Group #22569960 — subgroups

Subgroups — C₂×#16863686 #22569960

Order 8576.

480 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#14524043	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863686	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523974	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14523964	C₂	1	2	yes	maximal
6	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₃₄	—	2	1	no	—
8	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#14524043	—	2	2	no	—
10	#14524043	C₂²	1	4	yes	complemented
11	C₂×C₄⋊Dic₆₇	—	2	1	no	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	#14523974	C₂²	1	8	yes	complemented
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
15	#14523964	C₂²	1	8	yes	—
16	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	—	2	1	no	—
18	C₄×D₄×C₆₇	—	2	2	no	—
19	C₄×D₄×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
22	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	4	no	—
24	C₂×Dic₁₃₄	—	4	4	no	—
25	C₂²×Dic₆₇	—	2	2	no	—
26	C₂×C₆₇⋊C₈	—	2	4	no	—
27	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
28	C₄⋊Dic₆₇	—	2	4	no	—
29	C₄⋊Dic₆₇	—	2	2	no	—
30	C₄⋊Dic₆₇	C₂³	1	4	yes	—
31	C₂⁴×C₆₇	—	2	1	no	—
32	C₂×D₄×C₆₇	—	2	1	no	—
33	C₂×D₄×C₆₇	—	2	2	no	—
34	C₂×D₄×C₆₇	Q₈	1	4	yes	complemented
35	C₂×D₄×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₆₇	—	2	1	no	—
37	C₂²×C₄×C₆₇	—	2	4	no	—
38	C₂²×C₄×C₆₇	—	4	4	no	—
39	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
44	C₂²⋊C₄×C₆₇	—	2	4	no	—
45	C₄²×C₆₇	D₄	1	2	yes	—
46	C₄²×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₆₇	—	2	4	no	—
48	C₂³×C₆₇	—	2	2	no	—
49	C₂³×C₆₇	—	4	2	no	—
50	C₂³×C₆₇	—	4	2	no	—
51	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
52	D₄×C₆₇	—	2	4	no	—
53	D₄×C₆₇	—	2	2	no	—
54	D₄×C₆₇	C₂×Q₈	1	4	yes	complemented
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	4	2	no	—
58	C₂×C₄×C₆₇	—	4	8	no	—
59	C₂×C₄×C₆₇	SD₁₆	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×D₄	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	#41	1	2	yes	—
65	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₆₇	—	2	4	no	—
67	C₂×Dic₆₇	—	4	4	no	—
68	Dic₁₃₄	—	4	8	no	—
69	C₆₇⋊C₈	—	2	4	no	—
70	C₂²×C₆₇	—	2	4	no	—
71	C₂²×C₆₇	—	4	2	no	—
72	C₂²×C₆₇	—	4	2	no	—
73	C₂²×C₆₇	—	4	8	no	—
74	C₂²×C₆₇	C₂²⋊Q₈	1	2	yes	—
75	C₂²×C₆₇	C₂×SD₁₆	1	2	yes	—
76	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	—	4	2	no	—
80	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
81	C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
82	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
84	Dic₆₇	—	4	4	no	—
85	C₂×C₆₇	—	4	4	no	—
86	C₂×C₆₇	#476	1	4	yes	complemented
87	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
89	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×#476	—	67	1	no	maximal, Sylow
91	C₆₇	C₂×#476	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄⋊Q₈	—	67	1	no	—
93	C₂×C₄×D₄	D₆₇	1	1	yes	characteristic
94	#476	—	67	8	no	—
95	C₂×#106	—	67	2	no	—
96	C₂×C₄⋊C₈	—	67	1	no	—
97	C₂×#102	—	67	2	no	—
98	C₂²×Q₈	—	134	1	no	—
99	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
100	C₂³×C₄	—	2	1	no	—
101	C₂²×C₈	—	67	2	no	—
102	C₄⋊Q₈	—	67	4	no	—
103	C₄⋊Q₈	—	134	2	no	—
104	C₄×D₄	—	2	2	no	—
105	C₄×D₄	C₂×D₆₇	1	4	yes	—
106	C₂×C₄⋊C₄	—	67	2	no	—
107	C₂×C₄⋊C₄	—	134	1	no	—
108	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
109	C₂×C₂²⋊C₄	—	2	1	no	—
110	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
111	#106	—	67	8	no	—
112	C₄⋊C₈	—	67	4	no	—
113	#102	—	67	8	no	—
114	C₂⁴	—	2	1	no	—
115	C₂×Q₈	—	268	4	no	—
116	C₂×Q₈	—	134	4	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	Dic₁₃₄	1	4	yes	complemented
120	C₂×D₄	C₂²×D₆₇	1	2	yes	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	4	4	no	—
124	C₂²×C₄	—	134	2	no	—
125	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
128	C₂×C₈	—	67	4	no	—
129	C₂×C₈	—	134	4	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	67	4	no	—
132	C₄⋊C₄	—	134	2	no	—
133	C₄⋊C₄	—	134	4	no	—
134	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	C₂²⋊D₆₇	1	2	yes	—
137	C₄²	C₂²×D₆₇	1	2	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
143	Q₈	—	268	8	no	—
144	D₄	—	2	4	no	—
145	D₄	—	2	2	no	—
146	D₄	C₂×Dic₁₃₄	1	4	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	—	268	4	no	—
152	C₂×C₄	—	134	4	no	—
153	C₂×C₄	#11761616	1	4	yes	—
154	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
157	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
158	C₂×C₄	#11761628	1	2	yes	—
159	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
160	C₈	—	134	4	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	8	no	—
165	C₂²	#14524022	1	2	yes	—
166	C₂²	C₂×#11761616	1	2	yes	—
167	C₂²	#14524068	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	268	4	no	—
172	C₄	#14524022	1	2	yes	—
173	C₄	C₂×#11761616	1	4	yes	—
174	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
175	C₄	C₂×#11761628	1	1	yes	characteristic
176	C₂	—	4	4	no	—
177	C₂	#16863686	1	4	yes	direct product factor
178	C₂	C₂×#14524022	1	1	yes	characteristic
179	C₂	C₂²×#11761616	1	1	yes	characteristic
180	C₂	C₂×#14524068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic, complemented

