Group #141827 — subgroups

Subgroups — #141827

Order 512.

273 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 74 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	#19668	C₂	1	1	yes	maximal, characteristic
2	#11679	C₂	1	2	yes	maximal
3	#11676	C₂	1	2	yes	maximal
4	#11406	C₂	1	1	yes	maximal, characteristic
5	#8796	C₂	1	1	yes	maximal, characteristic
6	#3231	—	2	2	no	—
7	#3170	C₂²	1	2	yes	—
8	C₄×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×#346	C₂²	1	1	yes	characteristic
10	#1718	—	2	2	no	—
11	#1713	C₄	1	1	yes	characteristic
12	#1710	C₄	1	1	yes	characteristic
13	#1707	C₄	1	2	yes	—
14	#1568	C₂²	1	2	yes	—
15	#1450	C₂²	1	1	yes	characteristic
16	#1334	—	2	4	no	—
17	#1332	—	2	4	no	—
18	#1313	—	2	2	no	—
19	#1287	—	2	2	no	—
20	#533	—	2	2	no	—
21	C₂×M₃₂	—	2	1	no	—
22	C₂²×C₁₆	—	2	1	no	—
23	#500	—	2	2	no	—
24	#500	C₂×C₄	1	2	yes	—
25	#498	—	2	2	no	—
26	#494	C₂×C₄	1	2	yes	—
27	#487	—	2	2	no	—
28	#487	—	2	2	no	—
29	#487	—	2	2	no	—
30	#485	—	4	4	no	—
31	#479	—	4	4	no	—
32	#467	—	2	2	no	—
33	#467	—	2	2	no	—
34	C₄×Q₁₆	—	2	2	no	—
35	C₄×Q₁₆	—	2	2	no	—
36	#432	—	2	2	no	—
37	#431	D₄	1	2	yes	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#346	—	2	1	no	—
40	#346	—	2	1	no	—
41	#346	D₄	1	2	yes	—
42	C₄×C₁₆	D₄	1	2	yes	—
43	#335	—	2	2	no	—
44	#334	—	2	2	no	—
45	#322	D₄	1	2	yes	—
46	C₈²	C₂×C₄	1	1	yes	characteristic
47	C₈²	C₂×C₄	1	1	yes	characteristic
48	C₂×Q₁₆	—	2	2	no	—
49	C₂×Q₁₆	—	2	2	no	—
50	C₂²×C₈	#41	1	1	yes	characteristic
51	C₂²×C₈	#41	1	1	yes	characteristic
52	C₄⋊Q₈	—	2	2	no	—
53	C₄⋊Q₈	—	2	2	no	—
54	#125	—	4	2	no	—
55	C₂²⋊Q₈	—	4	2	no	—
56	C₂²⋊Q₈	—	4	2	no	—
57	C₄×Q₈	—	4	2	no	—
58	C₄×Q₈	—	4	2	no	—
59	#117	—	4	2	no	—
60	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
61	M₃₂	—	2	4	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	4	2	no	—
65	#107	—	2	2	no	—
66	#107	—	2	2	no	—
67	#107	—	4	4	no	—
68	#106	—	4	4	no	—
69	C₄⋊C₈	—	2	2	no	—
70	C₄⋊C₈	—	4	2	no	—
71	#103	—	4	4	no	—
72	#103	—	4	4	no	—
73	#103	—	4	2	no	—
74	#103	—	4	2	no	—
75	C₂²⋊C₈	—	2	2	no	—
76	C₂²⋊C₈	—	4	2	no	—
77	#97	—	4	2	no	—
78	C₄×C₈	C₂²⋊C₄	1	2	yes	—
79	C₄×C₈	C₂²⋊C₄	1	2	yes	—
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂×D₄	1	1	yes	characteristic
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂²×C₄	#124	1	1	yes	characteristic
87	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
88	Q₁₆	—	4	4	no	—
89	Q₁₆	—	4	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	8	2	no	—
95	C₂×C₈	C₄×D₄	1	1	yes	characteristic
96	C₂×C₈	C₄×D₄	1	1	yes	characteristic
97	C₂×C₈	#121	1	1	yes	characteristic
98	C₂×C₈	#121	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₂²⋊C₄	—	8	2	no	—
106	C₄²	—	8	2	no	—
107	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₄²	#127	1	1	yes	characteristic
109	C₄²	AGL(1,ℤ/8)	1	2	yes	—
110	C₁₆	—	2	4	no	—
111	C₁₆	—	4	2	no	—
112	C₂³	#490	1	1	yes	characteristic
113	Q₈	—	8	4	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	8	2	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	#390	1	1	yes	characteristic
122	C₂×C₄	#418	1	2	yes	—
123	C₂×C₄	#490	1	1	yes	characteristic
124	C₂×C₄	#496	1	2	yes	—
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	4	2	no	—
128	C₈	—	4	2	no	—
129	C₈	—	16	2	no	—
130	C₈	#443	1	2	yes	—
131	C₈	#444	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	2	no	—
134	C₂²	#1961	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	16	2	no	—
139	C₄	—	16	2	no	—
140	C₄	#1974	1	1	yes	characteristic
141	C₄	#1975	1	1	yes	characteristic
142	C₂	—	4	1	no	—
143	C₂	#9860	1	1	yes	characteristic
144	C₂	#13216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19668

C₂

yes

maximal, characteristic

#11679

C₂

yes

maximal

#11676

C₂

yes

maximal

#11406

C₂

yes

maximal, characteristic

#8796

C₂

yes

maximal, characteristic

#3231

—

#3170

C₂²

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1718

—

#1713

C₄

yes

characteristic

#1710

C₄

yes

characteristic

#1707

C₄

yes

—

#1568

C₂²

yes

—

#1450

C₂²

yes

—

—

—

—

—

C₂×M₃₂

—

C₂²×C₁₆

—

#500

—

#500

C₂×C₄

yes

—

#498

—

#494

C₂×C₄

yes

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#432

—

#431

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

D₄

yes

—

C₄×C₁₆

D₄

yes

—

#335

—

#334

—

#322

D₄

yes

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

yes

C₂²×C₈

yes

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#124

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

C₂×C₂²⋊C₄

yes

characteristic

108

C₄²

#127

yes

characteristic

109

C₄²

AGL(1,ℤ/8)

yes

—

110

C₁₆

—

111

C₁₆

—

112

C₂³

#490

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#390

yes

characteristic

122

C₂×C₄

#418

yes

—

123

C₂×C₄

#490

yes

characteristic

124

C₂×C₄

#496

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#443

yes

—

131

C₈

#444

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

#1961

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#1974

yes

characteristic

141

C₄

#1975

yes

characteristic

142

C₂

—

143

C₂

#9860

yes

characteristic

144

C₂

#13216

yes

—

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#490
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

#1961

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#490

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
#19668	2
#11679	2
#11676	2
#11406	2
#8796	2

subgroup

index

Subgroups — #141827

Conjugacy classes of subgroups

Maximal subgroups