Group #22569864 — subgroups

Subgroups — #22569864

Order 8576.

460 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16863967	C₂	1	2	yes	maximal
3	#16863654	C₂	1	4	yes	maximal, complemented
4	#16863637	C₂	1	4	yes	maximal, complemented
5	#16863434	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863406	C₂	1	2	yes	maximal, complemented
7	C₂×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
9	C₂×Dic₂₆₈	—	2	4	no	—
10	C₂×Dic₂₆₈	C₂²	1	2	yes	—
11	C₂×Dic₂₆₈	C₂²	1	2	yes	—
12	#14524041	—	2	2	no	—
13	#14524030	—	2	2	no	—
14	#14524024	—	2	4	no	—
15	#14524022	C₂²	1	8	yes	—
16	#14524002	C₂²	1	2	yes	—
17	#14523977	C₂²	1	2	yes	—
18	#14523975	C₂²	1	2	yes	—
19	#14523968	C₂²	1	8	yes	—
20	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
21	#131×C₆₇	—	2	4	no	—
22	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×Dic₁₃₄	—	2	4	no	—
27	C₂×Dic₁₃₄	—	2	4	no	—
28	C₂×Dic₁₃₄	—	2	2	no	—
29	C₂×Dic₁₃₄	C₂³	1	4	yes	—
30	C₂²×Dic₆₇	—	2	2	no	—
31	Dic₂₆₈	—	2	4	no	—
32	Dic₂₆₈	—	2	4	no	—
33	C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	Dic₆₇⋊C₄	—	2	8	no	—
35	C₂²⋊Dic₆₇	—	2	2	no	—
36	C₂²⋊Dic₆₇	—	2	8	no	—
37	C₄×Dic₆₇	—	2	2	no	—
38	#41×C₆₇	—	2	2	no	—
39	#41×C₆₇	—	2	2	no	—
40	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂×D₄×C₆₇	D₄	1	2	yes	—
42	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	M₁₆×C₆₇	—	2	2	no	—
46	M₁₆×C₆₇	—	2	4	no	—
47	C₂×C₈×C₆₇	—	2	4	no	—
48	C₂×C₈×C₆₇	D₄	1	4	yes	—
49	C₂×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂³×C₆₇	C₂×D₄	1	2	yes	—
53	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₆₇	—	2	2	no	—
55	D₄×C₆₇	—	2	4	no	—
56	D₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#41	1	2	yes	—
63	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₆₇	—	2	4	no	—
65	C₂×Dic₆₇	—	2	4	no	—
66	C₂×Dic₆₇	—	4	4	no	—
67	Dic₁₃₄	—	2	4	no	—
68	Dic₁₃₄	—	2	4	no	—
69	Dic₁₃₄	—	4	4	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	C₂×D₄	1	2	yes	—
72	C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂²×C₆₇	—	2	2	no	—
74	C₂²×C₆₇	—	2	2	no	—
75	C₂²×C₆₇	—	2	4	no	—
76	C₂²×C₆₇	#121	1	2	yes	—
77	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₆₇	—	2	2	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	—	4	4	no	—
83	Dic₆₇	—	4	4	no	—
84	C₂×C₆₇	—	2	2	no	—
85	C₂×C₆₇	—	4	2	no	—
86	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
87	C₂×C₆₇	#578	1	2	yes	—
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	D₆₇	1	1	yes	characteristic
92	#536	—	67	2	no	—
93	#470	—	67	4	no	—
94	#467	—	67	4	no	—
95	#427	—	67	1	no	—
96	#419	—	67	2	no	—
97	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
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	—	2	4	no	—
103	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
104	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
105	C₂²×C₈	C₂×D₆₇	1	2	yes	—
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	—	67	2	no	—
112	#107	—	67	2	no	—
113	#106	—	67	2	no	—
114	#103	—	67	8	no	—
115	#41	—	2	2	no	—
116	#41	—	2	2	no	—
117	C₂×Q₈	—	67	4	no	—
118	C₂×Q₈	—	134	4	no	—
119	C₂×Q₈	—	134	4	no	—
120	C₂×Q₈	—	134	2	no	—
121	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
122	C₂×D₄	D₂₆₈	1	2	yes	—
123	C₂×D₄	D₂₆₈	1	1	yes	characteristic
124	C₂²×C₄	—	134	2	no	—
125	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
126	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
127	Q₁₆	—	134	4	no	—
128	Q₁₆	—	134	4	no	—
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	C₂²⋊D₆₇	1	4	yes	—
133	C₂×C₈	C₂²×D₆₇	1	2	yes	—
134	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄⋊C₄	—	67	4	no	—
137	C₄⋊C₄	—	134	8	no	—
138	C₂²⋊C₄	—	134	2	no	—
139	C₂²⋊C₄	—	134	8	no	—
140	C₄²	—	134	2	no	—
141	C₂³	C₂×D₂₆₈	1	2	yes	—
142	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
143	Q₈	—	2	2	no	—
144	Q₈	—	268	4	no	—
145	Q₈	—	134	4	no	—
146	Q₈	—	134	4	no	—
147	D₄	—	2	4	no	—
148	D₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	268	4	no	—
152	C₂×C₄	—	134	4	no	—
153	C₂×C₄	—	134	4	no	—
154	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
155	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
156	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
157	C₂×C₄	#11761628	1	2	yes	—
158	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
161	C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	#14524017	1	2	yes	—
166	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	268	4	no	—
169	C₄	—	268	4	no	—
170	C₄	#14524017	1	2	yes	—
171	C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₄	C₂×#11761628	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	4	2	no	—
175	C₂	C₂×#14524017	1	1	yes	characteristic
176	C₂	#16864213	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic, complemented

