Group #22569865 — subgroups

Subgroups — #22569865

Order 8576.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14524049	C₂	1	1	yes	maximal, characteristic
3	#16863972	C₂	1	2	yes	maximal
4	#16863655	C₂	1	4	yes	maximal, complemented
5	#16863638	C₂	1	4	yes	maximal, complemented
6	#16863404	C₂	1	2	yes	maximal, complemented
7	#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761628	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
10	C₂×Dic₂₆₈	—	2	4	no	—
11	C₂×Dic₂₆₈	C₂²	1	2	yes	—
12	C₂×Dic₂₆₈	C₂²	1	2	yes	—
13	#14524049	—	2	4	no	—
14	C₂×#11761612	C₂²	1	1	yes	characteristic
15	C₂×#11761611	C₂²	1	2	yes	—
16	C₂×C₈×D₆₇	C₂²	1	2	yes	—
17	#14524043	—	2	2	no	—
18	#14524031	—	2	2	no	—
19	#14524027	—	2	4	no	—
20	#14524023	C₂²	1	8	yes	—
21	#14523966	C₂²	1	8	yes	—
22	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#117×C₆₇	C₂²	1	2	yes	—
24	#107×C₆₇	C₂²	1	2	yes	—
25	#106×C₆₇	C₂²	1	2	yes	—
26	#11761628	—	2	2	no	—
27	#11761628	—	2	2	no	—
28	C₂×Dic₁₃₄	—	2	4	no	—
29	C₂×Dic₁₃₄	—	2	4	no	—
30	C₂×Dic₁₃₄	—	2	2	no	—
31	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
32	C₂×Dic₁₃₄	C₂³	1	4	yes	—
33	C₂×D₂₆₈	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₆₇	D₄	1	2	yes	—
35	C₂²×Dic₆₇	—	2	2	no	—
36	C₂×C₄×D₆₇	C₂³	1	2	yes	—
37	Dic₂₆₈	—	2	4	no	—
38	Dic₂₆₈	—	2	4	no	—
39	#11761612	—	2	2	no	—
40	#11761611	—	2	4	no	—
41	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₈×D₆₇	—	2	4	no	—
43	C₄⋊Dic₆₇	—	2	8	no	—
44	(C₂×D₆₇)⋊C₄	—	2	8	no	—
45	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₆₇	D₄	1	4	yes	—
47	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
50	C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	—	2	2	no	—
52	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₆₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₆₇	—	2	4	no	—
55	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₆₇	#41	1	2	yes	—
58	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₆₇	—	2	4	no	—
60	C₂×Dic₆₇	—	2	4	no	—
61	C₂×Dic₆₇	—	4	4	no	—
62	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
63	D₂₆₈	—	2	2	no	—
64	C₄×D₆₇	—	2	2	no	—
65	C₄×D₆₇	—	2	2	no	—
66	Dic₁₃₄	—	2	2	no	—
67	Dic₁₃₄	—	2	4	no	—
68	Dic₁₃₄	—	2	4	no	—
69	Dic₁₃₄	—	4	4	no	—
70	C₈×C₆₇	C₂×D₄	1	2	yes	—
71	C₈×C₆₇	C₂×D₄	1	2	yes	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₂²×C₆₇	—	2	2	no	—
74	C₂²×C₆₇	#121	1	2	yes	—
75	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₆₇	—	2	2	no	—
77	C₂×D₆₇	—	2	4	no	—
78	C₄×C₆₇	—	4	4	no	—
79	C₄×C₆₇	#121	1	2	yes	—
80	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
82	Dic₆₇	—	2	2	no	—
83	Dic₆₇	—	4	4	no	—