#16863686

C₂

yes

maximal, direct product factor

C₂×#14523974

C₂

yes

maximal, complemented

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×#14523964

C₂

yes

maximal

C₂×C₄×D₄×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

C₂²×C₆₇⋊C₈

C₂²

yes

—

#14524043

—

#14524043

C₂²

yes

complemented

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523974

C₂²

yes

complemented

C₄⋊(C₆₇⋊C₈)

C₂²

yes

complemented

#14523964

C₂²

yes

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂³×C₄×C₆₇

—

C₄×D₄×C₆₇

—

C₄×D₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₂⁴×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

Q₈

yes

complemented

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

C₂×Q₈

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

C₂×SD₁₆

yes

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#476

yes

complemented

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×D₄

D₆₇

yes

characteristic

#476

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₂²×Q₈

—

134

—

C₂²×D₄

C₂×D₆₇

yes

characteristic

100

C₂³×C₄

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

134

—

104

C₄×D₄

—

105

C₄×D₄

C₂×D₆₇

yes

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

134

—

108

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

109

C₂×C₂²⋊C₄

—

110

C₂×C₄²

C₂×D₆₇

yes

characteristic

111

#106

—

112

C₄⋊C₈

—

113

#102

—

114

C₂⁴

—

115

C₂×Q₈

—

268

—

116

C₂×Q₈

—

134

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

Dic₁₃₄

yes

complemented

120

C₂×D₄

C₂²×D₆₇

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

134

—

125

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

126

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

127

C₂²×C₄

C₂²×D₆₇

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

134

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

134

—

133

C₄⋊C₄

—

134

—

134

C₄⋊C₄

C₂²×D₆₇

yes

—

135

C₂²⋊C₄

—

136

C₄²

C₂²⋊D₆₇

yes

—

137

C₄²

C₂²×D₆₇

yes

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

143

Q₈

—

268

—

144

D₄

—

145

D₄

—

146

D₄

C₂×Dic₁₃₄

yes

complemented

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

134

—

153

C₂×C₄

#11761616

yes

—

154

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

155

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

157

C₂×C₄

C₂×Dic₁₃₄

yes

—

158

C₂×C₄

#11761628

yes

—

159

C₂×C₄

C₂³×D₆₇

yes

characteristic

160

C₈

—

134

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14524022

yes

—

166

C₂²

C₂×#11761616

yes

—

167

C₂²

#14524068

yes

—

168

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

268

—

172

C₄

#14524022

yes

—

173

C₄

C₂×#11761616

yes

—

174

C₄

C₂²×Dic₁₃₄

yes

characteristic

175

C₄

C₂×#11761628

yes

characteristic

176

C₂

—

177

C₂

#16863686

yes

direct product factor

178

C₂

C₂×#14524022

yes

characteristic

179

C₂

C₂²×#11761616

yes

characteristic

180

C₂

C₂×#14524068

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄×D₄×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄×D₄	D₆₇
hypocenter	C₆₇	C₂×#476
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄×D₄×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄×D₄

D₆₇

hypocenter

C₆₇

C₂×#476

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×#14524043	C₂	semidirect
2. #16863686	C₂	direct (N × H)
3. C₂×#14523974	C₂	semidirect
4. C₂×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
10. #14524043	C₂²	semidirect
13. #14523974	C₂²	semidirect
14. C₄⋊(C₆₇⋊C₈)	C₂²	semidirect
34. C₂×D₄×C₆₇	Q₈	semidirect
54. D₄×C₆₇	C₂×Q₈	semidirect
86. C₂×C₆₇	#476	semidirect
91. C₆₇	C₂×#476	semidirect
119. C₂×D₄	Dic₁₃₄	semidirect
146. D₄	C₂×Dic₁₃₄	semidirect
177. C₂	#16863686	direct (N × H)

H ≅ G/N

type

1. C₂×#14524043

C₂

semidirect

2. #16863686

C₂

direct (N × H)

3. C₂×#14523974

C₂

semidirect

4. C₂×C₄⋊(C₆₇⋊C₈)

C₂

semidirect

10. #14524043

C₂²

semidirect

13. #14523974

C₂²

semidirect

14. C₄⋊(C₆₇⋊C₈)

C₂²

semidirect

34. C₂×D₄×C₆₇

Q₈

semidirect

54. D₄×C₆₇

C₂×Q₈

semidirect

86. C₂×C₆₇

#476

semidirect

91. C₆₇

C₂×#476

semidirect

119. C₂×D₄

Dic₁₃₄

semidirect

146. D₄

C₂×Dic₁₃₄

semidirect

177. C₂

#16863686

direct (N × H)

Subgroups — C₂×#16863686 #22569960

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863686 #22569960

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863686 #22569960