Group #153365 — subgroups

Subgroups — #153365

Order 512.

188 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 62 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×#1420	C₂	1	1	yes	maximal, characteristic
2	#13618	C₂	1	1	yes	maximal, characteristic
3	#13244	C₂	1	1	yes	maximal, characteristic
4	#12911	C₂	1	1	yes	maximal, characteristic
5	#7480	C₂	1	1	yes	maximal, characteristic
6	#7479	C₂	1	1	yes	maximal, characteristic
7	#7469	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	1	yes	characteristic
12	C₂×#368	C₂²	1	1	yes	characteristic
13	#2158	C₂²	1	1	yes	characteristic
14	#2155	—	2	1	no	—
15	C₂×#362	—	2	1	no	—
16	#2108	C₂²	1	1	yes	characteristic
17	#1979	—	2	1	no	—
18	#1833	—	2	1	no	—
19	#1420	—	2	2	no	—
20	#1416	—	2	2	no	—
21	C₂²×Q₁₆	—	2	1	no	—
22	C₂×Q₃₂	—	2	1	no	—
23	C₂×Q₃₂	—	2	1	no	—
24	C₂×Q₃₂	—	2	1	no	—
25	C₂×Q₃₂	—	2	1	no	—
26	C₂×Q₃₂	—	2	1	no	—
27	C₂×Q₃₂	D₄	1	2	yes	—
28	C₂×M₃₂	D₄	1	1	yes	characteristic
29	C₂²×C₁₆	C₂³	1	1	yes	characteristic
30	C₂×#108	—	2	1	no	—
31	C₂×#108	—	2	1	no	—
32	#427	—	2	1	no	—
33	#419	—	4	1	no	—
34	C₂×#101	—	4	1	no	—
35	#405	D₄	1	1	yes	characteristic
36	M₆₄	—	2	1	no	—
37	M₆₄	—	2	1	no	—
38	M₆₄	—	2	1	no	—
39	M₆₄	—	2	1	no	—
40	C₂×C₃₂	—	2	1	no	—
41	C₂×C₃₂	—	2	1	no	—
42	#368	—	2	1	no	—
43	#368	—	2	1	no	—
44	#368	—	2	1	no	—
45	#368	—	2	1	no	—
46	#367	—	2	1	no	—
47	#366	—	2	1	no	—
48	#365	—	4	2	no	—
49	#362	—	4	2	no	—
50	#346	—	2	1	no	—
51	C₄×C₁₆	—	2	1	no	—
52	#344	—	4	1	no	—
53	H(3,ℤ/4)	—	4	1	no	—
54	C₂²×Q₈	—	4	1	no	—
55	C₂×Q₁₆	—	2	2	no	—
56	C₂×Q₁₆	—	4	1	no	—
57	C₂×Q₁₆	—	4	1	no	—
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×Q₁₆	—	4	1	no	—
60	C₂×Q₁₆	—	4	1	no	—
61	C₂×M₁₆	—	4	1	no	—
62	C₂×M₁₆	—	4	1	no	—
63	C₂×M₁₆	D₈	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	#117	—	4	1	no	—
66	#117	D₈	1	1	yes	characteristic
67	Q₃₂	—	2	2	no	—
68	Q₃₂	—	2	2	no	—
69	Q₃₂	—	4	1	no	—
70	Q₃₂	—	4	1	no	—
71	M₃₂	—	2	1	no	—
72	M₃₂	SD₁₆	1	2	yes	—
73	C₂×C₁₆	—	2	1	no	—
74	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	#41	1	1	yes	characteristic
76	C₂×C₁₆	#41	1	1	yes	characteristic
77	C₂×C₁₆	#41	1	1	yes	characteristic
78	C₂×C₁₆	#41	1	1	yes	characteristic
79	C₂×C₁₆	#41	1	1	yes	characteristic
80	#108	—	4	1	no	—
81	#108	—	4	1	no	—
82	#108	—	4	1	no	—
83	#108	—	4	1	no	—
84	#107	—	4	1	no	—
85	#106	—	4	1	no	—
86	#103	—	8	2	no	—
87	#101	—	8	2	no	—
88	#97	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₃₂	—	4	1	no	—
91	C₃₂	—	4	1	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	8	1	no	—
94	C₂×Q₈	—	8	1	no	—
95	C₂×Q₈	—	8	1	no	—
96	C₂×Q₈	—	8	1	no	—
97	C₂×Q₈	—	8	1	no	—
98	C₂²×C₄	—	8	1	no	—
99	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
100	Q₁₆	—	4	2	no	—
101	Q₁₆	—	4	2	no	—
102	Q₁₆	—	8	1	no	—
103	Q₁₆	—	8	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	2	1	no	—
106	M₁₆	—	8	1	no	—
107	M₁₆	—	8	1	no	—
108	M₁₆	—	8	1	no	—
109	M₁₆	—	8	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	8	1	no	—
112	C₂×C₈	—	8	1	no	—
113	C₂×C₈	#121	1	1	yes	characteristic
114	C₂×C₈	#123	1	1	yes	characteristic
115	C₂×C₈	#124	1	1	yes	characteristic
116	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₈	#137	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	8	1	no	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	8	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	8	1	no	—
125	C₁₆	—	2	2	no	—
126	C₁₆	#123	1	1	yes	characteristic
127	C₁₆	#123	1	1	yes	characteristic
128	C₁₆	#126	1	1	yes	characteristic
129	C₁₆	#126	1	1	yes	characteristic
130	C₂³	#508	1	1	yes	characteristic
131	Q₈	—	8	2	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	16	1	no	—
134	Q₈	—	16	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	16	1	no	—
138	C₂×C₄	—	16	1	no	—
139	C₂×C₄	—	16	1	no	—
140	C₂×C₄	#459	1	1	yes	characteristic
141	C₂×C₄	#480	1	1	yes	characteristic
142	C₂×C₄	#486	1	1	yes	characteristic
143	C₂×C₄	#508	1	1	yes	characteristic
144	C₂×C₄	#509	1	1	yes	characteristic
145	C₂×C₄	#509	1	1	yes	characteristic
146	C₈	—	4	1	no	—
147	C₈	—	16	1	no	—
148	C₈	—	16	1	no	—
149	C₈	#397	1	1	yes	characteristic
150	C₈	#461	1	1	yes	characteristic
151	C₈	#481	1	1	yes	characteristic
152	C₈	#489	1	1	yes	characteristic
153	C₂²	—	4	1	no	—
154	C₂²	#2210	1	1	yes	characteristic
155	C₂²	#2234	1	1	yes	characteristic
156	C₂²	#2242	1	1	yes	characteristic
157	C₄	—	8	1	no	—
158	C₄	—	16	2	no	—
159	C₄	—	32	1	no	—
160	C₄	#2071	1	1	yes	characteristic
161	C₄	#2210	1	1	yes	characteristic
162	C₄	#2234	1	1	yes	characteristic
163	C₄	#2244	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#13482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1420

