Group #72230720 — subgroups

Subgroups — #72230720

Order 4096.

295 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 95 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	#72230718	C₂	1	2	yes	maximal, complemented
2	#72230715	C₂	1	2	yes	maximal, complemented
3	#72230713	—	2	2	no	—
4	#72230710	—	2	2	no	—
5	#72230706	C₂	1	1	yes	maximal, characteristic
6	#72230705	C₄	1	1	yes	characteristic
7	#72230702	C₄	1	2	yes	—
8	#72230701	C₄	1	1	yes	characteristic
9	#72230700	C₂²	1	2	yes	—
10	#72230696	C₂	1	1	yes	maximal, characteristic, complemented
11	#72230695	C₂²	1	2	yes	—
12	#72230692	C₂	1	1	yes	maximal, characteristic, complemented
13	#72230689	C₂²	1	1	yes	characteristic
14	#72230688	C₂²	1	1	yes	characteristic
15	#72225140	C₂²	1	1	yes	characteristic
16	#153594	C₈	1	1	yes	characteristic
17	#151101	—	4	2	no	—
18	#151080	C₂×C₄	1	2	yes	—
19	#122946	—	2	2	no	—
20	#94752	C₈	1	1	yes	characteristic
21	#94609	C₈	1	2	yes	—
22	#94508	—	2	1	no	—
23	#94505	C₂×C₄	1	1	yes	characteristic
24	#94505	C₂×C₄	1	1	yes	characteristic
25	#93884	C₂×C₄	1	2	yes	—
26	#93876	—	2	2	no	—
27	#93755	—	4	2	no	—
28	#93749	—	2	2	no	—
29	#93667	D₄	1	2	yes	—
30	#93666	C₂³	1	1	yes	characteristic
31	C₂×D₆₄	C₁₆	1	1	yes	characteristic
32	C₄×D₃₂	C₂×C₈	1	2	yes	—
33	D₄×C₃₂	—	8	2	no	—
34	C₈×D₁₆	—	2	2	no	—
35	C₁₆×D₈	—	4	2	no	—
36	#7546	C₁₆	1	1	yes	characteristic
37	#7537	C₁₆	1	2	yes	—
38	#7510	—	2	1	no	—
39	#7510	—	4	1	no	—
40	#7510	C₂×C₈	1	1	yes	characteristic
41	#7510	C₂×C₈	1	1	yes	characteristic
42	#7461	C₂×C₈	1	2	yes	—
43	#7455	—	4	2	no	—
44	#7383	—	2	2	no	—
45	D₄⋊C₃₂	—	8	2	no	—
46	#7331	—	2	2	no	—
47	#7329	C₂²×C₄	1	1	yes	characteristic
48	#7329	#41	1	1	yes	characteristic
49	#7101	—	2	2	no	—
50	#7068	—	4	2	no	—
51	#7052	C₂×D₄	1	1	yes	characteristic
52	C₂×D₃₂	C₂×C₁₆	1	2	yes	—
53	C₂²×C₃₂	—	16	2	no	—
54	C₄×D₁₆	—	2	2	no	—
55	D₄×C₁₆	—	8	2	no	—
56	C₈×D₈	—	4	2	no	—
57	D₆₄	C₃₂	1	4	yes	complemented
58	C₂×C₆₄	—	2	1	no	—
59	C₂×C₆₄	—	4	1	no	—
60	C₂×C₆₄	—	8	1	no	—
61	C₂×C₆₄	C₂×C₁₆	1	1	yes	characteristic
62	C₂×C₆₄	C₂×C₁₆	1	1	yes	characteristic
63	#1423	C₂×C₁₆	1	2	yes	—
64	C₄⋊C₃₂	—	8	2	no	—
65	#1415	—	2	2	no	—
66	C₂²⋊C₃₂	—	16	2	no	—
67	C₄×C₃₂	—	2	1	no	—
68	C₄×C₃₂	—	4	2	no	—
69	C₄×C₃₂	C₂²×C₈	1	1	yes	characteristic
70	C₄×C₃₂	#131	1	1	yes	characteristic
71	#1371	—	2	2	no	—
72	#1370	—	4	2	no	—
73	#1331	—	4	2	no	—
74	D₄⋊C₁₆	—	8	2	no	—
75	C₈×C₁₆	C₄×D₄	1	1	yes	characteristic
76	C₈×C₁₆	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×D₁₆	—	2	2	no	—
78	C₂²×C₁₆	—	16	2	no	—
