Group #144019 — subgroups

Subgroups — #144019

Order 512.

373 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2254	C₂	1	1	yes	maximal, characteristic, complemented
2	#19192	C₂	1	2	yes	maximal
3	#13191	C₂	1	1	yes	maximal, characteristic
4	#11941	C₂	1	2	yes	maximal, complemented
5	#7155	C₂	1	1	yes	maximal, characteristic, complemented
6	#3206	—	2	2	no	—
7	C₂×#500	C₂²	1	2	yes	—
8	#2931	C₂²	1	1	yes	characteristic
9	#2254	—	2	4	no	—
10	C₂×#367	—	2	2	no	—
11	#2150	C₂²	1	1	yes	characteristic
12	C₂×#350	—	2	2	no	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1693	—	2	4	no	—
15	#1673	—	2	4	no	—
16	#1564	C₂²	1	2	yes	—
17	#1549	—	2	2	no	—
18	#1365	—	2	4	no	—
19	C₂×#131	—	2	1	no	—
20	#532	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×M₃₂	—	2	1	no	—
23	C₂²×C₁₆	D₄	1	2	yes	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	D₄	1	4	yes	—
28	#492	—	4	2	no	—
29	#487	—	4	2	no	—
30	#461	—	2	8	no	—
31	#448	—	4	4	no	—
32	#436	—	2	2	no	—
33	C₈×D₄	—	2	2	no	—
34	#431	—	2	2	no	—
35	C₂×#107	—	2	2	no	—
36	C₂×#107	—	2	2	no	—
37	#423	D₄	1	1	yes	characteristic
38	#408	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#367	—	2	8	no	—
41	C₄⋊C₁₆	—	2	2	no	—
42	#350	—	4	4	no	—
43	C₄×C₁₆	—	2	2	no	—
44	#335	D₄	1	4	yes	—
45	#334	—	2	2	no	—
46	#332	—	2	8	no	—
47	C₂×#41	—	2	1	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂×SD₁₆	—	4	8	no	—
50	#131	—	4	2	no	—
51	#131	—	4	2	no	—
52	C₂×M₁₆	—	2	1	no	—
53	C₂×M₁₆	—	4	1	no	—
54	C₂²×C₈	—	4	1	no	—
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₈	#41	1	1	yes	characteristic
57	C₄⋊Q₈	—	2	4	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	C₄⋊Q₈	—	4	2	no	—
60	C₄⋊Q₈	—	4	2	no	—
61	C₄⋊Q₈	D₈	1	1	yes	characteristic
62	#127	D₈	1	1	yes	characteristic
63	#124	—	2	1	no	—
64	#121	—	2	2	no	—
65	C₄×D₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	M₃₂	—	4	2	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	4	1	no	—
73	C₂×C₁₆	SD₁₆	1	4	yes	complemented
74	#107	—	2	4	no	—
75	#107	—	2	4	no	—
76	#107	—	4	2	no	—
77	#107	—	4	2	no	—
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	2	4	no	—
80	#103	—	4	8	no	—
81	C₂²⋊C₈	—	4	2	no	—
82	C₂²⋊C₈	—	4	2	no	—
83	C₄×C₈	—	4	2	no	—
84	C₄×C₈	C₂×D₄	1	2	yes	—
85	C₄×C₈	C₂×D₄	1	2	yes	—
86	#41	—	4	2	no	—
87	#41	—	4	2	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×Q₈	—	4	4	no	—
90	C₂×Q₈	—	8	2	no	—
91	C₂×Q₈	—	16	4	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	4	1	no	—
95	C₂²×C₄	—	4	1	no	—
96	C₂²×C₄	—	8	2	no	—
97	C₂²×C₄	#121	1	1	yes	characteristic
98	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
99	SD₁₆	—	16	8	no	—
100	M₁₆	—	4	2	no	—
101	M₁₆	—	8	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	4	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	—	8	1	no	—
107	C₂×C₈	#120	1	1	yes	characteristic
108	C₂×C₈	#121	1	2	yes	—
109	C₂×C₈	#127	1	1	yes	characteristic
110	C₂×C₈	C₂×SD₁₆	1	2	yes	—
111	C₂×C₈	#137	1	2	yes	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₄²	#120	1	2	yes	—
119	C₄²	C₂×D₈	1	1	yes	characteristic
120	C₄²	#135	1	1	yes	characteristic
121	C₁₆	—	2	4	no	—
122	C₁₆	—	8	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	#469	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	8	no	—
127	Q₈	—	16	4	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	4	no	—
130	D₄	—	8	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	16	4	no	—
137	C₂×C₄	#392	1	1	yes	characteristic
138	C₂×C₄	#447	1	2	yes	—
139	C₂×C₄	#449	1	2	yes	—
140	C₂×C₄	#461	1	2	yes	—
141	C₂×C₄	#466	1	1	yes	characteristic
142	C₂×C₄	#473	1	2	yes	—
143	C₈	—	2	2	no	—
144	C₈	—	8	4	no	—
145	C₈	—	8	1	no	—
146	C₈	#448	1	2	yes	—
147	C₈	#492	1	1	yes	characteristic
148	C₈	#497	1	1	yes	characteristic
149	C₂²	—	2	2	no	—
150	C₂²	—	8	2	no	—
151	C₂²	—	8	1	no	—
152	C₂²	#1693	1	2	yes	—
153	C₂²	#1999	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	8	1	no	—
156	C₄	—	16	4	no	—
157	C₄	#1673	1	2	yes	—
158	C₄	#2013	1	1	yes	characteristic
159	C₄	#2020	1	1	yes	characteristic
160	C₄	#2194	1	4	yes	—
161	C₂	—	2	2	no	—
162	C₂	—	16	1	no	—
163	C₂	#10264	1	1	yes	characteristic
164	C₂	#13321	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2254

