Group #22559377 — subgroups

Subgroups — #22559377

Order 8576.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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	D₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#16863747	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863692	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863686	C₂	1	1	yes	maximal, characteristic
5	#16863143	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863128	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	—	2	1	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14524043	C₂²	1	1	yes	characteristic
11	#14524042	—	2	1	no	—
12	#14524022	—	2	1	no	—
13	#14524020	—	2	2	no	—
14	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
15	D₄×Dic₆₇	—	2	1	no	—
16	#14523977	—	2	1	no	—
17	#14523974	—	2	1	no	—
18	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	#14523968	—	2	1	no	—
20	#14523968	—	2	2	no	—
21	#14523964	—	2	1	no	—
22	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₃₄	—	2	2	no	—
26	C₂×Dic₁₃₄	—	2	1	no	—
27	C₂×Dic₁₃₄	—	2	1	no	—
28	C₂×Dic₁₃₄	—	4	2	no	—
29	C₂×Dic₁₃₄	—	4	2	no	—
30	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
31	C₂²×Dic₆₇	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	1	no	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₄⋊Dic₆₇	—	2	1	no	—
35	C₄⋊Dic₆₇	—	2	1	no	—
36	C₄⋊Dic₆₇	D₄	1	2	yes	—
37	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
38	Dic₆₇⋊C₄	—	2	2	no	—
39	Dic₆₇⋊C₄	—	2	1	no	—
40	C₂²⋊Dic₆₇	—	2	2	no	—
41	C₄×Dic₆₇	—	2	1	no	—
42	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂²⋊C₄×C₆₇	—	2	1	no	—
48	C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₆₇	—	2	1	no	—
50	D₄×C₆₇	—	2	1	no	—
51	D₄×C₆₇	Q₁₆	1	2	yes	—
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	4	1	no	—
54	C₂×C₄×C₆₇	—	4	1	no	—
55	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₆₇	—	2	1	no	—
59	C₂×Dic₆₇	—	2	1	no	—
60	C₂×Dic₆₇	—	2	2	no	—
61	C₂×Dic₆₇	—	4	4	no	—
62	C₂×Dic₆₇	—	4	1	no	—
63	Dic₁₃₄	—	2	1	no	—
64	Dic₁₃₄	—	4	4	no	—
65	Dic₁₃₄	—	4	1	no	—
66	Dic₁₃₄	—	4	1	no	—
67	Dic₁₃₄	—	4	2	no	—
68	Dic₁₃₄	SD₁₆	1	2	yes	—
69	C₈×C₆₇	—	4	1	no	—
70	C₈×C₆₇	—	4	1	no	—
71	C₆₇⋊C₈	—	4	1	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	—	4	1	no	—
74	C₂²×C₆₇	—	4	1	no	—
75	C₂²×C₆₇	#120	1	1	yes	characteristic
76	C₄×C₆₇	—	2	1	no	—
77	C₄×C₆₇	—	4	1	no	—
78	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₄×C₆₇	#137	1	1	yes	characteristic
82	Dic₆₇	—	2	2	no	—
83	Dic₆₇	—	4	2	no	—
84	Dic₆₇	—	4	1	no	—
85	Dic₆₇	—	8	1	no	—
86	C₂×C₆₇	—	4	2	no	—
87	C₂×C₆₇	#450	1	1	yes	characteristic
88	C₂×C₆₇	#451	1	1	yes	characteristic
89	C₂×C₆₇	#454	1	1	yes	characteristic
90	#1637	—	67	1	no	maximal, Sylow
91	C₆₇	#1637	1	1	yes	characteristic, complemented, Sylow
92	D₄×Q₈	—	67	1	no	—
93	#487	—	67	1	no	—
94	#477	—	67	1	no	—
95	#476	—	67	1	no	—
96	#332	—	67	1	no	—
97	Q₈⋊C₈	—	67	1	no	—
98	D₄⋊C₈	D₆₇	1	1	yes	characteristic
99	C₂²×Q₈	—	134	1	no	—
100	C₄⋊Q₈	—	67	1	no	—
101	C₄⋊Q₈	—	67	1	no	—
102	C₄⋊Q₈	—	134	1	no	—
103	C₂²⋊Q₈	—	134	2	no	—
104	C₂²⋊Q₈	—	134	1	no	—
105	C₄×Q₈	—	67	1	no	—
106	C₄×D₄	—	134	1	no	—
107	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
108	#107	—	134	1	no	—
109	#106	—	134	1	no	—
110	C₄⋊C₈	—	67	1	no	—
111	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
112	#103	—	134	2	no	—
113	#103	—	134	1	no	—
114	#102	—	134	1	no	—
115	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
116	C₂×Q₈	—	268	2	no	—
117	C₂×Q₈	—	268	2	no	—
118	C₂×Q₈	—	67	1	no	—
119	C₂×Q₈	—	134	2	no	—
120	C₂×Q₈	—	134	1	no	—
121	C₂×Q₈	—	134	1	no	—
122	C₂×D₄	D₂₆₈	1	1	yes	characteristic
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	134	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	134	1	no	—
128	C₄⋊C₄	—	67	2	no	—
129	C₄⋊C₄	—	67	1	no	—
130	C₄⋊C₄	—	134	2	no	—
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	D₂₆₈	1	1	yes	characteristic
136	C₂²⋊C₄	—	2	1	no	—
137	C₂²⋊C₄	—	134	2	no	—
138	C₄²	—	134	1	no	—
139	C₄²	C₂²×D₆₇	1	1	yes	characteristic
140	C₂³	—	2	1	no	—
141	Q₈	—	268	2	no	—
142	Q₈	—	268	4	no	—
143	Q₈	—	268	1	no	—
144	Q₈	—	268	1	no	—
145	Q₈	—	67	2	no	—
146	Q₈	—	134	1	no	—
147	D₄	—	2	1	no	—
148	D₄	Dic₂₆₈	1	2	yes	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	268	4	no	—
153	C₂×C₄	—	268	1	no	—
154	C₂×C₄	—	134	1	no	—
155	C₂×C₄	—	134	2	no	—
156	C₂×C₄	—	134	1	no	—
157	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
158	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
159	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	—	268	1	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#14524013	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	268	2	no	—
170	C₄	—	268	1	no	—
171	C₄	—	536	1	no	—
172	C₄	—	134	2	no	—
173	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
174	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
175	C₄	#14524067	1	1	yes	characteristic
176	C₄	#14524076	1	1	yes	characteristic
177	C₂	—	4	2	no	—
178	C₂	#16863548	1	1	yes	characteristic
179	C₂	#16863554	1	1	yes	characteristic
180	C₂	#16863570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×Dic₁₃₄

