Group #153309 — subgroups

Subgroups — #153309

Order 512.

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

normal subgroups only

This group has 302 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×D₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1415	C₂	1	2	yes	maximal, complemented
4	#13235	C₂	1	2	yes	maximal, complemented
5	#7471	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₆	C₂²	1	2	yes	—
7	C₂×D₃₂	—	2	4	no	—
8	C₂×D₃₂	—	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×#357	—	2	2	no	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1970	—	2	2	no	—
15	C₄⋊C₃₂	—	2	2	no	—
16	#1415	—	2	4	no	—
17	C₂²×D₈	—	2	2	no	—
18	C₂×D₁₆	—	2	2	no	—
19	C₂×D₁₆	—	2	2	no	—
20	C₂×D₁₆	—	2	2	no	—
21	C₂×D₁₆	—	2	2	no	—
22	C₂×D₁₆	—	2	2	no	—
23	C₂×D₁₆	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×#102	—	4	2	no	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	#390	—	4	2	no	—
30	D₃₂	—	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	#357	—	4	4	no	—
38	C₄×C₁₆	—	2	1	no	—
39	C₄×C₁₆	—	2	1	no	—
40	C₂²×D₄	—	4	2	no	—
41	C₂×D₈	—	2	4	no	—
42	C₂×D₈	—	4	2	no	—
43	C₂×D₈	—	4	2	no	—
44	C₂×D₈	—	4	2	no	—
45	C₂×D₈	—	4	2	no	—
46	C₂×D₈	—	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×C₂²⋊C₄	—	8	2	no	—
51	C₂×C₂²⋊C₄	—	8	2	no	—
52	C₂×C₄²	D₈	1	1	yes	characteristic
53	D₁₆	—	2	4	no	—
54	D₁₆	—	2	4	no	—
55	D₁₆	—	4	2	no	—
56	D₁₆	—	4	2	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	—	2	2	no	—
59	C₂×C₁₆	C₂×D₄	1	2	yes	—
60	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
61	C₂×C₁₆	#41	1	1	yes	characteristic
62	C₂×C₁₆	#41	1	2	yes	—
63	#107	—	4	2	no	—
64	#107	—	4	2	no	—
65	#102	—	8	4	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₃₂	—	2	2	no	—
69	C₃₂	—	4	1	no	—
70	C₂⁴	—	8	2	no	—
71	C₂×D₄	—	4	4	no	—
72	C₂×D₄	—	8	2	no	—
73	C₂×D₄	—	8	2	no	—
74	C₂×D₄	—	8	2	no	—
75	C₂×D₄	—	8	2	no	—
76	C₂×D₄	—	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	D₈	—	4	4	no	—
82	D₈	—	4	4	no	—
83	D₈	—	8	2	no	—
84	D₈	—	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	C₂²⋊C₄	—	16	4	no	—
94	C₂²⋊C₄	—	16	4	no	—
95	C₄²	—	2	1	no	—
96	C₄²	—	2	1	no	—
97	C₁₆	—	4	1	no	—
98	C₁₆	#120	1	1	yes	characteristic
99	C₁₆	#123	1	2	yes	—
100	C₁₆	#124	1	1	yes	characteristic
101	C₂³	—	8	4	no	—
102	C₂³	—	16	2	no	—
103	C₂³	—	16	2	no	—
104	C₂³	—	16	2	no	—
105	C₂³	—	16	2	no	—
106	C₂³	—	16	2	no	—
107	C₂³	—	16	2	no	—
108	C₂³	C₂×D₁₆	1	1	yes	characteristic
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	16	2	no	—
112	D₄	—	16	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	16	2	no	—
118	C₂×C₄	—	16	2	no	—
119	C₂×C₄	—	16	2	no	—
120	C₂×C₄	D₃₂	1	2	yes	complemented
121	C₂×C₄	#459	1	2	yes	—
122	C₂×C₄	#466	1	1	yes	characteristic
123	C₂×C₄	#508	1	1	yes	characteristic
124	C₂×C₄	#509	1	2	yes	—
125	C₈	—	4	1	no	—
126	C₈	#394	1	1	yes	characteristic
127	C₈	#463	1	2	yes	—
128	C₈	#469	1	1	yes	characteristic
129	C₂²	—	16	4	no	—
130	C₂²	—	16	4	no	—
131	C₂²	—	16	4	no	—
132	C₂²	—	32	2	no	—
133	C₂²	—	32	2	no	—
134	C₂²	—	32	2	no	—
135	C₂²	—	32	2	no	—
136	C₂²	#2206	1	2	yes	—
137	C₂²	#2215	1	1	yes	characteristic
138	C₂²	C₂×D₃₂	1	1	yes	characteristic
139	C₂²	#2262	1	1	yes	characteristic
140	C₂²	#2263	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	32	2	no	—
144	C₄	#2054	1	1	yes	characteristic
145	C₄	#2210	1	2	yes	—
146	C₄	#2218	1	1	yes	characteristic
147	C₂	—	32	4	no	—
148	C₂	#13448	1	1	yes	characteristic
149	C₂	#13685	1	2	yes	—
150	C₂	#13689	1	2	yes	—
151	C₂	#13694	1	1	yes	characteristic
152	C₂	#13697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₃₂

