Group #140713 — subgroups

Subgroups — #140713

Order 512.

294 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 110 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	#17958	C₂	1	1	yes	maximal, characteristic
2	#13266	C₂	1	2	yes	maximal
3	#12004	C₂	1	1	yes	maximal, characteristic
4	#11608	C₂	1	2	yes	maximal
5	C₂×#1311	C₂	1	1	yes	maximal, characteristic
6	#3170	C₄	1	1	yes	characteristic
7	#2960	C₂²	1	2	yes	—
8	#2150	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1591	—	2	2	no	—
12	C₈×Q₁₆	—	2	2	no	—
13	#1567	C₄	1	1	yes	characteristic
14	#1493	C₄	1	2	yes	—
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1379	—	2	4	no	—
17	#1370	—	2	2	no	—
18	Q₈⋊C₁₆	—	2	4	no	—
19	#1311	—	2	2	no	—
20	#533	C₂×C₄	1	2	yes	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂²×C₁₆	—	2	1	no	—
23	C₂²×C₁₆	D₄	1	2	yes	—
24	#500	C₈	1	1	yes	characteristic
25	#494	C₈	1	1	yes	characteristic
26	#487	C₈	1	2	yes	—
27	#467	—	2	2	no	—
28	#446	—	2	2	no	—
29	#446	—	2	2	no	—
30	C₈×Q₈	—	2	2	no	—
31	C₈×Q₈	—	2	2	no	—
32	C₄×Q₁₆	—	2	2	no	—
33	#424	—	2	2	no	—
34	#423	C₂×C₄	1	2	yes	—
35	#405	—	2	2	no	—
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₁₆	—	2	2	no	—
40	C₄⋊C₁₆	—	2	2	no	—
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	D₄	1	2	yes	—
45	#335	—	2	2	no	—
46	Q₈⋊C₈	—	2	4	no	—
47	C₈²	D₄	1	2	yes	—
48	C₈²	D₄	1	2	yes	—
49	C₂×Q₁₆	—	2	2	no	—
50	C₂×M₁₆	—	2	2	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	M₁₆	1	2	yes	—
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	#41	1	1	yes	characteristic
55	C₄⋊Q₈	C₂×C₈	1	2	yes	—
56	C₄⋊Q₈	C₂×C₈	1	2	yes	—
57	#125	—	2	2	no	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₂²⋊Q₈	—	2	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	—	2	2	no	—
62	#117	—	2	2	no	—
63	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
64	M₃₂	—	4	4	no	—
65	C₂×C₁₆	—	2	4	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	4	2	no	—
70	C₂×C₁₆	Q₁₆	1	4	yes	complemented
71	#107	—	2	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	4	2	no	—
75	C₄⋊C₈	—	4	2	no	—
76	#103	—	2	4	no	—
77	#97	—	4	2	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	4	2	no	—
80	C₄×C₈	C₂²⋊C₄	1	2	yes	—
81	C₄×C₈	C₂²⋊C₄	1	2	yes	—
82	C₄×C₈	C₂×C₈	1	1	yes	characteristic
83	C₄×C₈	C₂×C₈	1	1	yes	characteristic
84	C₄×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄×C₈	C₂×D₄	1	1	yes	characteristic
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
92	C₂²×C₄	#131	1	1	yes	characteristic
93	Q₁₆	—	4	4	no	—
94	M₁₆	—	4	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₂²⋊C₈	1	2	yes	—
101	C₂×C₈	C₂²⋊C₈	1	2	yes	—
102	C₂×C₈	#121	1	1	yes	characteristic
103	C₂×C₈	#121	1	1	yes	characteristic
104	C₂×C₈	#124	1	1	yes	characteristic
105	C₂×C₈	#127	1	1	yes	characteristic
106	C₂×C₈	C₂×Q₁₆	1	2	yes	—
107	C₂×C₈	#135	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₂²⋊C₄	—	4	2	no	—
115	C₄²	—	4	2	no	—
116	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₄²	C₄×D₄	1	1	yes	characteristic
118	C₄²	C₂²×C₈	1	1	yes	characteristic
119	C₄²	#131	1	1	yes	characteristic
120	C₁₆	—	2	4	no	—
121	C₁₆	—	2	4	no	—
122	C₁₆	—	8	2	no	—
123	C₂³	#436	1	1	yes	characteristic
124	Q₈	—	4	4	no	—
125	Q₈	—	4	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	#390	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₄	#408	1	1	yes	characteristic
133	C₂×C₄	C₈×D₄	1	1	yes	characteristic
134	C₂×C₄	C₈×D₄	1	1	yes	characteristic
135	C₂×C₄	#436	1	1	yes	characteristic
136	C₂×C₄	C₄×Q₁₆	1	2	yes	—
137	C₂×C₄	#443	1	2	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	4	2	no	—
143	C₈	—	8	2	no	—
144	C₈	#467	1	2	yes	—
145	C₈	#494	1	1	yes	characteristic
146	C₈	#495	1	1	yes	characteristic
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#1591	1	2	yes	—
150	C₂²	#1954	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	8	2	no	—
154	C₄	—	8	2	no	—
155	C₄	C₈×Q₁₆	1	2	yes	—
156	C₄	#1969	1	1	yes	characteristic
157	C₄	#1974	1	1	yes	characteristic
158	C₄	#2117	1	2	yes	—
159	C₄	#2171	1	2	yes	—
160	C₂	—	2	2	no	—
161	C₂	#9656	1	1	yes	characteristic
162	C₂	#13192	1	1	yes	characteristic
163	C₂	#13196	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17958

