Group #141729 — subgroups

Subgroups — #141729

Order 512.

380 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 98 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	#33713	C₂	1	1	yes	maximal, characteristic
2	#13211	C₂	1	2	yes	maximal, complemented
3	#12889	C₂	1	1	yes	maximal, characteristic, complemented
4	#11401	C₂	1	1	yes	maximal, characteristic, complemented
5	#9756	C₂	1	2	yes	maximal, complemented
6	C₂×#508	—	2	2	no	—
7	#3147	C₂²	1	2	yes	—
8	#2249	—	2	2	no	—
9	#2248	C₄	1	2	yes	complemented
10	#2247	C₄	1	1	yes	characteristic, complemented
11	#2246	C₄	1	1	yes	characteristic, complemented
12	C₂×#360	—	2	4	no	—
13	C₄×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1846	C₂²	1	2	yes	—
16	#1517	—	2	4	no	—
17	#1450	C₂²	1	1	yes	characteristic
18	#1380	—	2	2	no	—
19	#1312	—	2	2	no	—
20	C₂×#135	—	2	2	no	—
21	C₂×#135	—	2	2	no	—
22	#532	—	2	2	no	—
23	#508	—	4	8	no	—
24	C₂×Q₃₂	—	2	2	no	—
25	C₂×SD₃₂	—	2	4	no	—
26	C₂×D₁₆	—	2	2	no	—
27	C₂×M₃₂	D₄	1	2	yes	—
28	C₂²×C₁₆	D₄	1	2	yes	—
29	#495	—	2	2	no	—
30	#495	—	2	2	no	—
31	#494	C₂×C₄	1	2	yes	—
32	#493	C₂×C₄	1	2	yes	—
33	#492	—	2	2	no	—
34	#431	D₄	1	2	yes	—
35	C₂×#107	—	2	2	no	—
36	C₂×#106	—	2	2	no	—
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#382	—	2	2	no	—
39	#360	—	4	8	no	—
40	#346	—	2	1	no	—
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	2	yes	—
45	#327	—	4	8	no	—
46	#322	—	2	1	no	—
47	C₈²	—	2	1	no	—
48	C₂×#41	—	4	2	no	—
49	#135	—	4	4	no	—
50	#135	—	4	4	no	—
51	#135	—	4	4	no	—
52	#135	—	4	4	no	—
53	C₂×Q₁₆	—	2	2	no	—
54	C₂×Q₁₆	—	2	2	no	—
55	C₂×SD₁₆	—	4	2	no	—
56	C₂×SD₁₆	—	4	2	no	—
57	C₂×D₈	—	2	2	no	—
58	C₂×D₈	—	2	2	no	—
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Q₈	—	2	2	no	—
62	#127	—	2	2	no	—
63	#124	—	4	2	no	—
64	#121	—	4	4	no	—
65	C₄×D₄	—	4	4	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	Q₃₂	—	4	4	no	—
69	SD₃₂	—	4	8	no	—
70	D₁₆	—	4	4	no	—
71	M₃₂	—	2	4	no	—
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
75	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
76	#107	—	4	4	no	—
77	#106	—	4	4	no	—
78	C₄⋊C₈	—	2	2	no	—
79	C₂²⋊C₈	D₈	1	4	yes	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₄×C₈	#41	1	1	yes	characteristic
85	#95	—	4	4	no	—
86	#41	—	8	2	no	—
87	#41	—	8	2	no	—
88	#41	—	8	2	no	—
89	#41	—	8	2	no	—
90	C₂×Q₈	—	4	2	no	—
91	C₂×D₄	—	4	2	no	—
92	C₂×D₄	—	4	4	no	—
93	C₂×D₄	—	8	2	no	—
94	C₂²×C₄	—	8	2	no	—
95	C₂²×C₄	—	8	2	no	—
96	C₂²×C₄	#127	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₈	1	2	yes	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	SD₁₆	—	8	4	no	—
101	SD₁₆	—	8	4	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈	C₄×D₄	1	1	yes	characteristic
107	C₂×C₈	#121	1	1	yes	characteristic
108	C₂×C₈	#124	1	1	yes	characteristic
109	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
110	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄⋊C₄	—	8	2	no	—
114	C₂²⋊C₄	—	8	4	no	—
115	C₄²	C₄×D₄	1	1	yes	characteristic
116	C₄²	#121	1	1	yes	characteristic
117	C₄²	#135	1	2	yes	—
118	C₁₆	—	2	2	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	2	no	—
121	C₂³	—	8	2	no	—
122	C₂³	#493	1	1	yes	characteristic
123	Q₈	—	8	2	no	—
124	Q₈	—	8	2	no	—
125	D₄	—	8	2	no	—
126	D₄	—	8	2	no	—
127	D₄	—	16	4	no	—
128	D₄	—	16	2	no	—
129	D₄	—	16	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	16	2	no	—
133	C₂×C₄	—	16	2	no	—
134	C₂×C₄	—	16	4	no	—
135	C₂×C₄	—	16	2	no	—
136	C₂×C₄	#390	1	1	yes	characteristic
137	C₂×C₄	C₄×D₈	1	2	yes	—
138	C₂×C₄	#466	1	2	yes	—
139	C₂×C₄	#495	1	1	yes	characteristic
140	C₂×C₄	#496	1	2	yes	—
141	C₂×C₄	#496	1	2	yes	—
142	C₈	—	4	4	no	—
143	C₈	#418	1	2	yes	—
144	C₈	#442	1	2	yes	—
145	C₈	#469	1	2	yes	—
146	C₈	#490	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	—	16	2	no	—
149	C₂²	—	16	4	no	—
150	C₂²	#1721	1	2	yes	—
151	C₂²	#1970	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	16	2	no	—
155	C₄	—	32	2	no	—
156	C₄	#1717	1	2	yes	—
157	C₄	#1961	1	1	yes	characteristic
158	C₄	#1972	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	—	32	2	no	—
161	C₂	#9832	1	1	yes	characteristic
162	C₂	#13243	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33713

