Group #10313 — subgroups

Subgroups — #10313

Order 256.

277 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 85 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×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×#487	C₂	1	1	yes	maximal, characteristic
3	C₂×#487	C₂	1	1	yes	maximal, characteristic
4	#1990	C₂	1	1	yes	maximal, characteristic
5	#1990	C₂	1	1	yes	maximal, characteristic
6	#1956	C₂	1	1	yes	maximal, characteristic
7	#1273	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	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	#500	—	2	2	no	—
12	#500	—	2	2	no	—
13	#487	—	2	2	no	—
14	#487	—	2	2	no	—
15	#487	—	2	2	no	—
16	#487	—	2	2	no	—
17	C₂×#107	—	2	2	no	—
18	C₂×#103	—	2	2	no	—
19	C₂×#103	—	2	2	no	—
20	C₂×#103	—	2	2	no	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#384	C₂²	1	1	yes	characteristic
25	#340	—	2	2	no	—
26	#340	—	2	2	no	—
27	C₂²×Q₈	—	2	1	no	—
28	C₂²×Q₈	—	2	1	no	—
29	C₂²×Q₈	—	2	1	no	—
30	C₂²×C₈	D₄	1	2	yes	—
31	C₂²×C₈	D₄	1	2	yes	—
32	C₂²×C₈	D₄	1	2	yes	—
33	C₄⋊Q₈	—	2	1	no	—
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	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	—	2	1	no	—
40	C₄⋊Q₈	—	2	1	no	—
41	C₄⋊Q₈	—	2	1	no	—
42	C₄⋊Q₈	D₄	1	2	yes	—
43	C₄⋊Q₈	D₄	1	2	yes	—
44	C₄⋊Q₈	D₄	1	2	yes	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄⋊C₄	—	2	1	no	—
48	C₂×C₄⋊C₄	—	2	1	no	—
49	C₂×C₄⋊C₄	—	2	1	no	—
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄⋊C₄	—	2	1	no	—
52	C₂×C₄⋊C₄	—	2	1	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#107	—	2	8	no	—
56	#103	—	2	8	no	—
57	#103	—	2	8	no	—
58	#103	—	4	4	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	—	4	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	—	4	1	no	—
71	C₂×Q₈	—	4	2	no	—
72	C₂×Q₈	—	4	2	no	—
73	C₂×Q₈	—	4	2	no	—
74	C₂×Q₈	—	4	2	no	—
75	C₂×Q₈	—	4	2	no	—
76	C₂×Q₈	—	4	2	no	—
77	C₂²×C₄	—	4	1	no	—
78	C₂²×C₄	—	4	1	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	—	4	1	no	—
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	Q₁₆	1	4	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂×D₄	1	1	yes	characteristic
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	#41	1	1	yes	characteristic
112	C₂³	#127	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	Q₈	—	4	4	no	—
115	Q₈	—	4	4	no	—
116	Q₈	—	8	2	no	—
117	Q₈	—	8	2	no	—
118	Q₈	—	8	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	8	1	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	#120	1	1	yes	characteristic
132	C₂×C₄	#120	1	1	yes	characteristic
133	C₂×C₄	#120	1	1	yes	characteristic
134	C₂×C₄	#121	1	1	yes	characteristic
135	C₂×C₄	#121	1	1	yes	characteristic
136	C₂×C₄	#121	1	1	yes	characteristic
137	C₂×C₄	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄	#135	1	2	yes	—
139	C₂×C₄	#135	1	2	yes	—
140	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
141	C₂×C₄	#137	1	2	yes	—
142	C₂×C₄	#137	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	4	2	no	—
145	C₈	—	4	2	no	—
146	C₂²	#392	1	1	yes	characteristic
147	C₂²	#494	1	1	yes	characteristic
148	C₂²	#495	1	1	yes	characteristic
149	C₂²	#495	1	1	yes	characteristic
150	C₂²	#496	1	1	yes	characteristic
151	C₂²	#497	1	1	yes	characteristic
152	C₂²	#497	1	1	yes	characteristic
153	C₄	—	8	2	no	—
154	C₄	—	8	2	no	—
155	C₄	—	8	2	no	—
156	C₄	—	16	1	no	—
157	C₄	#449	1	2	yes	—
158	C₄	#451	1	2	yes	—
159	C₄	#452	1	2	yes	—
160	C₄	#462	1	2	yes	—
161	C₄	#463	1	2	yes	—
162	C₄	#464	1	2	yes	—
163	C₂	#1710	1	1	yes	characteristic
164	C₂	#1718	1	1	yes	characteristic
165	C₂	#1719	1	1	yes	characteristic
166	C₂	#1723	1	1	yes	characteristic
167	C₂	#2012	1	1	yes	characteristic
168	C₂	#2015	1	1	yes	characteristic
169	C₂	#2016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

C₂×#487

C₂

yes

maximal, characteristic

C₂×#487

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal, characteristic

#1956

C₂

yes

maximal, characteristic

#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#340

—

#340

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

C₄⋊Q₈

D₄

yes

—

C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×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

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂×D₄

yes

characteristic

109

C₄²

C₂×D₄

yes

characteristic

110

C₄²

C₂×D₄

yes

characteristic

111

C₄²

#41

yes

characteristic

112

C₂³

#127

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#120

yes

characteristic

132

C₂×C₄

#120

yes

characteristic

133

C₂×C₄

#120

yes

characteristic

134

C₂×C₄

#121

yes

characteristic

135

C₂×C₄

#121

yes

characteristic

136

C₂×C₄

#121

yes

characteristic

137

C₂×C₄

C₂×Q₁₆

yes

—

138

C₂×C₄

#135

yes

—

139

C₂×C₄

#135

yes

—

140

C₂×C₄

AGL(1,ℤ/8)

yes

—

141

C₂×C₄

#137

yes

—

142

C₂×C₄

#137

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

#392

yes

characteristic

147

C₂²

#494

yes

characteristic

148

C₂²

#495

yes

characteristic

149

C₂²

#495

yes

characteristic

150

C₂²

#496

yes

characteristic

151

C₂²

#497

yes

characteristic

152

C₂²

#497

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#449

yes

—

158

C₄

#451

yes

—

159

C₄

#452

yes

—

160

C₄

#462

yes

—

161

C₄

#463

yes

—

162

C₄

#464

yes

—

163

C₂

#1710

yes

characteristic

164

C₂

#1718

yes

characteristic

165

C₂

#1719

yes

characteristic

166

C₂

#1723

yes

characteristic

167

C₂

#2012

yes

characteristic

168

C₂

#2015

yes

characteristic

169

C₂

#2016

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ 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×#500	2
C₂×#487	2
#1990	2
#1956	2
#1273	2

subgroup

index

Subgroups — #10313

Conjugacy classes of subgroups

Maximal subgroups