79	C₄×D₈	—	4	2	no	—
80	C₈×D₄	—	8	2	no	—
81	D₃₂	—	2	2	no	—
82	D₃₂	C₂×C₃₂	1	4	yes	complemented
83	C₂×C₃₂	—	2	1	no	—
84	C₂×C₃₂	—	4	1	no	—
85	C₂×C₃₂	—	8	2	no	—
86	C₂×C₃₂	—	16	2	no	—
87	C₂×C₃₂	—	32	4	no	—
88	C₂×C₃₂	C₂²×C₁₆	1	1	yes	characteristic
89	C₂×C₃₂	#504	1	1	yes	characteristic
90	#366	—	2	2	no	—
91	C₄⋊C₁₆	—	8	2	no	—
92	#357	—	4	2	no	—
93	C₂²⋊C₁₆	—	16	2	no	—
94	C₄×C₁₆	—	2	2	no	—
95	C₄×C₁₆	C₈×D₄	1	1	yes	characteristic
96	C₄×C₁₆	#444	1	1	yes	characteristic
97	#335	—	4	2	no	—
98	D₄⋊C₈	—	8	2	no	—
99	C₈²	#442	1	1	yes	characteristic
100	C₆₄	—	2	1	no	—
101	C₆₄	—	4	1	no	—
102	C₆₄	—	8	1	no	—
103	C₆₄	—	16	1	no	—
104	C₆₄	C₂×C₃₂	1	2	yes	complemented
105	C₂×D₈	—	4	2	no	—
106	C₂²×C₈	—	16	2	no	—
107	C₄×D₄	—	8	2	no	—
108	D₁₆	—	2	4	no	—
109	D₁₆	—	4	2	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	4	2	no	—
112	C₂×C₁₆	—	16	2	no	—
113	C₂×C₁₆	—	32	2	no	—
114	C₂×C₁₆	—	32	2	no	—
115	C₂×C₁₆	D₄×C₁₆	1	1	yes	characteristic
116	C₂×C₁₆	#2171	1	1	yes	characteristic
117	#107	—	4	2	no	—
118	C₄⋊C₈	—	8	2	no	—
119	#102	—	8	2	no	—
120	C₂²⋊C₈	—	16	2	no	—
121	C₄×C₈	#1580	1	1	yes	characteristic
122	C₄×C₈	AGL(1,ℤ/16)	1	2	yes	—
123	C₃₂	—	2	1	no	—
124	C₃₂	—	4	1	no	—
125	C₃₂	—	8	1	no	—
126	C₃₂	—	16	2	no	—
127	C₃₂	—	32	2	no	—
128	C₃₂	C₂²×C₃₂	1	1	yes	characteristic
129	C₃₂	#2258	1	1	yes	characteristic
130	C₂×D₄	—	8	2	no	—
131	C₂²×C₄	—	16	2	no	—
132	D₈	—	4	4	no	—
133	D₈	—	8	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	16	2	no	—
136	C₂×C₈	—	32	2	no	—
137	C₂×C₈	—	32	2	no	—
138	C₂×C₈	#12062	1	1	yes	characteristic
139	C₂×C₈	#12068	1	1	yes	characteristic
140	C₄⋊C₄	—	8	2	no	—
141	C₂²⋊C₄	—	16	2	no	—
142	C₄²	#12067	1	1	yes	characteristic
143	C₁₆	—	2	1	no	—
144	C₁₆	—	4	1	no	—
145	C₁₆	—	8	2	no	—
146	C₁₆	—	32	2	no	—
147	C₁₆	D₄×C₃₂	1	1	yes	characteristic
148	C₁₆	#13648	1	1	yes	characteristic
149	C₂³	—	16	2	no	—
150	D₄	—	8	4	no	—
151	D₄	—	16	2	no	—
152	C₂×C₄	—	16	2	no	—
153	C₂×C₄	—	32	2	no	—
154	C₂×C₄	—	32	2	no	—
155	C₂×C₄	#122963	1	1	yes	characteristic
156	C₂×C₄	AGL(1,ℤ/32)	1	2	yes	—
157	C₈	—	2	1	no	—
158	C₈	—	4	2	no	—
159	C₈	—	32	2	no	—
160	C₈	#151101	1	1	yes	characteristic
161	C₈	#151103	1	1	yes	characteristic
162	C₂²	—	16	4	no	—
163	C₂²	—	32	2	no	—
164	C₂²	—	32	2	no	—
165	C₂²	#72224639	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	32	2	no	—
168	C₄	#72224647	1	1	yes	characteristic
169	C₄	#72230710	1	1	yes	characteristic
170	C₂	—	32	4	no	—
171	C₂	#72224662	1	2	yes	—
172	C₂	#72230715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72230718

