Group #22559778 — subgroups

Subgroups — #22559778

Order 8576.

162 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16863743	C₂	1	1	yes	maximal, characteristic
2	#16863607	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863597	C₂	1	1	yes	maximal, characteristic
4	#16863526	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863143	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863139	C₂	1	1	yes	maximal, characteristic
7	#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₆₈	—	2	1	no	—
9	C₂×#11761617	—	2	1	no	—
10	#14524043	C₂²	1	1	yes	characteristic
11	#14524040	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₆₇⋊C₈	—	2	1	no	—
15	#14523968	—	2	1	no	—
16	#14523965	—	2	1	no	—
17	#14523965	—	2	1	no	—
18	#14523965	—	2	1	no	—
19	#14523953	—	2	1	no	—
20	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
25	C₂×D₂₆₈	—	2	1	no	—
26	C₂×D₂₆₈	—	2	1	no	—
27	C₂×D₂₆₈	—	2	1	no	—
28	Dic₂₆₈	—	4	2	no	—
29	#11761617	—	4	2	no	—
30	C₂×C₆₇⋊C₈	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	—	2	1	no	—
33	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
34	Dic₆₇⋊C₄	—	2	1	no	—
35	C₄×Dic₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	—	2	1	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	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×D₆₇	—	4	1	no	—
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
45	C₂×Dic₆₇	—	2	1	no	—
46	C₂×Dic₆₇	—	2	1	no	—
47	C₂×Dic₆₇	—	4	1	no	—
48	D₂₆₈	—	4	1	no	—
49	D₂₆₈	—	4	1	no	—
50	D₂₆₈	—	4	1	no	—
51	D₂₆₈	—	4	1	no	—
52	D₂₆₈	—	8	1	no	—
53	Dic₁₃₄	—	2	1	no	—
54	Dic₁₃₄	—	2	1	no	—
55	Dic₁₃₄	—	4	1	no	—
56	Dic₁₃₄	—	4	1	no	—
57	C₈×C₆₇	—	2	1	no	—
58	C₈×C₆₇	—	2	1	no	—
59	C₈×C₆₇	—	4	1	no	—
60	C₈×C₆₇	—	4	1	no	—
61	C₆₇⋊C₈	—	4	1	no	—
62	C₂²×C₆₇	#121	1	1	yes	characteristic
63	C₂×D₆₇	—	8	1	no	—
64	C₂×D₆₇	—	8	1	no	—
65	C₂×D₆₇	—	8	1	no	—
66	C₄×C₆₇	—	2	1	no	—
67	C₄×C₆₇	#135	1	1	yes	characteristic
68	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₆₇	#137	1	1	yes	characteristic
71	Dic₆₇	—	4	1	no	—
72	Dic₆₇	—	4	1	no	—
73	Dic₆₇	—	8	1	no	—
74	C₂×C₆₇	#464	1	1	yes	characteristic
75	C₂×C₆₇	#469	1	1	yes	characteristic
76	C₂×C₆₇	#471	1	1	yes	characteristic
77	D₆₇	—	16	1	no	—
78	#1702	—	67	1	no	maximal, Sylow
79	C₆₇	#1702	1	1	yes	characteristic, complemented, Sylow
80	#486	—	67	1	no	—
81	#462	—	67	1	no	—
82	#460	—	67	1	no	—
83	#446	—	67	1	no	—
84	#334	D₆₇	1	1	yes	characteristic, complemented
85	#332	—	67	1	no	—
86	#331	—	67	1	no	—
87	C₂×Q₁₆	—	134	1	no	—
88	C₂×SD₁₆	—	134	1	no	—
89	C₄⋊Q₈	—	67	1	no	—
90	#127	—	67	1	no	—
91	C₄×Q₈	—	67	1	no	—
92	C₄⋊C₈	—	67	1	no	—
93	C₄⋊C₈	—	134	1	no	—
94	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
95	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
96	#103	—	134	1	no	—
97	#102	—	134	1	no	—
98	#102	—	134	1	no	—
99	#102	—	134	1	no	—
100	#97	—	134	1	no	—
101	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
102	C₂×Q₈	—	67	1	no	—
103	C₂×Q₈	—	134	1	no	—
104	C₂×D₄	—	134	1	no	—
105	C₂×D₄	—	134	1	no	—
106	C₂×D₄	—	134	1	no	—
107	Q₁₆	—	268	2	no	—
108	SD₁₆	—	268	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	134	1	no	—
112	C₂×C₈	D₂₆₈	1	1	yes	characteristic
113	C₂×C₈	D₂₆₈	1	1	yes	characteristic
114	C₄⋊C₄	—	67	1	no	—
115	C₄⋊C₄	—	134	1	no	—
116	C₄⋊C₄	—	134	1	no	—
117	C₄⋊C₄	—	134	1	no	—
118	C₄²	—	134	1	no	—
119	C₄²	C₂²×D₆₇	1	1	yes	characteristic
120	C₂³	—	268	1	no	—
121	Q₈	—	268	1	no	—
122	Q₈	—	268	1	no	—
123	Q₈	—	134	1	no	—
124	Q₈	—	134	1	no	—
125	D₄	—	268	1	no	—
126	D₄	—	268	1	no	—
127	D₄	—	268	1	no	—
128	D₄	—	268	1	no	—
129	D₄	—	536	1	no	—
130	C₂×C₄	—	268	1	no	—
131	C₂×C₄	—	134	1	no	—
132	C₂×C₄	—	134	1	no	—
133	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
134	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
135	C₂×C₄	#11761629	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	4	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	268	1	no	—
141	C₂²	—	536	1	no	—
142	C₂²	—	536	1	no	—
143	C₂²	—	536	1	no	—
144	C₂²	#14524018	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	268	1	no	—
147	C₄	—	268	1	no	—
148	C₄	—	536	1	no	—
149	C₄	#14524065	1	1	yes	characteristic
150	C₄	#14524070	1	1	yes	characteristic
151	C₄	#14524070	1	1	yes	characteristic
152	C₄	#14524076	1	1	yes	characteristic
153	C₂	—	1072	1	no	—
154	C₂	#16863619	1	1	yes	characteristic
155	C₂	#16863650	1	1	yes	characteristic
156	C₂	#16863660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863743