C₂

yes

maximal, characteristic

#13618

C₂

yes

maximal, characteristic

#13244

C₂

yes

maximal, characteristic

#12911

C₂

yes

maximal, characteristic

#7480

C₂

yes

maximal, characteristic

#7479

C₂

yes

maximal, characteristic

#7469

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

#2158

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₄

yes

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

D₄

yes

characteristic

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

—

#344

—

H(3,ℤ/4)

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

#117

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

Q₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

#121

yes

characteristic

114

C₂×C₈

#123

yes

characteristic

115

C₂×C₈

#124

yes

characteristic

116

C₂×C₈

C₂×SD₁₆

yes

characteristic

117

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₈

#137

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₁₆

—

126

C₁₆

#123

yes

characteristic

127

C₁₆

#123

yes

characteristic

128

C₁₆

#126

yes

characteristic

129

C₁₆

#126

yes

characteristic

130

C₂³

#508

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#459

yes

characteristic

141

C₂×C₄

#480

yes

characteristic

142

C₂×C₄

#486

yes

characteristic

143

C₂×C₄

#508

yes

characteristic

144

C₂×C₄

#509

yes

characteristic

145

C₂×C₄

#509

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

#397

yes

characteristic

150

C₈

#461

yes

characteristic

151

C₈

#481

yes

characteristic

152

C₈

#489

yes

characteristic

153

C₂²

—

154

C₂²

#2210

yes

characteristic

155

C₂²

#2234

yes

characteristic

156

C₂²

#2242

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#2071

yes

characteristic

161

C₄

#2210

yes

characteristic

162

C₄

#2234

yes

characteristic

163

C₄

#2244

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

167

C₂

#13482

yes

characteristic

	subgroup	G/H
center, socle	C₂	#13482
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#13482

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#508

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#1420	2
#13618	2
#13244	2
#12911	2
#7480	2
#7479	2
#7469	2

subgroup

index

Subgroups — #153365

Conjugacy classes of subgroups

Maximal subgroups