Group #46811644 — subgroups

Subgroups — #46811644

Order 34432.

256 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#22575387	C₂	1	2	yes	maximal
2	#31770475	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#31770368	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₂₆₉	—	2	1	no	—
7	C₂×C₄⋊Dic₂₆₉	—	2	1	no	—
8	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₆₉	—	2	1	no	—
12	#22575387	—	2	4	no	—
13	#22575360	—	2	1	no	—
14	C₂²×C₈×C₂₆₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₆₉	C₄	1	4	yes	complemented
17	C₂²×Dic₂₆₉	—	2	1	no	—
18	C₂²×Dic₂₆₉	—	2	1	no	—
19	C₂²×Dic₂₆₉	—	2	2	no	—
20	C₄⋊Dic₂₆₉	—	2	2	no	—
21	C₄⋊Dic₂₆₉	—	2	1	no	—
22	C₄⋊Dic₂₆₉	—	2	1	no	—
23	C₄⋊Dic₂₆₉	—	2	2	no	—
24	C₄⋊Dic₂₆₉	—	4	2	no	—
25	C₄⋊Dic₂₆₉	—	4	2	no	—
26	C₄⋊Dic₂₆₉	D₄	1	2	yes	—
27	C₄⋊Dic₂₆₉	Q₈	1	2	yes	—
28	C₄×Dic₂₆₉	—	4	2	no	—
29	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₂₆₉	—	2	2	no	—
33	C₂×C₈×C₂₆₉	—	2	2	no	—
34	C₂×C₈×C₂₆₉	C₂×C₄	1	4	yes	—
35	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
36	C₄²×C₂₆₉	Q₈	1	2	yes	—
37	C₂×Dic₂₆₉	—	2	2	no	—
38	C₂×Dic₂₆₉	—	2	2	no	—
39	C₂×Dic₂₆₉	—	4	1	no	—
40	C₂×Dic₂₆₉	—	4	1	no	—
41	C₂×Dic₂₆₉	—	4	1	no	—
42	C₂×Dic₂₆₉	—	4	1	no	—
43	C₂×Dic₂₆₉	—	4	2	no	—
44	C₂×Dic₂₆₉	—	4	2	no	—
45	C₂×Dic₂₆₉	—	4	2	no	—
46	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₆₉	—	2	1	no	—
48	C₂×C₄×C₂₆₉	—	2	1	no	—
49	C₂×C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₂₆₉	D₈	1	2	yes	—
52	C₂×C₄×C₂₆₉	Q₁₆	1	2	yes	—
53	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
58	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
59	C₈×C₂₆₉	—	2	4	no	—
60	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₆₉	#121	1	1	yes	characteristic
62	C₂²×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₂₆₉	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₂₆₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×C₂₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₂₆₉	#137	1	1	yes	characteristic
67	C₄×C₂₆₉	—	2	2	no	—
68	C₄×C₂₆₉	#107	1	4	yes	complemented
69	C₄×C₂₆₉	C₄×D₄	1	1	yes	characteristic
70	C₄×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
71	C₄×C₂₆₉	#125	1	1	yes	characteristic
72	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₂₆₉	—	4	2	no	—
74	Dic₂₆₉	—	4	2	no	—
75	Dic₂₆₉	—	8	2	no	—
76	C₂×C₂₆₉	#389	1	1	yes	characteristic
77	C₂×C₂₆₉	C₂×#107	1	1	yes	characteristic
78	C₂×C₂₆₉	#428	1	1	yes	characteristic
79	C₂×C₂₆₉	#459	1	1	yes	characteristic
80	C₂×C₂₆₉	#462	1	1	yes	characteristic
81	C₂×C₂₆₉	#474	1	1	yes	characteristic
82	C₂×C₂₆₉	#477	1	1	yes	characteristic
