Group #153357 — subgroups

Subgroups — #153357

Order 512.

215 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 76 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×#1416	C₂	1	2	yes	maximal
2	#13236	C₂	1	1	yes	maximal, characteristic
3	#12907	C₂	1	1	yes	maximal, characteristic
4	#7471	C₂	1	2	yes	maximal
5	#7465	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	2	yes	—
8	C₂×#367	—	2	1	no	—
9	C₂×#366	C₂²	1	2	yes	—
10	C₂×#366	C₂²	1	1	yes	characteristic
11	C₂×#358	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1969	—	2	1	no	—
14	#1846	—	2	1	no	—
15	#1416	—	2	4	no	—
16	C₂²×Q₁₆	—	2	1	no	—
17	C₂×Q₃₂	—	2	2	no	—
18	C₂×Q₃₂	—	2	2	no	—
19	C₂×Q₃₂	—	2	1	no	—
20	C₂×Q₃₂	D₄	1	2	yes	—
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂³	1	1	yes	characteristic
23	C₂×#107	—	2	2	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#106	—	4	1	no	—
26	C₂×#103	—	4	1	no	—
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	#391	—	4	1	no	—
29	#382	—	4	1	no	—
30	C₂×C₃₂	—	2	2	no	—
31	C₂×C₃₂	—	2	2	no	—
32	C₂×C₃₂	—	2	2	no	—
33	#367	—	4	2	no	—
34	#366	—	2	2	no	—
35	#366	—	2	2	no	—
36	#366	—	2	2	no	—
37	#358	—	4	2	no	—
38	C₄×C₁₆	—	2	2	no	—
39	C₂²×Q₈	—	4	1	no	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	—	4	1	no	—
42	C₂×Q₁₆	—	4	1	no	—
43	C₂×Q₁₆	—	4	1	no	—
44	C₂×Q₁₆	—	4	1	no	—
45	C₂×Q₁₆	—	4	1	no	—
46	C₂²×C₈	D₈	1	1	yes	characteristic
47	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	1	no	—
50	C₂×C₄²	D₈	1	1	yes	characteristic
51	Q₃₂	—	2	4	no	—
52	Q₃₂	—	4	2	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	SD₁₆	1	2	yes	—
56	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
57	C₂×C₁₆	#41	1	2	yes	—
58	C₂×C₁₆	#41	1	2	yes	—
59	C₂×C₁₆	#41	1	1	yes	characteristic
60	#107	—	4	2	no	—
61	#107	—	4	1	no	—
62	#107	—	4	1	no	—
63	#107	—	4	2	no	—
64	#106	—	8	2	no	—
65	#103	—	8	2	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	#95	—	8	2	no	—
69	#95	—	8	2	no	—
70	C₃₂	—	4	2	no	—
71	C₂×Q₈	—	4	2	no	—
72	C₂×Q₈	—	8	1	no	—
73	C₂×Q₈	—	8	1	no	—
74	C₂×Q₈	—	8	1	no	—
75	C₂×Q₈	—	8	1	no	—
76	C₂×Q₈	—	8	1	no	—
77	C₂²×C₄	—	8	1	no	—
78	C₂²×C₄	—	8	2	no	—
79	C₂²×C₄	—	8	1	no	—
80	C₂²×C₄	D₁₆	1	2	yes	—
81	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
82	Q₁₆	—	4	2	no	—
83	Q₁₆	—	4	2	no	—
84	Q₁₆	—	8	1	no	—
85	Q₁₆	—	8	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	SD₃₂	1	2	yes	—
89	C₂×C₈	#121	1	1	yes	characteristic
90	C₂×C₈	#123	1	1	yes	characteristic
91	C₂×C₈	#124	1	1	yes	characteristic
92	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₈	#137	1	1	yes	characteristic
95	C₄⋊C₄	—	8	2	no	—
96	C₄⋊C₄	—	8	1	no	—
97	C₄⋊C₄	—	8	1	no	—
98	C₄⋊C₄	—	8	2	no	—
99	C₄²	—	2	1	no	—
100	C₄²	—	2	1	no	—
101	C₁₆	—	2	2	no	—
102	C₁₆	#123	1	2	yes	—
103	C₁₆	#126	1	2	yes	—
104	C₂³	C₂×D₁₆	1	1	yes	characteristic
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	16	1	no	—
108	Q₈	—	16	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	16	1	no	—
114	C₂×C₄	—	16	1	no	—
115	C₂×C₄	—	16	2	no	—
116	C₂×C₄	—	16	1	no	—
117	C₂×C₄	—	16	2	no	—
118	C₂×C₄	—	16	2	no	—
119	C₂×C₄	—	16	2	no	—
120	C₂×C₄	#459	1	1	yes	characteristic
121	C₂×C₄	#480	1	1	yes	characteristic
122	C₂×C₄	#486	1	1	yes	characteristic
123	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
124	C₂×C₄	#509	1	1	yes	characteristic
125	C₂×C₄	#510	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	#397	1	1	yes	characteristic
128	C₈	#461	1	1	yes	characteristic
129	C₈	#481	1	1	yes	characteristic
130	C₈	#489	1	1	yes	characteristic
131	C₂²	#2206	1	1	yes	characteristic
132	C₂²	#2232	1	1	yes	characteristic
133	C₂²	#2240	1	1	yes	characteristic
134	C₂²	#2262	1	2	yes	—
135	C₂²	#2264	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	16	2	no	—
138	C₄	—	32	2	no	—
139	C₄	—	32	1	no	—
140	C₄	#2071	1	1	yes	characteristic
141	C₄	#2208	1	1	yes	characteristic
142	C₄	#2233	1	1	yes	characteristic
143	C₄	#2243	1	1	yes	characteristic
144	C₂	#13480	1	1	yes	characteristic
145	C₂	#13690	1	2	yes	—
146	C₂	#13716	1	2	yes	—
147	C₂	#13721	1	1	yes	characteristic
148	C₂	#13724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1416

