Group #22561672 — subgroups

Subgroups — #22561672

Order 8576.

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16863444	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863402	C₂	1	1	yes	maximal, characteristic
3	#16863400	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523959	C₂	1	1	yes	maximal, characteristic
5	#16863168	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863152	C₂	1	1	yes	maximal, characteristic, complemented
7	#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×#11761612	C₂²	1	1	yes	characteristic
10	C₂×#11761612	C₂²	1	1	yes	characteristic
11	#14524002	C₂²	1	1	yes	characteristic
12	#14523978	—	2	2	no	—
13	#14523965	—	2	2	no	—
14	#14523963	—	2	2	no	—
15	#14523959	—	2	2	no	—
16	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	#117×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×D₆₇	—	2	1	no	—
20	C₂×D₂₆₈	—	2	1	no	—
21	C₂×D₂₆₈	—	2	1	no	—
22	C₂×D₂₆₈	—	2	1	no	—
23	C₂×D₂₆₈	—	2	1	no	—
24	C₂×D₂₆₈	—	2	1	no	—
25	C₂×D₂₆₈	D₄	1	2	yes	—
26	#11761612	—	2	1	no	—
27	#11761612	—	2	1	no	—
28	#11761612	—	2	1	no	—
29	#11761612	—	2	1	no	—
30	C₂×C₆₇⋊C₈	—	2	1	no	—
31	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₂²⋊Dic₆₇	—	2	1	no	—
34	C₄×Dic₆₇	—	2	1	no	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₆₇	—	2	1	no	—
37	M₁₆×C₆₇	—	2	1	no	—
38	M₁₆×C₆₇	—	2	1	no	—
39	M₁₆×C₆₇	—	2	1	no	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂²⋊C₄×C₆₇	—	2	1	no	—
44	C₄²×C₆₇	—	2	1	no	—
45	C₂³×C₆₇	#41	1	1	yes	characteristic
46	C₂²×D₆₇	—	2	2	no	—
47	C₂²×D₆₇	—	4	1	no	—
48	C₂²×D₆₇	—	4	1	no	—
49	C₂²×D₆₇	—	8	2	no	—
50	C₂×C₄×C₆₇	—	4	1	no	—
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
57	C₂×Dic₆₇	—	4	1	no	—
58	D₂₆₈	—	2	2	no	—
59	D₂₆₈	—	2	2	no	—
60	D₂₆₈	—	4	1	no	—
61	D₂₆₈	—	4	1	no	—
62	C₈×C₆₇	—	4	1	no	—
63	C₈×C₆₇	—	4	1	no	—
64	C₆₇⋊C₈	—	2	2	no	—
65	C₆₇⋊C₈	—	4	1	no	—
66	C₂²×C₆₇	—	4	1	no	—
67	C₂²×C₆₇	#123	1	1	yes	characteristic
68	C₂²×C₆₇	#124	1	1	yes	characteristic
69	C₂²×C₆₇	#126	1	1	yes	characteristic
70	C₂×D₆₇	—	4	2	no	—
71	C₂×D₆₇	—	8	2	no	—
72	C₂×D₆₇	—	8	2	no	—
73	C₂×D₆₇	—	8	2	no	—
74	C₄×C₆₇	—	8	1	no	—
75	C₄×C₆₇	#121	1	1	yes	characteristic
76	C₄×C₆₇	#123	1	1	yes	characteristic
77	C₄×C₆₇	#123	1	1	yes	characteristic
78	C₄×C₆₇	#126	1	1	yes	characteristic
79	Dic₆₇	—	8	1	no	—
80	C₂×C₆₇	—	2	1	no	—
81	C₂×C₆₇	—	2	1	no	—
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	#397	1	1	yes	characteristic
84	D₆₇	—	8	2	no	—
85	#2081	—	67	1	no	maximal, Sylow
86	C₆₇	#2081	1	1	yes	characteristic, complemented, Sylow
87	#430	—	67	1	no	—
88	#418	—	67	1	no	—
89	#418	—	67	1	no	—
90	C₂×#100	—	67	1	no	—
91	#344	—	67	1	no	—
92	#344	D₆₇	1	1	yes	characteristic, complemented
93	H(3,ℤ/4)	—	67	1	no	—
94	C₂²×D₄	—	67	1	no	—
95	C₂×M₁₆	—	67	1	no	—
96	C₂×M₁₆	—	67	1	no	—
97	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
98	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
99	#117	—	67	1	no	—
100	#117	C₂×D₆₇	1	1	yes	characteristic
101	#108	—	134	2	no	—
102	#102	—	134	2	no	—
103	#102	—	134	2	no	—
104	#100	—	134	2	no	—
105	C₂⁴	—	134	1	no	—
106	C₂×D₄	—	67	2	no	—
107	C₂×D₄	—	134	1	no	—
108	C₂×D₄	—	134	1	no	—
109	C₂×D₄	—	134	1	no	—
110	C₂×D₄	—	134	1	no	—
111	C₂×D₄	—	134	1	no	—
112	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	134	1	no	—
118	M₁₆	—	134	1	no	—
119	M₁₆	—	134	1	no	—
120	M₁₆	—	134	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	67	2	no	—
124	C₂×C₈	—	134	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	134	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	134	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	134	1	no	—
131	C₂³	—	268	1	no	—
132	C₂³	—	268	1	no	—
133	C₂³	—	536	2	no	—
134	C₂³	—	134	2	no	—
135	C₂³	#11761628	1	1	yes	characteristic
136	D₄	—	268	1	no	—
137	D₄	—	268	1	no	—
138	D₄	—	134	2	no	—
139	D₄	—	134	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	268	1	no	—
142	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
144	C₂×C₄	#11761627	1	1	yes	characteristic
145	C₂×C₄	#11761628	1	1	yes	characteristic
146	C₂×C₄	#11761628	1	1	yes	characteristic
147	C₂×C₄	#11761629	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	268	1	no	—
151	C₈	—	134	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	268	2	no	—
154	C₂²	—	536	2	no	—
155	C₂²	—	536	2	no	—
156	C₂²	—	536	2	no	—
157	C₂²	#14524028	1	1	yes	characteristic
158	C₂²	#14524029	1	1	yes	characteristic
159	C₂²	#14524037	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	536	1	no	—
162	C₄	#14524016	1	1	yes	characteristic
163	C₄	#14524025	1	1	yes	characteristic
164	C₄	#14524028	1	1	yes	characteristic
165	C₄	#14524038	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	536	2	no	—
170	C₂	#16863325	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863444

