Group #153311 — subgroups

Subgroups — #153311

Order 512.

266 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 78 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×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#1416	C₂	1	2	yes	maximal
4	#13236	C₂	1	2	yes	maximal
5	#7471	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	C₂²	1	2	yes	—
7	C₂×Q₆₄	—	2	4	no	—
8	C₂×Q₆₄	—	2	2	no	—
9	C₂²×C₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	C₂×#366	C₂²	1	2	yes	—
12	C₂×#358	—	2	2	no	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1969	—	2	2	no	—
15	C₄⋊C₃₂	—	2	2	no	—
16	#1416	—	2	4	no	—
17	C₂²×Q₁₆	—	2	2	no	—
18	C₂×Q₃₂	—	2	2	no	—
19	C₂×Q₃₂	—	2	2	no	—
20	C₂×Q₃₂	—	2	2	no	—
21	C₂×Q₃₂	—	2	2	no	—
22	C₂×Q₃₂	—	2	2	no	—
23	C₂×Q₃₂	D₄	1	4	yes	—
24	C₂²×C₁₆	D₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂³	1	1	yes	characteristic
26	C₂×#107	—	2	2	no	—
27	C₂×#103	—	4	2	no	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	#391	—	4	2	no	—
30	Q₆₄	—	2	8	no	—
31	C₂×C₃₂	—	2	2	no	—
32	C₂×C₃₂	—	2	2	no	—
33	C₂×C₃₂	—	2	1	no	—
34	C₂×C₃₂	D₄	1	2	yes	—
35	#366	—	2	2	no	—
36	#366	—	2	2	no	—
37	#358	—	4	4	no	—
38	C₄×C₁₆	—	2	1	no	—
39	C₄×C₁₆	—	2	1	no	—
40	C₂²×Q₈	—	4	2	no	—
41	C₂×Q₁₆	—	2	4	no	—
42	C₂×Q₁₆	—	4	2	no	—
43	C₂×Q₁₆	—	4	2	no	—
44	C₂×Q₁₆	—	4	2	no	—
45	C₂×Q₁₆	—	4	2	no	—
46	C₂×Q₁₆	—	4	2	no	—
47	C₂²×C₈	D₈	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄²	D₈	1	1	yes	characteristic
51	Q₃₂	—	2	4	no	—
52	Q₃₂	—	2	4	no	—
53	Q₃₂	—	4	2	no	—
54	Q₃₂	—	4	2	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	C₂×D₄	1	2	yes	—
58	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	#41	1	1	yes	characteristic
60	C₂×C₁₆	#41	1	2	yes	—
61	#107	—	4	2	no	—
62	#107	—	4	2	no	—
63	#103	—	8	4	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	#95	—	8	2	no	—
67	#95	—	8	2	no	—
68	C₃₂	—	2	2	no	—
69	C₃₂	—	4	1	no	—
70	C₂×Q₈	—	4	4	no	—
71	C₂×Q₈	—	8	2	no	—
72	C₂×Q₈	—	8	2	no	—
73	C₂×Q₈	—	8	2	no	—
74	C₂×Q₈	—	8	2	no	—
75	C₂×Q₈	—	8	2	no	—
76	C₂²×C₄	—	8	2	no	—
77	C₂²×C₄	—	8	2	no	—
78	C₂²×C₄	D₁₆	1	1	yes	characteristic
79	C₂²×C₄	D₁₆	1	1	yes	characteristic
80	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
81	Q₁₆	—	4	4	no	—
82	Q₁₆	—	4	4	no	—
83	Q₁₆	—	8	2	no	—
84	Q₁₆	—	8	2	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	#121	1	2	yes	—
88	C₂×C₈	#121	1	1	yes	characteristic
89	C₂×C₈	#135	1	1	yes	characteristic
90	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
91	C₄⋊C₄	—	8	2	no	—
92	C₄⋊C₄	—	8	2	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	1	no	—
95	C₁₆	—	4	1	no	—
96	C₁₆	#120	1	1	yes	characteristic
97	C₁₆	#123	1	2	yes	—
98	C₁₆	#124	1	1	yes	characteristic
99	C₂³	C₂×D₁₆	1	1	yes	characteristic
100	Q₈	—	8	4	no	—
101	Q₈	—	8	4	no	—
102	Q₈	—	16	2	no	—
103	Q₈	—	16	2	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	16	2	no	—
110	C₂×C₄	—	16	2	no	—
111	C₂×C₄	—	16	2	no	—
112	C₂×C₄	—	16	2	no	—
113	C₂×C₄	—	16	2	no	—
114	C₂×C₄	Q₆₄	1	2	yes	complemented
115	C₂×C₄	#459	1	2	yes	—
116	C₂×C₄	#466	1	1	yes	characteristic
117	C₂×C₄	#508	1	1	yes	characteristic
118	C₂×C₄	#509	1	2	yes	—
119	C₈	—	4	1	no	—
120	C₈	#394	1	1	yes	characteristic
121	C₈	#463	1	2	yes	—
122	C₈	#469	1	1	yes	characteristic
123	C₂²	#2206	1	2	yes	—
124	C₂²	#2215	1	1	yes	characteristic
125	C₂²	C₂×Q₆₄	1	1	yes	characteristic
126	C₂²	#2262	1	1	yes	characteristic
127	C₂²	#2264	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	—	4	1	no	—
130	C₄	—	16	4	no	—
131	C₄	—	32	2	no	—
132	C₄	#2054	1	1	yes	characteristic
133	C₄	#2210	1	2	yes	—
134	C₄	#2218	1	1	yes	characteristic
135	C₂	#13448	1	1	yes	characteristic
136	C₂	#13688	1	2	yes	—
137	C₂	#13690	1	2	yes	—
138	C₂	#13695	1	1	yes	characteristic
139	C₂	#13698	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₆₄