C₂

yes

maximal

#13236

C₂

yes

maximal, characteristic

#12907

C₂

yes

maximal, characteristic

#7471

C₂

yes

maximal

#7465

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

C₂×#367

—

C₂×#366

C₂²

yes

—

C₂×#366

C₂²

yes

characteristic

C₂×#358

—

C₂×C₄×C₁₆

C₂²

yes

—

—

—

C₂²×Q₁₆

—

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

#391

—

#382

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

—

—

—

—

—

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×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

#41

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×SD₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

—

101

C₁₆

—

102

C₁₆

#123

yes

—

103

C₁₆

#126

yes

—

104

C₂³

C₂×D₁₆

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#459

yes

characteristic

121

C₂×C₄

#480

yes

characteristic

122

C₂×C₄

#486

yes

characteristic

123

C₂×C₄

C₂×SD₃₂

yes

characteristic

124

C₂×C₄

#509

yes

characteristic

125

C₂×C₄

#510

yes

characteristic

126

C₈

—

127

C₈

#397

yes

characteristic

128

C₈

#461

yes

characteristic

129

C₈

#481

yes

characteristic

130

C₈

#489

yes

characteristic

131

C₂²

#2206

yes

characteristic

132

C₂²

#2232

yes

characteristic

133

C₂²

#2240

yes

characteristic

134

C₂²

#2262

yes

—

135

C₂²

#2264

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#2071

yes

characteristic

141

C₄

#2208

yes

characteristic

142

C₄

#2233

yes

characteristic

143

C₄

#2243

yes

characteristic

144

C₂

#13480

yes

characteristic

145

C₂

#13690

yes

—

146

C₂

#13716

yes

—

147

C₂

#13721

yes

characteristic

148

C₂

#13724

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×#1416	2
#13236	2
#12907	2
#7471	2
#7465	2

subgroup

index

Subgroups — #153357

Conjugacy classes of subgroups

Maximal subgroups