Group #22560243 — subgroups

Subgroups — #22560243

Order 8576.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×#11761611	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523975	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863253	C₂	1	2	yes	maximal
4	#16863158	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₆₇	C₂²	1	1	yes	characteristic
7	C₂×#11761611	—	2	2	no	—
8	C₂×#11761611	—	2	2	no	—
9	C₂×#11761611	C₄	1	4	yes	complemented
10	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
13	#14523975	—	2	2	no	—
14	C₆₇⋊#95	—	2	2	no	—
15	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	#107×C₆₇	—	2	2	no	—
18	C₂³×D₆₇	D₄	1	1	yes	characteristic
19	C₂²×Dic₆₇	—	2	2	no	—
20	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄×D₆₇	—	2	1	no	—
22	C₂×C₄×D₆₇	—	2	1	no	—
23	C₂×C₄×D₆₇	—	2	1	no	—
24	C₂×C₄×D₆₇	—	2	1	no	—
25	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₆₇	Q₈	1	2	yes	—
27	#11761611	—	2	8	no	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	—	2	1	no	—
30	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₆₇	—	2	2	no	—
32	C₄⋊Dic₆₇	—	2	2	no	—
33	(C₂×D₆₇)⋊C₄	—	4	4	no	—
34	C₂²×C₄×C₆₇	—	2	2	no	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₆₇	D₄	1	2	yes	—
38	C₂×C₈×C₆₇	D₄	1	2	yes	—
39	C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₆₇	—	2	1	no	—
43	C₂²×D₆₇	—	2	1	no	—
44	C₂²×D₆₇	—	2	2	no	—
45	C₂²×D₆₇	—	2	2	no	—
46	C₂²×D₆₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₆₇	—	4	2	no	—
48	C₂×C₄×C₆₇	—	4	2	no	—
49	C₂×C₄×C₆₇	—	4	2	no	—
50	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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×Dic₆₇	—	2	1	no	—
57	C₂×Dic₆₇	—	2	1	no	—
58	C₂×Dic₆₇	—	4	2	no	—
59	C₂×Dic₆₇	—	4	2	no	—
60	C₂×Dic₆₇	—	4	2	no	—
61	C₂×Dic₆₇	C₄⋊C₄	1	2	yes	—
62	C₄×D₆₇	—	2	2	no	—
63	C₄×D₆₇	—	2	2	no	—
64	C₄×D₆₇	—	2	2	no	—
65	C₄×D₆₇	—	2	2	no	—
66	C₈×C₆₇	C₂²⋊C₄	1	4	yes	complemented
67	C₆₇⋊C₈	—	2	2	no	—
68	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₆₇	#121	1	1	yes	characteristic
70	C₂²×C₆₇	#121	1	1	yes	characteristic
71	C₂²×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²×C₆₇	#137	1	2	yes	—
73	C₂×D₆₇	—	2	2	no	—
74	C₂×D₆₇	—	2	4	no	—
75	C₂×D₆₇	—	4	2	no	—
76	C₂×D₆₇	—	4	1	no	—
77	C₂×D₆₇	—	4	1	no	—
78	C₄×C₆₇	—	8	2	no	—
79	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₆₇	#117	1	1	yes	characteristic
81	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
83	Dic₆₇	—	2	2	no	—
84	Dic₆₇	—	8	2	no	—
85	C₂×C₆₇	#380	1	1	yes	characteristic
86	C₂×C₆₇	#428	1	2	yes	—
87	C₂×C₆₇	#468	1	2	yes	—
88	C₂×C₆₇	#469	1	1	yes	characteristic
89	C₂×C₆₇	#470	1	1	yes	characteristic
90	D₆₇	—	4	2	no	—
