Group #153270 — subgroups

Subgroups — #153270

Order 512.

300 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 72 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×#2247	C₂	1	1	yes	maximal, characteristic
2	C₂×#1420	C₂	1	2	yes	maximal
3	#13618	C₂	1	2	yes	maximal
4	#13169	C₂	1	1	yes	maximal, characteristic
5	#7467	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	—	2	1	no	—
7	C₂²×Q₃₂	C₂²	1	2	yes	—
8	C₂×#494	—	2	1	no	—
9	C₂×M₆₄	C₂²	1	2	yes	—
10	#2247	—	2	4	no	—
11	C₂×#368	C₂²	1	1	yes	characteristic
12	#2158	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1945	—	2	1	no	—
15	#1420	—	2	4	no	—
16	#1416	—	2	4	no	—
17	#1382	—	2	1	no	—
18	C₂²×Q₁₆	—	2	2	no	—
19	C₂²×Q₁₆	—	4	1	no	—
20	C₂×C₄⋊Q₈	—	4	1	no	—
21	C₂×Q₃₂	—	2	4	no	—
22	C₂×Q₃₂	—	2	2	no	—
23	C₂×Q₃₂	—	2	2	no	—
24	C₂×Q₃₂	—	2	2	no	—
25	C₂×Q₃₂	—	2	2	no	—
26	C₂×Q₃₂	—	2	2	no	—
27	C₂×Q₃₂	—	4	2	no	—
28	C₂×Q₃₂	—	4	2	no	—
29	C₂×Q₃₂	D₄	1	4	yes	—
30	C₂²×C₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	#494	—	2	4	no	—
33	#494	—	4	2	no	—
34	C₂×#108	—	2	1	no	—
35	#427	—	2	1	no	—
36	#424	—	4	1	no	—
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	M₆₄	—	2	2	no	—
39	M₆₄	—	2	2	no	—
40	C₂×C₃₂	—	2	2	no	—
41	#368	—	2	2	no	—
42	#367	—	2	1	no	—
43	#366	D₄	1	2	yes	—
44	C₄×C₁₆	—	2	2	no	—
45	#339	—	4	1	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂×Q₁₆	—	2	4	no	—
48	C₂×Q₁₆	—	4	4	no	—
49	C₂×Q₁₆	—	4	2	no	—
50	C₂×Q₁₆	—	4	2	no	—
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	8	2	no	—
55	C₂×Q₁₆	—	8	2	no	—
56	C₂×M₁₆	—	4	1	no	—
57	C₂²×C₈	D₈	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₄⋊Q₈	—	4	4	no	—
60	C₄⋊Q₈	—	8	2	no	—
61	#117	—	4	1	no	—
62	C₂×C₄⋊C₄	—	8	2	no	—
63	C₂×C₄²	D₈	1	1	yes	characteristic
64	Q₃₂	—	2	4	no	—
65	Q₃₂	—	2	4	no	—
66	Q₃₂	—	4	8	no	—
67	Q₃₂	—	4	2	no	—
68	Q₃₂	—	4	2	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	SD₁₆	1	2	yes	—
72	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
73	C₂×C₁₆	C₂×D₄	1	2	yes	—
74	C₂×C₁₆	#41	1	2	yes	—
75	C₂×C₁₆	#41	1	1	yes	characteristic
76	#108	—	4	1	no	—
77	#108	—	4	1	no	—
78	#107	—	2	2	no	—
79	#106	—	4	1	no	—
80	C₄⋊C₈	—	8	2	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₃₂	—	4	2	no	—
84	C₂×Q₈	—	4	4	no	—
85	C₂×Q₈	—	8	2	no	—
86	C₂×Q₈	—	8	2	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂×Q₈	—	8	2	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	8	2	no	—
92	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
93	Q₁₆	—	4	4	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	8	8	no	—
96	Q₁₆	—	8	2	no	—
97	Q₁₆	—	8	2	no	—
98	M₁₆	—	8	1	no	—
99	M₁₆	—	8	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	D₁₆	1	2	yes	—
104	C₂×C₈	#120	1	1	yes	characteristic
105	C₂×C₈	#121	1	1	yes	characteristic
106	C₂×C₈	#124	1	1	yes	characteristic
107	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	8	8	no	—
112	C₂²⋊C₄	—	8	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	8	1	no	—
115	C₄²	SD₃₂	1	2	yes	—
116	C₁₆	—	2	2	no	—
117	C₁₆	#121	1	2	yes	—
118	C₁₆	#123	1	2	yes	—
119	C₂³	#509	1	1	yes	characteristic
120	Q₈	—	8	4	no	—
121	Q₈	—	8	4	no	—
122	Q₈	—	16	2	no	—
123	Q₈	—	16	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	8	4	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	16	2	no	—
129	C₂×C₄	—	16	2	no	—
130	C₂×C₄	#447	1	1	yes	characteristic
131	C₂×C₄	#459	1	1	yes	characteristic
132	C₂×C₄	#486	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
134	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
135	C₂×C₄	#509	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	16	1	no	—
138	C₈	#394	1	1	yes	characteristic
139	C₈	#450	1	1	yes	characteristic
140	C₈	#460	1	1	yes	characteristic
141	C₈	#490	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	#2187	1	1	yes	characteristic
144	C₂²	#2206	1	1	yes	characteristic
145	C₂²	#2244	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	16	4	no	—
148	C₄	—	16	2	no	—
149	C₄	#2041	1	1	yes	characteristic
150	C₄	#2184	1	1	yes	characteristic
151	C₄	#2207	1	1	yes	characteristic
152	C₄	#2240	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	#13407	1	1	yes	characteristic
156	C₂	#13693	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2247

