Group #22559674 — subgroups

Subgroups — #22559674

Order 8576.

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

normal subgroups only

This group has 182 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	#16863769	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863676	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863591	C₂	1	1	yes	maximal, characteristic
4	#16863149	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863143	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863139	C₂	1	1	yes	maximal, characteristic, complemented
7	#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761617	—	2	1	no	—
9	C₂×#11761617	—	2	1	no	—
10	C₂×#11761613	—	2	1	no	—
11	#14524043	C₂²	1	1	yes	characteristic
12	#14524040	C₂²	1	1	yes	characteristic
13	#14523976	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	#14523964	—	2	1	no	—
17	#14523963	—	2	1	no	—
18	C₂×M₁₆×C₆₇	—	2	1	no	—
19	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²⋊C₈×C₆₇	—	2	1	no	—
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×D₂₆₈	—	2	1	no	—
25	C₂×D₂₆₈	—	2	1	no	—
26	C₂×D₂₆₈	—	2	1	no	—
27	#11761617	—	4	2	no	—
28	#11761617	—	4	2	no	—
29	#11761613	—	4	2	no	—
30	C₂×C₆₇⋊C₈	—	2	1	no	—
31	C₂×C₆₇⋊C₈	—	2	1	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	—	2	1	no	—
36	M₁₆×C₆₇	—	4	2	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂²⋊C₄×C₆₇	—	2	1	no	—
42	C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₆₇	—	2	1	no	—
44	C₂²×D₆₇	—	4	1	no	—
45	D₄×C₆₇	—	2	1	no	—
46	D₄×C₆₇	—	2	1	no	—
47	C₂×C₄×C₆₇	—	2	1	no	—
48	C₂×C₄×C₆₇	—	4	1	no	—
49	C₂×C₄×C₆₇	—	4	1	no	—
50	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
53	C₂×Dic₆₇	—	4	1	no	—
54	D₂₆₈	—	4	1	no	—
55	D₂₆₈	—	4	1	no	—
56	D₂₆₈	—	4	1	no	—
57	D₂₆₈	—	4	1	no	—
58	D₂₆₈	—	8	1	no	—
59	Dic₁₃₄	—	4	1	no	—
60	Dic₁₃₄	—	4	1	no	—
61	C₈×C₆₇	—	2	1	no	—
62	C₈×C₆₇	—	4	1	no	—
63	C₈×C₆₇	D₈	1	2	yes	complemented
64	C₆₇⋊C₈	—	4	1	no	—
65	C₆₇⋊C₈	—	4	1	no	—
66	C₂²×C₆₇	—	4	1	no	—
67	C₂²×C₆₇	—	4	1	no	—
68	C₂²×C₆₇	—	4	1	no	—
69	C₂²×C₆₇	#121	1	1	yes	characteristic
70	C₂×D₆₇	—	8	1	no	—
71	C₂×D₆₇	—	8	1	no	—
72	C₂×D₆₇	—	8	1	no	—
73	C₄×C₆₇	—	2	1	no	—
74	C₄×C₆₇	—	4	1	no	—
75	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
76	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₆₇	#137	1	1	yes	characteristic
79	Dic₆₇	—	8	1	no	—
80	C₂×C₆₇	—	8	1	no	—
81	C₂×C₆₇	#459	1	1	yes	characteristic
82	C₂×C₆₇	#468	1	1	yes	characteristic
83	C₂×C₆₇	#471	1	1	yes	characteristic
84	D₆₇	—	16	1	no	—
85	#1685	—	67	1	no	maximal, Sylow
86	C₆₇	#1685	1	1	yes	characteristic, complemented, Sylow
87	#492	—	67	1	no	—
88	#474	—	67	1	no	—
89	#459	—	67	1	no	—
90	#436	D₆₇	1	1	yes	characteristic, complemented
91	#335	—	67	1	no	—
92	#332	—	67	1	no	—
93	#331	—	67	1	no	—
94	C₂×SD₁₆	—	134	1	no	—
95	C₂×SD₁₆	—	134	1	no	—
96	C₂×D₈	—	134	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₄⋊Q₈	—	67	1	no	—
99	#127	—	67	1	no	—
100	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
101	#107	—	134	1	no	—
102	C₄⋊C₈	—	67	1	no	—
103	C₄⋊C₈	—	67	1	no	—
104	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
105	#102	—	134	1	no	—
106	#102	—	134	1	no	—
107	C₂²⋊C₈	—	2	1	no	—
108	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
109	C₂×Q₈	—	134	1	no	—
110	C₂×D₄	—	134	1	no	—
111	C₂×D₄	—	134	1	no	—
112	C₂×D₄	—	134	1	no	—
113	C₂×D₄	D₂₆₈	1	1	yes	characteristic
114	C₂²×C₄	—	2	1	no	—
115	SD₁₆	—	268	2	no	—
116	SD₁₆	—	268	2	no	—
117	D₈	—	268	2	no	—
118	M₁₆	—	4	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	134	1	no	—
121	C₂×C₈	—	134	1	no	—
122	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
124	C₄⋊C₄	—	134	1	no	—
125	C₄⋊C₄	—	134	1	no	—
126	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
127	C₂²⋊C₄	—	2	1	no	—
128	C₄²	C₂²×D₆₇	1	1	yes	characteristic
129	C₂³	—	2	1	no	—
130	C₂³	—	268	1	no	—
131	Q₈	—	268	1	no	—
132	Q₈	—	268	1	no	—
133	D₄	—	2	1	no	—
134	D₄	—	2	1	no	—
135	D₄	—	268	1	no	—
136	D₄	—	268	1	no	—
137	D₄	—	268	1	no	—
138	D₄	—	268	1	no	—
139	D₄	—	536	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	268	1	no	—
144	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
146	C₂×C₄	#11761628	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	268	1	no	—
150	C₈	—	268	1	no	—
151	C₈	#11761613	1	2	yes	complemented
152	C₂²	—	4	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	536	1	no	—
156	C₂²	—	536	1	no	—
157	C₂²	—	536	1	no	—
158	C₂²	#14524017	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	536	1	no	—
162	C₄	C₂×#11761613	1	1	yes	characteristic
163	C₄	#14524068	1	1	yes	characteristic
164	C₄	#14524071	1	1	yes	characteristic
165	C₄	#14524076	1	1	yes	characteristic
166	C₂	—	8	1	no	—
167	C₂	—	1072	1	no	—
168	C₂	#16863591	1	1	yes	characteristic
169	C₂	#16863642	1	1	yes	characteristic
170	C₂	#16863659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863769