C₂

yes

maximal, characteristic

#16863607

C₂

yes

maximal, characteristic, complemented

#16863597

C₂

yes

maximal, characteristic

#16863526

C₂

yes

maximal, characteristic, complemented

#16863143

C₂

yes

maximal, characteristic, complemented

#16863139

C₂

yes

maximal, characteristic

#334×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₈

—

C₂×#11761617

—

#14524043

C₂²

yes

characteristic

#14524040

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

—

—

—

—

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

Dic₂₆₈

—

#11761617

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1702

—

maximal, Sylow

C₆₇

#1702

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic, complemented

#332

—

#331

—

C₂×Q₁₆

—

134

—

C₂×SD₁₆

—

134

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

#102

—

134

—

#102

—

134

—

#102

—

134

—

100

#97

—

134

—

101

C₄×C₈

C₂×D₆₇

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

134

—

104

C₂×D₄

—

134

—

105

C₂×D₄

—

134

—

106

C₂×D₄

—

134

—

107

Q₁₆

—

268

—

108

SD₁₆

—

268

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

134

—

112

C₂×C₈

D₂₆₈

yes

characteristic

113

C₂×C₈

D₂₆₈

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

134

—

116

C₄⋊C₄

—

134

—

117

C₄⋊C₄

—

134

—

118

C₄²

—

134

—

119

C₄²

C₂²×D₆₇

yes

characteristic

120

C₂³

—

268

—

121

Q₈

—

268

—

122

Q₈

—

268

—

123

Q₈

—

134

—

124

Q₈

—

134

—

125

D₄

—

268

—

126

D₄

—

268

—

127

D₄

—

268

—

128

D₄

—

268

—

129

D₄

—

536

—

130

C₂×C₄

—

268

—

131

C₂×C₄

—

134

—

132

C₂×C₄

—

134

—

133

C₂×C₄

D₄×D₆₇

yes

characteristic

134

C₂×C₄

C₂×D₂₆₈

yes

characteristic

135

C₂×C₄

#11761629

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

268

—

141

C₂²

—

536

—

142

C₂²

—

536

—

143

C₂²

—

536

—

144

C₂²

#14524018

yes

characteristic

145

C₄

—

146

C₄

—

268

—

147

C₄

—

268

—

148

C₄

—

536

—

149

C₄

#14524065

yes

characteristic

150

C₄

#14524070

yes

characteristic

151

C₄

#14524070

yes

characteristic

152

C₄

#14524076

yes

characteristic

153

C₂

—

1072

—

154

C₂

#16863619

yes

characteristic

155

C₂

#16863650

yes

characteristic

156

C₂

#16863660

yes

characteristic

	subgroup	G/H
center	C₂²	#14524018
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#334×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#334	D₆₇
hypocenter	C₆₇	#1702
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14524018

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#334×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #16863607	C₂	semidirect
4. #16863526	C₂	semidirect
5. #16863143	C₂	semidirect
7. #334×C₆₇	C₂	semidirect
79. C₆₇	#1702	semidirect
84. #334	D₆₇	semidirect

H ≅ G/N

type

2. #16863607

C₂

semidirect

4. #16863526

C₂

semidirect

5. #16863143

C₂

semidirect

7. #334×C₆₇

C₂

semidirect

79. C₆₇

#1702

semidirect

84. #334

D₆₇

semidirect

Subgroups — #22559778

Conjugacy classes of subgroups

Complements