Group #507093 — subgroups

Subgroups — #507093

Order 512.

315 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 144 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	#29230	C₂	1	2	yes	maximal
2	#13513	C₂	1	4	yes	maximal
3	#13188	C₂	1	2	yes	maximal
4	#13173	C₂	1	2	yes	maximal
5	#12911	C₂	1	2	yes	maximal
6	#12910	C₂	1	1	yes	maximal, characteristic
7	#12671	C₂	1	1	yes	maximal, characteristic
8	#12665	C₂	1	1	yes	maximal, characteristic
9	C₂×#368	C₂²	1	2	yes	—
10	C₂×#368	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	2	yes	—
12	#2158	C₂²	1	1	yes	characteristic
13	#2155	C₂²	1	4	yes	—
14	C₂×#364	C₂²	1	1	yes	characteristic
15	#2108	C₂²	1	2	yes	—
16	C₂×#346	C₂²	1	1	yes	characteristic
17	#2102	—	2	4	no	—
18	#1953	—	2	2	no	—
19	#1953	C₂²	1	4	yes	—
20	#1848	C₂²	1	2	yes	—
21	#1833	—	2	2	no	—
22	#1833	C₂²	1	4	yes	—
23	#1772	C₂²	1	2	yes	—
24	#1772	C₂²	1	1	yes	characteristic
25	#1388	C₂²	1	4	yes	—
26	#1382	C₂²	1	2	yes	—
27	#1379	C₂²	1	2	yes	—
28	C₂×M₃₂	C₂³	1	2	yes	—
29	C₂²×C₁₆	C₂³	1	1	yes	characteristic
30	C₂²×C₁₆	C₂³	1	1	yes	characteristic
31	#430	—	2	4	no	—
32	C₂×#108	—	2	2	no	—
33	C₂×#108	D₄	1	1	yes	characteristic
34	C₂×#108	C₂³	1	4	yes	—
35	C₂×#108	C₂³	1	2	yes	—
36	#427	D₄	1	1	yes	characteristic
37	#427	C₂³	1	2	yes	—
38	#423	—	2	2	no	—
39	#405	—	2	2	no	—
40	#405	C₂³	1	2	yes	—
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#368	—	2	2	no	—
43	#368	—	2	2	no	—
44	#368	—	2	2	no	—
45	#368	—	4	2	no	—
46	#367	—	2	1	no	—
47	#366	—	2	1	no	—
48	#365	—	4	4	no	—
49	#364	Q₈	1	4	yes	—
50	#346	—	2	2	no	—
51	#346	Q₈	1	4	yes	—
52	C₄×C₁₆	—	2	2	no	—
53	#344	—	2	4	no	—
54	#344	—	2	4	no	—
55	#339	—	2	2	no	—
56	#339	D₄	1	2	yes	—
57	H(3,ℤ/4)	—	2	4	no	—
58	C₂×M₁₆	—	2	4	no	—
59	C₂×M₁₆	—	2	2	no	—
60	C₂×M₁₆	C₂×D₄	1	2	yes	—
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	C₂⁴	1	1	yes	characteristic
63	#117	—	2	2	no	—
64	#117	C₂×D₄	1	2	yes	—
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	M₃₂	—	2	2	no	—
67	M₃₂	C₂×Q₈	1	4	yes	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	C₂×Q₈	1	2	yes	—
72	C₂×C₁₆	C₂×Q₈	1	2	yes	—
73	C₂×C₁₆	#41	1	2	yes	—
74	C₂×C₁₆	#41	1	1	yes	characteristic
75	C₂×C₁₆	#41	1	1	yes	characteristic
76	#108	—	2	2	no	—
77	#108	—	2	2	no	—
78	#108	—	2	4	no	—
79	#108	—	2	4	no	—
80	#108	—	2	2	no	—
81	#108	—	4	2	no	—
82	#108	—	4	2	no	—
83	#108	—	8	4	no	—
84	#107	—	2	1	no	—
85	#107	—	2	2	no	—
86	#106	—	2	1	no	—
87	#106	—	2	2	no	—
88	C₄⋊C₈	—	8	2	no	—
89	#97	—	2	2	no	—
90	#97	—	4	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	4	2	no	—
93	C₄×C₈	C₂×Q₈	1	2	yes	—
94	C₄×C₈	C₂×Q₈	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	—	2	2	no	—
99	M₁₆	—	4	2	no	—
100	M₁₆	—	4	4	no	—
101	M₁₆	—	4	4	no	—
102	M₁₆	—	4	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	C₄⋊Q₈	1	2	yes	—
107	C₂×C₈	#135	1	2	yes	—
108	C₂×C₈	#135	1	1	yes	characteristic
109	C₂×C₈	#135	1	1	yes	characteristic
110	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₈	C₂×#41	1	1	yes	characteristic
113	C₂×C₈	C₂×#41	1	1	yes	characteristic
114	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	4	2	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	4	2	no	—
123	C₄²	C₄⋊Q₈	1	2	yes	—
124	C₁₆	—	2	2	no	—
125	C₁₆	—	2	4	no	—
126	C₁₆	#125	1	2	yes	—
127	C₁₆	#125	1	2	yes	—
128	C₂³	#578	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	8	1	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂×C₄	#547	1	1	yes	characteristic
135	C₂×C₄	#550	1	1	yes	characteristic
136	C₂×C₄	C₂×#135	1	1	yes	characteristic
137	C₂×C₄	C₂×#135	1	1	yes	characteristic
138	C₂×C₄	#576	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	8	2	no	—
141	C₈	—	8	4	no	—
142	C₈	#499	1	2	yes	—
143	C₈	#499	1	2	yes	—
144	C₈	C₂×#125	1	1	yes	characteristic
145	C₈	#544	1	1	yes	characteristic
146	C₈	#554	1	1	yes	characteristic
147	C₈	#557	1	1	yes	characteristic
148	C₂²	—	4	1	no	—
149	C₂²	#3164	1	1	yes	characteristic
150	C₂²	#3342	1	1	yes	characteristic
151	C₂²	#3393	1	1	yes	characteristic
152	C₄	—	4	2	no	—
153	C₄	—	8	2	no	—
154	C₄	—	16	2	no	—
155	C₄	#2552	1	1	yes	characteristic
156	C₄	C₂×#499	1	1	yes	characteristic
157	C₄	#3341	1	1	yes	characteristic
158	C₄	#3396	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	#28828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29230