C₂

yes

maximal, characteristic

C₂×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₂×#1416

C₂

yes

maximal

#13236

C₂

yes

maximal

#7471

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

—

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂²×C₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#366

C₂²

yes

—

C₂×#358

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1969

—

C₄⋊C₃₂

—

#1416

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₄

yes

characteristic

#391

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

#366

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

#107

—

#107

—

#103

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

#120

yes

characteristic

C₁₆

#123

yes

—

C₁₆

#124

yes

characteristic

C₂³

C₂×D₁₆

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

Q₆₄

yes

complemented

115

C₂×C₄

#459

yes

—

116

C₂×C₄

#466

yes

characteristic

117

C₂×C₄

#508

yes

characteristic

118

C₂×C₄

#509

yes

—

119

C₈

—

120

C₈

#394

yes

characteristic

121

C₈

#463

yes

—

122

C₈

#469

yes

characteristic

123

C₂²

#2206

yes

—

124

C₂²

#2215

yes

characteristic

125

C₂²

C₂×Q₆₄

yes

characteristic

126

C₂²

#2262

yes

characteristic

127

C₂²

#2264

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#2054

yes

characteristic

133

C₄

#2210

yes

—

134

C₄

#2218

yes

characteristic

135

C₂

#13448

yes

characteristic

136

C₂

#13688

yes

—

137

C₂

#13690

yes

—

138

C₂

#13695

yes

characteristic

139

C₂

#13698

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

N	H ≅ G/N	type
114. C₂×C₄	Q₆₄	semidirect

H ≅ G/N

type

114. C₂×C₄

Q₆₄

semidirect

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

subgroup	index
C₂²×Q₆₄	2
C₂×C₄⋊C₃₂	2
C₂×#1416	2
#13236	2
#7471	2

subgroup

index

C₂²×Q₆₄

C₂×C₄⋊C₃₂

C₂×#1416

#13236

#7471

Subgroups — #153311

Conjugacy classes of subgroups

Complements

Maximal subgroups