Group #153323 — subgroups

Subgroups — #153323

Order 512.

225 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 84 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×#2253	C₂	1	1	yes	maximal, characteristic
2	C₂×#1424	C₂	1	1	yes	maximal, characteristic
3	C₂×#1423	C₂	1	1	yes	maximal, characteristic
4	#13165	C₂	1	1	yes	maximal, characteristic
5	#7471	C₂	1	1	yes	maximal, characteristic
6	#7471	C₂	1	1	yes	maximal, characteristic
7	#7465	C₂	1	1	yes	maximal, characteristic
8	C₂×#499	—	2	1	no	—
9	C₂²×C₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	#2253	—	2	2	no	—
12	#2253	—	2	2	no	—
13	C₂×#367	—	2	1	no	—
14	C₂×#366	C₂²	1	1	yes	characteristic
15	C₂×#366	C₂²	1	1	yes	characteristic
16	C₂×#366	C₂²	1	1	yes	characteristic
17	C₂×#366	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#1942	—	2	1	no	—
20	#1424	—	2	2	no	—
21	#1423	—	2	2	no	—
22	#1380	—	2	1	no	—
23	C₂×#125	—	4	1	no	—
24	C₂²×C₁₆	D₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂³	1	1	yes	characteristic
26	#499	—	2	4	no	—
27	#499	—	4	2	no	—
28	C₂×#107	—	2	1	no	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	—	2	1	no	—
32	C₂×#106	—	4	1	no	—
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	#389	—	4	1	no	—
35	C₂×C₃₂	—	2	1	no	—
36	C₂×C₃₂	—	2	1	no	—
37	C₂×C₃₂	—	2	1	no	—
38	C₂×C₃₂	—	2	1	no	—
39	C₂×C₃₂	Q₈	1	2	yes	—
40	C₂×C₃₂	Q₈	1	2	yes	—
41	#367	—	2	4	no	—
42	#366	—	2	1	no	—
43	#366	—	2	1	no	—
44	#366	—	2	1	no	—
45	#366	—	2	1	no	—
46	#366	—	2	1	no	—
47	#366	D₄	1	2	yes	—
48	#366	D₄	1	2	yes	—
49	#366	Q₈	1	2	yes	—
50	C₄×C₁₆	—	2	1	no	—
51	C₄×C₁₆	—	2	1	no	—
52	#340	—	4	1	no	—
53	C₂²×C₈	D₈	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	#125	—	4	4	no	—
56	#125	—	8	2	no	—
57	C₂×C₄⋊C₄	—	4	1	no	—
58	C₂×C₄⋊C₄	—	4	1	no	—
59	C₂×C₄⋊C₄	—	4	1	no	—
60	C₂×C₄⋊C₄	—	4	1	no	—
61	C₂×C₄⋊C₄	—	8	1	no	—
62	C₂×C₄⋊C₄	—	8	1	no	—
63	C₂×C₄⋊C₄	—	8	1	no	—
64	C₂×C₄²	D₈	1	1	yes	characteristic
65	C₂×C₁₆	—	2	1	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	Q₁₆	1	2	yes	—
68	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
70	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₁₆	#41	1	1	yes	characteristic
74	#107	—	2	2	no	—
75	#107	—	2	2	no	—
76	#107	—	2	2	no	—
77	#107	—	4	1	no	—
78	#107	—	4	1	no	—
79	#107	—	4	1	no	—
80	#107	—	4	1	no	—
81	#107	—	4	1	no	—
82	#106	—	4	4	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	2	1	no	—
85	#95	—	8	1	no	—
86	C₃₂	—	2	2	no	—
87	C₃₂	—	2	2	no	—
88	C₂²×C₄	—	8	1	no	—
89	C₂²×C₄	—	8	1	no	—
90	C₂²×C₄	—	8	1	no	—
91	C₂²×C₄	—	8	1	no	—
92	C₂²×C₄	D₁₆	1	1	yes	characteristic
93	C₂²×C₄	D₁₆	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	Q₃₂	1	2	yes	—
98	C₂×C₈	#120	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₈	#137	1	1	yes	characteristic
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	8	4	no	—
108	C₄⋊C₄	—	8	4	no	—
109	C₄⋊C₄	—	8	1	no	—
110	C₄⋊C₄	—	8	1	no	—
111	C₄⋊C₄	—	8	1	no	—
112	C₄⋊C₄	—	8	1	no	—
113	C₄⋊C₄	—	8	1	no	—
114	C₄⋊C₄	—	16	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₁₆	—	2	2	no	—
118	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
119	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
120	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
121	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
122	C₂³	C₂×D₁₆	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	16	1	no	—
133	C₂×C₄	—	16	1	no	—
134	C₂×C₄	—	16	1	no	—
135	C₂×C₄	—	16	1	no	—
136	C₂×C₄	—	16	1	no	—
137	C₂×C₄	—	16	1	no	—
138	C₂×C₄	—	16	1	no	—
139	C₂×C₄	—	16	1	no	—
140	C₂×C₄	—	16	1	no	—
141	C₂×C₄	#447	1	1	yes	characteristic
142	C₂×C₄	#474	1	1	yes	characteristic
143	C₂×C₄	#500	1	1	yes	characteristic
144	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
145	C₂×C₄	#509	1	1	yes	characteristic
146	C₂×C₄	#510	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	#395	1	1	yes	characteristic
149	C₈	#451	1	1	yes	characteristic
150	C₈	#477	1	1	yes	characteristic
151	C₈	#501	1	1	yes	characteristic
152	C₂²	#2184	1	1	yes	characteristic
153	C₂²	#2224	1	1	yes	characteristic
154	C₂²	#2252	1	1	yes	characteristic
155	C₂²	#2262	1	1	yes	characteristic
156	C₂²	#2262	1	1	yes	characteristic
157	C₂²	#2263	1	1	yes	characteristic
158	C₂²	#2264	1	1	yes	characteristic
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	16	2	no	—
162	C₄	—	16	2	no	—
163	C₄	—	16	2	no	—
164	C₄	—	32	1	no	—
165	C₄	#2057	1	1	yes	characteristic
166	C₄	#2188	1	1	yes	characteristic
167	C₄	#2227	1	1	yes	characteristic
168	C₄	#2255	1	1	yes	characteristic
169	C₂	#13451	1	1	yes	characteristic
170	C₂	#13663	1	1	yes	characteristic
171	C₂	#13666	1	1	yes	characteristic
172	C₂	#13707	1	1	yes	characteristic
173	C₂	#13708	1	1	yes	characteristic
174	C₂	#13732	1	1	yes	characteristic
175	C₂	#13734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2253

