Group #22557521 — subgroups

Subgroups — C₂×#484×C₆₇ #22557521

Order 8576.

460 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 216 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×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#484×C₆₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#107×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#103×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₆₇	—	2	1	no	—
9	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂²⋊Q₈×C₆₇	—	2	2	no	—
12	C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
15	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
20	#107×C₆₇	C₂²	1	8	yes	complemented
21	#103×C₆₇	C₂²	1	8	yes	complemented
22	C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
23	C₂⁴×C₆₇	D₄	1	1	yes	characteristic
24	C₂×Q₈×C₆₇	—	2	1	no	—
25	C₂×Q₈×C₆₇	—	2	4	no	—
26	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₆₇	—	2	4	no	—
28	C₂²×C₄×C₆₇	—	2	4	no	—
29	C₂²×C₄×C₆₇	—	2	1	no	—
30	C₂²×C₄×C₆₇	—	2	1	no	—
31	C₂²×C₄×C₆₇	—	2	2	no	—
32	C₂²×C₄×C₆₇	—	4	4	no	—
33	C₂²×C₄×C₆₇	D₄	1	2	yes	—
34	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₆₇	—	2	4	no	—
38	C₂×C₈×C₆₇	C₂³	1	4	yes	—
39	C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
45	C₂²⋊C₄×C₆₇	—	2	4	no	—
46	C₂³×C₆₇	—	2	2	no	—
47	C₂³×C₆₇	—	2	2	no	—
48	C₂³×C₆₇	Q₁₆	1	4	yes	—
49	C₂³×C₆₇	C₂×D₄	1	2	yes	—
50	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₆₇	—	2	4	no	—
52	Q₈×C₆₇	—	4	2	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	4	no	—
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	4	no	—
59	C₂×C₄×C₆₇	—	4	8	no	—
60	C₂×C₄×C₆₇	—	4	2	no	—
61	C₂×C₄×C₆₇	—	4	2	no	—
62	C₂×C₄×C₆₇	—	4	4	no	—
63	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#41	1	4	yes	—
66	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₈×C₆₇	—	2	4	no	—
68	C₂²×C₆₇	—	2	2	no	—
69	C₂²×C₆₇	—	2	2	no	—
70	C₂²×C₆₇	—	2	8	no	—
71	C₂²×C₆₇	#123	1	2	yes	—
72	C₂²×C₆₇	C₂×Q₁₆	1	4	yes	—
73	C₂²×C₆₇	C₂×Q₁₆	1	2	yes	—
74	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
75	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	—	4	2	no	—
78	C₄×C₆₇	—	4	2	no	—
79	C₄×C₆₇	—	4	4	no	—
80	C₄×C₆₇	#123	1	2	yes	—
81	C₄×C₆₇	C₂×#41	1	2	yes	—
82	C₂×C₆₇	—	2	4	no	—
83	C₂×C₆₇	#484	1	4	yes	direct product factor
84	C₂×C₆₇	C₂×#123	1	1	yes	characteristic
85	C₂×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
86	C₂×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂×#484	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₆₇	C₂×#484	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
90	C₂²×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
91	#484	C₂×C₆₇	1	8	yes	direct product factor
92	C₂×#107	C₂×C₆₇	1	2	yes	complemented
93	C₂×#103	C₂×C₆₇	1	2	yes	complemented
94	C₂×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
95	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
96	C₂³×C₄	—	2	1	no	—
97	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₆₇	1	2	yes	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
101	C₂×C₄⋊C₄	—	2	1	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	2	1	no	—
104	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
106	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
107	C₂×C₂²⋊C₄	—	2	1	no	—
108	#107	C₂²×C₆₇	1	8	yes	complemented
109	#103	C₂²×C₆₇	1	8	yes	complemented
110	C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
111	C₂⁴	D₄×C₆₇	1	1	yes	characteristic
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	C₂³×C₆₇	1	2	yes	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
122	C₂²×C₄	D₄×C₆₇	1	2	yes	—
123	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
124	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	C₂³×C₆₇	1	4	yes	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
132	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
133	C₂²⋊C₄	—	2	4	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	Q₁₆×C₆₇	1	4	yes	—
137	C₂³	C₂×D₄×C₆₇	1	2	yes	—
138	C₂³	C₂×D₄×C₆₇	1	1	yes	characteristic
139	Q₈	—	2	4	no	—
140	Q₈	—	4	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	4	8	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
152	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
153	C₂×C₄	#41×C₆₇	1	4	yes	—
154	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₂²	—	2	8	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	#123×C₆₇	1	2	yes	—
160	C₂²	C₂×Q₁₆×C₆₇	1	4	yes	—
161	C₂²	C₂×Q₁₆×C₆₇	1	2	yes	—
162	C₂²	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
163	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	4	no	—
167	C₄	—	4	2	no	—
168	C₄	#123×C₆₇	1	2	yes	—
169	C₄	C₂×#41×C₆₇	1	2	yes	—
170	C₂	—	2	4	no	—
171	C₂	#484×C₆₇	1	4	yes	direct product factor
172	C₂	C₂×#123×C₆₇	1	1	yes	characteristic
173	C₂	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
174	C₂	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₆₇