91	#1809	—	67	1	no	maximal, Sylow
92	C₆₇	#1809	1	1	yes	characteristic, complemented, Sylow
93	C₂²×M₁₆	—	67	1	no	—
94	C₂×#107	D₆₇	1	1	yes	characteristic, complemented
95	C₂×#106	—	67	1	no	—
96	#380	—	67	2	no	—
97	#340	—	67	2	no	—
98	C₂³×C₄	—	67	1	no	—
99	C₂×M₁₆	—	67	4	no	—
100	C₂×M₁₆	—	134	2	no	—
101	C₂×M₁₆	—	134	2	no	—
102	C₂²×C₈	—	67	1	no	—
103	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	67	2	no	—
105	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
106	C₂×C₂²⋊C₄	—	134	2	no	—
107	#107	—	2	2	no	—
108	#106	—	134	2	no	—
109	#95	—	134	2	no	—
110	C₂⁴	—	67	1	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	67	1	no	—
113	C₂²×C₄	—	67	2	no	—
114	C₂²×C₄	—	67	2	no	—
115	C₂²×C₄	—	134	2	no	—
116	C₂²×C₄	—	134	1	no	—
117	C₂²×C₄	—	134	1	no	—
118	C₂²×C₄	—	134	1	no	—
119	C₂²×C₄	—	134	1	no	—
120	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
121	M₁₆	—	134	8	no	—
122	C₂×C₈	—	67	2	no	—
123	C₂×C₈	—	134	1	no	—
124	C₂×C₈	—	134	1	no	—
125	C₂×C₈	C₄×D₆₇	1	2	yes	—
126	C₂×C₈	D₂₆₈	1	2	yes	—
127	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	134	2	no	—
131	C₄⋊C₄	—	134	2	no	—
132	C₂²⋊C₄	—	268	4	no	—
133	C₂³	—	67	2	no	—
134	C₂³	—	134	1	no	—
135	C₂³	—	134	1	no	—
136	C₂³	—	134	2	no	—
137	C₂³	—	134	2	no	—
138	C₂³	D₄×D₆₇	1	1	yes	characteristic
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	268	2	no	—
143	C₂×C₄	—	268	2	no	—
144	C₂×C₄	—	268	2	no	—
145	C₂×C₄	—	67	2	no	—
146	C₂×C₄	—	134	1	no	—
147	C₂×C₄	—	134	1	no	—
148	C₂×C₄	—	134	2	no	—
149	C₂×C₄	—	134	2	no	—
150	C₂×C₄	—	134	2	no	—
151	C₂×C₄	—	134	2	no	—
152	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
154	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
155	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
156	C₂×C₄	#11761627	1	1	yes	characteristic
157	C₂×C₄	#11761629	1	1	yes	characteristic
158	C₈	—	134	2	no	—
159	C₈	(C₂×D₆₇)⋊C₄	1	4	yes	complemented
160	C₂²	—	268	2	no	—
161	C₂²	—	268	1	no	—
162	C₂²	—	268	1	no	—
163	C₂²	—	134	2	no	—
164	C₂²	—	134	4	no	—
165	C₂²	C₄⋊C₄×D₆₇	1	1	yes	characteristic
166	C₂²	#14524014	1	1	yes	characteristic
167	C₂²	#14524018	1	1	yes	characteristic
168	C₂²	#14524067	1	2	yes	—
169	C₂²	#14524075	1	2	yes	—
170	C₄	—	8	2	no	—
171	C₄	—	536	2	no	—
172	C₄	—	134	2	no	—
173	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
174	C₄	#14524001	1	1	yes	characteristic
175	C₄	#14524019	1	1	yes	characteristic
176	C₄	#14524023	1	1	yes	characteristic
177	C₂	—	268	2	no	—
178	C₂	#16863253	1	1	yes	characteristic
179	C₂	#16863437	1	2	yes	—
180	C₂	#16863643	1	2	yes	—
181	C₂	#16863646	1	1	yes	characteristic
182	C₂	#16863651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761611

