Group #22561756 — subgroups

Subgroups — #22561756

Order 8576.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14523977	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523975	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523968	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523968	C₂	1	1	yes	maximal, characteristic
5	#16863302	C₂	1	1	yes	maximal, characteristic
6	#16863262	C₂	1	1	yes	maximal, characteristic
7	#336×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	#14523977	—	2	2	no	—
13	#14523975	—	2	2	no	—
14	#14523968	—	2	2	no	—
15	#14523968	—	2	2	no	—
16	C₆₇⋊#95	—	2	2	no	—
17	C₆₇⋊#95	—	2	2	no	—
18	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	2	1	no	—
22	C₂×Dic₁₃₄	—	2	1	no	—
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	—	2	1	no	—
25	C₂×Dic₁₃₄	—	2	1	no	—
26	C₂×Dic₁₃₄	Q₈	1	2	yes	—
27	C₂²×Dic₆₇	—	2	1	no	—
28	C₂²×Dic₆₇	—	2	1	no	—
29	C₂²×Dic₆₇	—	2	1	no	—
30	C₂²×Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₄⋊Dic₆₇	—	2	1	no	—
35	C₄⋊Dic₆₇	—	2	1	no	—
36	C₄⋊Dic₆₇	—	2	1	no	—
37	C₂²×C₄×C₆₇	D₄	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₂×C₈×C₆₇	—	2	1	no	—
43	C₂×C₈×C₆₇	D₄	1	2	yes	—
44	C₂×C₈×C₆₇	D₄	1	2	yes	—
45	C₄²×C₆₇	—	2	1	no	—
46	C₄²×C₆₇	—	2	1	no	—
47	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
57	C₂×Dic₆₇	—	2	2	no	—
58	C₂×Dic₆₇	—	4	1	no	—
59	C₂×Dic₆₇	—	4	1	no	—
60	C₂×Dic₆₇	—	4	1	no	—
61	C₂×Dic₆₇	—	4	1	no	—
62	C₂×Dic₆₇	—	4	1	no	—
63	C₂×Dic₆₇	—	4	2	no	—
64	C₂×Dic₆₇	—	4	2	no	—
65	C₂×Dic₆₇	—	4	2	no	—
66	Dic₁₃₄	—	2	2	no	—
67	Dic₁₃₄	—	2	2	no	—
68	Dic₁₃₄	—	4	1	no	—
69	Dic₁₃₄	—	4	1	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	—	2	2	no	—
72	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂²×C₆₇	#123	1	1	yes	characteristic
74	C₂²×C₆₇	#127	1	1	yes	characteristic
75	C₂²×C₆₇	#135	1	2	yes	—
76	C₂²×C₆₇	#137	1	2	yes	—
77	C₄×C₆₇	—	4	2	no	—
78	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄×C₆₇	#123	1	1	yes	characteristic
81	C₄×C₆₇	#123	1	1	yes	characteristic
82	Dic₆₇	—	4	2	no	—
83	Dic₆₇	—	8	1	no	—
84	Dic₆₇	—	8	1	no	—
85	Dic₆₇	—	8	1	no	—
86	C₂×C₆₇	#399	1	1	yes	characteristic
87	C₂×C₆₇	#479	1	2	yes	—
88	C₂×C₆₇	#485	1	2	yes	—
89	C₂×C₆₇	#495	1	1	yes	characteristic
90	C₂×C₆₇	#497	1	1	yes	characteristic
91	#2097	—	67	1	no	maximal, Sylow
92	C₆₇	#2097	1	1	yes	characteristic, complemented, Sylow
93	C₂×#107	—	67	1	no	—
94	C₂×#106	—	67	1	no	—
95	C₂×#103	—	67	1	no	—
96	C₂×#103	—	67	1	no	—
97	#391	—	67	1	no	—
98	#382	—	67	1	no	—
99	#336	D₆₇	1	1	yes	characteristic
100	C₂²×Q₈	—	67	1	no	—
101	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	67	1	no	—
104	C₂×C₄⋊C₄	—	67	1	no	—
105	C₂×C₄⋊C₄	—	67	1	no	—
106	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
107	#107	—	134	2	no	—
108	#106	—	134	2	no	—
109	#103	—	134	2	no	—
110	#103	—	134	2	no	—
111	#95	—	134	2	no	—
112	#95	—	134	2	no	—
113	C₂×Q₈	—	67	2	no	—
114	C₂×Q₈	—	134	1	no	—
115	C₂×Q₈	—	134	1	no	—
116	C₂×Q₈	—	134	1	no	—
117	C₂×Q₈	—	134	1	no	—
118	C₂×Q₈	—	134	1	no	—
119	C₂²×C₄	—	134	1	no	—
120	C₂²×C₄	—	134	1	no	—
121	C₂²×C₄	—	134	1	no	—
122	C₂²×C₄	—	134	1	no	—
123	C₂²×C₄	D₂₆₈	1	2	yes	—
124	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	D₂₆₈	1	2	yes	—
130	C₂×C₈	D₂₆₈	1	2	yes	—
131	C₄⋊C₄	—	134	1	no	—
132	C₄⋊C₄	—	134	1	no	—
133	C₄⋊C₄	—	134	1	no	—
134	C₄⋊C₄	—	134	1	no	—
135	C₄⋊C₄	—	134	1	no	—
136	C₄⋊C₄	—	134	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
140	Q₈	—	268	1	no	—
141	Q₈	—	268	1	no	—
142	Q₈	—	134	2	no	—
143	Q₈	—	134	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	268	1	no	—
148	C₂×C₄	—	268	1	no	—
149	C₂×C₄	—	268	1	no	—
150	C₂×C₄	—	268	1	no	—
151	C₂×C₄	—	268	1	no	—
152	C₂×C₄	—	268	2	no	—
153	C₂×C₄	—	268	2	no	—
154	C₂×C₄	—	268	2	no	—
155	C₂×C₄	—	134	2	no	—
156	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
157	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
158	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
159	C₂×C₄	#11761627	1	1	yes	characteristic
160	C₂×C₄	#11761627	1	1	yes	characteristic
161	C₂×C₄	#11761627	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₂²	#14524023	1	1	yes	characteristic
165	C₂²	#14524027	1	1	yes	characteristic
166	C₂²	#14524040	1	1	yes	characteristic
167	C₂²	#14524065	1	2	yes	—
168	C₂²	#14524076	1	2	yes	—
169	C₄	—	4	2	no	—
170	C₄	—	268	2	no	—
171	C₄	—	536	1	no	—
172	C₄	—	536	1	no	—
173	C₄	—	536	1	no	—
174	C₄	#14524023	1	1	yes	characteristic
175	C₄	#14524023	1	1	yes	characteristic
176	C₄	#14524027	1	1	yes	characteristic
177	C₄	#14524027	1	1	yes	characteristic
178	C₂	#16863333	1	1	yes	characteristic
179	C₂	#16863704	1	2	yes	—
180	C₂	#16863738	1	2	yes	—
181	C₂	#16863780	1	1	yes	characteristic
182	C₂	#16863790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523977