C₂

yes

maximal, characteristic

C₂×#1424

C₂

yes

maximal, characteristic

C₂×#1423

C₂

yes

maximal, characteristic

#13165

C₂

yes

maximal, characteristic

#7471

C₂

yes

maximal, characteristic

#7471

C₂

yes

maximal, characteristic

#7465

C₂

yes

maximal, characteristic

C₂×#499

—

C₂²×C₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#366

C₂²

yes

characteristic

C₂×#366

C₂²

yes

characteristic

C₂×#366

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#389

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

Q₈

yes

—

C₂×C₃₂

Q₈

yes

—

—

—

—

—

—

—

D₄

yes

—

#366

D₄

yes

—

#366

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

#120

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

100

C₂×C₈

C₄⋊Q₈

yes

characteristic

101

C₂×C₈

C₂×Q₁₆

yes

characteristic

102

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

103

C₂×C₈

#137

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₁₆

—

118

C₁₆

C₂²⋊Q₈

yes

characteristic

119

C₁₆

C₂²⋊Q₈

yes

characteristic

120

C₁₆

C₄⋊Q₈

yes

characteristic

121

C₁₆

C₄⋊Q₈

yes

characteristic

122

C₂³

C₂×D₁₆

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#447

yes

characteristic

142

C₂×C₄

#474

yes

characteristic

143

C₂×C₄

#500

yes

characteristic

144

C₂×C₄

C₂×Q₃₂

yes

characteristic

145

C₂×C₄

#509

yes

characteristic

146

C₂×C₄

#510

yes

characteristic

147

C₈

—

148

C₈

#395

yes

characteristic

149

C₈

#451

yes

characteristic

150

C₈

#477

yes

characteristic

151

C₈

#501

yes

characteristic

152

C₂²

#2184

yes

characteristic

153

C₂²

#2224

yes

characteristic

154

C₂²

#2252

yes

characteristic

155

C₂²

#2262

yes

characteristic

156

C₂²

#2262

yes

characteristic

157

C₂²

#2263

yes

characteristic

158

C₂²

#2264

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#2057

yes

characteristic

166

C₄

#2188

yes

characteristic

167

C₄

#2227

yes

characteristic

168

C₄

#2255

yes

characteristic

169

C₂

#13451

yes

characteristic

170

C₂

#13663

yes

characteristic

171

C₂

#13666

yes

characteristic

172

C₂

#13707

yes

characteristic

173

C₂

#13708

yes

characteristic

174

C₂

#13732

yes

characteristic

175

C₂

#13734

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#2253	2
C₂×#1424	2
C₂×#1423	2
#13165	2
#7471	2
#7465	2

subgroup

index

Subgroups — #153323

Conjugacy classes of subgroups

Maximal subgroups