Group #10447 — subgroups

Subgroups — #10447

Order 256.

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

normal subgroups only

This group has 204 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×#462	C₂	1	1	yes	maximal, characteristic
2	#1801	C₂	1	1	yes	maximal, characteristic
3	#1570	C₂	1	1	yes	maximal, characteristic
4	#1546	C₂	1	1	yes	maximal, characteristic
5	#1546	C₂	1	1	yes	maximal, characteristic
6	#1479	C₂	1	1	yes	maximal, characteristic
7	#1301	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	—	2	1	no	—
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
11	#462	—	2	2	no	—
12	#462	—	2	2	no	—
13	#431	C₂²	1	1	yes	characteristic
14	#431	C₂²	1	1	yes	characteristic
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	#334	—	2	2	no	—
22	#332	—	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×Q₁₆	—	2	4	no	—
28	C₂×Q₁₆	—	4	2	no	—
29	C₂×Q₁₆	—	4	2	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	D₄	1	2	yes	—
37	C₄×Q₈	—	2	1	no	—
38	C₄×Q₈	—	2	1	no	—
39	C₄×Q₈	—	2	1	no	—
40	C₄×Q₈	—	2	1	no	—
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄⋊C₄	—	2	1	no	—
44	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₄²	—	2	1	no	—
46	C₂×C₄²	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	D₄	1	2	yes	—
54	C₄⋊C₈	D₄	1	2	yes	—
55	#103	—	2	4	no	—
56	#101	—	4	2	no	—
57	C₂²⋊C₈	—	2	1	no	—
58	C₂²⋊C₈	—	2	1	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	2	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×Q₈	—	4	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	—	4	1	no	—
71	C₂×Q₈	—	4	1	no	—
72	C₂×Q₈	SD₁₆	1	2	yes	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	4	1	no	—
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	#41	1	1	yes	characteristic
78	C₂²×C₄	#41	1	1	yes	characteristic
79	Q₁₆	—	4	8	no	—
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	1	no	—
82	M₁₆	—	4	1	no	—
83	M₁₆	—	4	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄²	—	4	1	no	—
100	C₄²	—	4	1	no	—
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂×D₄	1	1	yes	characteristic
103	C₄²	C₂×D₄	1	1	yes	characteristic
104	C₄²	#41	1	1	yes	characteristic
105	C₂³	#123	1	1	yes	characteristic
106	Q₈	—	2	2	no	—
107	Q₈	—	2	2	no	—
108	Q₈	—	4	1	no	—
109	Q₈	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	8	1	no	—
113	Q₈	—	8	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	#120	1	1	yes	characteristic
125	C₂×C₄	#121	1	1	yes	characteristic
126	C₂×C₄	#121	1	1	yes	characteristic
127	C₂×C₄	#121	1	1	yes	characteristic
128	C₂×C₄	#123	1	1	yes	characteristic
129	C₂×C₄	#124	1	1	yes	characteristic
130	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄	#135	1	1	yes	characteristic
132	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	4	2	no	—
136	C₈	—	8	1	no	—
137	C₈	—	8	1	no	—
138	C₈	—	8	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#394	1	1	yes	characteristic
142	C₂²	#481	1	1	yes	characteristic
143	C₂²	#482	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	1	no	—
149	C₄	—	8	2	no	—
150	C₄	#449	1	1	yes	characteristic
151	C₄	#450	1	1	yes	characteristic
152	C₄	#465	1	1	yes	characteristic
153	C₄	#469	1	1	yes	characteristic
154	C₄	#473	1	2	yes	—
155	C₂	—	2	1	no	—
156	C₂	—	2	1	no	—
157	C₂	#2034	1	1	yes	characteristic
158	C₂	#2047	1	1	yes	characteristic
159	C₂	#2049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#462

C₂

yes

maximal, characteristic

#1801

C₂

yes

maximal, characteristic

#1570

C₂

yes

maximal, characteristic

#1546

C₂

yes

maximal, characteristic

#1546

C₂

yes

maximal, characteristic

#1479

C₂

yes

maximal, characteristic

#1301

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

—

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

—

—

—

—

—

Q₈⋊C₈

—

C₂²×Q₈

—

C₂²×Q₈

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

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

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

SD₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

Q₁₆

—

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

—

100

C₄²

—

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

C₂×D₄

yes

characteristic

103

C₄²

C₂×D₄

yes

characteristic

104

C₄²

#41

yes

characteristic

105

C₂³

#123

yes

characteristic

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#120

yes

characteristic

125

C₂×C₄

#121

yes

characteristic

126

C₂×C₄

#121

yes

characteristic

127

C₂×C₄

#121

yes

characteristic

128

C₂×C₄

#123

yes

characteristic

129

C₂×C₄

#124

yes

characteristic

130

C₂×C₄

C₂×SD₁₆

yes

characteristic

131

C₂×C₄

#135

yes

characteristic

132

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

133

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

#394

yes

characteristic

142

C₂²

#481

yes

characteristic

143

C₂²

#482

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#449

yes

characteristic

151

C₄

#450

yes

characteristic

152

C₄

#465

yes

characteristic

153

C₄

#469

yes

characteristic

154

C₄

#473

yes

—

155

C₂

—

156

C₂

—

157

C₂

#2034

yes

characteristic

158

C₂

#2047

yes

characteristic

159

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×#462	2
#1801	2
#1570	2
#1546	2
#1479	2
#1301	2

subgroup

index

Subgroups — #10447

Conjugacy classes of subgroups

Maximal subgroups