Group #22560756 — subgroups

Subgroups — #22560756

Order 8576.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16863960	C₂	1	1	yes	maximal, characteristic
2	#16863380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523959	C₂	1	2	yes	maximal, complemented
4	#16863168	C₂	1	2	yes	maximal, complemented
5	#568×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×#11761612	C₂²	1	2	yes	—
8	#14524039	—	2	1	no	—
9	#14524032	—	2	1	no	—
10	#14524017	—	2	2	no	—
11	#14524011	C₄	1	4	yes	—
12	#14524002	C₂²	1	1	yes	characteristic
13	#14523962	—	2	1	no	—
14	#14523960	—	2	1	no	—
15	#14523959	—	2	4	no	—
16	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
17	#131×C₆₇	—	2	4	no	—
18	C₂×M₁₆×C₆₇	—	2	2	no	—
19	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂³×D₆₇	C₂×C₄	1	2	yes	—
21	C₂×D₂₆₈	—	2	2	no	—
22	C₂×D₂₆₈	—	2	1	no	—
23	C₂×D₂₆₈	—	2	1	no	—
24	C₂×D₂₆₈	—	2	2	no	—
25	C₂×C₂²⋊D₆₇	—	2	4	no	—
26	#11761612	—	2	2	no	—
27	#11761612	—	2	2	no	—
28	C₂×C₆₇⋊C₈	—	2	2	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
31	(C₂×D₆₇)⋊C₄	—	2	4	no	—
32	C₄×Dic₆₇	—	2	1	no	—
33	#41×C₆₇	—	2	2	no	—
34	#41×C₆₇	—	2	2	no	—
35	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	D₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	M₁₆×C₆₇	—	2	4	no	—
41	M₁₆×C₆₇	—	2	2	no	—
42	M₁₆×C₆₇	D₄	1	4	yes	complemented
43	C₂×C₈×C₆₇	—	2	4	no	—
44	C₂×C₈×C₆₇	—	2	2	no	—
45	C₂³×C₆₇	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₆₇	—	2	2	no	—
48	C₂²×D₆₇	—	2	4	no	—
49	C₂²×D₆₇	—	8	2	no	—
50	Q₈×C₆₇	—	2	2	no	—
51	D₄×C₆₇	—	2	4	no	—
52	D₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₆₇	#41	1	2	yes	—
60	C₂²⋊D₆₇	—	8	4	no	—
61	C₂×Dic₆₇	—	2	2	no	—
62	D₂₆₈	—	4	1	no	—
63	D₂₆₈	—	4	2	no	—
64	D₂₆₈	—	4	1	no	—
65	C₈×C₆₇	—	2	4	no	—
66	C₆₇⋊C₈	—	4	2	no	—
67	C₂²×C₆₇	—	2	2	no	—
68	C₂²×C₆₇	—	4	1	no	—
69	C₂²×C₆₇	—	4	2	no	—
70	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₆₇	C₄×D₄	1	2	yes	—
72	C₂×D₆₇	—	4	4	no	—
73	C₂×D₆₇	—	4	2	no	—
74	C₂×D₆₇	—	8	4	no	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	#120	1	1	yes	characteristic
77	C₄×C₆₇	#121	1	2	yes	—
78	C₄×C₆₇	#123	1	1	yes	characteristic
79	Dic₆₇	—	4	2	no	—
80	C₂×C₆₇	—	2	1	no	—
81	C₂×C₆₇	—	2	2	no	—
82	C₂×C₆₇	—	4	2	no	—
83	C₂×C₆₇	#386	1	1	yes	characteristic
84	D₆₇	—	8	2	no	—
85	#1914	—	67	1	no	maximal, Sylow
86	C₆₇	#1914	1	1	yes	characteristic, complemented, Sylow
87	#568	D₆₇	1	1	yes	characteristic, complemented
88	#535	—	67	1	no	—
89	#413	—	67	1	no	—
90	C₂×#100	—	67	2	no	—
91	#344	—	67	2	no	—
92	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
93	C₂²×D₄	—	67	1	no	—
94	#131	—	2	4	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	67	2	no	—
97	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
98	#127	—	134	1	no	—
99	#124	—	134	1	no	—
100	#121	—	134	2	no	—
101	#120	—	67	4	no	—
102	#117	—	67	1	no	—
103	#101	—	134	1	no	—
104	#100	—	134	1	no	—
105	#100	—	134	4	no	—
106	C₂⁴	—	67	2	no	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
110	C₂×D₄	—	134	4	no	—
111	C₂×D₄	—	134	2	no	—
112	C₂×D₄	—	134	1	no	—
113	C₂×D₄	—	134	1	no	—
114	C₂×D₄	—	134	2	no	—
115	C₂×D₄	C₄×D₆₇	1	2	yes	—
116	C₂×D₄	D₂₆₈	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
118	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
119	M₁₆	—	2	4	no	—
120	M₁₆	—	2	2	no	—
121	M₁₆	—	134	2	no	—
122	M₁₆	—	134	2	no	—
123	M₁₆	C₂²⋊D₆₇	1	4	yes	complemented
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	134	2	no	—
127	C₄⋊C₄	—	134	1	no	—
128	C₂²⋊C₄	—	67	2	no	—
129	C₂²⋊C₄	—	134	4	no	—
130	C₄²	—	134	1	no	—
131	C₂³	—	536	2	no	—
132	C₂³	—	134	2	no	—
133	C₂³	—	134	4	no	—
134	C₂³	(C₂×D₆₇)⋊C₄	1	2	yes	—
135	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	268	1	no	—
140	D₄	—	268	2	no	—
141	D₄	—	268	1	no	—
142	D₄	—	536	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	134	2	no	—
146	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
150	C₂×C₄	#11761628	1	2	yes	—
151	C₈	—	2	4	no	—
152	C₈	—	268	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	268	4	no	—
157	C₂²	—	268	2	no	—
158	C₂²	—	536	4	no	—
159	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
160	C₂²	C₄×C₂²⋊D₆₇	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	268	2	no	—
163	C₄	#14524012	1	1	yes	characteristic
164	C₄	#14524017	1	2	yes	—
165	C₄	#14524026	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	536	2	no	—
170	C₂	#16863275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863960