C₂

yes

maximal, characteristic

C₂×#14523975

C₂

yes

maximal, characteristic, complemented

#16863253

C₂

yes

maximal

#16863158

C₂

yes

maximal, complemented

C₂×#107×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₆₇

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

—

#14523975

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#107×C₆₇

—

C₂³×D₆₇

D₄

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

characteristic

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

C₂×C₄×D₆₇

Q₈

yes

—

#11761611

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

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×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₄⋊C₄

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

C₂²⋊C₄

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#137

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#380

yes

characteristic

C₂×C₆₇

#428

yes

—

C₂×C₆₇

#468

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1809

—

maximal, Sylow

C₆₇

#1809

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×#107

D₆₇

yes

characteristic, complemented

C₂×#106

—

#380

—

#340

—

C₂³×C₄

—

C₂×M₁₆

—

100

C₂×M₁₆

—

134

—

101

C₂×M₁₆

—

134

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂×D₆₇

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

106

C₂×C₂²⋊C₄

—

134

—

107

#107

—

108

#106

—

134

—

109

#95

—

134

—

110

C₂⁴

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

134

—

116

C₂²×C₄

—

134

—

117

C₂²×C₄

—

134

—

118

C₂²×C₄

—

134

—

119

C₂²×C₄

—

134

—

120

C₂²×C₄

C₂²×D₆₇

yes

characteristic

121

M₁₆

—

134

—

122

C₂×C₈

—

123

C₂×C₈

—

134

—

124

C₂×C₈

—

134

—

125

C₂×C₈

C₄×D₆₇

yes

—

126

C₂×C₈

D₂₆₈

yes

—

127

C₂×C₈

C₂²⋊D₆₇

yes

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

134

—

131

C₄⋊C₄

—

134

—

132

C₂²⋊C₄

—

268

—

133

C₂³

—

134

C₂³

—

134

—

135

C₂³

—

134

—

136

C₂³

—

134

—

137

C₂³

—

134

—

138

C₂³

D₄×D₆₇

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

268

—

143

C₂×C₄

—

268

—

144

C₂×C₄

—

268

—

145

C₂×C₄

—

146

C₂×C₄

—

134

—

147

C₂×C₄

—

134

—

148

C₂×C₄

—

134

—

149

C₂×C₄

—

134

—

150

C₂×C₄

—

134

—

151

C₂×C₄

—

134

—

152

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

153

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

154

C₂×C₄

C₂×D₂₆₈

yes

characteristic

155

C₂×C₄

Q₈×D₆₇

yes

characteristic

156

C₂×C₄

#11761627

yes

characteristic

157

C₂×C₄

#11761629

yes

characteristic

158

C₈

—

134

—

159

C₈

(C₂×D₆₇)⋊C₄

yes

complemented

160

C₂²

—

268

—

161

C₂²

—

268

—

162

C₂²

—

268

—

163

C₂²

—

134

—

164

C₂²

—

134

—

165

C₂²

C₄⋊C₄×D₆₇

yes

characteristic

166

C₂²

#14524014

yes

characteristic

167

C₂²

#14524018

yes

characteristic

168

C₂²

#14524067

yes

—

169

C₂²

#14524075

yes

—

170

C₄

—

171

C₄

—

536

—

172

C₄

—

134

—

173

C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

174

C₄

#14524001

yes

characteristic

175

C₄

#14524019

yes

characteristic

176

C₄

#14524023

yes

characteristic

177

C₂

—

268

—

178

C₂

#16863253

yes

characteristic

179

C₂

#16863437

yes

—

180

C₂

#16863643

yes

—

181

C₂

#16863646

yes

characteristic

182

C₂

#16863651

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#107×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#107	D₆₇
hypocenter	C₆₇	#1809
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#107×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#14523975	C₂	semidirect
4. #16863158	C₂	semidirect
5. C₂×#107×C₆₇	C₂	semidirect
9. C₂×#11761611	C₄	semidirect
66. C₈×C₆₇	C₂²⋊C₄	semidirect
92. C₆₇	#1809	semidirect
94. C₂×#107	D₆₇	semidirect
159. C₈	(C₂×D₆₇)⋊C₄	semidirect

H ≅ G/N

type

2. C₂×#14523975

C₂

semidirect

4. #16863158

C₂

semidirect

5. C₂×#107×C₆₇

C₂

semidirect

9. C₂×#11761611

C₄

semidirect

66. C₈×C₆₇

C₂²⋊C₄

semidirect

92. C₆₇

#1809

semidirect

94. C₂×#107

D₆₇

semidirect

159. C₈

(C₂×D₆₇)⋊C₄

semidirect

Subgroups — #22560243

Conjugacy classes of subgroups

Complements