Group #22557060 — subgroups

Subgroups — C₆₇×#2629 #22557060

Order 8576.

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

normal subgroups only

This group has 472 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×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#398×C₆₇	C₂	1	4	yes	maximal, complemented
4	#396×C₆₇	C₂	1	4	yes	maximal
5	#389×C₆₇	C₂	1	2	yes	maximal, complemented
6	#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#380×C₆₇	C₂	1	2	yes	maximal
8	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	#125×C₆₇	—	2	4	no	—
10	#117×C₆₇	—	2	4	no	—
11	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#95×C₆₇	C₂²	1	4	yes	—
22	#95×C₆₇	C₂²	1	4	yes	—
23	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	—	2	2	no	—
25	C₂²×C₄×C₆₇	—	2	2	no	—
26	C₂²×C₄×C₆₇	Q₈	1	4	yes	—
27	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
30	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₆₇	—	2	4	no	—
34	C₄⋊C₄×C₆₇	—	2	8	no	—
35	C₄⋊C₄×C₆₇	—	2	4	no	—
36	C₄⋊C₄×C₆₇	—	2	4	no	—
37	C₄⋊C₄×C₆₇	—	4	4	no	—
38	C₄⋊C₄×C₆₇	—	4	4	no	—
39	C₂²⋊C₄×C₆₇	—	2	4	no	—
40	C₂²⋊C₄×C₆₇	—	4	4	no	—
41	C₄²×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	D₄	1	4	yes	—
43	C₂³×C₆₇	—	2	1	no	—
44	C₂³×C₆₇	—	2	1	no	—
45	C₂³×C₆₇	—	2	2	no	—
46	C₂³×C₆₇	—	2	2	no	—
47	C₂³×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	1	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	8	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	4	no	—
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	—	2	4	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	4	no	—
63	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₆₇	#41	1	2	yes	—
68	C₂²×C₆₇	—	2	2	no	—
69	C₂²×C₆₇	—	2	4	no	—
70	C₂²×C₆₇	—	4	2	no	—
71	C₂²×C₆₇	—	4	1	no	—
72	C₂²×C₆₇	—	4	1	no	—
73	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
76	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
77	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	—	2	4	no	—
80	C₄×C₆₇	—	4	2	no	—
81	C₄×C₆₇	—	4	4	no	—
82	C₄×C₆₇	—	4	4	no	—
83	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
84	C₂×C₆₇	—	4	2	no	—
85	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₆₇	#537	1	2	yes	—
87	C₂×C₆₇	#540	1	1	yes	characteristic
88	C₂×C₆₇	#541	1	1	yes	characteristic
89	C₂×C₆₇	#544	1	2	yes	—
90	#2629	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₆₇	#2629	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
93	C₂×#117	C₂×C₆₇	1	1	yes	characteristic
94	#398	C₂×C₆₇	1	4	yes	complemented
95	#396	C₂×C₆₇	1	4	yes	—
96	#389	C₂×C₆₇	1	2	yes	complemented
97	#384	C₂×C₆₇	1	1	yes	characteristic, complemented
98	#380	C₂×C₆₇	1	2	yes	—
99	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
100	#125	—	2	4	no	—
101	#117	—	2	4	no	—
102	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
103	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
104	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
105	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
107	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
110	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
111	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
112	#95	C₂²×C₆₇	1	4	yes	—
113	#95	C₂²×C₆₇	1	4	yes	—
114	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	Q₈×C₆₇	1	4	yes	—
118	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
119	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
120	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
121	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
122	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	2	8	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	4	4	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	4	4	no	—
132	C₄²	—	2	2	no	—
133	C₄²	D₄×C₆₇	1	4	yes	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	1	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₂×Q₈×C₆₇	1	2	yes	—
139	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
155	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
156	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
157	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
158	C₂×C₄	#41×C₆₇	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	4	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
165	C₂²	C₂²×Q₈×C₆₇	1	1	yes	characteristic
166	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
167	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
168	C₂²	2₋¹⁺⁴×C₆₇	1	2	yes	—
169	C₄	—	2	2	no	—
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	4	no	—
173	C₄	—	4	4	no	—
174	C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
175	C₂	—	4	2	no	—
176	C₂	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
177	C₂	#537×C₆₇	1	2	yes	—
178	C₂	#540×C₆₇	1	1	yes	characteristic
179	C₂	#541×C₆₇	1	1	yes	characteristic
180	C₂	#544×C₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#117×C₆₇

C₂

yes

maximal, characteristic

#398×C₆₇

C₂

yes

maximal, complemented

#396×C₆₇

C₂

yes

maximal

#389×C₆₇

C₂

yes

maximal, complemented

#384×C₆₇

C₂

yes

maximal, characteristic, complemented

#380×C₆₇

C₂

yes

maximal

C₂³×C₄×C₆₇

C₂²

yes

characteristic

#125×C₆₇

—

#117×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

C₂⁴×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

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×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₇

