Group #72225410 — subgroups

Subgroups — #72225410

Order 2048.

248 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 106 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	#72216226	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×#7052	C₄	1	1	yes	characteristic
7	#94748	C₄	1	4	yes	—
8	#94746	—	2	2	no	—
9	#93666	—	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	D₄	1	2	yes	—
21	C₈×C₃₂	—	2	1	no	—
22	C₈×C₃₂	C₂×C₄	1	2	yes	—
23	#7052	—	2	1	no	—
24	#7052	C₈	1	2	yes	—
25	C₂×M₆₄	—	2	1	no	—
26	C₂×M₆₄	—	2	1	no	—
27	C₂×M₆₄	C₁₆	1	2	yes	—
28	C₂×M₆₄	SD₁₆	1	2	yes	—
29	C₂×C₄×C₁₆	—	2	2	no	—
30	C₂×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈²	M₁₆	1	1	yes	characteristic
33	M₁₂₈	—	4	4	no	—
34	#1397	—	2	1	no	—
35	#1397	—	2	1	no	—
36	#1397	C₁₆	1	2	yes	—
37	#1397	D₈	1	1	yes	characteristic
38	#1397	Q₁₆	1	1	yes	characteristic
39	C₄×C₃₂	—	2	1	no	—
40	C₄×C₃₂	—	2	1	no	—
41	C₄×C₃₂	C₂²⋊C₄	1	2	yes	—
42	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
43	C₈×C₁₆	—	2	2	no	—
44	C₈×C₁₆	—	2	2	no	—
45	C₈×C₁₆	C₄²	1	1	yes	characteristic
46	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
47	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
48	C₈×C₁₆	M₁₆	1	1	yes	characteristic
49	C₂²×C₁₆	—	4	2	no	—
50	C₂²×C₁₆	#106	1	1	yes	characteristic
51	C₂²×C₁₆	#108	1	1	yes	characteristic
52	C₂²×C₁₆	C₂×C₁₆	1	1	yes	characteristic
53	C₂²×C₁₆	M₃₂	1	1	yes	characteristic
54	C₂×C₄×C₈	—	2	1	no	—
55	C₂×C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
56	M₆₄	—	2	4	no	—
57	M₆₄	—	2	4	no	—
58	M₆₄	—	4	2	no	—
59	M₆₄	—	4	2	no	—
60	C₂×C₃₂	—	2	2	no	—
61	C₂×C₃₂	—	2	2	no	—
62	C₂×C₃₂	—	2	2	no	—
63	C₂×C₃₂	#102	1	2	yes	—
64	C₂×C₃₂	#103	1	2	yes	—
65	C₄×C₁₆	—	2	4	no	—
66	C₄×C₁₆	—	4	2	no	—
67	C₄×C₁₆	#95	1	1	yes	characteristic
68	C₄×C₁₆	C₄×C₈	1	1	yes	characteristic
69	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
70	C₄×C₁₆	C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₁₆	#107	1	1	yes	characteristic
72	C₄×C₁₆	#108	1	1	yes	characteristic
73	C₄×C₁₆	C₂×C₁₆	1	1	yes	characteristic
74	C₄×C₁₆	M₃₂	1	1	yes	characteristic
75	C₈²	—	2	1	no	—
76	C₈²	#97	1	1	yes	characteristic
77	C₈²	C₂²⋊C₈	1	1	yes	characteristic
78	C₆₄	—	8	4	no	—
79	C₂²×C₈	—	2	2	no	—
80	C₂²×C₈	#334	1	1	yes	characteristic
81	C₂²×C₈	C₄⋊C₁₆	1	1	yes	characteristic
82	C₂×C₄²	#364	1	1	yes	characteristic
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	4	4	no	—
88	C₂×C₁₆	—	8	4	no	—
89	C₂×C₁₆	#334	1	1	yes	characteristic
90	C₂×C₁₆	#335	1	1	yes	characteristic
91	C₂×C₁₆	#340	1	1	yes	characteristic
92	C₂×C₁₆	#341	1	1	yes	characteristic
93	C₂×C₁₆	C₄×C₁₆	1	1	yes	characteristic
94	C₂×C₁₆	#346	1	1	yes	characteristic
95	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
96	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	#335	1	1	yes	characteristic
101	C₄×C₈	#336	1	1	yes	characteristic
102	C₄×C₈	C₄⋊C₁₆	1	1	yes	characteristic
103	C₄×C₈	#364	1	1	yes	characteristic
104	C₃₂	—	2	4	no	—
105	C₃₂	—	2	2	no	—
106	C₃₂	—	4	2	no	—
107	C₃₂	—	4	2	no	—
108	C₃₂	#359	1	4	yes	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	#1367	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	4	no	—
118	C₂×C₈	#1277	1	1	yes	characteristic
119	C₂×C₈	#1367	1	1	yes	characteristic
120	C₂×C₈	#1370	1	1	yes	characteristic
121	C₂×C₈	#1374	1	1	yes	characteristic
122	C₄²	—	2	1	no	—
123	C₄²	#1370	1	1	yes	characteristic
124	C₄²	#1376	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	1	no	—
127	C₁₆	—	2	1	no	—
128	C₁₆	—	2	1	no	—
129	C₁₆	—	8	4	no	—
130	C₁₆	#1373	1	2	yes	—
131	C₁₆	#1382	1	2	yes	—
132	C₁₆	#1398	1	2	yes	—
133	C₁₆	#1400	1	2	yes	—
134	C₂³	#7456	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	#7151	1	1	yes	characteristic
141	C₂×C₄	#7456	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	2	no	—
146	C₈	#7155	1	1	yes	characteristic
147	C₈	#7160	1	1	yes	characteristic
148	C₈	#7466	1	1	yes	characteristic
149	C₈	#7468	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	#94019	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	#92912	1	1	yes	characteristic
157	C₄	#94016	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	4	1	no	—
160	C₂	#72225409	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72225395

