Group #513156 — subgroups

Subgroups — #513156

Order 512.

319 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 110 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	#33735	C₂	1	2	yes	maximal, complemented
2	#13708	C₂	1	2	yes	maximal, complemented
3	#13707	C₂	1	2	yes	maximal, complemented
4	#13699	C₂	1	2	yes	maximal, complemented
5	#13698	C₂	1	1	yes	maximal, characteristic, complemented
6	#13697	C₂	1	1	yes	maximal, characteristic
7	#13690	C₂	1	2	yes	maximal, complemented
8	#13689	C₂	1	2	yes	maximal, complemented
9	#13627	C₂	1	1	yes	maximal, characteristic, complemented
10	#3168	—	2	2	no	—
11	C₂×Q₆₄	C₂²	1	1	yes	characteristic
12	C₂×SD₆₄	C₂²	1	2	yes	—
13	C₂×D₃₂	C₂²	1	1	yes	characteristic
14	C₂×M₆₄	C₂²	1	1	yes	characteristic
15	C₂×M₆₄	C₂²	1	1	yes	characteristic
16	#2253	C₂²	1	2	yes	—
17	#2249	C₂²	1	2	yes	—
18	#2242	—	2	2	no	—
19	#2217	—	2	2	no	—
20	#2216	C₂²	1	2	yes	—
21	#2215	C₂²	1	2	yes	—
22	C₄×Q₃₂	C₂²	1	2	yes	—
23	C₄×SD₃₂	—	2	2	no	—
24	C₄×D₁₆	C₂²	1	2	yes	—
25	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
26	#1424	C₂²	1	1	yes	characteristic
27	#1424	C₂²	1	1	yes	characteristic
28	#1423	C₂²	1	2	yes	—
29	C₄⋊C₃₂	C₂²	1	2	yes	—
30	C₄⋊C₃₂	C₂²	1	2	yes	—
31	#1416	C₂²	1	2	yes	—
32	#1416	C₂²	1	2	yes	—
33	#1415	C₂²	1	2	yes	—
34	#1415	C₂²	1	2	yes	—
35	#529	—	4	2	no	—
36	C₂×Q₃₂	C₂³	1	2	yes	—
37	C₂×SD₃₂	—	2	2	no	—
38	C₂×D₁₆	C₂³	1	2	yes	—
39	C₂²×C₁₆	D₄	1	1	yes	characteristic
40	C₂²×C₁₆	C₂³	1	1	yes	characteristic
41	#499	—	2	2	no	—
42	#495	—	2	2	no	—
43	#488	—	4	2	no	—
44	#467	—	2	2	no	—
45	#466	—	2	2	no	—
46	#465	—	4	2	no	—
47	C₄×Q₁₆	—	2	2	no	—
48	C₄×SD₁₆	—	4	2	no	—
49	C₄×D₈	—	2	2	no	—
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	Q₆₄	—	4	1	no	—
52	SD₆₄	—	4	2	no	—
53	D₃₂	—	4	1	no	—
54	M₆₄	—	4	1	no	—
55	M₆₄	—	4	1	no	—
56	C₂×C₃₂	C₂³	1	1	yes	characteristic
57	C₂×C₃₂	C₂³	1	2	yes	—
58	C₂×C₃₂	C₂³	1	1	yes	characteristic
59	#367	—	2	2	no	—
60	#366	C₂³	1	2	yes	—
61	#366	C₂³	1	2	yes	—
62	#358	—	2	2	no	—
63	#358	—	2	2	no	—
64	#357	—	2	2	no	—
65	#357	—	2	2	no	—
66	C₄×C₁₆	D₄	1	2	yes	—
67	C₄×C₁₆	C₂³	1	2	yes	—
68	C₂×Q₁₆	—	2	2	no	—
69	C₂×SD₁₆	—	4	2	no	—
70	C₂×D₈	—	2	2	no	—
71	C₂²×C₈	D₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	#126	—	8	2	no	—
74	#125	—	4	2	no	—
75	#124	—	4	2	no	—
76	#123	—	8	2	no	—
77	C₂²⋊Q₈	—	4	2	no	—
78	#121	—	4	2	no	—
79	C₄×Q₈	—	4	2	no	—
80	C₄×D₄	—	4	2	no	—
81	C₄×D₄	—	8	2	no	—
82	#117	—	8	2	no	—
83	C₂×C₄²	D₈	1	1	yes	characteristic
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	SD₃₂	—	4	4	no	—
87	D₁₆	—	2	2	no	—
88	D₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	4	1	no	—
92	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
93	C₂×C₁₆	C₂×D₄	1	2	yes	—
94	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#106	—	4	2	no	—
98	#103	—	4	2	no	—
99	#103	—	4	2	no	—
100	#102	—	4	2	no	—
101	#102	—	4	2	no	—
102	C₄×C₈	D₈	1	2	yes	—
103	C₄×C₈	C₂×D₄	1	2	yes	—
104	C₃₂	—	2	1	no	—
105	C₃₂	—	2	2	no	—
106	C₃₂	—	2	1	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×D₄	—	4	2	no	—
109	C₂×D₄	—	8	2	no	—
110	C₂²×C₄	—	8	2	no	—
111	C₂²×C₄	D₁₆	1	2	yes	—
112	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	4	2	no	—
115	SD₁₆	—	8	4	no	—
116	D₈	—	4	2	no	—
117	D₈	—	4	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	C₂×D₈	1	1	yes	characteristic
122	C₂×C₈	C₂×D₈	1	2	yes	—
123	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	8	2	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₄⋊C₄	—	8	2	no	—
129	C₄⋊C₄	—	8	2	no	—
130	C₂²⋊C₄	—	8	2	no	—
131	C₂²⋊C₄	—	8	2	no	—
132	C₂²⋊C₄	—	8	2	no	—
133	C₂²⋊C₄	—	8	2	no	—
134	C₂²⋊C₄	—	8	2	no	—
135	C₄²	—	8	2	no	—
136	C₄²	D₁₆	1	1	yes	characteristic
137	C₄²	D₁₆	1	1	yes	characteristic
138	C₄²	C₂×D₈	1	2	yes	—
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
142	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂³	—	8	2	no	—
144	C₂³	C₂×D₁₆	1	1	yes	characteristic
145	Q₈	—	8	2	no	—
146	Q₈	—	8	2	no	—
147	D₄	—	8	2	no	—
148	D₄	—	8	2	no	—
149	D₄	—	16	4	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	16	4	no	—
157	C₂×C₄	C₂×D₁₆	1	2	yes	—
158	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
159	C₂×C₄	C₂×D₁₆	1	2	yes	—
160	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	1	no	—
163	C₈	#537	1	1	yes	characteristic
164	C₈	#575	1	1	yes	characteristic
165	C₂²	—	2	1	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	16	2	no	—
168	C₂²	—	16	2	no	—
169	C₂²	—	16	2	no	—
170	C₂²	C₂²×D₁₆	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	16	2	no	—
175	C₄	—	16	2	no	—
176	C₄	—	16	2	no	—
177	C₄	#3245	1	1	yes	characteristic
178	C₄	#3414	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	—	32	2	no	—
181	C₂	#33811	1	1	yes	characteristic
182	C₂	#33981	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33735