C₂

yes

maximal, characteristic, complemented

#16863402

C₂

yes

maximal, characteristic

#16863400

C₂

yes

maximal, characteristic

C₂×#14523959

C₂

yes

maximal, characteristic

#16863168

C₂

yes

maximal, characteristic, complemented

#16863152

C₂

yes

maximal, characteristic, complemented

#344×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

#14524002

C₂²

yes

—

—

—

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

D₄

yes

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#397

yes

characteristic

D₆₇

—

#2081

—

maximal, Sylow

C₆₇

#2081

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic, complemented

H(3,ℤ/4)

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

characteristic

#117

—

100

#117

C₂×D₆₇

yes

characteristic

101

#108

—

134

—

102

#102

—

134

—

103

#102

—

134

—

104

#100

—

134

—

105

C₂⁴

—

134

—

106

C₂×D₄

—

107

C₂×D₄

—

134

—

108

C₂×D₄

—

134

—

109

C₂×D₄

—

134

—

110

C₂×D₄

—

134

—

111

C₂×D₄

—

134

—

112

C₂²×C₄

C₂²×D₆₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

134

—

118

M₁₆

—

134

—

119

M₁₆

—

134

—

120

M₁₆

—

134

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

134

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

134

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

134

—

129

C₄²

—

130

C₄²

—

134

—

131

C₂³

—

268

—

132

C₂³

—

268

—

133

C₂³

—

536

—

134

C₂³

—

134

—

135

C₂³

#11761628

yes

characteristic

136

D₄

—

268

—

137

D₄

—

268

—

138

D₄

—

134

—

139

D₄

—

134

—

140

C₂×C₄

—

141

C₂×C₄

—

268

—

142

C₂×C₄

D₄×D₆₇

yes

characteristic

143

C₂×C₄

D₄×D₆₇

yes

characteristic

144

C₂×C₄

#11761627

yes

characteristic

145

C₂×C₄

#11761628

yes

characteristic

146

C₂×C₄

#11761628

yes

characteristic

147

C₂×C₄

#11761629

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

268

—

151

C₈

—

134

—

152

C₂²

—

153

C₂²

—

268

—

154

C₂²

—

536

—

155

C₂²

—

536

—

156

C₂²

—

536

—

157

C₂²

#14524028

yes

characteristic

158

C₂²

#14524029

yes

characteristic

159

C₂²

#14524037

yes

characteristic

160

C₄

—

161

C₄

—

536

—

162

C₄

#14524016

yes

characteristic

163

C₄

#14524025

yes

characteristic

164

C₄

#14524028

yes

characteristic

165

C₄

#14524038

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

536

—

170

C₂

#16863325

yes

characteristic

	subgroup	G/H
center	C₂	#16863325
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#344×C₆₇	C₂
socle	C₂×C₆₇	#397
hypercenter	#344	D₆₇
hypocenter	C₆₇	#2081
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#16863325

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#344×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
1. #16863444	C₂	semidirect
5. #16863168	C₂	semidirect
6. #16863152	C₂	semidirect
7. #344×C₆₇	C₂	semidirect
86. C₆₇	#2081	semidirect
92. #344	D₆₇	semidirect

H ≅ G/N

type

1. #16863444

C₂

semidirect

5. #16863168

C₂

semidirect

6. #16863152

C₂

semidirect

7. #344×C₆₇

C₂

semidirect

86. C₆₇

#2081

semidirect

92. #344

D₆₇

semidirect

Subgroups — #22561672

Conjugacy classes of subgroups

Complements