Group #141731 — subgroups

Subgroups — #141731

Order 512.

420 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2246	C₂	1	1	yes	maximal, characteristic
2	#13213	C₂	1	2	yes	maximal, complemented
3	#12891	C₂	1	1	yes	maximal, characteristic, complemented
4	#11401	C₂	1	1	yes	maximal, characteristic, complemented
5	#9773	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₆	—	2	2	no	—
7	C₂×#493	C₂²	1	2	yes	—
8	#2246	—	2	2	no	—
9	#2246	C₄	1	4	yes	—
10	C₂×#361	—	2	4	no	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	2	yes	—
14	#1545	—	2	4	no	—
15	#1450	C₂²	1	1	yes	characteristic
16	#1381	—	2	2	no	—
17	#1312	—	2	2	no	—
18	C₂²×D₈	—	2	2	no	—
19	C₂²×D₈	—	2	2	no	—
20	C₂×#127	—	2	2	no	—
21	C₂×D₁₆	—	2	8	no	—
22	C₂×D₁₆	—	4	8	no	—
23	C₂×M₃₂	D₄	1	2	yes	—
24	C₂²×C₁₆	D₄	1	2	yes	—
25	#493	—	2	2	no	—
26	#493	—	2	2	no	—
27	#493	—	2	2	no	—
28	#493	C₂×C₄	1	4	yes	—
29	#431	D₄	1	2	yes	—
30	C₂×#108	—	2	2	no	—
31	C₂×#108	—	2	2	no	—
32	#423	—	2	2	no	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	#361	—	4	8	no	—
35	#346	—	2	1	no	—
36	C₄×C₁₆	—	2	1	no	—
37	C₄×C₁₆	C₂×C₄	1	2	yes	—
38	C₄×C₁₆	D₄	1	2	yes	—
39	#331	—	4	8	no	—
40	#322	—	2	1	no	—
41	C₈²	—	2	1	no	—
42	C₂²×D₄	—	4	2	no	—
43	C₂²×D₄	—	4	4	no	—
44	C₂×D₈	—	2	4	no	—
45	C₂×D₈	—	2	4	no	—
46	C₂×D₈	—	4	2	no	—
47	C₂×D₈	—	4	2	no	—
48	C₂×D₈	—	4	4	no	—
49	C₂×D₈	—	4	4	no	—
50	C₂×D₈	—	4	4	no	—
51	C₂×D₈	—	4	4	no	—
52	C₂×M₁₆	—	4	2	no	—
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	#127	—	2	4	no	—
56	#127	—	4	2	no	—
57	#127	—	4	4	no	—
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	D₁₆	—	4	16	no	—
60	M₃₂	—	2	4	no	—
61	C₂×C₁₆	—	2	4	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
64	#108	—	4	4	no	—
65	#108	—	4	4	no	—
66	C₄⋊C₈	—	4	4	no	—
67	C₄⋊C₈	D₈	1	4	yes	—
68	C₂²⋊C₈	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	C₂⁴	—	8	2	no	—
75	C₂×D₄	—	4	4	no	—
76	C₂×D₄	—	4	8	no	—
77	C₂×D₄	—	8	4	no	—
78	C₂×D₄	—	8	2	no	—
79	C₂×D₄	—	8	2	no	—
80	C₂×D₄	—	8	2	no	—
81	C₂×D₄	—	8	2	no	—
82	C₂×D₄	—	8	2	no	—
83	C₂×D₄	—	16	8	no	—
84	C₂²×C₄	#127	1	1	yes	characteristic
85	C₂²×C₄	#135	1	2	yes	—
86	D₈	—	4	8	no	—
87	D₈	—	4	8	no	—
88	D₈	—	8	4	no	—
89	D₈	—	8	4	no	—
90	M₁₆	—	8	2	no	—
91	M₁₆	—	8	2	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	8	2	no	—
94	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈	C₄×D₄	1	1	yes	characteristic
96	C₂×C₈	#121	1	1	yes	characteristic
97	C₂×C₈	#124	1	1	yes	characteristic
98	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
100	C₄²	C₄×D₄	1	1	yes	characteristic
101	C₄²	#121	1	1	yes	characteristic
102	C₄²	C₂×D₈	1	2	yes	—
103	C₁₆	—	2	4	no	—
104	C₁₆	—	4	2	no	—
105	C₂³	—	8	4	no	—
106	C₂³	—	16	8	no	—
107	C₂³	—	16	2	no	—
108	C₂³	—	16	2	no	—
109	C₂³	#495	1	1	yes	characteristic
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	D₄	—	16	8	no	—
113	D₄	—	16	8	no	—
114	D₄	—	16	2	no	—
115	D₄	—	16	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	#390	1	1	yes	characteristic
118	C₂×C₄	C₄×D₈	1	2	yes	—
119	C₂×C₄	#466	1	2	yes	—
120	C₂×C₄	#493	1	1	yes	characteristic
121	C₂×C₄	#496	1	2	yes	—
122	C₂×C₄	#496	1	2	yes	—
123	C₈	—	4	4	no	—
124	C₈	—	16	2	no	—
125	C₈	#418	1	2	yes	—
126	C₈	#442	1	2	yes	—
127	C₈	#469	1	2	yes	—
128	C₈	#490	1	2	yes	—
129	C₂²	—	2	2	no	—
130	C₂²	—	16	4	no	—
131	C₂²	—	16	8	no	—
132	C₂²	—	32	4	no	—
133	C₂²	—	32	4	no	—
134	C₂²	#1717	1	2	yes	—
135	C₂²	#1970	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	#1721	1	2	yes	—
138	C₄	#1961	1	1	yes	characteristic
139	C₄	#1972	1	1	yes	characteristic
140	C₄	#2221	1	4	yes	—
141	C₂	—	2	2	no	—
142	C₂	—	32	4	no	—
143	C₂	#9832	1	1	yes	characteristic
144	C₂	#13245	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2246