#537

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2629

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#2629

yes

characteristic, direct product factor, Sylow

C₂×#125

C₂×C₆₇

yes

characteristic, complemented

C₂×#117

C₂×C₆₇

yes

characteristic

#398

C₂×C₆₇

yes

complemented

#396

C₂×C₆₇

yes

—

#389

C₂×C₆₇

yes

complemented

#384

C₂×C₆₇

yes

characteristic, complemented

#380

C₂×C₆₇

yes

—

C₂³×C₄

C₂²×C₆₇

yes

characteristic

100

#125

—

101

#117

—

102

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

103

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

104

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

105

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

106

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

107

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

108

C₂×C₂²⋊C₄

C₂²×C₆₇

yes

—

109

C₂×C₂²⋊C₄

C₂²×C₆₇

yes

—

110

C₂×C₄²

C₂²×C₆₇

yes

characteristic

111

C₂×C₄²

C₂²×C₆₇

yes

characteristic

112

#95

C₂²×C₆₇

yes

—

113

#95

C₂²×C₆₇

yes

—

114

C₂⁴

C₂³×C₆₇

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

Q₈×C₆₇

yes

—

118

C₂²×C₄

C₂³×C₆₇

yes

—

119

C₂²×C₄

C₂³×C₆₇

yes

—

120

C₂²×C₄

C₂³×C₆₇

yes

—

121

C₂²×C₄

C₂³×C₆₇

yes

—

122

C₂²×C₄

C₂³×C₆₇

yes

characteristic

123

C₂²×C₄

C₂³×C₆₇

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

D₄×C₆₇

yes

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂×Q₈×C₆₇

yes

—

139

C₂³

C₂⁴×C₆₇

yes

characteristic

140

C₂×C₄

—

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

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×D₄×C₆₇

yes

—

155

C₂×C₄

C₂×D₄×C₆₇

yes

—

156

C₂×C₄

C₂×Q₈×C₆₇

yes

—

157

C₂×C₄

C₂×Q₈×C₆₇

yes

—

158

C₂×C₄

#41×C₆₇

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂²×D₄×C₆₇

yes

characteristic

165

C₂²

C₂²×Q₈×C₆₇

yes

characteristic

166

C₂²

C₂×#41×C₆₇

yes

characteristic

167

C₂²

H(5,𝔽₂)×C₆₇

yes

—

168

C₂²

2₋¹⁺⁴×C₆₇

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

C₂²⋊Q₈×C₆₇

yes

—

175

C₂

—

176

C₂

C₂×C₂²⋊Q₈×C₆₇

yes

characteristic

177

C₂

#537×C₆₇

yes

—

178

C₂

#540×C₆₇

yes

characteristic

179

C₂

#541×C₆₇

yes

characteristic

180

C₂

#544×C₆₇

yes

—

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

subgroup

G/H

center, socle

C₂³×C₆₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₆₇

prime radical

C₂⁴×C₆₇

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ G

N	H ≅ G/N	type
1. C₂×#125×C₆₇	C₂	semidirect
3. #398×C₆₇	C₂	semidirect
5. #389×C₆₇	C₂	semidirect
6. #384×C₆₇	C₂	semidirect
90. #2629	C₆₇	direct (N × H)
91. C₆₇	#2629	direct (N × H)
92. C₂×#125	C₂×C₆₇	semidirect
94. #398	C₂×C₆₇	semidirect
96. #389	C₂×C₆₇	semidirect
97. #384	C₂×C₆₇	semidirect

H ≅ G/N

type

1. C₂×#125×C₆₇

C₂

semidirect

3. #398×C₆₇

C₂

semidirect

5. #389×C₆₇

C₂

semidirect

6. #384×C₆₇

C₂

semidirect

90. #2629

C₆₇

direct (N × H)

91. C₆₇

#2629

direct (N × H)

92. C₂×#125

C₂×C₆₇

semidirect

94. #398

C₂×C₆₇

semidirect

96. #389

C₂×C₆₇

semidirect

97. #384

C₂×C₆₇

semidirect

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

subgroup	index
C₂×#125×C₆₇	2
C₂×#117×C₆₇	2
#398×C₆₇	2
#396×C₆₇	2
#389×C₆₇	2
#384×C₆₇	2
#380×C₆₇	2
#2629	67

subgroup

index

C₂×#125×C₆₇

C₂×#117×C₆₇

#398×C₆₇

#396×C₆₇

#389×C₆₇

#384×C₆₇

#380×C₆₇

#2629

Subgroups — C₆₇×#2629 #22557060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#2629 #22557060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2629 #22557060