83	C₂₆₉	#1947	1	1	yes	characteristic, complemented, Sylow
84	#1947	—	269	1	no	maximal, Sylow
85	C₂×#107	—	269	2	no	—
86	C₂×C₄⋊C₈	D₂₆₉	1	1	yes	characteristic
87	#384	—	269	1	no	—
88	C₄×C₄⋊C₄	—	269	1	no	—
89	#340	—	269	2	no	—
90	C₂²×C₈	C₂×D₂₆₉	1	2	yes	—
91	C₂×C₄⋊C₄	—	269	1	no	—
92	C₂×C₄⋊C₄	—	269	1	no	—
93	C₂×C₄⋊C₄	—	269	2	no	—
94	C₂×C₄⋊C₄	—	538	1	no	—
95	C₂×C₄⋊C₄	—	538	1	no	—
96	C₂×C₄²	—	538	1	no	—
97	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
98	#107	—	538	4	no	—
99	C₄⋊C₈	Dic₂₆₉	1	4	yes	complemented
100	#95	—	538	1	no	—
101	C₂²×C₄	—	538	1	no	—
102	C₂²×C₄	—	538	1	no	—
103	C₂²×C₄	—	538	2	no	—
104	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
105	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₂×Dic₂₆₉	1	4	yes	—
110	C₄⋊C₄	—	269	2	no	—
111	C₄⋊C₄	—	269	2	no	—
112	C₄⋊C₄	—	538	1	no	—
113	C₄⋊C₄	—	538	1	no	—
114	C₄⋊C₄	—	538	2	no	—
115	C₄⋊C₄	—	538	2	no	—
116	C₄⋊C₄	—	1076	2	no	—
117	C₄⋊C₄	—	1076	2	no	—
118	C₄²	—	1076	2	no	—
119	C₄²	C₂×Dic₂₆₉	1	2	yes	—
120	C₄²	Dic₅₃₈	1	2	yes	—
121	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	538	2	no	—
125	C₂×C₄	—	538	2	no	—
126	C₂×C₄	—	1076	1	no	—
127	C₂×C₄	—	1076	1	no	—
128	C₂×C₄	—	1076	1	no	—
129	C₂×C₄	—	1076	1	no	—
130	C₂×C₄	—	1076	2	no	—
131	C₂×C₄	—	1076	2	no	—
132	C₂×C₄	—	1076	2	no	—
133	C₂×C₄	C₄⋊Dic₂₆₉	1	2	yes	—
134	C₂×C₄	C₄⋊Dic₂₆₉	1	2	yes	—
135	C₂×C₄	D₂₁₅₂	1	2	yes	—
136	C₂×C₄	Dic₁₀₇₆	1	2	yes	—
137	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₂₆₉	1	1	yes	characteristic
140	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
141	C₂×C₄	#16864401	1	1	yes	characteristic
142	C₂×C₄	#16864403	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₂²	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
145	C₂²	#22575428	1	1	yes	characteristic
146	C₂²	#22575433	1	1	yes	characteristic
147	C₂²	C₂×D₂₁₅₂	1	1	yes	characteristic
148	C₂²	C₂×Dic₁₀₇₆	1	1	yes	characteristic
149	C₂²	#22575481	1	1	yes	characteristic
150	C₂²	#22575486	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	1076	2	no	—
153	C₄	—	1076	2	no	—
154	C₄	—	2152	2	no	—
155	C₄	#22575387	1	4	yes	complemented
156	C₄	D₄×Dic₂₆₉	1	1	yes	characteristic
157	C₄	Q₈×Dic₂₆₉	1	1	yes	characteristic
158	C₄	#22575444	1	1	yes	characteristic
159	C₄	#22575452	1	1	yes	characteristic
160	C₂	#31770504	1	1	yes	characteristic
161	C₂	C₂×#22575387	1	1	yes	characteristic
162	C₂	#31770647	1	1	yes	characteristic
163	C₂	#31770801	1	1	yes	characteristic
164	C₂	#31770816	1	1	yes	characteristic
165	C₂	#31770886	1	1	yes	characteristic
166	C₂	#31770901	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575387

