Group #46819330 — subgroups

Subgroups — #46819330

Order 34432.

764 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 293 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	C₂²×#16864386	C₂	1	2	yes	maximal, complemented
2	#31770572	C₂	1	8	yes	maximal, complemented
3	#31770569	C₂	1	4	yes	maximal, complemented
4	C₂²×#41×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#16864386	—	2	8	no	—
6	C₂×#16864386	C₂²	1	4	yes	—
7	C₂×#16864386	C₂²	1	4	yes	complemented
8	C₂²×#14524130	C₂²	1	4	yes	complemented
9	C₂²⋊#14524130	C₂²	1	16	yes	complemented
10	C₂×#41×C₂₆₉	—	2	8	no	—
11	C₂×#41×C₂₆₉	C₄	1	4	yes	—
12	C₂×#41×C₂₆₉	C₂²	1	4	yes	—
13	C₂²×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
14	C₂²×D₄×C₂₆₉	C₄	1	2	yes	—
15	C₂²×D₄×C₂₆₉	C₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₆₉	C₂²	1	2	yes	—
17	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
18	#16864386	—	2	8	no	—
19	#16864386	—	2	8	no	—
20	C₂×#14524130	—	2	8	no	—
21	C₂×#14524130	C₂³	1	8	yes	—
22	C₂⁴×C₂₆₉	C₂×C₄	1	2	yes	—
23	C₂⁴×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	#41×C₂₆₉	—	2	16	no	—
25	#41×C₂₆₉	—	2	8	no	—
26	#41×C₂₆₉	—	2	8	no	—
27	C₂×Q₈×C₂₆₉	—	2	4	no	—
28	C₂×Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₂₆₉	—	2	8	no	—
30	C₂×D₄×C₂₆₉	—	2	4	no	—
31	C₂×D₄×C₂₆₉	C₂×C₄	1	8	yes	—
32	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₂₆₉	—	2	8	no	—
34	C₂²×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₂₆₉	D₄	1	8	yes	—
39	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
42	#14524130	—	2	8	no	—
43	C₂³×C₂₆₉	—	2	4	no	—
44	C₂³×C₂₆₉	—	4	2	no	—
45	C₂³×C₂₆₉	—	4	4	no	—
46	C₂³×C₂₆₉	C₂²⋊C₄	1	4	yes	—
47	C₂³×C₂₆₉	C₂²×C₄	1	4	yes	—
48	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
50	Q₈×C₂₆₉	—	2	8	no	—
51	D₄×C₂₆₉	—	2	16	no	—
52	D₄×C₂₆₉	—	2	8	no	—
53	C₂×C₄×C₂₆₉	—	2	4	no	—
54	C₂×C₄×C₂₆₉	—	2	8	no	—
55	C₂×C₄×C₂₆₉	—	2	8	no	—
56	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	4	yes	—
57	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	8	yes	—
58	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
63	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
65	C₂²×C₂₆₉	—	2	2	no	—
66	C₂²×C₂₆₉	—	2	4	no	—
67	C₂²×C₂₆₉	—	2	4	no	—
68	C₂²×C₂₆₉	—	2	8	no	—
69	C₂²×C₂₆₉	—	4	4	no	—
70	C₂²×C₂₆₉	—	4	8	no	—
71	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂²×C₂₆₉	C₂×C₂²⋊C₄	1	4	yes	—
73	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₂₆₉	—	2	4	no	—
75	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	2	yes	—
76	C₄×C₂₆₉	C₂×C₂²⋊C₄	1	4	yes	—
77	C₄×C₂₆₉	C₂²×D₄	1	2	yes	—
78	C₂×C₂₆₉	—	2	2	no	—
79	C₂×C₂₆₉	—	2	4	no	—
80	C₂×C₂₆₉	—	4	4	no	—
81	C₂×C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₂₆₉	#568	1	2	yes	—
83	C₂₆₉	#2879	1	1	yes	characteristic, complemented, Sylow
84	#2879	—	269	1	no	maximal, Sylow
85	C₂²×#41	D₂₆₉	1	1	yes	characteristic
86	C₂²×M₁₆	—	269	2	no	—
87	#408	—	269	8	no	—
88	#407	—	269	4	no	—
89	C₂×#41	—	2	8	no	—
90	C₂×#41	Dic₂₆₉	1	4	yes	—
91	C₂×#41	C₂×D₂₆₉	1	4	yes	—
92	C₂²×Q₈	Dic₂₆₉	1	1	yes	characteristic
93	C₂²×D₄	Dic₂₆₉	1	2	yes	—
94	C₂²×D₄	Dic₂₆₉	1	1	yes	characteristic
95	C₂³×C₄	C₂×D₂₆₉	1	2	yes	—
96	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
97	C₂×M₁₆	—	269	4	no	—
98	C₂×M₁₆	—	269	4	no	—
99	C₂×M₁₆	—	538	8	no	—
100	C₂²×C₈	—	269	4	no	—
101	C₂²⋊C₈	—	269	16	no	—
102	C₂⁴	C₂×Dic₂₆₉	1	2	yes	—
103	C₂⁴	C₂×Dic₂₆₉	1	1	yes	characteristic
104	#41	—	2	16	no	—
105	#41	—	2	8	no	—
106	#41	—	2	8	no	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	C₂×Dic₂₆₉	1	4	yes	—
109	C₂×D₄	—	2	8	no	—
110	C₂×D₄	—	2	4	no	—
111	C₂×D₄	C₂×Dic₂₆₉	1	8	yes	—
112	C₂×D₄	C₂×Dic₂₆₉	1	4	yes	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	C₂×Dic₂₆₉	1	2	yes	—
115	C₂²×C₄	C₂×Dic₂₆₉	1	4	yes	—
116	C₂²×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
117	C₂²×C₄	C₂×Dic₂₆₉	1	2	yes	—
118	C₂²×C₄	C₂²⋊D₂₆₉	1	8	yes	—
119	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
120	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
121	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
122	M₁₆	—	538	8	no	—
123	M₁₆	—	538	8	no	—
124	C₂×C₈	—	269	8	no	—
125	C₂×C₈	—	538	8	no	—
126	C₂³	—	2	4	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	4	4	no	—
129	C₂³	C₂²⋊Dic₂₆₉	1	4	yes	—
130	C₂³	C₂²×Dic₂₆₉	1	4	yes	—
131	C₂³	C₂²×Dic₂₆₉	1	1	yes	characteristic
132	C₂³	C₂²×Dic₂₆₉	1	2	yes	—
133	Q₈	—	2	8	no	—
134	D₄	—	2	16	no	—
135	D₄	—	2	8	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	C₂²⋊Dic₂₆₉	1	4	yes	—
140	C₂×C₄	C₂²⋊Dic₂₆₉	1	8	yes	—
141	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
142	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
143	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
145	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	4	yes	—
147	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
148	C₈	—	538	8	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	8	no	—
155	C₂²	C₂×C₂²⋊Dic₂₆₉	1	2	yes	—
156	C₂²	C₂×C₂²⋊Dic₂₆₉	1	4	yes	—
157	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	C₂×C₂²⋊Dic₂₆₉	1	2	yes	—
160	C₄	C₂×C₂²⋊Dic₂₆₉	1	4	yes	—
161	C₄	C₂²×C₂²⋊D₂₆₉	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	2	4	no	—
164	C₂	—	4	4	no	—
165	C₂	C₂²×C₂²⋊Dic₂₆₉	1	1	yes	characteristic
166	C₂	#31771372	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#16864386

