Group #22569308 — subgroups

Subgroups — #22569308

Order 8576.

330 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 161 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	#16863949	C₂	1	1	yes	maximal, characteristic
2	#16863525	C₂	1	2	yes	maximal, complemented
3	#16863524	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
5	#16863520	C₂	1	2	yes	maximal, complemented
6	#16863457	C₂	1	1	yes	maximal, characteristic
7	#16863454	C₂	1	2	yes	maximal
8	#16863446	C₂	1	2	yes	maximal
9	#433×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#14524042	C₄	1	2	yes	—
11	#14524034	C₄	1	2	yes	—
12	#14524020	C₄	1	2	yes	—
13	#14524020	C₄	1	2	yes	—
14	C₄×Dic₁₃₄	C₂²	1	2	yes	—
15	C₄×Dic₁₃₄	C₂²	1	2	yes	—
16	#14524002	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	Dic₆₇⋊C₈	C₂²	1	2	yes	—
20	Dic₆₇⋊C₈	C₂²	1	2	yes	—
21	Dic₆₇⋊C₈	C₂²	1	2	yes	—
22	Dic₆₇⋊C₈	C₂²	1	2	yes	—
23	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
24	#14523954	C₂²	1	1	yes	characteristic
25	#14523953	C₂²	1	2	yes	—
26	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₈×Dic₆₇	C₂²	1	2	yes	—
28	C₈×Dic₆₇	C₂²	1	2	yes	—
29	C₈×Dic₆₇	C₂²	1	2	yes	—
30	#117×C₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
33	#97×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
36	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
37	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
43	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
44	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
45	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
46	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
47	C₂²⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₆₇	C₂³	1	2	yes	—
52	C₄×Dic₆₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂×C₈×C₆₇	C₂³	1	2	yes	—
56	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₆₇	—	2	4	no	—
67	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
68	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
70	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
71	Dic₁₃₄	—	2	4	no	—
72	Dic₁₃₄	—	2	4	no	—
73	C₈×C₆₇	—	2	2	no	—
74	C₈×C₆₇	—	2	2	no	—
75	C₆₇⋊C₈	—	2	2	no	—
76	C₆₇⋊C₈	—	2	2	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	—	2	2	no	—
79	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
84	Dic₆₇	—	2	2	no	—
85	Dic₆₇	—	2	2	no	—
86	Dic₆₇	—	2	2	no	—
87	Dic₆₇	#131	1	4	yes	—
88	C₂×C₆₇	—	4	1	no	—
89	C₂×C₆₇	#519	1	1	yes	characteristic
90	C₂×C₆₇	C₂×#131	1	2	yes	—
91	#2986	—	67	1	no	maximal, Sylow
92	C₆₇	#2986	1	1	yes	characteristic, complemented, Sylow
93	#533	—	67	1	no	—
94	#446	—	67	2	no	—
95	#446	—	67	2	no	—
96	C₈×Q₈	—	67	2	no	—
97	C₈×Q₈	—	67	2	no	—
98	#433	—	67	1	no	—
99	#433	—	67	2	no	—
100	#433	D₆₇	1	1	yes	characteristic
101	#431	—	67	2	no	—
102	C₄⋊Q₈	—	67	2	no	—
103	#125	—	67	2	no	—
104	C₂²⋊Q₈	—	67	2	no	—
105	C₂²⋊Q₈	—	67	2	no	—
106	C₄×Q₈	—	67	2	no	—
107	C₄×Q₈	—	67	2	no	—
108	#117	—	67	1	no	—
109	#117	C₂×D₆₇	1	1	yes	characteristic
110	C₂×C₄²	—	67	1	no	—
111	C₄⋊C₈	—	67	2	no	—
112	C₄⋊C₈	—	67	2	no	—
113	C₄⋊C₈	—	67	2	no	—
114	C₄⋊C₈	—	67	2	no	—
115	C₄⋊C₈	—	67	2	no	—
116	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
117	C₂²⋊C₈	—	67	2	no	—
118	C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
119	#97	—	67	2	no	—
120	#97	—	67	1	no	—
121	#97	C₂×D₆₇	1	1	yes	characteristic
122	C₄×C₈	—	67	2	no	—
123	C₄×C₈	—	67	2	no	—
124	C₄×C₈	—	67	2	no	—
125	C₄×C₈	—	67	1	no	—
126	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
127	C₂×Q₈	—	67	2	no	—
128	C₂×Q₈	—	67	2	no	—
129	C₂²×C₄	—	67	2	no	—
130	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
131	C₂×C₈	—	67	2	no	—
132	C₂×C₈	—	67	2	no	—
133	C₂×C₈	C₂²×D₆₇	1	2	yes	—
134	C₂×C₈	C₂²×D₆₇	1	2	yes	—
135	C₄⋊C₄	—	67	2	no	—
136	C₄⋊C₄	—	67	2	no	—
137	C₄⋊C₄	—	67	2	no	—
138	C₄⋊C₄	—	67	2	no	—
139	C₄⋊C₄	—	67	2	no	—
140	C₄⋊C₄	—	67	2	no	—
141	C₄⋊C₄	—	67	2	no	—
142	C₄⋊C₄	C₄×D₆₇	1	2	yes	—
143	C₂²⋊C₄	—	67	1	no	—
144	C₂²⋊C₄	—	67	1	no	—
145	C₂²⋊C₄	C₄×D₆₇	1	2	yes	—
146	C₄²	—	67	1	no	—
147	C₄²	—	67	1	no	—
148	C₄²	—	67	2	no	—
149	C₄²	—	67	2	no	—
150	C₄²	C₂²×D₆₇	1	1	yes	characteristic
151	C₄²	C₂²×D₆₇	1	1	yes	characteristic
152	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
153	Q₈	—	134	4	no	—
154	Q₈	—	134	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	67	2	no	—
157	C₂×C₄	—	67	2	no	—
158	C₂×C₄	—	67	2	no	—
159	C₂×C₄	—	67	2	no	—
160	C₂×C₄	—	134	4	no	—
161	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
163	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
164	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	134	2	no	—
168	C₈	—	134	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	67	4	no	—
176	C₄	—	134	2	no	—
177	C₄	—	134	2	no	—
178	C₄	—	134	2	no	—
179	C₄	#14524090	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#16863873	1	1	yes	characteristic
182	C₂	#131×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863949

