Group #22561576 — subgroups

Subgroups — #22561576

Order 8576.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14523978	C₂	1	1	yes	maximal, characteristic
2	#16863439	C₂	1	1	yes	maximal, characteristic
3	#16863423	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523967	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523962	C₂	1	1	yes	maximal, characteristic
6	#16863164	C₂	1	1	yes	maximal, characteristic
7	#419×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#11761612	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	#14523978	—	2	2	no	—
13	#14523976	—	2	1	no	—
14	#14523974	—	2	1	no	—
15	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
16	#14523967	—	2	2	no	—
17	#14523962	—	2	2	no	—
18	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	#117×C₆₇	C₂²	1	1	yes	characteristic
21	#103×C₆₇	—	2	1	no	—
22	#103×C₆₇	—	2	1	no	—
23	C₂²×Dic₆₇	—	2	1	no	—
24	#11761612	—	2	1	no	—
25	#11761612	—	2	1	no	—
26	#11761612	—	2	1	no	—
27	#11761612	D₄	1	2	yes	—
28	C₂×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₆₇⋊C₈	—	2	1	no	—
30	C₂×C₆₇⋊C₈	—	2	1	no	—
31	C₂×C₆₇⋊C₈	—	2	1	no	—
32	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₄⋊Dic₆₇	—	2	1	no	—
35	C₂×Q₈×C₆₇	—	2	1	no	—
36	C₂×Q₈×C₆₇	—	2	1	no	—
37	C₂×Q₈×C₆₇	—	2	1	no	—
38	C₂×Q₈×C₆₇	—	2	1	no	—
39	C₂×Q₈×C₆₇	—	2	1	no	—
40	C₂×Q₈×C₆₇	Q₈	1	2	yes	—
41	C₂²×C₄×C₆₇	—	2	1	no	—
42	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	M₁₆×C₆₇	—	2	1	no	—
44	M₁₆×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₆₇	—	2	1	no	—
49	C₄²×C₆₇	—	2	1	no	—
50	C₂³×C₆₇	#41	1	1	yes	characteristic
51	Q₈×C₆₇	—	2	2	no	—
52	Q₈×C₆₇	—	2	2	no	—
53	Q₈×C₆₇	—	4	1	no	—
54	Q₈×C₆₇	—	4	1	no	—
55	C₂×C₄×C₆₇	—	2	1	no	—
56	C₂×C₄×C₆₇	—	2	1	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	4	1	no	—
59	C₂×C₄×C₆₇	—	4	1	no	—
60	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
66	C₂×Dic₆₇	—	4	1	no	—
67	C₂×Dic₆₇	—	8	1	no	—
68	C₈×C₆₇	—	4	1	no	—
69	C₆₇⋊C₈	—	2	2	no	—
70	C₆₇⋊C₈	—	2	2	no	—
71	C₆₇⋊C₈	—	4	1	no	—
72	C₂²×C₆₇	—	2	1	no	—
73	C₂²×C₆₇	—	2	1	no	—
74	C₂²×C₆₇	#121	1	1	yes	characteristic
75	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₆₇	#126	1	1	yes	characteristic
77	C₄×C₆₇	—	4	2	no	—
78	C₄×C₆₇	—	4	1	no	—
79	C₄×C₆₇	—	4	1	no	—
80	C₄×C₆₇	#121	1	1	yes	characteristic
81	C₄×C₆₇	#121	1	1	yes	characteristic
82	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₆₇	—	8	1	no	—
85	C₂×C₆₇	—	2	1	no	—
86	C₂×C₆₇	—	2	1	no	—
87	C₂×C₆₇	#396	1	1	yes	characteristic
88	C₂×C₆₇	#471	1	1	yes	characteristic
89	C₂×C₆₇	#473	1	1	yes	characteristic
90	#2064	—	67	1	no	maximal, Sylow
91	C₆₇	#2064	1	1	yes	characteristic, complemented, Sylow
92	C₂×#108	—	67	1	no	—
93	#428	—	67	1	no	—
94	#424	—	67	1	no	—
95	#419	D₆₇	1	1	yes	characteristic
96	C₂×#103	—	67	1	no	—
97	C₂×#101	—	67	1	no	—
98	#343	—	67	1	no	—
99	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
100	C₂×M₁₆	—	67	1	no	—
101	C₂×M₁₆	—	67	1	no	—
102	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
103	C₂²×C₈	—	67	1	no	—
104	#117	C₂×D₆₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	67	1	no	—
106	#108	—	134	2	no	—
107	#107	—	134	1	no	—
108	#106	—	134	1	no	—
109	C₄⋊C₈	—	134	2	no	—
110	#103	—	2	1	no	—
111	#103	—	2	1	no	—
112	#103	—	134	2	no	—
113	#101	—	134	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	Dic₁₃₄	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	134	1	no	—
122	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	67	2	no	—
126	M₁₆	—	134	1	no	—
127	M₁₆	—	134	1	no	—
128	M₁₆	—	134	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	67	2	no	—
131	C₂×C₈	—	134	1	no	—
132	C₂×C₈	—	134	1	no	—
133	C₂×C₈	—	134	1	no	—
134	C₂×C₈	—	134	1	no	—
135	C₄⋊C₄	—	134	1	no	—
136	C₄⋊C₄	—	134	1	no	—
137	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
138	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₂³	#11761627	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	268	1	no	—
152	C₂×C₄	—	536	1	no	—
153	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
154	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
157	C₂×C₄	#11761627	1	1	yes	characteristic
158	C₂×C₄	#11761628	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	268	1	no	—
161	C₈	—	134	2	no	—
162	C₈	—	134	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#14524015	1	1	yes	characteristic
166	C₂²	#14524020	1	1	yes	characteristic
167	C₂²	#14524036	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	536	1	no	—
172	C₄	#14524014	1	1	yes	characteristic
173	C₄	#14524016	1	1	yes	characteristic
174	C₄	#14524020	1	1	yes	characteristic
175	C₄	#14524022	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#16863319	1	1	yes	characteristic
179	C₂	#16863661	1	1	yes	characteristic
180	C₂	#16863669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523978