C₂

yes

maximal, characteristic

#484×C₆₇

C₂

yes

maximal, direct product factor

C₂×#107×C₆₇

C₂

yes

maximal, complemented

C₂×#103×C₆₇

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂³×C₄×C₆₇

—

C₂³×C₄×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₆₇

—

#107×C₆₇

C₂²

yes

complemented

#103×C₆₇

C₂²

yes

complemented

C₂²⋊C₈×C₆₇

C₂²

yes

—

C₂⁴×C₆₇

D₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

Q₁₆

yes

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#123

yes

—

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#123

yes

—

C₄×C₆₇

C₂×#41

yes

—

C₂×C₆₇

—

C₂×C₆₇

#484

yes

direct product factor

C₂×C₆₇

C₂×#123

yes

characteristic

C₂×C₆₇

C₂²×Q₁₆

yes

C₂×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

C₂×#484

yes

characteristic, direct product factor, Sylow

C₂×C₂²⋊Q₈

C₂×C₆₇

yes

characteristic

C₂²×C₄⋊C₄

C₂×C₆₇

yes

characteristic

#484

C₂×C₆₇

yes

direct product factor

C₂×#107

C₂×C₆₇

yes

complemented

C₂×#103

C₂×C₆₇

yes

complemented

C₂×C₂²⋊C₈

C₂×C₆₇

yes

characteristic

C₂²×Q₈

C₂²×C₆₇

yes

characteristic

C₂³×C₄

—

C₂³×C₄

C₂²×C₆₇

yes

characteristic

C₂²×C₈

C₂²×C₆₇

yes

—

C₂²⋊Q₈

—

100

C₂²⋊Q₈

C₂²×C₆₇

yes

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

105

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

106

C₂×C₄⋊C₄

C₂²×C₆₇

yes

characteristic

107

C₂×C₂²⋊C₄

—

108

#107

C₂²×C₆₇

yes

complemented

109

#103

C₂²×C₆₇

yes

complemented

110

C₂²⋊C₈

C₂²×C₆₇

yes

—

111

C₂⁴

D₄×C₆₇

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

C₂³×C₆₇

yes

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

D₄×C₆₇

yes

characteristic

122

C₂²×C₄

D₄×C₆₇

yes

—

123

C₂²×C₄

C₂³×C₆₇

yes

—

124

C₂²×C₄

C₂³×C₆₇

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

C₂³×C₆₇

yes

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

C₂³×C₆₇

yes

—

132

C₄⋊C₄

C₂³×C₆₇

yes

—

133

C₂²⋊C₄

—

134

C₂³

—

135

C₂³

—

136

C₂³

Q₁₆×C₆₇

yes

—

137

C₂³

C₂×D₄×C₆₇

yes

—

138

C₂³

C₂×D₄×C₆₇

yes

characteristic

139

Q₈

—

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

C₂×D₄×C₆₇

yes

—

152

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

153

C₂×C₄

#41×C₆₇

yes

—

154

C₂×C₄

C₂⁴×C₆₇

yes

characteristic

155

C₈

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#123×C₆₇

yes

—

160

C₂²

C₂×Q₁₆×C₆₇

yes

—

161

C₂²

C₂×Q₁₆×C₆₇

yes

—

162

C₂²

AGL(1,ℤ/8)×C₆₇

yes

—

163

C₂²

C₂²×D₄×C₆₇

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

#123×C₆₇

yes

—

169

C₄

C₂×#41×C₆₇

yes

—

170

C₂

—

171

C₂

#484×C₆₇

yes

direct product factor

172

C₂

C₂×#123×C₆₇

yes

characteristic

173

C₂

C₂²×Q₁₆×C₆₇

yes

characteristic

174

C₂

C₂×AGL(1,ℤ/8)×C₆₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₇

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₆₇

prime radical

C₂⁴×C₆₇

D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

N	H ≅ G/N	type
3. #484×C₆₇	C₂	direct (N × H)
4. C₂×#107×C₆₇	C₂	semidirect
5. C₂×#103×C₆₇	C₂	semidirect
20. #107×C₆₇	C₂²	semidirect
21. #103×C₆₇	C₂²	semidirect
83. C₂×C₆₇	#484	direct (N × H)
87. C₂×#484	C₆₇	direct (N × H)
88. C₆₇	C₂×#484	direct (N × H)
91. #484	C₂×C₆₇	direct (N × H)
92. C₂×#107	C₂×C₆₇	semidirect
93. C₂×#103	C₂×C₆₇	semidirect
108. #107	C₂²×C₆₇	semidirect
109. #103	C₂²×C₆₇	semidirect
171. C₂	#484×C₆₇	direct (N × H)

H ≅ G/N

type

3. #484×C₆₇

C₂

direct (N × H)

4. C₂×#107×C₆₇

C₂

semidirect

5. C₂×#103×C₆₇

C₂

semidirect

20. #107×C₆₇

C₂²

semidirect

21. #103×C₆₇

C₂²

semidirect

83. C₂×C₆₇

#484

direct (N × H)

87. C₂×#484

C₆₇

direct (N × H)

88. C₆₇

C₂×#484

direct (N × H)

91. #484

C₂×C₆₇

direct (N × H)

92. C₂×#107

C₂×C₆₇

semidirect

93. C₂×#103

C₂×C₆₇

semidirect

108. #107

C₂²×C₆₇

semidirect

109. #103

C₂²×C₆₇

semidirect

171. C₂

#484×C₆₇

direct (N × H)

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

subgroup	index
C₂×C₂²⋊Q₈×C₆₇	2
C₂²×C₄⋊C₄×C₆₇	2
#484×C₆₇	2
C₂×#107×C₆₇	2
C₂×#103×C₆₇	2
C₂×C₂²⋊C₈×C₆₇	2
C₂×#484	67

subgroup

index

C₂×C₂²⋊Q₈×C₆₇

C₂²×C₄⋊C₄×C₆₇

#484×C₆₇

C₂×#107×C₆₇

C₂×#103×C₆₇

C₂×C₂²⋊C₈×C₆₇

C₂×#484

Subgroups — C₂×#484×C₆₇ #22557521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#484×C67 #22557521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#484×C₆₇ #22557521