C₂

yes

maximal, characteristic

#16863525

C₂

yes

maximal, complemented

#16863524

C₂

yes

maximal, complemented

C₈×Dic₁₃₄

C₂

yes

maximal, complemented

#16863520

C₂

yes

maximal, complemented

#16863457

C₂

yes

maximal, characteristic

#16863454

C₂

yes

maximal

#16863446

C₂

yes

maximal

#433×C₆₇

C₂

yes

maximal, characteristic

#14524042

C₄

yes

—

#14524034

C₄

yes

—

#14524020

C₄

yes

—

#14524020

C₄

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524002

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523954

C₂²

yes

characteristic

#14523953

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

C₈×Dic₆₇

C₂²

yes

—

#117×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂²⋊C₈×C₆₇

C₂²

yes

—

#97×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂²×Dic₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

Dic₆₇⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₆₇

C₂×C₄

yes

characteristic

C₂²⋊Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₄×Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

2₋¹⁺⁴

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

#131

yes

—

C₂×C₆₇

—

C₂×C₆₇

#519

yes

characteristic

C₂×C₆₇

C₂×#131

yes

—

#2986

—

maximal, Sylow

C₆₇

#2986

yes

characteristic, complemented, Sylow

#533

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

#433

—

#433

—

100

#433

D₆₇

yes

characteristic

101

#431

—

102

C₄⋊Q₈

—

103

#125

—

104

C₂²⋊Q₈

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

C₄×Q₈

—

108

#117

—

109

#117

C₂×D₆₇

yes

characteristic

110

C₂×C₄²

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

C₂×D₆₇

yes

—

117

C₂²⋊C₈

—

118

C₂²⋊C₈

C₂×D₆₇

yes

—

119

#97

—

120

#97

—

121

#97

C₂×D₆₇

yes

characteristic

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₂×D₆₇

yes

characteristic

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂²×D₆₇

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

C₂²×D₆₇

yes

—

134

C₂×C₈

C₂²×D₆₇

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

C₄×D₆₇

yes

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

—

145

C₂²⋊C₄

C₄×D₆₇

yes

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

C₂²×D₆₇

yes

characteristic

151

C₄²

C₂²×D₆₇

yes

characteristic

152

C₂³

C₂×C₄×D₆₇

yes

characteristic

153

Q₈

—

134

—

154

Q₈

—

134

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

134

—

161

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

162

C₂×C₄

C₂×C₄×D₆₇

yes

—

163

C₂×C₄

C₂×C₄×D₆₇

yes

—

164

C₂×C₄

C₂³×D₆₇

yes

characteristic

165

C₈

—

166

C₈

—

167

C₈

—

134

—

168

C₈

—

134

—

169

C₂²

—

170

C₂²

—

171

C₂²

C₂²×C₄×D₆₇

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

134

—

177

C₄

—

134

—

178

C₄

—

134

—

179

C₄

#14524090

yes

—

180

C₂

—

181

C₂

#16863873

yes

characteristic

182

C₂

#131×D₆₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#433×C₆₇	C₂
hypercenter	#433	D₆₇
hypocenter	C₆₇	#2986
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₇

commutator, socle

C₂²×C₆₇

C₂³×C₄

Fitting

#433×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #433

N	H ≅ G/N	type
2. #16863525	C₂	semidirect
3. #16863524	C₂	semidirect
4. C₈×Dic₁₃₄	C₂	semidirect
5. #16863520	C₂	semidirect
92. C₆₇	#2986	semidirect

H ≅ G/N

type

2. #16863525

C₂

semidirect

3. #16863524

C₂

semidirect

4. C₈×Dic₁₃₄

C₂

semidirect

5. #16863520

C₂

semidirect

92. C₆₇

#2986

semidirect

Subgroups — #22569308

Conjugacy classes of subgroups

Complements