Group #72227191 — subgroups

Subgroups — #72227191

Order 2048.

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

normal subgroups only

This group has 260 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	#72225395	C₂	1	2	yes	maximal
2	#72216220	C₂	1	1	yes	maximal, characteristic
3	#153582	—	2	2	no	—
4	C₈×M₆₄	C₄	1	1	yes	characteristic
5	C₈×M₆₄	C₂²	1	1	yes	characteristic
6	C₂×C₁₆²	C₄	1	1	yes	characteristic
7	#94748	—	2	2	no	—
8	#94746	C₄	1	4	yes	—
9	#93665	—	2	2	no	—
10	C₄×M₆₄	C₈	1	1	yes	characteristic
11	C₄×M₆₄	C₈	1	1	yes	characteristic
12	C₄×M₆₄	D₄	1	1	yes	characteristic
13	C₄×M₆₄	Q₈	1	1	yes	characteristic
14	C₂×C₈×C₁₆	—	2	1	no	—
15	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
16	#7544	—	2	4	no	—
17	#7513	—	4	2	no	—
18	#7511	—	4	2	no	—
19	#7329	—	2	1	no	—
20	#7329	C₂×C₄	1	2	yes	—
21	C₈×C₃₂	—	2	1	no	—
22	C₈×C₃₂	D₄	1	2	yes	—
23	C₁₆²	—	2	1	no	—
24	C₁₆²	C₈	1	1	yes	characteristic
25	C₁₆²	C₈	1	1	yes	characteristic
26	C₂×M₆₄	—	2	1	no	—
27	C₂×M₆₄	—	2	1	no	—
28	C₂×M₆₄	C₁₆	1	2	yes	—
29	C₂×M₆₄	D₈	1	1	yes	characteristic
30	C₂×M₆₄	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₆	—	2	2	no	—
32	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈²	M₁₆	1	1	yes	characteristic
35	M₁₂₈	—	4	4	no	—
36	#1397	—	2	1	no	—
37	#1397	—	2	1	no	—
38	#1397	C₁₆	1	2	yes	—
39	#1397	SD₁₆	1	2	yes	—
40	C₄×C₃₂	—	2	1	no	—
41	C₄×C₃₂	—	2	1	no	—
42	C₄×C₃₂	C₂²⋊C₄	1	2	yes	—
43	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
44	C₈×C₁₆	—	2	2	no	—
45	C₈×C₁₆	—	2	1	no	—
46	C₈×C₁₆	—	2	1	no	—
47	C₈×C₁₆	C₄²	1	1	yes	characteristic
48	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
49	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
50	C₈×C₁₆	M₁₆	1	1	yes	characteristic
51	C₂²×C₁₆	—	2	4	no	—
52	C₂²×C₁₆	#107	1	1	yes	characteristic
53	C₂²×C₁₆	#108	1	1	yes	characteristic
54	C₂²×C₁₆	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×C₁₆	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₈	—	2	1	no	—
57	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
58	M₆₄	—	4	1	no	—
59	M₆₄	—	4	1	no	—
60	M₆₄	—	4	2	no	—
61	M₆₄	—	4	2	no	—
62	M₆₄	—	4	2	no	—
63	C₂×C₃₂	—	2	2	no	—
64	C₂×C₃₂	—	2	2	no	—
65	C₂×C₃₂	—	2	2	no	—
66	C₂×C₃₂	#102	1	2	yes	—
67	C₂×C₃₂	#103	1	2	yes	—
68	C₄×C₁₆	—	2	2	no	—
69	C₄×C₁₆	—	2	2	no	—
70	C₄×C₁₆	—	2	4	no	—
71	C₄×C₁₆	#95	1	1	yes	characteristic
72	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