C₂

yes

maximal

#31770475

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₆₉

C₂

yes

maximal, characteristic

#31770368

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₆₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₆₉

C₂²

yes

characteristic

C₂×C₄×Dic₂₆₉

—

#22575387

—

#22575360

—

C₂²×C₈×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

C₄

yes

complemented

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

D₄

yes

—

C₄⋊Dic₂₆₉

Q₈

yes

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

Q₈

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₆₉

D₈

yes

—

C₂×C₄×C₂₆₉

Q₁₆

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

C₂×C₄×C₂₆₉

yes

C₂×C₄×C₂₆₉

yes

C₈×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₉

#121

yes

characteristic

C₂²×C₂₆₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₆₉

C₂×D₈

yes

characteristic

C₂²×C₂₆₉

C₂×Q₁₆

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₄×C₂₆₉

—

C₄×C₂₆₉

#107

yes

complemented

C₄×C₂₆₉

C₄×D₄

yes

characteristic

C₄×C₂₆₉

C₄×Q₈

yes

characteristic

C₄×C₂₆₉

#125

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1947

—

269

maximal, Sylow

C₂×#107

—

269

—

C₂×C₄⋊C₈

D₂₆₉

yes

characteristic

#384

—

269

—

C₄×C₄⋊C₄

—

269

—

#340

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄²

—

538

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#107

—

538

—

C₄⋊C₈

Dic₂₆₉

yes

complemented

100

#95

—

538

—

101

C₂²×C₄

—

538

—

102

C₂²×C₄

—

538

—

103

C₂²×C₄

—

538

—

104

C₂²×C₄

D₁₀₇₆

yes

characteristic

105

C₂²×C₄

D₁₀₇₆

yes

characteristic

106

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×Dic₂₆₉

yes

—

110

C₄⋊C₄

—

269

—

111

C₄⋊C₄

—

269

—

112

C₄⋊C₄

—

538

—

113

C₄⋊C₄

—

538

—

114

C₄⋊C₄

—

538

—

115

C₄⋊C₄

—

538

—

116

C₄⋊C₄

—

1076

—

117

C₄⋊C₄

—

1076

—

118

C₄²

—

1076

—

119

C₄²

C₂×Dic₂₆₉

yes

—

120

C₄²

Dic₅₃₈

yes

—

121

C₂³

C₂×D₁₀₇₆

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

538

—

125

C₂×C₄

—

538

—

126

C₂×C₄

—

1076

—

127

C₂×C₄

—

1076

—

128

C₂×C₄

—

1076

—

129

C₂×C₄

—

1076

—

130

C₂×C₄

—

1076

—

131

C₂×C₄

—

1076

—

132

C₂×C₄

—

1076

—

133

C₂×C₄

C₄⋊Dic₂₆₉

yes

—

134

C₂×C₄

C₄⋊Dic₂₆₉

yes

—

135

C₂×C₄

D₂₁₅₂

yes

—

136

C₂×C₄

Dic₁₀₇₆

yes

—

137

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

138

C₂×C₄

D₄×D₂₆₉

yes

characteristic

139

C₂×C₄

Q₈×D₂₆₉

yes

characteristic

140

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

141

C₂×C₄

#16864401

yes

characteristic

142

C₂×C₄

#16864403

yes

characteristic

143

C₈

—

144

C₂²

C₂×C₄⋊Dic₂₆₉

yes

characteristic

145

C₂²

#22575428

yes

characteristic

146

C₂²

#22575433

yes

characteristic

147

C₂²

C₂×D₂₁₅₂

yes

characteristic

148

C₂²

C₂×Dic₁₀₇₆

yes

characteristic

149

C₂²

#22575481

yes

characteristic

150

C₂²

#22575486

yes

characteristic

151

C₄

—

152

C₄

—

1076

—

153

C₄

—

1076

—

154

C₄

—

2152

—

155

C₄

#22575387

yes

complemented

156

C₄

D₄×Dic₂₆₉

yes

characteristic

157

C₄

Q₈×Dic₂₆₉

yes

characteristic

158

C₄

#22575444

yes

characteristic

159

C₄

#22575452

yes

characteristic

160

C₂

#31770504

yes

characteristic

161

C₂

C₂×#22575387

yes

characteristic

162

C₂

#31770647

yes

characteristic

163

C₂

#31770801

yes

characteristic

164

C₂

#31770816

yes

characteristic

165

C₂

#31770886

yes

characteristic

166

C₂

#31770901

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₇₆
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂×C₄⋊C₈×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₆₉
hypocenter	C₂₆₉	#1947
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₇₆

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

C₂×C₄⋊C₈×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

cosocle

C₂²×C₄×C₂₆₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₆₉

hypocenter

C₂₆₉

#1947

LCS

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
16. C₄⋊C₈×C₂₆₉	C₄	semidirect
68. C₄×C₂₆₉	#107	semidirect
83. C₂₆₉	#1947	semidirect
99. C₄⋊C₈	Dic₂₆₉	semidirect
155. C₄	#22575387	semidirect

H ≅ G/N

type

16. C₄⋊C₈×C₂₆₉

C₄

semidirect

68. C₄×C₂₆₉

#107

semidirect

83. C₂₆₉

#1947

semidirect

99. C₄⋊C₈

Dic₂₆₉

semidirect

155. C₄

#22575387

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#22575387	2
#31770475	2
C₄×C₄⋊Dic₂₆₉	2
#31770368	2
C₂×C₄⋊C₈×C₂₆₉	2
#1947	269

subgroup

index

C₄×C₄⋊Dic₂₆₉

C₂×C₄⋊C₈×C₂₆₉

269

Subgroups — #46811644

Conjugacy classes of subgroups

Complements

Maximal subgroups