C₂

yes

maximal, complemented

#72230715

C₂

yes

maximal, complemented

#72230713

—

#72230710

—

#72230706

C₂

yes

maximal, characteristic

#72230705

C₄

yes

characteristic

#72230702

C₄

yes

—

#72230701

C₄

yes

characteristic

#72230700

C₂²

yes

—

#72230696

C₂

yes

maximal, characteristic, complemented

#72230695

C₂²

yes

—

#72230692

C₂

yes

maximal, characteristic, complemented

#72230689

C₂²

yes

characteristic

#72230688

C₂²

yes

characteristic

#72225140

C₂²

yes

characteristic

#153594

C₈

yes

characteristic

#151101

—

#151080

C₂×C₄

yes

—

#122946

—

#94752

C₈

yes

characteristic

#94609

C₈

yes

—

#94508

—

#94505

C₂×C₄

yes

characteristic

#94505

C₂×C₄

yes

characteristic

#93884

C₂×C₄

yes

—

—

—

—

D₄

yes

—

#93666

C₂³

yes

characteristic

C₂×D₆₄

C₁₆

yes

characteristic

C₄×D₃₂

C₂×C₈

yes

—

D₄×C₃₂

—

C₈×D₁₆

—

C₁₆×D₈

—

#7546

C₁₆

yes

characteristic

#7537

C₁₆

yes

—

#7510

—

#7510

—

#7510

C₂×C₈

yes

characteristic

#7510

C₂×C₈

yes

characteristic

#7461

C₂×C₈

yes

—

#7455

—

#7383

—

D₄⋊C₃₂

—

#7331

—

#7329

C₂²×C₄

yes

yes

—

—

C₂×D₄

yes

characteristic

C₂×D₃₂

C₂×C₁₆

yes

—

C₂²×C₃₂

—

C₄×D₁₆

—

D₄×C₁₆

—

C₈×D₈

—

D₆₄

C₃₂

yes

complemented

C₂×C₆₄

—

C₂×C₆₄

—

C₂×C₆₄

—

C₂×C₆₄

C₂×C₁₆

yes

characteristic

C₂×C₆₄

C₂×C₁₆

yes

characteristic

#1423

C₂×C₁₆

yes

—

C₄⋊C₃₂

—

#1415

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

C₂²×C₈

yes

C₄×C₃₂

yes

—

—

—

D₄⋊C₁₆

—

C₈×C₁₆

C₄×D₄

yes

characteristic

C₈×C₁₆

AGL(1,ℤ/8)

yes

—

C₂×D₁₆

—

C₂²×C₁₆

—

C₄×D₈

—

C₈×D₄

—

D₃₂

—

D₃₂

C₂×C₃₂

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₁₆

yes

C₂×C₃₂

yes

—

C₄⋊C₁₆

—

#357

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈×D₄

yes

C₄×C₁₆

yes

—

D₄⋊C₈

—

C₈²

#442

yes

characteristic

100

C₆₄

—

101

C₆₄

—

102

C₆₄

—

103

C₆₄

—

104

C₆₄

C₂×C₃₂

yes

complemented

105

C₂×D₈

—

106

C₂²×C₈

—

107

C₄×D₄

—

108

D₁₆

—

109

D₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₂×C₁₆

D₄×C₁₆

yes

characteristic

116

C₂×C₁₆

#2171

yes

characteristic

117

#107

—

118

C₄⋊C₈

—

119

#102

—

120

C₂²⋊C₈

—

121

C₄×C₈

#1580

yes

characteristic

122

C₄×C₈

AGL(1,ℤ/16)

yes

—

123

C₃₂

—

124

C₃₂

—

125

C₃₂

—

126

C₃₂

—

127

C₃₂

—

128

C₃₂

C₂²×C₃₂

yes

characteristic

129

C₃₂

#2258

yes

characteristic

130

C₂×D₄

—

131

C₂²×C₄

—

132

D₈

—

133

D₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

#12062

yes

characteristic

139

C₂×C₈

#12068

yes

characteristic

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

#12067

yes

characteristic

143

C₁₆

—

144

C₁₆

—

145

C₁₆

—

146

C₁₆

—

147

C₁₆

D₄×C₃₂

yes

characteristic

148

C₁₆

#13648

yes

characteristic

149

C₂³

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

#122963

yes

characteristic

156

C₂×C₄

AGL(1,ℤ/32)

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

#151101

yes

characteristic

161

C₈

#151103

yes

characteristic

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#72224639

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

#72224647

yes

characteristic

169

C₄

#72230710

yes

characteristic

170

C₂

—

171

C₂

#72224662

yes

—

172

C₂

#72230715

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#72224639
commutator	C₃₂	C₂²×C₃₂
Frattini, cosocle	#93666	C₂³
prime radical	C₂×D₆₄	C₁₆
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #7052 ◁ #72225140 ◁ G
LECS	G ▷ #93666 ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#72224639

commutator

C₃₂

C₂²×C₃₂

Frattini, cosocle

#93666

C₂³

prime radical

C₂×D₆₄

C₁₆

LCS

G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #7052 ◁ #72225140 ◁ G

LECS

G ▷ #93666 ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #72230718	C₂	semidirect
2. #72230715	C₂	semidirect
10. #72230696	C₂	semidirect
12. #72230692	C₂	semidirect
57. D₆₄	C₃₂	semidirect
82. D₃₂	C₂×C₃₂	semidirect
104. C₆₄	C₂×C₃₂	semidirect

H ≅ G/N

type

1. #72230718

C₂

semidirect

2. #72230715

C₂

semidirect

10. #72230696

C₂

semidirect

12. #72230692

C₂

semidirect

57. D₆₄

C₃₂

semidirect

82. D₃₂

C₂×C₃₂

semidirect

104. C₆₄

C₂×C₃₂

semidirect

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

subgroup	index
#72230718	2
#72230715	2
#72230706	2
#72230696	2
#72230692	2

subgroup

index

Subgroups — #72230720

Conjugacy classes of subgroups

Complements

Maximal subgroups