Group #153401 — subgroups

Subgroups — #153401

Order 512.

231 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 84 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×#1423	C₂	1	2	yes	maximal
2	C₂×#1416	C₂	1	2	yes	maximal
3	#13236	C₂	1	1	yes	maximal, characteristic
4	#12905	C₂	1	1	yes	maximal, characteristic
5	#7465	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	2	yes	—
8	C₂×#366	—	2	1	no	—
9	C₂×#366	C₂²	1	2	yes	—
10	C₂×#366	C₂²	1	1	yes	characteristic
11	C₂×#358	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1969	—	2	1	no	—
14	#1845	—	2	1	no	—
15	#1423	—	2	4	no	—
16	#1416	—	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×#107	—	2	2	no	—
25	C₂×#107	—	2	1	no	—
26	C₂×#107	—	4	1	no	—
27	C₂×#103	—	4	1	no	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	#391	—	4	1	no	—
30	#384	—	4	1	no	—
31	C₂×C₃₂	—	2	2	no	—
32	C₂×C₃₂	—	2	2	no	—
33	C₂×C₃₂	D₄	1	4	yes	—
34	#366	—	2	2	no	—
35	#366	—	2	2	no	—
36	#366	—	2	2	no	—
37	#366	—	4	2	no	—
38	#358	—	4	2	no	—
39	C₄×C₁₆	—	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	C₂²×C₈	D₈	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄⋊C₄	—	4	1	no	—
51	C₂×C₄⋊C₄	—	8	2	no	—
52	C₂×C₄²	D₈	1	1	yes	characteristic
53	Q₃₂	—	2	4	no	—
54	Q₃₂	—	4	2	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	D₈	1	2	yes	—
58	C₂×C₁₆	C₂×D₄	1	2	yes	—
59	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₁₆	#41	1	2	yes	—
61	C₂×C₁₆	#41	1	1	yes	characteristic
62	#107	—	4	2	no	—
63	#107	—	4	1	no	—
64	#107	—	4	1	no	—
65	#107	—	4	2	no	—
66	#107	—	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	—	8	2	no	—
81	C₂²×C₄	D₁₆	1	2	yes	—
82	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
83	Q₁₆	—	4	2	no	—
84	Q₁₆	—	4	2	no	—
85	Q₁₆	—	8	1	no	—
86	Q₁₆	—	8	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	D₁₆	1	2	yes	—
90	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₈	#123	1	1	yes	characteristic
92	C₂×C₈	#127	1	1	yes	characteristic
93	C₂×C₈	C₂×D₈	1	1	yes	characteristic
94	C₂×C₈	#137	1	1	yes	characteristic
95	C₂×C₈	#137	1	1	yes	characteristic
96	C₄⋊C₄	—	8	1	no	—
97	C₄⋊C₄	—	8	2	no	—
98	C₄⋊C₄	—	8	1	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	—	16	4	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₁₆	—	2	2	no	—
104	C₁₆	C₂²⋊Q₈	1	2	yes	—
105	C₁₆	#123	1	2	yes	—
106	C₂³	C₂×D₁₆	1	1	yes	characteristic
107	Q₈	—	8	2	no	—
108	Q₈	—	8	2	no	—
109	Q₈	—	16	1	no	—
110	Q₈	—	16	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	8	2	no	—
114	C₂×C₄	—	16	1	no	—
115	C₂×C₄	—	16	1	no	—
116	C₂×C₄	—	16	1	no	—
117	C₂×C₄	—	16	2	no	—
118	C₂×C₄	—	16	4	no	—
119	C₂×C₄	—	16	2	no	—
120	C₂×C₄	Q₆₄	1	4	yes	—
121	C₂×C₄	#474	1	1	yes	characteristic
122	C₂×C₄	#480	1	1	yes	characteristic
123	C₂×C₄	#493	1	1	yes	characteristic
124	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
125	C₂×C₄	#510	1	1	yes	characteristic
126	C₂×C₄	#510	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	#399	1	1	yes	characteristic
129	C₈	#474	1	1	yes	characteristic
130	C₈	#480	1	1	yes	characteristic
131	C₈	#497	1	1	yes	characteristic
132	C₂²	#2224	1	1	yes	characteristic
133	C₂²	#2232	1	1	yes	characteristic
134	C₂²	#2246	1	1	yes	characteristic
135	C₂²	C₂×Q₆₄	1	2	yes	—
136	C₂²	#2263	1	2	yes	—
137	C₄	—	2	4	no	—
138	C₄	—	16	2	no	—
139	C₄	—	32	2	no	—
140	C₄	—	32	1	no	—
141	C₄	#2093	1	1	yes	characteristic
142	C₄	#2224	1	1	yes	characteristic
143	C₄	#2232	1	1	yes	characteristic
144	C₄	#2251	1	1	yes	characteristic
145	C₂	#13526	1	1	yes	characteristic
146	C₂	#13706	1	2	yes	—
147	C₂	#13715	1	2	yes	—
148	C₂	#13726	1	1	yes	characteristic
149	C₂	#13729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1423

C₂

yes

maximal

C₂×#1416

C₂

yes

maximal

#13236

C₂

yes

maximal, characteristic

#12905

C₂

yes

maximal, characteristic

#7465

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

C₂×#366

—

C₂×#366

C₂²

yes

—

C₂×#366

C₂²

yes

characteristic

C₂×#358

—

C₂×C₄×C₁₆

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

—

#384

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

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

—

C₄×C₈

—

#95

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₆

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

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₁₆

—

104

C₁₆

C₂²⋊Q₈

yes

—

105

C₁₆

#123

yes

—

106

C₂³

C₂×D₁₆

yes

characteristic

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

Q₆₄

yes

—

121

C₂×C₄

#474

yes

characteristic

122

C₂×C₄

#480

yes

characteristic

123

C₂×C₄

#493

yes

characteristic

124

C₂×C₄

C₂×D₁₆

yes

characteristic

125

C₂×C₄

#510

yes

characteristic

126

C₂×C₄

#510

yes

characteristic

127

C₈

—

128

C₈

#399

yes

characteristic

129

C₈

#474

yes

characteristic

130

C₈

#480

yes

characteristic

131

C₈

#497

yes

characteristic

132

C₂²

#2224

yes

characteristic

133

C₂²

#2232

yes

characteristic

134

C₂²

#2246

yes

characteristic

135

C₂²

C₂×Q₆₄

yes

—

136

C₂²

#2263

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#2093

yes

characteristic

142

C₄

#2224

yes

characteristic

143

C₄

#2232

yes

characteristic

144

C₄

#2251

yes

characteristic

145

C₂

#13526

yes

characteristic

146

C₂

#13706

yes

—

147

C₂

#13715

yes

—

148

C₂

#13726

yes

characteristic

149

C₂

#13729

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ 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×#1423	2
C₂×#1416	2
#13236	2
#12905	2
#7465	2

subgroup

index

Subgroups — #153401

Conjugacy classes of subgroups

Maximal subgroups