C₂

yes

maximal, characteristic, complemented

#16863676

C₂

yes

maximal, characteristic, complemented

#16863591

C₂

yes

maximal, characteristic

#16863149

C₂

yes

maximal, characteristic, complemented

#16863143

C₂

yes

maximal, characteristic, complemented

#16863139

C₂

yes

maximal, characteristic, complemented

#436×C₆₇

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#14524040

C₂²

yes

characteristic

#14523976

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523964

—

#14523963

—

C₂×M₁₆×C₆₇

—

C₄×D₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

#11761617

—

#11761617

—

#11761613

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂²×D₆₇

—

D₄×C₆₇

—

D₄×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

#41

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

D₈

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1685

—

maximal, Sylow

C₆₇

#1685

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic, complemented

#335

—

#332

—

#331

—

C₂×SD₁₆

—

134

—

C₂×SD₁₆

—

134

—

C₂×D₈

—

134

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

100

C₄×D₄

C₂×D₆₇

yes

characteristic

101

#107

—

134

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₆₇

yes

characteristic

105

#102

—

134

—

106

#102

—

134

—

107

C₂²⋊C₈

—

108

C₄×C₈

C₂×D₆₇

yes

characteristic

109

C₂×Q₈

—

134

—

110

C₂×D₄

—

134

—

111

C₂×D₄

—

134

—

112

C₂×D₄

—

134

—

113

C₂×D₄

D₂₆₈

yes

characteristic

114

C₂²×C₄

—

115

SD₁₆

—

268

—

116

SD₁₆

—

268

—

117

D₈

—

268

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

134

—

121

C₂×C₈

—

134

—

122

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

123

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

124

C₄⋊C₄

—

134

—

125

C₄⋊C₄

—

134

—

126

C₄⋊C₄

D₂₆₈

yes

characteristic

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₆₇

yes

characteristic

129

C₂³

—

130

C₂³

—

268

—

131

Q₈

—

268

—

132

Q₈

—

268

—

133

D₄

—

134

D₄

—

135

D₄

—

268

—

136

D₄

—

268

—

137

D₄

—

268

—

138

D₄

—

268

—

139

D₄

—

536

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

268

—

144

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

145

C₂×C₄

C₂×D₂₆₈

yes

characteristic

146

C₂×C₄

#11761628

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

268

—

150

C₈

—

268

—

151

C₈

#11761613

yes

complemented

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

536

—

156

C₂²

—

536

—

157

C₂²

—

536

—

158

C₂²

#14524017

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

536

—

162

C₄

C₂×#11761613

yes

characteristic

163

C₄

#14524068

yes

characteristic

164

C₄

#14524071

yes

characteristic

165

C₄

#14524076

yes

characteristic

166

C₂

—

167

C₂

—

1072

—

168

C₂

#16863591

yes

characteristic

169

C₂

#16863642

yes

characteristic

170

C₂

#16863659

yes

characteristic

	subgroup	G/H
center	C₂²	#14524017
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#436×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#436	D₆₇
hypocenter	C₆₇	#1685
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #436

subgroup

G/H

center

C₂²

#14524017

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#436×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #436

N	H ≅ G/N	type
1. #16863769	C₂	semidirect
2. #16863676	C₂	semidirect
4. #16863149	C₂	semidirect
5. #16863143	C₂	semidirect
6. #16863139	C₂	semidirect
7. #436×C₆₇	C₂	semidirect
63. C₈×C₆₇	D₈	semidirect
86. C₆₇	#1685	semidirect
90. #436	D₆₇	semidirect
151. C₈	#11761613	semidirect

H ≅ G/N

type

1. #16863769

C₂

semidirect

2. #16863676

C₂

semidirect

4. #16863149

C₂

semidirect

5. #16863143

C₂

semidirect

6. #16863139

C₂

semidirect

7. #436×C₆₇

C₂

semidirect

63. C₈×C₆₇

D₈

semidirect

86. C₆₇

#1685

semidirect

90. #436

D₆₇

semidirect

151. C₈

#11761613

semidirect

Subgroups — #22559674

Conjugacy classes of subgroups

Complements