Group #153407 — subgroups

Subgroups — #153407

Order 512.

211 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 72 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×#1426	C₂	1	2	yes	maximal
2	C₂×#1420	C₂	1	2	yes	maximal
3	#13240	C₂	1	1	yes	maximal, characteristic
4	#12910	C₂	1	1	yes	maximal, characteristic
5	#7468	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
7	C₂×M₆₄	C₂²	1	2	yes	—
8	C₂×#368	—	2	1	no	—
9	C₂×#368	C₂²	1	2	yes	—
10	C₂×#367	C₂²	1	1	yes	characteristic
11	C₂×#358	—	2	1	no	—
12	C₂×#346	C₂²	1	1	yes	characteristic
13	#1969	—	2	1	no	—
14	#1848	—	2	1	no	—
15	#1426	—	2	4	no	—
16	#1420	—	2	4	no	—
17	C₂²×Q₁₆	—	2	1	no	—
18	C₂×Q₃₂	—	2	2	no	—
19	C₂×Q₃₂	—	2	2	no	—
20	C₂×Q₃₂	—	2	1	no	—
21	C₂×Q₃₂	Q₈	1	2	yes	—
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂³	1	1	yes	characteristic
24	C₂×#108	—	2	2	no	—
25	C₂×#108	—	4	1	no	—
26	C₂×#107	—	2	1	no	—
27	#423	—	4	1	no	—
28	C₂×#103	—	4	1	no	—
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	#391	—	4	1	no	—
31	M₆₄	—	2	2	no	—
32	M₆₄	D₄	1	4	yes	—
33	C₂×C₃₂	—	2	2	no	—
34	#368	—	2	2	no	—
35	#368	—	2	2	no	—
36	#368	—	4	2	no	—
37	#367	—	2	2	no	—
38	#358	—	4	2	no	—
39	#346	—	2	2	no	—
40	C₂²×Q₈	—	4	1	no	—
41	C₂×Q₁₆	—	2	2	no	—
42	C₂×Q₁₆	—	4	1	no	—
43	C₂×Q₁₆	—	4	1	no	—
44	C₂×Q₁₆	—	4	1	no	—
45	C₂×Q₁₆	—	4	1	no	—
46	C₂×Q₁₆	—	4	1	no	—
47	C₂×M₁₆	—	4	2	no	—
48	C₂²×C₈	D₈	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄	—	4	1	no	—
51	C₂×C₄²	D₈	1	1	yes	characteristic
52	Q₃₂	—	2	4	no	—
53	Q₃₂	—	4	2	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	D₈	1	2	yes	—
57	C₂×C₁₆	C₂×D₄	1	2	yes	—
58	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₁₆	#41	1	2	yes	—
60	C₂×C₁₆	#41	1	1	yes	characteristic
61	#108	—	4	2	no	—
62	#108	—	4	2	no	—
63	#108	—	8	2	no	—
64	#107	—	4	1	no	—
65	#107	—	4	1	no	—
66	C₄⋊C₈	—	8	2	no	—
67	#103	—	8	2	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	#95	—	8	2	no	—
71	C₃₂	—	2	4	no	—
72	C₂×Q₈	—	4	2	no	—
73	C₂×Q₈	—	8	1	no	—
74	C₂×Q₈	—	8	1	no	—
75	C₂×Q₈	—	8	1	no	—
76	C₂×Q₈	—	8	1	no	—
77	C₂×Q₈	—	8	1	no	—
78	C₂²×C₄	—	8	1	no	—
79	C₂²×C₄	—	8	1	no	—
80	C₂²×C₄	SD₃₂	1	2	yes	—
81	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
82	Q₁₆	—	4	2	no	—
83	Q₁₆	—	4	2	no	—
84	Q₁₆	—	8	1	no	—
85	Q₁₆	—	8	1	no	—
86	M₁₆	—	8	2	no	—
87	M₁₆	—	8	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	8	2	no	—
91	C₂×C₈	SD₃₂	1	2	yes	—
92	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₈	#123	1	1	yes	characteristic
94	C₂×C₈	#127	1	1	yes	characteristic
95	C₂×C₈	C₂×D₈	1	1	yes	characteristic
96	C₂×C₈	#137	1	1	yes	characteristic
97	C₂×C₈	#137	1	1	yes	characteristic
98	C₄⋊C₄	—	8	1	no	—
99	C₄⋊C₄	—	8	1	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₁₆	—	2	2	no	—
103	C₁₆	C₂²⋊Q₈	1	2	yes	—
104	C₁₆	#123	1	2	yes	—
105	C₂³	C₂×SD₃₂	1	1	yes	characteristic
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	16	1	no	—
109	Q₈	—	16	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	16	1	no	—
114	C₂×C₄	—	16	1	no	—
115	C₂×C₄	—	16	2	no	—
116	C₂×C₄	—	16	1	no	—
117	C₂×C₄	#474	1	1	yes	characteristic
118	C₂×C₄	#480	1	1	yes	characteristic
119	C₂×C₄	#493	1	1	yes	characteristic
120	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
121	C₂×C₄	#509	1	1	yes	characteristic
122	C₂×C₄	#509	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	16	2	no	—
125	C₈	#399	1	1	yes	characteristic
126	C₈	#474	1	1	yes	characteristic
127	C₈	#480	1	1	yes	characteristic
128	C₈	#497	1	1	yes	characteristic
129	C₂²	—	2	2	no	—
130	C₂²	#2226	1	1	yes	characteristic
131	C₂²	#2233	1	1	yes	characteristic
132	C₂²	#2248	1	1	yes	characteristic
133	C₄	—	4	2	no	—
134	C₄	—	16	2	no	—
135	C₄	—	32	1	no	—
136	C₄	#2093	1	1	yes	characteristic
137	C₄	#2226	1	1	yes	characteristic
138	C₄	#2233	1	1	yes	characteristic
139	C₄	#2250	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#13527	1	1	yes	characteristic
143	C₂	#13718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1426

C₂

yes

maximal

C₂×#1420

C₂

yes

maximal

#13240

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#7468

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

—

C₂×#368

—

C₂×#368

C₂²

yes

—

C₂×#367

C₂²

yes

characteristic

C₂×#358

—

C₂×#346

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

Q₈

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#391

—

M₆₄

—

M₆₄

D₄

yes

—

C₂×C₃₂

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

SD₃₂

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

C₂²⋊Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₁₆

—

103

C₁₆

C₂²⋊Q₈

yes

—

104

C₁₆

#123

yes

—

105

C₂³

C₂×SD₃₂

yes

characteristic

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#474

yes

characteristic

118

C₂×C₄

#480

yes

characteristic

119

C₂×C₄

#493

yes

characteristic

120

C₂×C₄

C₂×SD₃₂

yes

characteristic

121

C₂×C₄

#509

yes

characteristic

122

C₂×C₄

#509

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

#399

yes

characteristic

126

C₈

#474

yes

characteristic

127

C₈

#480

yes

characteristic

128

C₈

#497

yes

characteristic

129

C₂²

—

130

C₂²

#2226

yes

characteristic

131

C₂²

#2233

yes

characteristic

132

C₂²

#2248

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#2093

yes

characteristic

137

C₄

#2226

yes

characteristic

138

C₄

#2233

yes

characteristic

139

C₄

#2250

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#13527

yes

characteristic

143

C₂

#13718

yes

—

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2233

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

C₂×SD₃₂

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ 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×#1426	2
C₂×#1420	2
#13240	2
#12910	2
#7468	2

subgroup

index

Subgroups — #153407

Conjugacy classes of subgroups

Maximal subgroups