#16863967

C₂

yes

maximal

#16863654

C₂

yes

maximal, complemented

#16863637

C₂

yes

maximal, complemented

#16863434

C₂

yes

maximal, characteristic, complemented

#16863406

C₂

yes

maximal, complemented

C₂×#131×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

—

—

—

C₂²

yes

—

#14524002

C₂²

yes

—

#14523977

C₂²

yes

—

#14523975

C₂²

yes

—

#14523968

C₂²

yes

—

C₂×#41×C₆₇

C₂²

yes

characteristic

#131×C₆₇

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

D₄

yes

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₄

yes

—

C₈×C₆₇

C₂×D₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

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×C₆₇

C₂×#121

yes

characteristic

C₂×C₆₇

#578

yes

—

#3059

—

maximal, Sylow

C₆₇

#3059

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×#131

D₆₇

yes

—

—

—

—

—

C₂×#41

C₂×D₆₇

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

134

—

102

#131

—

103

C₂×M₁₆

C₂×D₆₇

yes

characteristic

104

C₂×M₁₆

C₂×D₆₇

yes

—

105

C₂²×C₈

C₂×D₆₇

yes

—

106

C₂²×C₈

C₂×D₆₇

yes

characteristic

107

C₄⋊Q₈

—

134

—

108

#124

—

134

—

109

#123

—

134

—

110

C₂²⋊Q₈

—

111

#117

—

112

#107

—

113

#106

—

114

#103

—

115

#41

—

116

#41

—

117

C₂×Q₈

—

118

C₂×Q₈

—

134

—

119

C₂×Q₈

—

134

—

120

C₂×Q₈

—

134

—

121

C₂×Q₈

D₂₆₈

yes

characteristic

122

C₂×D₄

D₂₆₈

yes

—

123

C₂×D₄

D₂₆₈

yes

characteristic

124

C₂²×C₄

—

134

—

125

C₂²×C₄

C₂²×D₆₇

yes

—

126

C₂²×C₄

C₂²×D₆₇

yes

characteristic

127

Q₁₆

—

134

—

128

Q₁₆

—

134

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

C₂²⋊D₆₇

yes

—

133

C₂×C₈

C₂²×D₆₇

yes

—

134

C₂×C₈

C₂²×D₆₇

yes

characteristic

135

C₂×C₈

C₂²×D₆₇

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

134

—

138

C₂²⋊C₄

—

134

—

139

C₂²⋊C₄

—

134

—

140

C₄²

—

134

—

141

C₂³

C₂×D₂₆₈

yes

—

142

C₂³

C₂×D₂₆₈

yes

characteristic

143

Q₈

—

144

Q₈

—

268

—

145

Q₈

—

134

—

146

Q₈

—

134

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

134

—

153

C₂×C₄

—

134

—

154

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

155

C₂×C₄

C₂×D₂₆₈

yes

—

156

C₂×C₄

C₂×D₂₆₈

yes

characteristic

157

C₂×C₄

#11761628

yes

—

158

C₂×C₄

C₂³×D₆₇

yes

characteristic

159

C₈

—

160

C₈

C₂×C₂²⋊D₆₇

yes

—

161

C₈

C₂×C₂²⋊D₆₇

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14524017

yes

—

166

C₂²

C₂²×D₂₆₈

yes

characteristic

167

C₄

—

168

C₄

—

268

—

169

C₄

—

268

—

170

C₄

#14524017

yes

—

171

C₄

C₂²×C₂²⋊D₆₇

yes

characteristic

172

C₄

C₂×#11761628

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

C₂×#14524017

yes

characteristic

176

C₂

#16864213

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×#131	D₆₇
hypocenter	C₆₇	#3059
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×#131×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
1. C₂²×Dic₂₆₈	C₂	semidirect
3. #16863654	C₂	semidirect
4. #16863637	C₂	semidirect
5. #16863434	C₂	semidirect
6. #16863406	C₂	semidirect
89. C₆₇	#3059	semidirect

H ≅ G/N

type

1. C₂²×Dic₂₆₈

C₂

semidirect

3. #16863654

C₂

semidirect

4. #16863637

C₂

semidirect

5. #16863434

C₂

semidirect

6. #16863406

C₂

semidirect

89. C₆₇

#3059

semidirect

Subgroups — #22569864

Conjugacy classes of subgroups

Complements