C₂

yes

maximal, complemented

#31770572

C₂

yes

maximal, complemented

#31770569

C₂

yes

maximal, complemented

C₂²×#41×C₂₆₉

C₂

yes

maximal, characteristic

C₂×#16864386

—

C₂×#16864386

C₂²

yes

—

C₂×#16864386

C₂²

yes

complemented

C₂²×#14524130

C₂²

yes

complemented

C₂²⋊#14524130

C₂²

yes

complemented

C₂×#41×C₂₆₉

—

C₂×#41×C₂₆₉

C₄

yes

—

C₂×#41×C₂₆₉

C₂²

yes

—

C₂²×Q₈×C₂₆₉

C₄

yes

characteristic

C₂²×D₄×C₂₆₉

C₄

yes

—

C₂²×D₄×C₂₆₉

C₄

yes

characteristic

C₂³×C₄×C₂₆₉

C₂²

yes

—

C₂³×C₄×C₂₆₉

C₂²

yes

—

—

—

C₂³

yes

—

C₂⁴×C₂₆₉

C₂×C₄

yes

—

C₂⁴×C₂₆₉

C₂×C₄

yes

characteristic

#41×C₂₆₉

—

#41×C₂₆₉

—

#41×C₂₆₉

—

C₂×Q₈×C₂₆₉

—

C₂×Q₈×C₂₆₉

C₂×C₄

yes

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂×D₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