73	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₁₆	#106	1	1	yes	characteristic
76	C₄×C₁₆	#108	1	1	yes	characteristic
77	C₄×C₁₆	C₂×C₁₆	1	1	yes	characteristic
78	C₄×C₁₆	M₃₂	1	1	yes	characteristic
79	C₈²	—	2	1	no	—
80	C₈²	#97	1	1	yes	characteristic
81	C₈²	C₂²⋊C₈	1	1	yes	characteristic
82	C₆₄	—	8	4	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	#335	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₁₆	1	1	yes	characteristic
86	C₂×C₄²	#364	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	4	8	no	—
95	C₂×C₁₆	#334	1	1	yes	characteristic
96	C₂×C₁₆	#335	1	1	yes	characteristic
97	C₂×C₁₆	#340	1	1	yes	characteristic
98	C₂×C₁₆	#341	1	1	yes	characteristic
99	C₂×C₁₆	C₄×C₁₆	1	1	yes	characteristic
100	C₂×C₁₆	#346	1	1	yes	characteristic
101	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	#334	1	1	yes	characteristic
107	C₄×C₈	#336	1	1	yes	characteristic
108	C₄×C₈	C₄⋊C₁₆	1	1	yes	characteristic
109	C₄×C₈	#364	1	1	yes	characteristic
110	C₃₂	—	2	2	no	—
111	C₃₂	—	2	2	no	—
112	C₃₂	—	4	2	no	—
113	C₃₂	—	4	2	no	—
114	C₃₂	—	4	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	#1370	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	4	no	—
124	C₂×C₈	#1277	1	1	yes	characteristic
125	C₂×C₈	#1367	1	1	yes	characteristic
126	C₂×C₈	#1370	1	1	yes	characteristic
127	C₂×C₈	#1374	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	#1367	1	1	yes	characteristic
130	C₄²	#1376	1	1	yes	characteristic
131	C₁₆	—	2	8	no	—
132	C₁₆	—	2	1	no	—
133	C₁₆	—	2	1	no	—
134	C₁₆	—	2	1	no	—
135	C₁₆	—	2	1	no	—
136	C₁₆	—	4	4	no	—
137	C₁₆	#1369	1	2	yes	—
138	C₁₆	#1384	1	1	yes	characteristic
139	C₁₆	#1385	1	1	yes	characteristic
140	C₁₆	#1398	1	2	yes	—
141	C₁₆	#1400	1	2	yes	—
142	C₂³	#7459	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	#7151	1	1	yes	characteristic
149	C₂×C₄	#7459	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	4	2	no	—
154	C₈	#7157	1	1	yes	characteristic
155	C₈	#7161	1	1	yes	characteristic
156	C₈	#7466	1	1	yes	characteristic
157	C₈	#7468	1	1	yes	characteristic
158	C₂²	—	2	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	#94018	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	1	no	—
163	C₄	—	4	1	no	—
164	C₄	#92914	1	1	yes	characteristic
165	C₄	#94016	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	4	1	no	—
168	C₂	#72227190	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72225395

