Group #22561718 — subgroups

Subgroups — #22561718

Order 8576.

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16863438	C₂	1	1	yes	maximal, characteristic
2	#16863436	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523976	C₂	1	1	yes	maximal, characteristic
4	#16863424	C₂	1	1	yes	maximal, characteristic
5	#16863164	C₂	1	1	yes	maximal, characteristic
6	#16863160	C₂	1	1	yes	maximal, characteristic
7	#343×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#14524002	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
12	#14523977	—	2	1	no	—
13	#14523976	—	2	2	no	—
14	#14523976	—	2	1	no	—
15	#14523975	—	2	1	no	—
16	#14523974	—	2	1	no	—
17	Dic₆₇⋊C₈	—	2	2	no	—
18	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×Dic₆₇	—	2	1	no	—
22	#11761612	—	2	1	no	—
23	#11761612	Q₈	1	2	yes	—
24	C₂×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
31	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
32	C₂²⋊Dic₆₇	—	2	1	no	—
33	C₄×Dic₆₇	—	2	1	no	—
34	C₂²×C₄×C₆₇	—	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	Q₈	1	2	yes	—
40	C₂×C₈×C₆₇	—	2	1	no	—
41	C₂×C₈×C₆₇	—	2	1	no	—
42	C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂³×C₆₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	—	8	1	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
53	C₂×Dic₆₇	—	2	1	no	—
54	C₂×Dic₆₇	—	2	1	no	—
55	C₂×Dic₆₇	—	4	1	no	—
56	C₂×Dic₆₇	—	8	1	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	—	4	1	no	—
59	C₆₇⋊C₈	—	2	2	no	—
60	C₆₇⋊C₈	—	2	2	no	—
61	C₂²×C₆₇	—	2	1	no	—
62	C₂²×C₆₇	—	2	1	no	—
63	C₂²×C₆₇	#121	1	1	yes	characteristic
64	C₂²×C₆₇	#125	1	1	yes	characteristic
65	C₂²×C₆₇	#125	1	1	yes	characteristic
66	C₄×C₆₇	—	8	1	no	—
67	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₄×C₆₇	#123	1	1	yes	characteristic
70	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
71	Dic₆₇	—	4	1	no	—
72	Dic₆₇	—	4	1	no	—
73	Dic₆₇	—	8	1	no	—
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	#398	1	1	yes	characteristic
77	C₂×C₆₇	#472	1	1	yes	characteristic
78	C₂×C₆₇	#473	1	1	yes	characteristic
79	#2089	—	67	1	no	maximal, Sylow
80	C₆₇	#2089	1	1	yes	characteristic, complemented, Sylow
81	#428	—	67	1	no	—
82	#428	—	67	1	no	—
83	C₂×#107	—	67	1	no	—
84	#424	—	67	1	no	—
85	#343	—	67	1	no	—
86	#343	D₆₇	1	1	yes	characteristic
87	#341	—	67	1	no	—
88	C₂×M₁₆	—	67	1	no	—
89	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
90	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
91	C₂²×C₈	—	67	1	no	—
92	#117	—	67	1	no	—
93	C₂×C₄⋊C₄	—	67	1	no	—
94	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
95	#107	—	134	1	no	—
96	#107	—	134	2	no	—
97	#107	—	134	1	no	—
98	#106	—	134	1	no	—
99	#106	—	134	1	no	—
100	C₄⋊C₈	—	134	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	134	1	no	—
103	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	2	1	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	67	2	no	—
108	M₁₆	—	134	1	no	—
109	M₁₆	Dic₁₃₄	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	—	134	1	no	—
114	C₂×C₈	—	134	1	no	—
115	C₂×C₈	—	134	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	67	1	no	—
119	C₄⋊C₄	—	67	1	no	—
120	C₄⋊C₄	—	134	1	no	—
121	C₄⋊C₄	—	134	1	no	—
122	C₂²⋊C₄	—	134	1	no	—
123	C₄²	—	134	1	no	—
124	C₂³	#11761628	1	1	yes	characteristic
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	—	268	1	no	—
128	C₂×C₄	—	536	1	no	—
129	C₂×C₄	—	134	1	no	—
130	C₂×C₄	—	134	1	no	—
131	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
132	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
135	C₂×C₄	#11761627	1	1	yes	characteristic
136	C₂×C₄	#11761628	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	4	1	no	—
139	C₈	—	134	2	no	—
140	C₈	—	134	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14524016	1	1	yes	characteristic
144	C₂²	#14524033	1	1	yes	characteristic
145	C₂²	#14524035	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	268	1	no	—
148	C₄	—	268	1	no	—
149	C₄	—	536	1	no	—
150	C₄	#14524020	1	1	yes	characteristic
151	C₄	#14524021	1	1	yes	characteristic
152	C₄	#14524025	1	1	yes	characteristic
153	C₄	#14524042	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16863330	1	1	yes	characteristic
157	C₂	#16863665	1	1	yes	characteristic
158	C₂	#16863670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863438

C₂

yes

maximal, characteristic

#16863436

C₂

yes

maximal, characteristic

C₂×#14523976

C₂

yes

maximal, characteristic

#16863424

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal, characteristic

#16863160

C₂

yes

maximal, characteristic

#343×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

—

—

—

—

Dic₆₇⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

#11761612

—

#11761612

Q₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

C₄⋊Dic₆₇

D₄

yes

characteristic

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

Q₈

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

#123

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

#107

—

134

—

#107

—

134

—

#107

—

134

—

#106

—

134

—

#106

—

134

—

100

C₄⋊C₈

—

134

—

101

C₂²×C₄

—

102

C₂²×C₄

—

134

—

103

C₂²×C₄

C₂²×D₆₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

134

—

109

M₁₆

Dic₁₃₄

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

134

—

114

C₂×C₈

—

134

—

115

C₂×C₈

—

134

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

134

—

121

C₄⋊C₄

—

134

—

122

C₂²⋊C₄

—

134

—

123

C₄²

—

134

—

124

C₂³

#11761628

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

268

—

128

C₂×C₄

—

536

—

129

C₂×C₄

—

134

—

130

C₂×C₄

—

134

—

131

C₂×C₄

D₄×D₆₇

yes

characteristic

132

C₂×C₄

D₄×D₆₇

yes

characteristic

133

C₂×C₄

Q₈×D₆₇

yes

characteristic

134

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

135

C₂×C₄

#11761627

yes

characteristic

136

C₂×C₄

#11761628

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

134

—

140

C₈

—

134

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14524016

yes

characteristic

144

C₂²

#14524033

yes

characteristic

145

C₂²

#14524035

yes

characteristic

146

C₄

—

147

C₄

—

268

—

148

C₄

—

268

—

149

C₄

—

536

—

150

C₄

#14524020

yes

characteristic

151

C₄

#14524021

yes

characteristic

152

C₄

#14524025

yes

characteristic

153

C₄

#14524042

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#16863330

yes

characteristic

157

C₂

#16863665

yes

characteristic

158

C₂

#16863670

yes

characteristic

	subgroup	G/H
center	C₂²	#14524016
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#343×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#343	D₆₇
hypocenter	C₆₇	#2089
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#14524016

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#343×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
80. C₆₇	#2089	semidirect

H ≅ G/N

type

80. C₆₇

#2089

semidirect

Subgroups — #22561718

Conjugacy classes of subgroups

Complements