C₂

yes

maximal, characteristic

#16863380

C₂

yes

maximal, characteristic, complemented

C₂×#14523959

C₂

yes

maximal, complemented

#16863168

C₂

yes

maximal, complemented

#568×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

—

—

—

—

C₄

yes

—

#14524002

C₂²

yes

—

—

—

C₂×#41×C₆₇

C₂²

yes

characteristic

#131×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂³×D₆₇

C₂×C₄

yes

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₂²⋊D₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

C₂×C₄

yes

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂×D₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

D₄

yes

complemented

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂³×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

C₄×D₄

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

#121

yes

—

C₄×C₆₇

#123

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#386

yes

characteristic

D₆₇

—

#1914

—

maximal, Sylow

C₆₇

#1914

yes

characteristic, complemented, Sylow

#568

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂×#41

C₂×D₆₇

yes

characteristic

C₂²×D₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

#127

—

134

—

#124

—

134

—

100

#121

—

134

—

101

#120

—

102

#117

—

103

#101

—

134

—

104

#100

—

134

—

105

#100

—

134

—

106

C₂⁴

—

107

#41

—

108

#41

—

109

C₂×Q₈

D₂₆₈

yes

characteristic

110

C₂×D₄

—

134

—

111

C₂×D₄

—

134

—

112

C₂×D₄

—

134

—

113

C₂×D₄

—

134

—

114

C₂×D₄

—

134

—

115

C₂×D₄

C₄×D₆₇

yes

—

116

C₂×D₄

D₂₆₈

yes

characteristic

117

C₂²×C₄

C₂²⋊D₆₇

yes

—

118

C₂²×C₄

C₂²×D₆₇

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

134

—

122

M₁₆

—

134

—

123

M₁₆

C₂²⋊D₆₇

yes

complemented

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

134

—

127

C₄⋊C₄

—

134

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

134

—

130

C₄²

—

134

—

131

C₂³

—

536

—

132

C₂³

—

134

—

133

C₂³

—

134

—

134

C₂³

(C₂×D₆₇)⋊C₄

yes

—

135

C₂³

C₂×C₄×D₆₇

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

268

—

140

D₄

—

268

—

141

D₄

—

268

—

142

D₄

—

536

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

134

—

146

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

147

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

148

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

149

C₂×C₄

C₂×D₂₆₈

yes

characteristic

150

C₂×C₄

#11761628

yes

—

151

C₈

—

152

C₈

—

268

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

268

—

157

C₂²

—

268

—

158

C₂²

—

536

—

159

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

160

C₂²

C₄×C₂²⋊D₆₇

yes

—

161

C₄

—

162

C₄

—

268

—

163

C₄

#14524012

yes

characteristic

164

C₄

#14524017

yes

—

165

C₄

#14524026

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

536

—

170

C₂

#16863275

yes

characteristic

	subgroup	G/H
center	C₂	#16863275
commutator	C₂³×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#568×C₆₇	C₂
socle	C₂×C₆₇	#386
hypercenter	#568	D₆₇
hypocenter	C₆₇	#1914
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#16863275

commutator

C₂³×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#568×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #16863380	C₂	semidirect
3. C₂×#14523959	C₂	semidirect
4. #16863168	C₂	semidirect
5. #568×C₆₇	C₂	semidirect
42. M₁₆×C₆₇	D₄	semidirect
86. C₆₇	#1914	semidirect
87. #568	D₆₇	semidirect
123. M₁₆	C₂²⋊D₆₇	semidirect

H ≅ G/N

type

2. #16863380

C₂

semidirect

3. C₂×#14523959

C₂

semidirect

4. #16863168

C₂

semidirect

5. #568×C₆₇

C₂

semidirect

42. M₁₆×C₆₇

D₄

semidirect

86. C₆₇

#1914

semidirect

87. #568

D₆₇

semidirect

123. M₁₆

C₂²⋊D₆₇

semidirect

Subgroups — #22560756

Conjugacy classes of subgroups

Complements