C₂

yes

maximal

#72216220

C₂

yes

maximal, characteristic

#153582

—

C₈×M₆₄

C₄

yes

characteristic

C₈×M₆₄

C₂²

yes

characteristic

C₂×C₁₆²

C₄

yes

characteristic

#94748

—

#94746

C₄

yes

—

#93665

—

C₄×M₆₄

C₈

yes

characteristic

C₄×M₆₄

C₈

yes

characteristic

C₄×M₆₄

D₄

yes

characteristic

C₄×M₆₄

Q₈

yes

characteristic

C₂×C₈×C₁₆

—

C₂×C₈×C₁₆

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₈×C₃₂

—

C₈×C₃₂

D₄

yes

—

C₁₆²

—

C₁₆²

C₈

yes

characteristic

C₁₆²

C₈

yes

characteristic

C₂×M₆₄

—

C₂×M₆₄

—

C₂×M₆₄

C₁₆

yes

—

C₂×M₆₄

D₈

yes

characteristic

C₂×M₆₄

Q₁₆

yes

characteristic

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆

C₂×C₈

yes

characteristic

C₂×C₈²

M₁₆

yes

characteristic

M₁₂₈

—

#1397

—

#1397

—

#1397

C₁₆

yes

—

#1397

SD₁₆

yes

—

C₄×C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

C₂²⋊C₄

yes

—

C₄×C₃₂

C₄⋊C₄

yes

—

C₈×C₁₆

—

C₈×C₁₆

—

C₈×C₁₆

—

C₈×C₁₆

C₄²

yes

characteristic

C₈×C₁₆

C₂²⋊C₄

yes

characteristic

C₈×C₁₆

C₂×C₈

yes

characteristic

C₈×C₁₆

M₁₆

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

yes

C₂²×C₁₆

yes

C₂²×C₁₆

C₂×C₁₆

yes

characteristic

C₂²×C₁₆

M₃₂

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₄⋊C₈

yes

characteristic

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

#102

yes

—

C₂×C₃₂

#103

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

#95

yes

characteristic

C₄×C₁₆

C₄×C₈

yes

characteristic

C₄×C₁₆

C₂²⋊C₈

yes

characteristic

C₄×C₁₆

C₄⋊C₈

yes

C₄×C₁₆

yes

C₄×C₁₆

yes

C₄×C₁₆

C₂×C₁₆

yes

characteristic

C₄×C₁₆

M₃₂

yes

characteristic

C₈²

—

C₈²

#97

yes

characteristic

C₈²

C₂²⋊C₈

yes

characteristic

C₆₄

—

C₂²×C₈

—

C₂²×C₈

#335

yes

characteristic

C₂²×C₈

C₄⋊C₁₆

yes

characteristic

C₂×C₄²

#364

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

C₄×C₁₆

yes

characteristic

100

C₂×C₁₆

#346

yes

characteristic

101

C₂×C₁₆

C₂²⋊C₁₆

yes

characteristic

102

C₂×C₁₆

C₂²⋊C₁₆

yes

characteristic

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

#334

yes

characteristic

107

C₄×C₈

#336

yes

characteristic

108

C₄×C₈

C₄⋊C₁₆

yes

characteristic

109

C₄×C₈

#364

yes

characteristic

110

C₃₂

—

111

C₃₂

—

112

C₃₂

—

113

C₃₂

—

114

C₃₂

—

115

C₂²×C₄

—

116

C₂²×C₄

#1370

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#1277

yes

characteristic

125

C₂×C₈

#1367

yes

characteristic

126

C₂×C₈

#1370

yes

characteristic

127

C₂×C₈

#1374

yes

characteristic

128

C₄²

—

129

C₄²

#1367

yes

characteristic

130

C₄²

#1376

yes

characteristic

131

C₁₆

—

132

C₁₆

—

133

C₁₆

—

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

137

C₁₆

#1369

yes

—

138

C₁₆

#1384

yes

characteristic

139

C₁₆

#1385

yes

characteristic

140

C₁₆

#1398

yes

—

141

C₁₆

#1400

yes

—

142

C₂³

#7459

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#7151

yes

characteristic

149

C₂×C₄

#7459

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

#7157

yes

characteristic

155

C₈

#7161

yes

characteristic

156

C₈

#7466

yes

characteristic

157

C₈

#7468

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

#94018

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

#92914

yes

characteristic

165

C₄

#94016

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#72227190

yes

characteristic

	subgroup	G/H
center	C₁₆	#1384
commutator	C₂×C₁₆	C₄×C₁₆
Frattini, cosocle	C₈×M₆₄	C₂²
socle	C₂	#72227190
prime radical	C₂³	#7459
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₄×C₁₆ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₁₆

#1384

commutator

C₂×C₁₆

C₄×C₁₆

Frattini, cosocle

C₈×M₆₄

C₂²

C₂

C₂³

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₆ ◁ C₄×C₁₆ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G

LECS

G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#72225395	2
#72216220	2

subgroup

index

#72225395

#72216220

Subgroups — #72227191

Conjugacy classes of subgroups

Maximal subgroups