84	C₂×C₆₇	—	2	2	no	—
85	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
86	C₂×C₆₇	#578	1	2	yes	—
87	D₆₇	—	4	2	no	—
88	#3059	—	67	1	no	maximal, Sylow
89	C₆₇	#3059	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₁₆	—	67	1	no	—
91	C₂×#131	—	67	1	no	—
92	#536	—	67	2	no	—
93	#470	—	67	4	no	—
94	#467	—	67	4	no	—
95	#427	D₆₇	1	1	yes	characteristic, complemented
96	#419	—	67	2	no	—
97	C₂×#41	—	67	1	no	—
98	C₂²×Q₈	—	67	2	no	—
99	C₂×Q₁₆	—	67	2	no	—
100	C₂×Q₁₆	—	67	2	no	—
101	C₂×Q₁₆	—	134	4	no	—
102	#131	—	134	4	no	—
103	C₂×M₁₆	—	67	1	no	—
104	C₂×M₁₆	—	67	2	no	—
105	C₂²×C₈	—	67	2	no	—
106	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
107	C₄⋊Q₈	—	134	2	no	—
108	#124	—	134	2	no	—
109	#123	—	134	4	no	—
110	C₂²⋊Q₈	—	67	8	no	—
111	#117	C₂×D₆₇	1	2	yes	—
112	#107	C₂×D₆₇	1	2	yes	—
113	#106	C₂×D₆₇	1	2	yes	—
114	#103	—	67	8	no	—
115	#41	—	134	2	no	—
116	#41	—	134	2	no	—
117	C₂×Q₈	—	67	1	no	—
118	C₂×Q₈	—	67	4	no	—
119	C₂×Q₈	—	134	4	no	—
120	C₂×Q₈	—	134	4	no	—
121	C₂×Q₈	—	134	2	no	—
122	C₂×D₄	—	67	2	no	—
123	C₂×D₄	—	67	1	no	—
124	C₂²×C₄	—	67	2	no	—
125	C₂²×C₄	—	134	2	no	—
126	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
127	Q₁₆	—	134	4	no	—
128	Q₁₆	—	134	4	no	—
129	M₁₆	—	134	2	no	—
130	M₁₆	—	134	4	no	—
131	C₂×C₈	—	67	2	no	—
132	C₂×C₈	—	134	4	no	—
133	C₂×C₈	D₂₆₈	1	4	yes	—
134	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄⋊C₄	—	134	8	no	—
137	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	134	8	no	—
140	C₄²	—	2	2	no	—
141	C₂³	—	67	2	no	—
142	C₂³	D₄×D₆₇	1	1	yes	characteristic
143	Q₈	—	268	4	no	—
144	Q₈	—	134	4	no	—
145	Q₈	—	134	4	no	—
146	Q₈	—	134	2	no	—
147	D₄	—	134	4	no	—
148	D₄	—	134	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	268	4	no	—
151	C₂×C₄	—	67	2	no	—
152	C₂×C₄	—	134	4	no	—
153	C₂×C₄	—	134	2	no	—
154	C₂×C₄	—	134	2	no	—
155	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
156	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
157	C₂×C₄	#11761629	1	2	yes	—
158	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
159	C₈	—	134	2	no	—
160	C₈	C₂×D₂₆₈	1	2	yes	—
161	C₈	C₂×D₂₆₈	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	134	2	no	—
164	C₂²	—	134	4	no	—
165	C₂²	#14524018	1	2	yes	—
166	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
167	C₄	—	4	4	no	—
168	C₄	—	268	4	no	—
169	C₄	—	134	2	no	—
170	C₄	#14524018	1	2	yes	—
171	C₄	C₂²×D₂₆₈	1	1	yes	characteristic
172	C₄	C₂×#11761629	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	268	2	no	—
175	C₂	C₂×#14524018	1	1	yes	characteristic
176	C₂	#16864210	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic, complemented