C₂

yes

maximal

#72216226

C₂

yes

maximal, characteristic

#153582

—

C₈×M₆₄

C₄

yes

characteristic

C₈×M₆₄

C₂²

yes

characteristic

C₂×#7052

C₄

yes

characteristic

#94748

C₄

yes

—

#94746

—

#93666

—

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

—

—

—

—

D₄

yes

—

C₈×C₃₂

—

C₈×C₃₂

C₂×C₄

yes

—

#7052

—

#7052

C₈

yes

—

C₂×M₆₄

—

C₂×M₆₄

—

C₂×M₆₄

C₁₆

yes

—

C₂×M₆₄

SD₁₆

yes

—

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

D₈

yes

characteristic

#1397

Q₁₆

yes

characteristic

C₄×C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

C₂²⋊C₄

yes

—

C₄×C₃₂

C₄⋊C₄

yes

—

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

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

#102

yes

—

C₂×C₃₂

#103

yes

—

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

#334

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

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

C₄×C₁₆

yes

characteristic

C₂×C₁₆

#346

yes

characteristic

C₂×C₁₆

C₂²⋊C₁₆

yes

characteristic

C₂×C₁₆

C₂²⋊C₁₆

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

#335

yes

characteristic

101

C₄×C₈

#336

yes

characteristic

102

C₄×C₈

C₄⋊C₁₆

yes

characteristic

103

C₄×C₈

#364

yes

characteristic

104

C₃₂

—

105

C₃₂

—

106

C₃₂

—

107

C₃₂

—

108

C₃₂

#359

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

#1367

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

#1277

yes

characteristic

119

C₂×C₈

#1367

yes

characteristic

120

C₂×C₈

#1370

yes

characteristic

121

C₂×C₈

#1374

yes

characteristic

122

C₄²

—

123

C₄²

#1370

yes

characteristic

124

C₄²

#1376

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

130

C₁₆

#1373

yes

—

131

C₁₆

#1382

yes

—

132

C₁₆

#1398

yes

—

133

C₁₆

#1400

yes

—

134

C₂³

#7456

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#7151

yes

characteristic

141

C₂×C₄

#7456

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#7155

yes

characteristic

147

C₈

#7160

yes

characteristic

148

C₈

#7466

yes

characteristic

149

C₈

#7468

yes

characteristic

150

C₂²

—

151

C₂²

—

152

C₂²

#94019

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#92912

yes

characteristic

157

C₄

#94016

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

#72225409

yes

characteristic

	subgroup	G/H
center	C₈	#7155
commutator	C₂×C₁₆	C₄×C₁₆
Frattini, cosocle	C₈×M₆₄	C₂²
socle	C₂	#72225409
prime radical	C₂³	#7456
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #1397 ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#7155

commutator

C₂×C₁₆

C₄×C₁₆

Frattini, cosocle

C₈×M₆₄

C₂²

C₂

C₂³

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ #1397 ◁ 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
#72216226	2

subgroup

index

#72225395

#72216226

Subgroups — #72225410

Conjugacy classes of subgroups

Maximal subgroups