C₂×C₄

yes

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

#14524130

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

—

C₂³×C₂₆₉

C₂²⋊C₄

yes

—

C₂³×C₂₆₉

C₂²×C₄

yes

—

C₂³×C₂₆₉

C₂²×C₄

yes

characteristic

C₂³×C₂₆₉

C₂²×C₄

yes

—

Q₈×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₆₉

C₂³×C₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₄×C₂₆₉

C₂×C₂²⋊C₄

yes

—

C₄×C₂₆₉

C₂²×D₄

yes

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₆₉

#568

yes

—

C₂₆₉

#2879

yes

characteristic, complemented, Sylow

#2879

—

269

maximal, Sylow

C₂²×#41

D₂₆₉

yes

characteristic

C₂²×M₁₆

—

269

—

#408

—

269

—

#407

—

269

—

C₂×#41

—

C₂×#41

Dic₂₆₉

yes

—

C₂×#41

C₂×D₂₆₉

yes

—

C₂²×Q₈

Dic₂₆₉

yes

characteristic

C₂²×D₄

Dic₂₆₉

yes

—

C₂²×D₄

Dic₂₆₉

yes

characteristic

C₂³×C₄

C₂×D₂₆₉

yes

—

C₂³×C₄

C₂×D₂₆₉

yes

characteristic

C₂×M₁₆

—

269

—

C₂×M₁₆

—

269

—

C₂×M₁₆

—

538

—

100

C₂²×C₈

—

269

—

101

C₂²⋊C₈

—

269

—

102

C₂⁴

C₂×Dic₂₆₉

yes

—

103

C₂⁴

C₂×Dic₂₆₉

yes

characteristic

104

#41

—

105

#41

—

106

#41

—

107

C₂×Q₈

—

108

C₂×Q₈

C₂×Dic₂₆₉

yes

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂×Dic₂₆₉

yes

—

112

C₂×D₄

C₂×Dic₂₆₉

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×Dic₂₆₉

yes

—

115

C₂²×C₄

C₂×Dic₂₆₉

yes

—

116

C₂²×C₄

C₂×Dic₂₆₉

yes

characteristic

117

C₂²×C₄

C₂×Dic₂₆₉

yes

—

118

C₂²×C₄

C₂²⋊D₂₆₉

yes

—

119

C₂²×C₄

C₂²×D₂₆₉

yes

—

120

C₂²×C₄

C₂²×D₂₆₉

yes

—

121

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

122

M₁₆

—

538

—

123

M₁₆

—

538

—

124

C₂×C₈

—

269

—

125

C₂×C₈

—

538

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²⋊Dic₂₆₉

yes

—

130

C₂³

C₂²×Dic₂₆₉

yes

—

131

C₂³

C₂²×Dic₂₆₉

yes

characteristic

132

C₂³

C₂²×Dic₂₆₉

yes

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊Dic₂₆₉

yes

—

140

C₂×C₄

C₂²⋊Dic₂₆₉

yes

—

141

C₂×C₄

C₂²×Dic₂₆₉

yes

—

142

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

143

C₂×C₄

C₂²×Dic₂₆₉

yes

—

144

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

145

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

146

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

—

147

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

148

C₈

—

538

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

C₂×C₂²⋊Dic₂₆₉

yes

—

156

C₂²

C₂×C₂²⋊Dic₂₆₉

yes

—

157

C₂²

C₂³×Dic₂₆₉

yes

characteristic

158

C₄

—

159

C₄

C₂×C₂²⋊Dic₂₆₉

yes

—

160

C₄

C₂×C₂²⋊Dic₂₆₉

yes

—

161

C₄

C₂²×C₂²⋊D₂₆₉

yes

—

162

C₂

—

163

C₂

—

164

C₂

—

165

C₂

C₂²×C₂²⋊Dic₂₆₉

yes

characteristic

166

C₂

#31771372

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting, prime radical	C₂²×#41×C₂₆₉	C₂
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	C₂²×#41	D₂₆₉
hypocenter	C₂₆₉	#2879
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₆₉

commutator, socle

C₂²×C₂₆₉

C₂³×C₄

Fitting, prime radical

C₂²×#41×C₂₆₉

C₂

cosocle

C₂×C₄×C₂₆₉

C₂⁴

hypercenter

C₂²×#41

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂×C₄ ◁ C₂²×#41

N	H ≅ G/N	type
1. C₂²×#16864386	C₂	semidirect
2. #31770572	C₂	semidirect
3. #31770569	C₂	semidirect
7. C₂×#16864386	C₂²	semidirect
8. C₂²×#14524130	C₂²	semidirect
9. C₂²⋊#14524130	C₂²	semidirect
83. C₂₆₉	#2879	semidirect

H ≅ G/N

type

1. C₂²×#16864386

C₂

semidirect

2. #31770572

C₂

semidirect

3. #31770569

C₂

semidirect

7. C₂×#16864386

C₂²

semidirect

8. C₂²×#14524130

C₂²

semidirect

9. C₂²⋊#14524130

C₂²

semidirect

83. C₂₆₉

#2879

semidirect

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

subgroup	index
C₂²×#16864386	2
#31770572	2
#31770569	2
C₂²×#41×C₂₆₉	2
#2879	269

subgroup

index

C₂²×#41×C₂₆₉

269

Subgroups — #46819330

Conjugacy classes of subgroups

Complements

Maximal subgroups