C₂

yes

maximal, characteristic

C₂×C₄⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#1415

C₂

yes

maximal, complemented

#13235

C₂

yes

maximal, complemented

#7471

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆

C₂²

yes

—

C₂×D₃₂

—

C₂×D₃₂

—

C₂²×C₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#366

C₂²

yes

—

C₂×#357

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1970

—

C₄⋊C₃₂

—

#1415

—

C₂²×D₈

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

D₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×#107

—

C₂×#102

—

C₂×C₄×C₈

D₄

yes

characteristic

#390

—

D₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

#366

—

#366

—

#357

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

D₁₆

—

D₁₆

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

#107

—

#107

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

D₈

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

#120

yes

characteristic

C₁₆

#123

yes

—

100

C₁₆

#124

yes

characteristic

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×D₁₆

yes

characteristic

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₃₂

yes

complemented

121

C₂×C₄

#459

yes

—

122

C₂×C₄

#466

yes

characteristic

123

C₂×C₄

#508

yes

characteristic

124

C₂×C₄

#509

yes

—

125

C₈

—

126

C₈

#394

yes

characteristic

127

C₈

#463

yes

—

128

C₈

#469

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#2206

yes

—

137

C₂²

#2215

yes

characteristic

138

C₂²

C₂×D₃₂

yes

characteristic

139

C₂²

#2262

yes

characteristic

140

C₂²

#2263

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#2054

yes

characteristic

145

C₄

#2210

yes

—

146

C₄

#2218

yes

characteristic

147

C₂

—

148

C₂

#13448

yes

characteristic

149

C₂

#13685

yes

—

150

C₂

#13689

yes

—

151

C₂

#13694

yes

characteristic

152

C₂

#13697

yes

characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₃₂	C₂
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

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂²×D₃₂

C₂

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
2. C₂×C₄⋊C₃₂	C₂	semidirect
3. C₂×#1415	C₂	semidirect
4. #13235	C₂	semidirect
5. #7471	C₂	semidirect
120. C₂×C₄	D₃₂	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₃₂

C₂

semidirect

3. C₂×#1415

C₂

semidirect

4. #13235

C₂

semidirect

5. #7471

C₂

semidirect

120. C₂×C₄

D₃₂

semidirect

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

subgroup	index
C₂²×D₃₂	2
C₂×C₄⋊C₃₂	2
C₂×#1415	2
#13235	2
#7471	2

subgroup

index

C₂²×D₃₂

C₂×C₄⋊C₃₂

C₂×#1415

#13235

#7471

Subgroups — #153309

Conjugacy classes of subgroups

Complements

Maximal subgroups