C₂

yes

maximal, characteristic

C₂×#1420

C₂

yes

maximal

#13618

C₂

yes

maximal

#13169

C₂

yes

maximal, characteristic

#7467

C₂

yes

maximal, characteristic

C₂²×Q₃₂

—

C₂²×Q₃₂

C₂²

yes

—

C₂×#494

—

C₂×M₆₄

C₂²

yes

—

#2247

—

C₂×#368

C₂²

yes

characteristic

#2158

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#368

—

#367

—

#366

D₄

yes

—

C₄×C₁₆

—

#339

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

D₁₆

yes

—

104

C₂×C₈

#120

yes

characteristic

105

C₂×C₈

#121

yes

characteristic

106

C₂×C₈

#124

yes

characteristic

107

C₂×C₈

C₂×SD₁₆

yes

characteristic

108

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

109

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

SD₃₂

yes

—

116

C₁₆

—

117

C₁₆

#121

yes

—

118

C₁₆

#123

yes

—

119

C₂³

#509

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#447

yes

characteristic

131

C₂×C₄

#459

yes

characteristic

132

C₂×C₄

#486

yes

characteristic

133

C₂×C₄

C₂×D₁₆

yes

characteristic

134

C₂×C₄

C₂×SD₃₂

yes

characteristic

135

C₂×C₄

#509

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

#394

yes

characteristic

139

C₈

#450

yes

characteristic

140

C₈

#460

yes

characteristic

141

C₈

#490

yes

characteristic

142

C₂²

—

143

C₂²

#2187

yes

characteristic

144

C₂²

#2206

yes

characteristic

145

C₂²

#2244

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#2041

yes

characteristic

150

C₄

#2184

yes

characteristic

151

C₄

#2207

yes

characteristic

152

C₄

#2240

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

#13407

yes

characteristic

156

C₂

#13693

yes

—

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2206

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#509

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ 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×#2247	2
C₂×#1420	2
#13618	2
#13169	2
#7467	2

subgroup

index

Subgroups — #153270

Conjugacy classes of subgroups

Maximal subgroups