C₂

yes

maximal, characteristic

#13213

C₂

yes

maximal, complemented

#12891

C₂

yes

maximal, characteristic, complemented

#11401

C₂

yes

maximal, characteristic, complemented

#9773

C₂

yes

maximal, complemented

C₂²×D₁₆

—

C₂×#493

C₂²

yes

—

#2246

—

#2246

C₄

yes

—

C₂×#361

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1545

—

#1450

C₂²

yes

characteristic

#1381

—

#1312

—

C₂²×D₈

—

C₂²×D₈

—

C₂×#127

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#431

D₄

yes

—

C₂×#108

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂³

yes

characteristic

#361

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

#331

—

#322

—

C₈²

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

C₂×C₄²

C₂×D₄

yes

characteristic

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

#135

yes

—

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

100

C₄²

C₄×D₄

yes

characteristic

101

C₄²

#121

yes

characteristic

102

C₄²

C₂×D₈

yes

—

103

C₁₆

—

104

C₁₆

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

#495

yes

characteristic

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

#390

yes

characteristic

118

C₂×C₄

C₄×D₈

yes

—

119

C₂×C₄

#466

yes

—

120

C₂×C₄

#493

yes

characteristic

121

C₂×C₄

#496

yes

—

122

C₂×C₄

#496

yes

—

123

C₈

—

124

C₈

—

125

C₈

#418

yes

—

126

C₈

#442

yes

—

127

C₈

#469

yes

—

128

C₈

#490

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#1717

yes

—

135

C₂²

#1970

yes

characteristic

136

C₄

—

137

C₄

#1721

yes

—

138

C₄

#1961

yes

characteristic

139

C₄

#1972

yes

characteristic

140

C₄

#2221

yes

—

141

C₂

—

142

C₂

—

143

C₂

#9832

yes

characteristic

144

C₂

#13245

yes

—

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#2246	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

C₂×#2246

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. #13213	C₂	semidirect
3. #12891	C₂	semidirect
4. #11401	C₂	semidirect
5. #9773	C₂	semidirect

H ≅ G/N

type

2. #13213

C₂

semidirect

3. #12891

C₂

semidirect

4. #11401

C₂

semidirect

5. #9773

C₂

semidirect

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

subgroup	index
C₂×#2246	2
#13213	2
#12891	2
#11401	2
#9773	2

subgroup

index

Subgroups — #141731

Conjugacy classes of subgroups

Complements

Maximal subgroups