C₂

yes

maximal, complemented

#13708

C₂

yes

maximal, complemented

#13707

C₂

yes

maximal, complemented

#13699

C₂

yes

maximal, complemented

#13698

C₂

yes

maximal, characteristic, complemented

#13697

C₂

yes

maximal, characteristic

#13690

C₂

yes

maximal, complemented

#13689

C₂

yes

maximal, complemented

#13627

C₂

yes

maximal, characteristic, complemented

#3168

—

C₂×Q₆₄

C₂²

yes

characteristic

C₂×SD₆₄

C₂²

yes

—

C₂×D₃₂

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

#2253

C₂²

yes

—

#2249

C₂²

yes

—

#2242

—

#2217

—

#2216

C₂²

yes

—

#2215

C₂²

yes

—

C₄×Q₃₂

C₂²

yes

—

C₄×SD₃₂

—

C₄×D₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1424

C₂²

yes

characteristic

#1424

C₂²

yes

characteristic

#1423

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

—

#1416

C₂²

yes

—

#1416

C₂²

yes

—

#1415

C₂²

yes

—

#1415

C₂²

yes

—

#529

—

C₂×Q₃₂

C₂³

yes

—

C₂×SD₃₂

—

C₂×D₁₆

C₂³

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

C₄×Q₁₆

—

C₄×SD₁₆

—

C₄×D₈

—

C₂×C₄×C₈

D₄

yes

characteristic

Q₆₄

—

SD₆₄

—

D₃₂

—

M₆₄

—

M₆₄

—

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

characteristic

#367

—

#366

C₂³

yes

—

#366

C₂³

yes

—

—

—

—

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

—

—

100

#102

—

101

#102

—

102

C₄×C₈

D₈

yes

—

103

C₄×C₈

C₂×D₄

yes

—

104

C₃₂

—

105

C₃₂

—

106

C₃₂

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

—

111

C₂²×C₄

D₁₆

yes

—

112

C₂²×C₄

C₂×D₈

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

115

SD₁₆

—

116

D₈

—

117

D₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂×D₈

yes

characteristic

122

C₂×C₈

C₂×D₈

yes

—

123

C₂×C₈

C₂²×D₄

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

D₁₆

yes

characteristic

137

C₄²

D₁₆

yes

characteristic

138

C₄²

C₂×D₈

yes

—

139

C₁₆

—

140

C₁₆

—

141

C₁₆

H(5,𝔽₂)

yes

characteristic

142

C₁₆

2₋¹⁺⁴

yes

characteristic

143

C₂³

—

144

C₂³

C₂×D₁₆

yes

characteristic

145

Q₈

—

146

Q₈

—

147

D₄

—

148

D₄

—

149

D₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×D₁₆

yes

—

158

C₂×C₄

C₂×D₁₆

yes

characteristic

159

C₂×C₄

C₂×D₁₆

yes

—

160

C₂×C₄

C₂²×D₈

yes

characteristic

161

C₈

—

162

C₈

—

163

C₈

#537

yes

characteristic

164

C₈

#575

yes

characteristic

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂²×D₁₆

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#3245

yes

characteristic

178

C₄

#3414

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

#33811

yes

characteristic

182

C₂

#33981

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#13697	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂⁴

prime radical

#13697

C₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #33735	C₂	semidirect
2. #13708	C₂	semidirect
3. #13707	C₂	semidirect
4. #13699	C₂	semidirect
5. #13698	C₂	semidirect
7. #13690	C₂	semidirect
8. #13689	C₂	semidirect
9. #13627	C₂	semidirect

H ≅ G/N

type

1. #33735

C₂

semidirect

2. #13708

C₂

semidirect

3. #13707

C₂

semidirect

4. #13699

C₂

semidirect

5. #13698

C₂

semidirect

7. #13690

C₂

semidirect

8. #13689

C₂

semidirect

9. #13627

C₂

semidirect

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

subgroup	index
#33735	2
#13708	2
#13707	2
#13699	2
#13698	2
#13697	2
#13690	2
#13689	2
#13627	2

subgroup

index

Subgroups — #513156

Conjugacy classes of subgroups

Complements

Maximal subgroups