C₂

yes

maximal, characteristic

#13266

C₂

yes

maximal

#12004

C₂

yes

maximal, characteristic

#11608

C₂

yes

maximal

C₂×#1311

C₂

yes

maximal, characteristic

#3170

C₄

yes

characteristic

#2960

C₂²

yes

—

#2150

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1591

—

C₈×Q₁₆

—

#1567

C₄

yes

characteristic

#1493

C₄

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1379

—

#1370

—

Q₈⋊C₁₆

—

#1311

—

#533

C₂×C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

#500

C₈

yes

characteristic

#494

C₈

yes

characteristic

#487

C₈

yes

—

#467

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

#424

—

#423

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

#335

—

Q₈⋊C₈

—

C₈²

D₄

yes

—

C₈²

D₄

yes

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

C₂×C₈

yes

—

C₄⋊Q₈

C₂×C₈

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

complemented

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂²×C₄

#131

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂²⋊C₈

yes

—

101

C₂×C₈

C₂²⋊C₈

yes

—

102

C₂×C₈

#121

yes

characteristic

103

C₂×C₈

#121

yes

characteristic

104

C₂×C₈

#124

yes

characteristic

105

C₂×C₈

#127

yes

characteristic

106

C₂×C₈

C₂×Q₁₆

yes

—

107

C₂×C₈

#135

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

C₂×C₂²⋊C₄

yes

characteristic

117

C₄²

C₄×D₄

yes

characteristic

118

C₄²

C₂²×C₈

yes

characteristic

119

C₄²

#131

yes

characteristic

120

C₁₆

—

121

C₁₆

—

122

C₁₆

—

123

C₂³

#436

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#390

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

132

C₂×C₄

#408

yes

characteristic

133

C₂×C₄

C₈×D₄

yes

characteristic

134

C₂×C₄

C₈×D₄

yes

characteristic

135

C₂×C₄

#436

yes

characteristic

136

C₂×C₄

C₄×Q₁₆

yes

—

137

C₂×C₄

#443

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

#467

yes

—

145

C₈

#494

yes

characteristic

146

C₈

#495

yes

characteristic

147

C₂²

—

148

C₂²

—

149

C₂²

#1591

yes

—

150

C₂²

#1954

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

C₈×Q₁₆

yes

—

156

C₄

#1969

yes

characteristic

157

C₄

#1974

yes

characteristic

158

C₄

#2117

yes

—

159

C₄

#2171

yes

—

160

C₂

—

161

C₂

#9656

yes

characteristic

162

C₂

#13192

yes

characteristic

163

C₂

#13196

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	#436
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
70. C₂×C₁₆	Q₁₆	semidirect

H ≅ G/N

type

70. C₂×C₁₆

Q₁₆

semidirect

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

subgroup	index
#17958	2
#13266	2
#12004	2
#11608	2
C₂×#1311	2

subgroup

index

Subgroups — #140713

Conjugacy classes of subgroups

Complements

Maximal subgroups