C₂

yes

maximal, characteristic, complemented

#19192

C₂

yes

maximal

#13191

C₂

yes

maximal, characteristic

#11941

C₂

yes

maximal, complemented

#7155

C₂

yes

maximal, characteristic, complemented

#3206

—

C₂×#500

C₂²

yes

—

#2931

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#350

—

C₂×C₄×C₁₆

C₂²

yes

—

—

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

C₈×D₄

—

—

—

—

D₄

yes

characteristic

#408

—

C₂×C₄×C₈

C₂³

yes

characteristic

#367

—

C₄⋊C₁₆

—

#350

—

C₄×C₁₆

—

#335

D₄

yes

—

#334

—

#332

—

C₂×#41

—

C₂²×Q₈

—

C₂×SD₁₆

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₈

yes

characteristic

#127

D₈

yes

characteristic

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

SD₁₆

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

#120

yes

characteristic

108

C₂×C₈

#121

yes

—

109

C₂×C₈

#127

yes

characteristic

110

C₂×C₈

C₂×SD₁₆

yes

—

111

C₂×C₈

#137

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

#120

yes

—

119

C₄²

C₂×D₈

yes

characteristic

120

C₄²

#135

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₂³

—

124

C₂³

#469

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#392

yes

characteristic

138

C₂×C₄

#447

yes

—

139

C₂×C₄

#449

yes

—

140

C₂×C₄

#461

yes

—

141

C₂×C₄

#466

yes

characteristic

142

C₂×C₄

#473

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#448

yes

—

147

C₈

#492

yes

characteristic

148

C₈

#497

yes

characteristic

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#1693

yes

—

153

C₂²

#1999

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#1673

yes

—

158

C₄

#2013

yes

characteristic

159

C₄

#2020

yes

characteristic

160

C₄

#2194

yes

—

161

C₂

—

162

C₂

—

163

C₂

#10264

yes

characteristic

164

C₂

#13321

yes

—

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1999

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

#532

D₄

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#2254	C₂	semidirect
4. #11941	C₂	semidirect
5. #7155	C₂	semidirect
73. C₂×C₁₆	SD₁₆	semidirect

H ≅ G/N

type

1. C₂×#2254

C₂

semidirect

4. #11941

C₂

semidirect

5. #7155

C₂

semidirect

73. C₂×C₁₆

SD₁₆

semidirect

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

subgroup	index
C₂×#2254	2
#19192	2
#13191	2
#11941	2
#7155	2

subgroup

index

Subgroups — #144019

Conjugacy classes of subgroups

Complements

Maximal subgroups