₆₇⋊C₈	C₂²×C₄	1	8	yes	complemented
70	C₂²×C₆₇	—	2	2	no	—
71	C₂²×C₆₇	—	2	2	no	—
72	C₂²×C₆₇	—	2	8	no	—
73	C₂²×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂²×C₆₇	#137	1	2	yes	—
77	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₆₇	—	4	4	no	—
79	C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
81	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
82	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	Dic₆₇	—	4	2	no	—
84	Dic₆₇	—	4	2	no	—
85	C₂×C₆₇	—	2	4	no	—
86	C₂×C₆₇	#428	1	4	yes	complemented
87	C₂×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₆₇	C₂×#137	1	1	yes	characteristic
90	C₂×#428	—	67	1	no	maximal, Sylow
91	C₆₇	C₂×#428	1	1	yes	characteristic, complemented, Sylow
92	C₂²×M₁₆	—	67	1	no	—
93	C₂×#117	—	67	1	no	—
94	C₂²×C₄⋊C₄	D₆₇	1	1	yes	characteristic
95	#428	—	67	8	no	—
96	C₂×#107	—	67	2	no	—
97	C₂×#106	—	67	2	no	—
98	C₂³×C₄	—	2	1	no	—
99	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
100	C₂×M₁₆	—	67	8	no	—
101	C₂×M₁₆	—	67	4	no	—
102	C₂²×C₈	—	67	2	no	—
103	#117	—	67	4	no	—
104	#117	—	134	2	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	67	1	no	—
108	C₂×C₄⋊C₄	—	67	1	no	—
109	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
111	C₂×C₂²⋊C₄	—	134	1	no	—
112	C₂×C₄²	—	134	1	no	—
113	#107	—	67	8	no	—
114	#106	—	67	8	no	—
115	C₂⁴	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	—	134	1	no	—
120	C₂²×C₄	—	134	1	no	—
121	C₂²×C₄	Dic₁₃₄	1	4	yes	—
122	C₂²×C₄	C₄×D₆₇	1	4	yes	—
123	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
126	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
127	M₁₆	—	67	16	no	—
128	C₂×C₈	—	67	8	no	—
129	C₂×C₈	—	67	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	67	2	no	—
133	C₄⋊C₄	—	67	2	no	—
134	C₄⋊C₄	—	134	1	no	—
135	C₄⋊C₄	—	134	1	no	—
136	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
137	C₂²⋊C₄	—	134	4	no	—
138	C₄²	—	134	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	Dic₆₇⋊C₄	1	4	yes	—
142	C₂³	C₂×C₂²⋊D₆₇	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	268	2	no	—
148	C₂×C₄	—	268	2	no	—
149	C₂×C₄	—	134	2	no	—
150	C₂×C₄	—	134	2	no	—
151	C₂×C₄	Dic₆₇⋊C₄	1	8	yes	—
152	C₂×C₄	Dic₆₇⋊C₄	1	4	yes	—
153	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₆₇	1	4	yes	—
155	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
158	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
159	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
160	C₈	—	67	8	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×Dic₆₇⋊C₄	1	4	yes	—
165	C₂²	C₂×Dic₆₇⋊C₄	1	2	yes	—
166	C₂²	#14524069	1	2	yes	—
167	C₂²	#14524074	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
169	C₄	—	4	4	no	—
170	C₄	—	268	2	no	—
171	C₄	—	268	2	no	—
172	C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
173	C₄	C₂×Dic₆₇⋊C₄	1	4	yes	—
174	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
175	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
176	C₂	—	2	4	no	—
177	C₂	#16863436	1	4	yes	direct product factor
178	C₂	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
179	C₂	C₂×#14524069	1	1	yes	characteristic
180	C₂	C₂×#14524074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761612

C₂

yes

maximal, characteristic

C₂×#14524002

C₂

yes

maximal, characteristic

#16863436

C₂

yes

maximal, direct product factor

C₂×#14523976

C₂

yes

maximal, complemented

C₂×#14523974

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₄

yes

complemented

C₂×#11761612

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

—

#14524002

—

#14524002

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₆₇

—

C₂×C₄×Dic₆₇

—

#14523976

C₂²

yes

complemented

#14523974

C₂²

yes

complemented

C₂³×C₄×C₆₇

—

C₂³×C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

C₂×C₄

yes

complemented

C₂×C₆₇⋊C₈

C₂×C₄

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂⁴×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₄⋊C₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

C₂²×C₄

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

#137

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₄×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