C₂

yes

maximal, characteristic

#13211

C₂

yes

maximal, complemented

#12889

C₂

yes

maximal, characteristic, complemented

#11401

C₂

yes

maximal, characteristic, complemented

#9756

C₂

yes

maximal, complemented

C₂×#508

—

#3147

C₂²

yes

—

#2249

—

#2248

C₄

yes

complemented

#2247

C₄

yes

characteristic, complemented

#2246

C₄

yes

characteristic, complemented

C₂×#360

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1846

C₂²

yes

—

#1517

—

#1450

C₂²

yes

—

—

—

—

—

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

#495

—

#495

—

#494

C₂×C₄

yes

—

#493

C₂×C₄

yes

—

#492

—

#431

D₄

yes

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

#327

—

#322

—

C₈²

—

C₂×#41

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

#107

—

#106

—

C₄⋊C₈

—

C₂²⋊C₈

D₈

yes

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

Q₁₆

—

100

SD₁₆

—

101

SD₁₆

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

106

C₂×C₈

C₄×D₄

yes

characteristic

107

C₂×C₈

#121

yes

characteristic

108

C₂×C₈

#124

yes

characteristic

109

C₂×C₈

AGL(1,ℤ/8)

yes

—

110

C₂×C₈

AGL(1,ℤ/8)

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

C₄×D₄

yes

characteristic

116

C₄²

#121

yes

characteristic

117

C₄²

#135

yes

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

—

122

C₂³

#493

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#390

yes

characteristic

137

C₂×C₄

C₄×D₈

yes

—

138

C₂×C₄

#466

yes

—

139

C₂×C₄

#495

yes

characteristic

140

C₂×C₄

#496

yes

—

141

C₂×C₄

#496

yes

—

142

C₈

—

143

C₈

#418

yes

—

144

C₈

#442

yes

—

145

C₈

#469

yes

—

146

C₈

#490

yes

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#1721

yes

—

151

C₂²

#1970

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#1717

yes

—

157

C₄

#1961

yes

characteristic

158

C₄

#1972

yes

characteristic

159

C₂

—

160

C₂

—

161

C₂

#9832

yes

characteristic

162

C₂

#13243

yes

—

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33713	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1970

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

#33713

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13211	C₂	semidirect
3. #12889	C₂	semidirect
4. #11401	C₂	semidirect
5. #9756	C₂	semidirect
9. #2248	C₄	semidirect
10. #2247	C₄	semidirect
11. #2246	C₄	semidirect

H ≅ G/N

type

2. #13211

C₂

semidirect

3. #12889

C₂

semidirect

4. #11401

C₂

semidirect

5. #9756

C₂

semidirect

9. #2248

C₄

semidirect

10. #2247

C₄

semidirect

11. #2246

C₄

semidirect

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

subgroup	index
#33713	2
#13211	2
#12889	2
#11401	2
#9756	2

subgroup

index

Subgroups — #141729

Conjugacy classes of subgroups

Complements

Maximal subgroups