C₂

yes

maximal, characteristic

C₂×#14523975

C₂

yes

maximal, characteristic

C₂×#14523968

C₂

yes

maximal, characteristic

C₂×#14523968

C₂

yes

maximal, characteristic

#16863302

C₂

yes

maximal, characteristic

#16863262

C₂

yes

maximal, characteristic

#336×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

—

—

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

Q₈

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

C₂²⋊Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#399

yes

characteristic

C₂×C₆₇

#479

yes

—

C₂×C₆₇

#485

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic

100

C₂²×Q₈

—

101

C₂²×C₈

C₂×D₆₇

yes

characteristic

102

C₂²×C₈

C₂×D₆₇

yes

characteristic

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂×D₆₇

yes

characteristic

107

#107

—

134

—

108

#106

—

134

—

109

#103

—

134

—

110

#103

—

134

—

111

#95

—

134

—

112

#95

—

134

—

113

C₂×Q₈

—

114

C₂×Q₈

—

134

—

115

C₂×Q₈

—

134

—

116

C₂×Q₈

—

134

—

117

C₂×Q₈

—

134

—

118

C₂×Q₈

—

134

—

119

C₂²×C₄

—

134

—

120

C₂²×C₄

—

134

—

121

C₂²×C₄

—

134

—

122

C₂²×C₄

—

134

—

123

C₂²×C₄

D₂₆₈

yes

—

124

C₂²×C₄

C₂²×D₆₇

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

D₂₆₈

yes

—

130

C₂×C₈

D₂₆₈

yes

—

131

C₄⋊C₄

—

134

—

132

C₄⋊C₄

—

134

—

133

C₄⋊C₄

—

134

—

134

C₄⋊C₄

—

134

—

135

C₄⋊C₄

—

134

—

136

C₄⋊C₄

—

134

—

137

C₄²

—

138

C₄²

—

139

C₂³

C₂×D₂₆₈

yes

characteristic

140

Q₈

—

268

—

141

Q₈

—

268

—

142

Q₈

—

134

—

143

Q₈

—

134

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

268

—

148

C₂×C₄

—

268

—

149

C₂×C₄

—

268

—

150

C₂×C₄

—

268

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

268

—

153

C₂×C₄

—

268

—

154

C₂×C₄

—

268

—

155

C₂×C₄

—

134

—

156

C₂×C₄

C₂×D₂₆₈

yes

characteristic

157

C₂×C₄

C₂×D₂₆₈

yes

characteristic

158

C₂×C₄

Q₈×D₆₇

yes

characteristic

159

C₂×C₄

#11761627

yes

characteristic

160

C₂×C₄

#11761627

yes

characteristic

161

C₂×C₄

#11761627

yes

characteristic

162

C₈

—

163

C₈

—

164

C₂²

#14524023

yes

characteristic

165

C₂²

#14524027

yes

characteristic

166

C₂²

#14524040

yes

characteristic

167

C₂²

#14524065

yes

—

168

C₂²

#14524076

yes

—

169

C₄

—

170

C₄

—

268

—

171

C₄

—

536

—

172

C₄

—

536

—

173

C₄

—

536

—

174

C₄

#14524023

yes

characteristic

175

C₄

#14524023

yes

characteristic

176

C₄

#14524027

yes

characteristic

177

C₄

#14524027

yes

characteristic

178

C₂

#16863333

yes

characteristic

179

C₂

#16863704

yes

—

180

C₂

#16863738

yes

—

181

C₂

#16863780

yes

characteristic

182

C₂

#16863790

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#336×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	#336	D₆₇
hypocenter	C₆₇	#2097
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#336×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
92. C₆₇	#2097	semidirect

H ≅ G/N

type

92. C₆₇

#2097

semidirect

Subgroups — #22561756

Conjugacy classes of subgroups

Complements