Group #141540 — subgroups

Subgroups — #141540

Order 512.

254 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 84 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×#2253	C₂	1	1	yes	maximal, characteristic
2	#12024	C₂	1	1	yes	maximal, characteristic
3	#11967	C₂	1	2	yes	maximal
4	#11408	C₂	1	1	yes	maximal, characteristic
5	#9788	C₂	1	2	yes	maximal
6	C₂×#499	C₂²	1	2	yes	—
7	#2253	—	2	1	no	—
8	#2253	—	2	1	no	—
9	#2253	—	2	1	no	—
10	#2253	—	2	1	no	—
11	#2162	C₂²	1	1	yes	characteristic
12	C₂×#367	C₄	1	2	yes	—
13	C₂×#366	C₄	1	1	yes	characteristic
14	C₂×#366	C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1568	C₂²	1	2	yes	—
17	#1528	—	2	2	no	—
18	#1470	—	2	2	no	—
19	C₈×M₁₆	C₂²	1	1	yes	characteristic
20	#1372	—	2	1	no	—
21	#1371	—	2	1	no	—
22	#1363	—	2	4	no	—
23	#1312	—	2	1	no	—
24	#1311	—	2	1	no	—
25	C₂×#125	—	2	2	no	—
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	#499	—	2	2	no	—
29	#499	—	2	2	no	—
30	#499	—	2	2	no	—
31	#499	—	2	2	no	—
32	#432	—	2	2	no	—
33	#431	D₄	1	1	yes	characteristic
34	C₂×#107	C₂×C₄	1	2	yes	—
35	C₂×#107	C₂×C₄	1	2	yes	—
36	C₂×#106	—	2	2	no	—
37	C₄×M₁₆	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#367	—	2	2	no	—
40	#367	—	2	2	no	—
41	#366	—	2	1	no	—
42	#366	—	2	1	no	—
43	#366	—	2	1	no	—
44	#366	—	2	1	no	—
45	C₄⋊C₁₆	D₄	1	2	yes	—
46	C₂²⋊C₁₆	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	D₄	1	1	yes	characteristic
49	C₄×C₁₆	D₄	1	1	yes	characteristic
50	C₄×C₁₆	D₄	1	1	yes	characteristic
51	C₄×C₁₆	D₄	1	1	yes	characteristic
52	#335	—	2	2	no	—
53	#334	—	2	2	no	—
54	#330	—	4	4	no	—
55	#324	—	4	4	no	—
56	#322	—	2	1	no	—
57	C₈²	—	2	1	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₈	#41	1	1	yes	characteristic
61	#125	—	4	2	no	—
62	#125	—	4	2	no	—
63	#125	—	4	2	no	—
64	#125	—	4	2	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	2	2	no	—
79	#107	—	2	2	no	—
80	#107	—	2	2	no	—
81	#106	—	4	2	no	—
82	#106	—	4	2	no	—
83	C₄⋊C₈	—	2	1	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₂²⋊C₈	—	4	2	no	—
86	C₂²⋊C₈	D₈	1	2	yes	—
87	#97	—	4	2	no	—
88	#97	D₈	1	2	yes	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂×D₄	1	1	yes	characteristic
92	C₄×C₈	C₂×D₄	1	1	yes	characteristic
93	C₄×C₈	C₂×D₄	1	1	yes	characteristic
94	C₄×C₈	#41	1	1	yes	characteristic
95	C₂²×C₄	—	4	2	no	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
99	C₂²×C₄	#135	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	8	2	no	—
107	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈	C₄×D₄	1	1	yes	characteristic
109	C₂×C₈	#121	1	1	yes	characteristic
110	C₂×C₈	#124	1	1	yes	characteristic
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄⋊C₄	—	8	2	no	—
119	C₄²	#121	1	1	yes	characteristic
120	C₄²	#127	1	1	yes	characteristic
121	C₄²	C₂×D₈	1	1	yes	characteristic
122	C₄²	#135	1	1	yes	characteristic
123	C₁₆	—	2	1	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	4	2	no	—
126	C₁₆	#101	1	2	yes	—
127	C₁₆	#101	1	2	yes	—
128	C₂³	C₄×D₈	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	16	2	no	—
137	C₂×C₄	#390	1	1	yes	characteristic
138	C₂×C₄	C₄×D₈	1	1	yes	characteristic
139	C₂×C₄	#466	1	1	yes	characteristic
140	C₂×C₄	#466	1	1	yes	characteristic
141	C₂×C₄	#493	1	1	yes	characteristic
142	C₂×C₄	#495	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	2	no	—
146	C₈	—	4	2	no	—
147	C₈	—	16	2	no	—
148	C₈	C₂×#101	1	1	yes	characteristic
149	C₈	C₂×#101	1	1	yes	characteristic
150	C₈	#473	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	#1970	1	1	yes	characteristic
154	C₂²	#2179	1	2	yes	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	16	2	no	—
161	C₄	—	16	2	no	—
162	C₄	#1966	1	1	yes	characteristic
163	C₄	#1979	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	#9799	1	1	yes	characteristic
166	C₂	#13242	1	1	yes	characteristic
167	C₂	#13243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2253

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal, characteristic

#11967

C₂

yes

maximal

#11408

C₂

yes

maximal, characteristic

#9788

C₂

yes

maximal

C₂×#499

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₂×#367

C₄

yes

—

C₂×#366

C₄

yes

characteristic

C₂×#366

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1568

C₂²

yes

—

#1528

—

#1470

—

C₈×M₁₆

C₂²

yes

—

—

—

—

—

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×#107

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₄×M₁₆

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

C₄⋊C₁₆

D₄

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

C₂²×C₄

yes

C₂²×C₈

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

D₈

yes

—

#97

—

#97

D₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

108

C₂×C₈

C₄×D₄

yes

characteristic

109

C₂×C₈

#121

yes

characteristic

110

C₂×C₈

#124

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

#121

yes

characteristic

120

C₄²

#127

yes

characteristic

121

C₄²

C₂×D₈

yes

characteristic

122

C₄²

#135

yes

characteristic

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

#101

yes

—

127

C₁₆

#101

yes

—

128

C₂³

C₄×D₈

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#390

yes

characteristic

138

C₂×C₄

C₄×D₈

yes

characteristic

139

C₂×C₄

#466

yes

characteristic

140

C₂×C₄

#466

yes

characteristic

141

C₂×C₄

#493

yes

characteristic

142

C₂×C₄

#495

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

C₂×#101

yes

characteristic

149

C₈

C₂×#101

yes

characteristic

150

C₈

#473

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

#1970

yes

characteristic

154

C₂²

#2179

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#1966

yes

characteristic

163

C₄

#1979

yes

characteristic

164

C₂

—

165

C₂

#9799

yes

characteristic

166

C₂

#13242

yes

characteristic

167

C₂

#13243

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1970

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₄×D₈

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#2253	2
#12024	2
#11967	2
#11408	2
#9788	2

subgroup

index

Subgroups — #141540

Conjugacy classes of subgroups

Maximal subgroups