Group #10451 — subgroups

Subgroups — #10451

Order 256.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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×#475	C₂	1	1	yes	maximal, characteristic
2	#1801	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic
4	#1549	C₂	1	1	yes	maximal, characteristic
5	#1548	C₂	1	1	yes	maximal, characteristic
6	#1479	C₂	1	1	yes	maximal, characteristic
7	#1301	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
10	#475	—	2	2	no	—
11	#475	—	2	2	no	—
12	#431	C₂²	1	1	yes	characteristic
13	#431	C₂²	1	1	yes	characteristic
14	C₂×#106	—	2	1	no	—
15	#423	C₂²	1	1	yes	characteristic
16	#423	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#103	—	2	1	no	—
19	C₂×#101	—	2	1	no	—
20	#343	—	2	1	no	—
21	#335	—	2	2	no	—
22	#333	—	2	2	no	—
23	#332	—	2	2	no	—
24	Q₈⋊C₈	—	2	2	no	—
25	C₂²×Q₈	—	2	1	no	—
26	C₂²×Q₈	D₄	1	1	yes	characteristic
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂²×C₈	—	2	1	no	—
30	C₄⋊Q₈	—	2	1	no	—
31	C₄⋊Q₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	D₄	1	2	yes	—
34	C₄×Q₈	—	2	1	no	—
35	C₄×Q₈	—	2	1	no	—
36	C₄×Q₈	—	2	1	no	—
37	C₄×Q₈	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄⋊C₄	—	2	1	no	—
40	C₂×C₄⋊C₄	—	2	1	no	—
41	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₂×C₄²	—	2	1	no	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	#106	—	2	4	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	—	2	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	D₄	1	2	yes	—
52	C₄⋊C₈	D₄	1	2	yes	—
53	#103	—	2	4	no	—
54	#101	—	4	2	no	—
55	C₂²⋊C₈	—	2	1	no	—
56	C₂²⋊C₈	—	2	1	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	—	2	1	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	2	no	—
63	C₂×Q₈	—	2	1	no	—
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	4	1	no	—
66	C₂×Q₈	—	8	2	no	—
67	C₂×Q₈	Q₁₆	1	2	yes	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	#41	1	1	yes	characteristic
73	C₂²×C₄	#41	1	1	yes	characteristic
74	M₁₆	—	4	1	no	—
75	M₁₆	—	4	1	no	—
76	M₁₆	—	4	1	no	—
77	M₁₆	—	4	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄⋊C₄	—	4	1	no	—
92	C₄⋊C₄	—	4	1	no	—
93	C₄²	—	4	1	no	—
94	C₄²	—	4	1	no	—
95	C₄²	C₂×D₄	1	1	yes	characteristic
96	C₄²	C₂×D₄	1	1	yes	characteristic
97	C₄²	C₂×D₄	1	1	yes	characteristic
98	C₄²	#41	1	1	yes	characteristic
99	C₂³	#123	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	8	2	no	—
105	Q₈	—	8	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	8	2	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	#120	1	1	yes	characteristic
116	C₂×C₄	#121	1	1	yes	characteristic
117	C₂×C₄	#121	1	1	yes	characteristic
118	C₂×C₄	#121	1	1	yes	characteristic
119	C₂×C₄	#123	1	1	yes	characteristic
120	C₂×C₄	#124	1	1	yes	characteristic
121	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄	#135	1	1	yes	characteristic
123	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄	#137	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₈	—	4	2	no	—
127	C₈	—	8	1	no	—
128	C₈	—	8	1	no	—
129	C₈	—	8	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#394	1	1	yes	characteristic
133	C₂²	#484	1	1	yes	characteristic
134	C₂²	#485	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	4	2	no	—
139	C₄	—	8	1	no	—
140	C₄	—	8	2	no	—
141	C₄	#449	1	1	yes	characteristic
142	C₄	#451	1	1	yes	characteristic
143	C₄	#467	1	1	yes	characteristic
144	C₄	#468	1	1	yes	characteristic
145	C₄	#471	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#2036	1	1	yes	characteristic
149	C₂	#2046	1	1	yes	characteristic
150	C₂	#2049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#475

C₂

yes

maximal, characteristic

#1801

C₂

yes

maximal, characteristic

#1571

C₂

yes

maximal, characteristic

#1549

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal, characteristic

#1479

C₂

yes

maximal, characteristic

#1301

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

—

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

C₂×#106

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

—

—

—

—

—

Q₈⋊C₈

—

C₂²×Q₈

—

C₂²×Q₈

D₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

C₄⋊C₈

D₄

yes

—

#103

—

#101

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₂³

yes

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#120

yes

characteristic

116

C₂×C₄

#121

yes

characteristic

117

C₂×C₄

#121

yes

characteristic

118

C₂×C₄

#121

yes

characteristic

119

C₂×C₄

#123

yes

characteristic

120

C₂×C₄

#124

yes

characteristic

121

C₂×C₄

C₂×Q₁₆

yes

characteristic

122

C₂×C₄

#135

yes

characteristic

123

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

124

C₂×C₄

#137

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

#394

yes

characteristic

133

C₂²

#484

yes

characteristic

134

C₂²

#485

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#449

yes

characteristic

142

C₄

#451

yes

characteristic

143

C₄

#467

yes

characteristic

144

C₄

#468

yes

characteristic

145

C₄

#471

yes

—

146

C₂

—

147

C₂

—

148

C₂

#2036

yes

characteristic

149

C₂

#2046

yes

characteristic

150

C₂

#2049

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

#123

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
C₂×#475	2
#1801	2
#1571	2
#1549	2
#1548	2
#1479	2
#1301	2

subgroup

index

Subgroups — #10451

Conjugacy classes of subgroups

Maximal subgroups