Group #22561606 — subgroups

Subgroups — #22561606

Order 8576.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16863445	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523959	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523959	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523959	C₂	1	1	yes	maximal, characteristic
6	C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
7	#430×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	C₂×#11761612	C₂²	1	1	yes	characteristic
12	#14523979	—	2	2	no	—
13	#14523978	—	2	2	no	—
14	#14523959	—	2	2	no	—
15	#14523959	—	2	2	no	—
16	#14523959	—	2	2	no	—
17	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	#108×C₆₇	—	2	2	no	—
21	C₂³×D₆₇	Q₈	1	2	yes	—
22	C₂×D₂₆₈	—	2	1	no	—
23	C₂×D₂₆₈	—	2	1	no	—
24	C₂×D₂₆₈	—	2	1	no	—
25	C₂×D₂₆₈	—	2	1	no	—
26	C₂×D₂₆₈	—	2	1	no	—
27	C₂×D₂₆₈	—	2	1	no	—
28	#11761612	—	2	1	no	—
29	#11761612	—	2	1	no	—
30	#11761612	—	2	1	no	—
31	#11761612	—	2	1	no	—
32	#11761612	—	2	1	no	—
33	#11761612	D₄	1	2	yes	—
34	C₂×C₆₇⋊C₈	—	2	1	no	—
35	C₂×C₆₇⋊C₈	—	2	1	no	—
36	C₂×C₆₇⋊C₈	—	2	1	no	—
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	M₁₆×C₆₇	—	2	1	no	—
39	M₁₆×C₆₇	—	2	1	no	—
40	M₁₆×C₆₇	—	2	1	no	—
41	M₁₆×C₆₇	—	2	1	no	—
42	M₁₆×C₆₇	D₄	1	2	yes	—
43	M₁₆×C₆₇	D₄	1	2	yes	—
44	C₂×C₈×C₆₇	—	2	1	no	—
45	C₂×C₈×C₆₇	—	2	1	no	—
46	C₂×C₈×C₆₇	—	2	1	no	—
47	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×D₆₇	—	2	2	no	—
49	C₂²×D₆₇	—	2	2	no	—
50	C₂²×D₆₇	—	2	2	no	—
51	C₂²×D₆₇	—	8	2	no	—
52	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
58	D₂₆₈	—	4	1	no	—
59	D₂₆₈	—	4	1	no	—
60	D₂₆₈	—	4	1	no	—
61	D₂₆₈	—	4	1	no	—
62	C₈×C₆₇	—	2	2	no	—
63	C₈×C₆₇	—	2	2	no	—
64	C₈×C₆₇	—	4	1	no	—
65	C₆₇⋊C₈	—	2	2	no	—
66	C₆₇⋊C₈	—	4	1	no	—
67	C₆₇⋊C₈	—	4	1	no	—
68	C₂²×C₆₇	—	4	1	no	—
69	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×D₆₇	—	4	2	no	—
73	C₂×D₆₇	—	8	2	no	—
74	C₂×D₆₇	—	8	2	no	—
75	C₂×D₆₇	—	8	2	no	—
76	C₄×C₆₇	#121	1	1	yes	characteristic
77	C₄×C₆₇	#121	1	1	yes	characteristic
78	C₄×C₆₇	#121	1	1	yes	characteristic
79	C₄×C₆₇	#126	1	1	yes	characteristic
80	C₂×C₆₇	—	2	1	no	—
81	C₂×C₆₇	—	2	1	no	—
82	C₂×C₆₇	—	2	1	no	—
83	C₂×C₆₇	#396	1	1	yes	characteristic
84	D₆₇	—	8	2	no	—
85	#2069	—	67	1	no	maximal, Sylow
86	C₆₇	#2069	1	1	yes	characteristic, complemented, Sylow
87	#430	—	67	1	no	—
88	#430	—	67	1	no	—
89	#430	D₆₇	1	1	yes	characteristic, complemented
90	C₂×#100	—	67	1	no	—
91	C₂×#100	—	67	1	no	—
92	C₂×#100	—	67	1	no	—
93	#338	—	67	1	no	—
94	C₂²×D₄	—	67	1	no	—
95	C₂×M₁₆	—	67	1	no	—
96	C₂×M₁₆	—	67	1	no	—
97	C₂×M₁₆	—	67	1	no	—
98	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
99	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
101	#108	—	2	2	no	—
102	#108	—	134	2	no	—
103	#108	—	134	2	no	—
104	#100	—	134	2	no	—
105	#100	—	134	2	no	—
106	#100	—	134	2	no	—
107	C₂⁴	—	67	2	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	C₂×D₄	—	134	1	no	—
113	C₂×D₄	—	134	1	no	—
114	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	2	1	no	—
119	M₁₆	—	67	2	no	—
120	M₁₆	—	134	1	no	—
121	M₁₆	—	134	1	no	—
122	M₁₆	—	134	1	no	—
123	M₁₆	—	134	1	no	—
124	M₁₆	—	134	1	no	—
125	M₁₆	D₂₆₈	1	2	yes	—
126	M₁₆	C₂²⋊D₆₇	1	2	yes	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	134	1	no	—
131	C₂×C₈	—	134	1	no	—
132	C₂×C₈	—	134	1	no	—
133	C₂³	—	536	2	no	—
134	C₂³	—	134	2	no	—
135	C₂³	—	134	2	no	—
136	C₂³	—	134	2	no	—
137	C₂³	Q₈×D₆₇	1	1	yes	characteristic
138	D₄	—	268	1	no	—
139	D₄	—	268	1	no	—
140	D₄	—	268	1	no	—
141	D₄	—	268	1	no	—
142	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
145	C₂×C₄	#11761627	1	1	yes	characteristic
146	C₂×C₄	#11761628	1	1	yes	characteristic
147	C₂×C₄	#11761629	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	—	268	1	no	—
152	C₈	—	268	1	no	—
153	C₈	—	134	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	268	2	no	—
156	C₂²	—	536	2	no	—
157	C₂²	—	536	2	no	—
158	C₂²	—	536	2	no	—
159	C₂²	#14524019	1	1	yes	characteristic
160	C₂²	#14524021	1	1	yes	characteristic
161	C₂²	#14524023	1	1	yes	characteristic
162	C₄	#14524015	1	1	yes	characteristic
163	C₄	#14524017	1	1	yes	characteristic
164	C₄	#14524018	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	#16863318	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863445