C₂

yes

maximal, characteristic

#16863439

C₂

yes

maximal, characteristic

#16863423

C₂

yes

maximal, characteristic

C₂×#14523967

C₂

yes

maximal, characteristic

C₂×#14523962

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal, characteristic

#419×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

—

—

C₄⋊(C₆₇⋊C₈)

—

#14523967

—

#14523962

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

#103×C₆₇

—

C₂²×Dic₆₇

—

—

—

—

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

#126

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

—

—

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₆₇

yes

characteristic

103

C₂²×C₈

—

104

#117

C₂×D₆₇

yes

characteristic

105

C₂×C₄⋊C₄

—

106

#108

—

134

—

107

#107

—

134

—

108

#106

—

134

—

109

C₄⋊C₈

—

134

—

110

#103

—

111

#103

—

112

#103

—

134

—

113

#101

—

134

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

Dic₁₃₄

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

134

—

122

C₂²×C₄

C₂²×D₆₇

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

134

—

127

M₁₆

—

134

—

128

M₁₆

—

134

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

134

—

132

C₂×C₈

—

134

—

133

C₂×C₈

—

134

—

134

C₂×C₈

—

134

—

135

C₄⋊C₄

—

134

—

136

C₄⋊C₄

—

134

—

137

C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

138

C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

#11761627

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

536

—

153

C₂×C₄

D₄×D₆₇

yes

characteristic

154

C₂×C₄

D₄×D₆₇

yes

characteristic

155

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

156

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

157

C₂×C₄

#11761627

yes

characteristic

158

C₂×C₄

#11761628

yes

characteristic

159

C₈

—

160

C₈

—

268

—

161

C₈

—

134

—

162

C₈

—

134

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14524015

yes

characteristic

166

C₂²

#14524020

yes

characteristic

167

C₂²

#14524036

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

536

—

172

C₄

#14524014

yes

characteristic

173

C₄

#14524016

yes

characteristic

174

C₄

#14524020

yes

characteristic

175

C₄

#14524022

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#16863319

yes

characteristic

179

C₂

#16863661

yes

characteristic

180

C₂

#16863669

yes

characteristic

	subgroup	G/H
center	C₂²	#14524015
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#419×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#419	D₆₇
hypocenter	C₆₇	#2064
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14524015

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#419×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
91. C₆₇	#2064	semidirect

H ≅ G/N

type

91. C₆₇

#2064

semidirect

Subgroups — #22561576

Conjugacy classes of subgroups

Complements