C₂×#14524049

C₂

yes

maximal, characteristic

#16863972

C₂

yes

maximal

#16863655

C₂

yes

maximal, complemented

#16863638

C₂

yes

maximal, complemented

#16863404

C₂

yes

maximal, complemented

#427×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

#14524049

—

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761611

C₂²

yes

—

C₂×C₈×D₆₇

C₂²

yes

—

—

—

—

C₂²

yes

—

#14523966

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

—

#107×C₆₇

C₂²

yes

—

#106×C₆₇

C₂²

yes

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×Dic₁₃₄

C₂³

yes

—

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₂²⋊D₆₇

D₄

yes

—

C₂²×Dic₆₇

—

C₂×C₄×D₆₇

C₂³

yes

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

#11761611

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₈×D₆₇

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂²×D₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

—

C₈×C₆₇

C₂×D₄

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

—

C₄×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#121

yes

characteristic

C₂×C₆₇

#578

yes

—

D₆₇

—

#3059

—

maximal, Sylow

C₆₇

#3059

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

—

—

—

—

D₆₇

yes

characteristic, complemented

#419

—

C₂×#41

—

C₂²×Q₈

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

134

—

102

#131

—

134

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₆₇

yes

characteristic

107

C₄⋊Q₈

—

134

—

108

#124

—

134

—

109

#123

—

134

—

110

C₂²⋊Q₈

—

111

#117

C₂×D₆₇

yes

—

112

#107

C₂×D₆₇

yes

—

113

#106

C₂×D₆₇

yes

—

114

#103

—

115

#41

—

134

—

116

#41

—

134

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

134

—

120

C₂×Q₈

—

134

—

121

C₂×Q₈

—

134

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

134

—

126

C₂²×C₄

C₂²×D₆₇

yes

characteristic

127

Q₁₆

—

134

—

128

Q₁₆

—

134

—

129

M₁₆

—

134

—

130

M₁₆

—

134

—

131

C₂×C₈

—

132

C₂×C₈

—

134

—

133

C₂×C₈

D₂₆₈

yes

—

134

C₂×C₈

C₂²×D₆₇

yes

characteristic

135

C₂×C₈

C₂²×D₆₇

yes

characteristic

136

C₄⋊C₄

—

134

—

137

C₄⋊C₄

C₂²×D₆₇

yes

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

134

—

140

C₄²

—

141

C₂³

—

142

C₂³

D₄×D₆₇

yes

characteristic

143

Q₈

—

268

—

144

Q₈

—

134

—

145

Q₈

—

134

—

146

Q₈

—

134

—

147

D₄

—

134

—

148

D₄

—

134

—

149

C₂×C₄

—

150

C₂×C₄

—

268

—

151

C₂×C₄

—

152

C₂×C₄

—

134

—

153

C₂×C₄

—

134

—

154

C₂×C₄

—

134

—

155

C₂×C₄

D₄×D₆₇

yes

characteristic

156

C₂×C₄

C₂×D₂₆₈

yes

—

157

C₂×C₄

#11761629

yes

—

158

C₂×C₄

C₂³×D₆₇

yes

characteristic

159

C₈

—

134

—

160

C₈

C₂×D₂₆₈

yes

—

161

C₈

C₂×D₂₆₈

yes

—

162

C₂²

—

163

C₂²

—

134

—

164

C₂²

—

134

—

165

C₂²

#14524018

yes

—

166

C₂²

C₂×D₄×D₆₇

yes

characteristic

167

C₄

—

168

C₄

—

268

—

169

C₄

—

134

—

170

C₄

#14524018

yes

—

171

C₄

C₂²×D₂₆₈

yes

characteristic

172

C₄

C₂×#11761629

yes

characteristic

173

C₂

—

174

C₂

—

268

—

175

C₂

C₂×#14524018

yes

characteristic

176

C₂

#16864210

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#427×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#427	D₆₇
hypocenter	C₆₇	#3059
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#427×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
1. C₂²×Dic₂₆₈	C₂	semidirect
4. #16863655	C₂	semidirect
5. #16863638	C₂	semidirect
6. #16863404	C₂	semidirect
7. #427×C₆₇	C₂	semidirect
89. C₆₇	#3059	semidirect
95. #427	D₆₇	semidirect

H ≅ G/N

type

1. C₂²×Dic₂₆₈

C₂

semidirect

4. #16863655

C₂

semidirect

5. #16863638

C₂

semidirect

6. #16863404

C₂

semidirect

7. #427×C₆₇

C₂

semidirect

89. C₆₇

#3059

semidirect

95. #427

D₆₇

semidirect

Subgroups — #22569865

Conjugacy classes of subgroups

Complements