C₂²×Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#428

yes

complemented

C₂×C₆₇

C₂²×C₄⋊C₄

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×#117

—

C₂²×C₄⋊C₄

D₆₇

yes

—

—

—

C₂³×C₄

—

C₂³×C₄

C₂×D₆₇

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

#117

—

104

#117

—

134

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

110

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

111

C₂×C₂²⋊C₄

—

134

—

112

C₂×C₄²

—

134

—

113

#107

—

114

#106

—

115

C₂⁴

C₂²⋊D₆₇

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

134

—

120

C₂²×C₄

—

134

—

121

C₂²×C₄

Dic₁₃₄

yes

—

122

C₂²×C₄

C₄×D₆₇

yes

—

123

C₂²×C₄

C₂²⋊D₆₇

yes

—

124

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

125

C₂²×C₄

C₂²×D₆₇

yes

—

126

C₂²×C₄

C₂²×D₆₇

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

—

134

—

135

C₄⋊C₄

—

134

—

136

C₄⋊C₄

C₂²×D₆₇

yes

—

137

C₂²⋊C₄

—

134

—

138

C₄²

—

134

—

139

C₂³

—

140

C₂³

—

141

C₂³

Dic₆₇⋊C₄

yes

—

142

C₂³

C₂×C₂²⋊D₆₇

yes

—

143

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

268

—

148

C₂×C₄

—

268

—

149

C₂×C₄

—

134

—

150

C₂×C₄

—

134

—

151

C₂×C₄

Dic₆₇⋊C₄

yes

—

152

C₂×C₄

Dic₆₇⋊C₄

yes

—

153

C₂×C₄

C₂×C₄×D₆₇

yes

—

154

C₂×C₄

C₂×C₄×D₆₇

yes

—

155

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

156

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

157

C₂×C₄

C₂×Dic₁₃₄

yes

—

158

C₂×C₄

C₂×Dic₁₃₄

yes

—

159

C₂×C₄

C₂³×D₆₇

yes

characteristic

160

C₈

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂×Dic₆₇⋊C₄

yes

—

165

C₂²

C₂×Dic₆₇⋊C₄

yes

—

166

C₂²

#14524069

yes

—

167

C₂²

#14524074

yes

—

168

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

169

C₄

—

170

C₄

—

268

—

171

C₄

—

268

—

172

C₄

C₂×Dic₆₇⋊C₄

yes

—

173

C₄

C₂×Dic₆₇⋊C₄

yes

—

174

C₄

C₂²×C₄×D₆₇

yes

characteristic

175

C₄

C₂²×Dic₁₃₄

yes

characteristic

176

C₂

—

177

C₂

#16863436

yes

direct product factor

178

C₂

C₂²×Dic₆₇⋊C₄

yes

characteristic

179

C₂

C₂×#14524069

yes

characteristic

180

C₂

C₂×#14524074

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂²×C₄⋊C₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂²×C₄⋊C₄	D₆₇
hypocenter	C₆₇	C₂×#428
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₄×C₆₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂²×C₄⋊C₄×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂²×C₄⋊C₄

D₆₇

hypocenter

C₆₇

C₂×#428

LCS

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #16863436	C₂	direct (N × H)
4. C₂×#14523976	C₂	semidirect
5. C₂×#14523974	C₂	semidirect
7. C₂×#11761612	C₄	semidirect
16. #14523976	C₂²	semidirect
17. #14523974	C₂²	semidirect
26. #11761612	C₂×C₄	semidirect
27. C₂×C₆₇⋊C₈	C₂×C₄	semidirect
69. C₆₇⋊C₈	C₂²×C₄	semidirect
86. C₂×C₆₇	#428	semidirect
91. C₆₇	C₂×#428	semidirect
177. C₂	#16863436	direct (N × H)

H ≅ G/N

type

3. #16863436

C₂

direct (N × H)

4. C₂×#14523976

C₂

semidirect

5. C₂×#14523974

C₂

semidirect

7. C₂×#11761612

C₄

semidirect

16. #14523976

C₂²

semidirect

17. #14523974

C₂²

semidirect

26. #11761612

C₂×C₄

semidirect

27. C₂×C₆₇⋊C₈

C₂×C₄

semidirect

69. C₆₇⋊C₈

C₂²×C₄

semidirect

86. C₂×C₆₇

#428

semidirect

91. C₆₇

C₂×#428

semidirect

177. C₂

#16863436

direct (N × H)

Subgroups — C₂×#16863436 #22568792

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863436 #22568792

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863436 #22568792