C₂

yes

maximal, characteristic, complemented

#16863444

C₂

yes

maximal, characteristic, complemented

C₂×#14523959

C₂

yes

maximal, characteristic

C₂×#14523959

C₂

yes

maximal, characteristic

C₂×#14523959

C₂

yes

maximal, characteristic

C₆₇⋊#338

C₂

yes

maximal, characteristic, complemented

#430×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#108×C₆₇

—

C₂³×D₆₇

Q₈

yes

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

—

—

—

—

—

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

D₄

yes

—

M₁₆×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂³×C₆₇

C₂×Q₈

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

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

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#396

yes

characteristic

D₆₇

—

#2069

—

maximal, Sylow

C₆₇

#2069

yes

characteristic, complemented, Sylow

#430

—

#430

—

#430

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

characteristic

100

C₂×M₁₆

C₂×D₆₇

yes

characteristic

101

#108

—

102

#108

—

134

—

103

#108

—

134

—

104

#100

—

134

—

105

#100

—

134

—

106

#100

—

134

—

107

C₂⁴

—

108

C₂×D₄

—

134

—

109

C₂×D₄

—

134

—

110

C₂×D₄

—

134

—

111

C₂×D₄

—

134

—

112

C₂×D₄

—

134

—

113

C₂×D₄

—

134

—

114

C₂²×C₄

C₂²×D₆₇

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

134

—

121

M₁₆

—

134

—

122

M₁₆

—

134

—

123

M₁₆

—

134

—

124

M₁₆

—

134

—

125

M₁₆

D₂₆₈

yes

—

126

M₁₆

C₂²⋊D₆₇

yes

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

134

—

131

C₂×C₈

—

134

—

132

C₂×C₈

—

134

—

133

C₂³

—

536

—

134

C₂³

—

134

—

135

C₂³

—

134

—

136

C₂³

—

134

—

137

C₂³

Q₈×D₆₇

yes

characteristic

138

D₄

—

268

—

139

D₄

—

268

—

140

D₄

—

268

—

141

D₄

—

268

—

142

C₂×C₄

D₄×D₆₇

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

144

C₂×C₄

C₂×D₂₆₈

yes

characteristic

145

C₂×C₄

#11761627

yes

characteristic

146

C₂×C₄

#11761628

yes

characteristic

147

C₂×C₄

#11761629

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

268

—

152

C₈

—

268

—

153

C₈

—

134

—

154

C₂²

—

155

C₂²

—

268

—

156

C₂²

—

536

—

157

C₂²

—

536

—

158

C₂²

—

536

—

159

C₂²

#14524019

yes

characteristic

160

C₂²

#14524021

yes

characteristic

161

C₂²

#14524023

yes

characteristic

162

C₄

#14524015

yes

characteristic

163

C₄

#14524017

yes

characteristic

164

C₄

#14524018

yes

characteristic

165

C₄

#14524038

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

536

—

170

C₂

#16863318

yes

characteristic

	subgroup	G/H
center	C₂	#16863318
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#430×C₆₇	C₂
socle	C₂×C₆₇	#396
hypercenter	#430	D₆₇
hypocenter	C₆₇	#2069
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₂

#16863318

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#430×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #16863445	C₂	semidirect
2. #16863444	C₂	semidirect
6. C₆₇⋊#338	C₂	semidirect
7. #430×C₆₇	C₂	semidirect
86. C₆₇	#2069	semidirect
89. #430	D₆₇	semidirect

H ≅ G/N

type

1. #16863445

C₂

semidirect

2. #16863444

C₂

semidirect

6. C₆₇⋊#338

C₂

semidirect

7. #430×C₆₇

C₂

semidirect

86. C₆₇

#2069

semidirect

89. #430

D₆₇

semidirect

Subgroups — #22561606

Conjugacy classes of subgroups

Complements