C₂

yes

maximal

#13513

C₂

yes

maximal

#13188

C₂

yes

maximal

#13173

C₂

yes

maximal

#12911

C₂

yes

maximal

#12910

C₂

yes

maximal, characteristic

#12671

C₂

yes

maximal, characteristic

#12665

C₂

yes

maximal, characteristic

C₂×#368

C₂²

yes

—

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

#2158

C₂²

yes

characteristic

#2155

C₂²

yes

—

C₂×#364

C₂²

yes

characteristic

#2108

C₂²

yes

—

C₂×#346

C₂²

yes

—

—

C₂²

yes

—

#1848

C₂²

yes

—

#1833

—

#1833

C₂²

yes

—

#1772

C₂²

yes

—

#1772

C₂²

yes

characteristic

#1388

C₂²

yes

—

#1382

C₂²

yes

—

#1379

C₂²

yes

—

C₂×M₃₂

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

D₄

yes

characteristic

C₂×#108

C₂³

yes

—

C₂×#108

C₂³

yes

—

#427

D₄

yes

characteristic

#427

C₂³

yes

—

#423

—

#405

—

#405

C₂³

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

Q₈

yes

—

#346

—

#346

Q₈

yes

—

C₄×C₁₆

—

—

—

—

D₄

yes

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

#117

—

#117

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

C₂×Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

—

—

—

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₄⋊Q₈

yes

—

107

C₂×C₈

#135

yes

—

108

C₂×C₈

#135

yes

characteristic

109

C₂×C₈

#135

yes

characteristic

110

C₂×C₈

C₂²×Q₈

yes

characteristic

111

C₂×C₈

C₂²×Q₈

yes

characteristic

112

C₂×C₈

C₂×#41

yes

characteristic

113

C₂×C₈

C₂×#41

yes

characteristic

114

C₂×C₈

H(5,𝔽₂)

yes

characteristic

115

C₂×C₈

2₋¹⁺⁴

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₄⋊Q₈

yes

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

#125

yes

—

127

C₁₆

#125

yes

—

128

C₂³

#578

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

134

C₂×C₄

#547

yes

characteristic

135

C₂×C₄

#550

yes

characteristic

136

C₂×C₄

C₂×#135

yes

characteristic

137

C₂×C₄

C₂×#135

yes

characteristic

138

C₂×C₄

#576

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#499

yes

—

143

C₈

#499

yes

—

144

C₈

C₂×#125

yes

characteristic

145

C₈

#544

yes

characteristic

146

C₈

#554

yes

characteristic

147

C₈

#557

yes

characteristic

148

C₂²

—

149

C₂²

#3164

yes

characteristic

150

C₂²

#3342

yes

characteristic

151

C₂²

#3393

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#2552

yes

characteristic

156

C₄

C₂×#499

yes

characteristic

157

C₄

#3341

yes

characteristic

158

C₄

#3396

yes

characteristic

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

#28828

yes

characteristic

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28828
prime radical	C₂³	#578
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#3341

commutator, Frattini, cosocle

C₂²×C₈

C₂⁴

C₂

C₂³

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#29230	2
#13513	2
#13188	2
#13173	2
#12911	2
#12910	2
#12671	2
#12665	2

subgroup

index

Subgroups — #507093

Conjugacy classes of subgroups

Maximal subgroups