C₂

yes

maximal, characteristic

#16863747

C₂

yes

maximal, characteristic, complemented

#16863692

C₂

yes

maximal, characteristic, complemented

#16863686

C₂

yes

maximal, characteristic

#16863143

C₂

yes

maximal, characteristic, complemented

#16863128

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

—

—

—

C₄×Dic₁₃₄

C₂²

yes

characteristic

D₄×Dic₆₇

—

#14523977

—

#14523974

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

—

—

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

Q₁₆

yes

—

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×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

SD₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

D₄×Q₈

—

—

—

—

—

Q₈⋊C₈

—

D₄⋊C₈

D₆₇

yes

characteristic

C₂²×Q₈

—

134

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

134

—

103

C₂²⋊Q₈

—

134

—

104

C₂²⋊Q₈

—

134

—

105

C₄×Q₈

—

106

C₄×D₄

—

134

—

107

C₄×D₄

C₂×D₆₇

yes

characteristic

108

#107

—

134

—

109

#106

—

134

—

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₆₇

yes

characteristic

112

#103

—

134

—

113

#103

—

134

—

114

#102

—

134

—

115

C₄×C₈

C₂×D₆₇

yes

characteristic

116

C₂×Q₈

—

268

—

117

C₂×Q₈

—

268

—

118

C₂×Q₈

—

119

C₂×Q₈

—

134

—

120

C₂×Q₈

—

134

—

121

C₂×Q₈

—

134

—

122

C₂×D₄

D₂₆₈

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

134

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

134

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

134

—

131

C₄⋊C₄

—

134

—

132

C₄⋊C₄

—

134

—

133

C₄⋊C₄

—

134

—

134

C₄⋊C₄

—

134

—

135

C₄⋊C₄

D₂₆₈

yes

characteristic

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

134

—

138

C₄²

—

134

—

139

C₄²

C₂²×D₆₇

yes

characteristic

140

C₂³

—

141

Q₈

—

268

—

142

Q₈

—

268

—

143

Q₈

—

268

—

144

Q₈

—

268

—

145

Q₈

—

146

Q₈

—

134

—

147

D₄

—

148

D₄

Dic₂₆₈

yes

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

268

—

153

C₂×C₄

—

268

—

154

C₂×C₄

—

134

—

155

C₂×C₄

—

134

—

156

C₂×C₄

—

134

—

157

C₂×C₄

D₄×D₆₇

yes

characteristic

158

C₂×C₄

D₄×D₆₇

yes

characteristic

159

C₂×C₄

C₂×D₂₆₈

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

268

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#14524013

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

268

—

170

C₄

—

268

—

171

C₄

—

536

—

172

C₄

—

134

—

173

C₄

SD₁₆×D₆₇

yes

characteristic

174

C₄

C₂×Dic₂₆₈

yes

characteristic

175

C₄

#14524067

yes

characteristic

176

C₄

#14524076

yes

characteristic

177

C₂

—

178

C₂

#16863548

yes

characteristic

179

C₂

#16863554

yes

characteristic

180

C₂

#16863570

yes

characteristic

	subgroup	G/H
center	C₂²	#14524013
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	D₄⋊C₈×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	D₄⋊C₈	D₆₇
hypocenter	C₆₇	#1637
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

subgroup

G/H

center

C₂²

#14524013

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

D₄⋊C₈×C₆₇

C₂

socle

C₂²×C₆₇

#120

hypercenter

D₄⋊C₈

D₆₇

hypocenter

C₆₇

#1637

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄⋊C₈

N	H ≅ G/N	type
2. #16863747	C₂	semidirect
3. #16863692	C₂	semidirect
5. #16863143	C₂	semidirect
6. #16863128	C₂	semidirect
91. C₆₇	#1637	semidirect

H ≅ G/N

type

2. #16863747

C₂

semidirect

3. #16863692

C₂

semidirect

5. #16863143

C₂

semidirect

6. #16863128

C₂

semidirect

91. C₆₇

#1637

semidirect

Subgroups — #22559377

Conjugacy classes of subgroups

Complements