Group #141534 — subgroups

Subgroups — #141534

Order 512.

280 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2252	C₂	1	1	yes	maximal, characteristic
2	#12022	C₂	1	1	yes	maximal, characteristic
3	#11972	C₂	1	2	yes	maximal
4	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#9801	C₂	1	2	yes	maximal
6	C₂×#500	C₂²	1	2	yes	—
7	#2252	—	2	1	no	—
8	#2252	—	2	1	no	—
9	#2252	—	2	2	no	—
10	#2162	C₂²	1	1	yes	characteristic
11	C₂×#366	C₄	1	4	yes	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1997	—	2	2	no	—
14	#1567	C₂²	1	2	yes	—
15	#1549	—	2	2	no	—
16	C₈×M₁₆	C₂²	1	1	yes	characteristic
17	#1371	—	2	2	no	—
18	#1365	—	2	4	no	—
19	#1312	—	2	1	no	—
20	C₈×C₁₆	—	2	1	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂²×C₁₆	C₂×C₄	1	2	yes	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#431	D₄	1	1	yes	characteristic
28	C₂×#107	—	2	2	no	—
29	C₂×#107	C₂×C₄	1	4	yes	—
30	#423	—	2	2	no	—
31	C₄×M₁₆	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	#366	—	2	4	no	—
34	#366	—	2	4	no	—
35	C₄⋊C₁₆	D₄	1	2	yes	—
36	C₂²⋊C₁₆	—	2	1	no	—
37	C₄×C₁₆	—	2	1	no	—
38	C₄×C₁₆	D₄	1	1	yes	characteristic
39	C₄×C₁₆	D₄	1	1	yes	characteristic
40	C₄×C₁₆	D₄	1	2	yes	—
41	#343	—	4	4	no	—
42	#335	—	2	2	no	—
43	#335	—	2	2	no	—
44	#332	—	4	4	no	—
45	#322	—	2	1	no	—
46	C₈²	—	2	1	no	—
47	C₂²×Q₈	—	4	2	no	—
48	C₂×M₁₆	—	4	2	no	—
49	C₂×M₁₆	D₈	1	2	yes	—
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	#41	1	1	yes	characteristic
52	C₄⋊Q₈	—	4	2	no	—
53	C₄⋊Q₈	—	4	2	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	—	4	2	no	—
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	—	2	4	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
62	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
63	#107	—	2	4	no	—
64	#107	—	2	4	no	—
65	#107	—	4	2	no	—
66	#107	—	4	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	C₄⋊C₈	—	4	2	no	—
69	C₄⋊C₈	D₈	1	2	yes	—
70	C₂²⋊C₈	—	2	1	no	—
71	#97	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×C₈	C₂×D₄	1	1	yes	characteristic
76	C₄×C₈	C₂×D₄	1	1	yes	characteristic
77	C₄×C₈	#41	1	1	yes	characteristic
78	C₂×Q₈	—	8	2	no	—
79	C₂×Q₈	—	8	2	no	—
80	C₂×Q₈	—	16	4	no	—
81	C₂²×C₄	—	4	4	no	—
82	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
84	C₂²×C₄	#135	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	8	4	no	—
87	M₁₆	Q₃₂	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	8	2	no	—
93	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈	C₄×D₄	1	1	yes	characteristic
95	C₂×C₈	#121	1	1	yes	characteristic
96	C₂×C₈	#124	1	1	yes	characteristic
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	4	4	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	—	8	2	no	—
101	C₄²	#121	1	1	yes	characteristic
102	C₄²	#127	1	1	yes	characteristic
103	C₄²	C₂×D₈	1	1	yes	characteristic
104	C₄²	#135	1	1	yes	characteristic
105	C₁₆	—	2	2	no	—
106	C₁₆	—	4	2	no	—
107	C₁₆	#101	1	4	yes	—
108	C₂³	C₄×D₈	1	1	yes	characteristic
109	Q₈	—	16	4	no	—
110	Q₈	—	16	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	8	4	no	—
115	C₂×C₄	—	16	4	no	—
116	C₂×C₄	#390	1	1	yes	characteristic
117	C₂×C₄	C₄×D₈	1	1	yes	characteristic
118	C₂×C₄	#466	1	1	yes	characteristic
119	C₂×C₄	#466	1	1	yes	characteristic
120	C₂×C₄	#493	1	1	yes	characteristic
121	C₂×C₄	#495	1	1	yes	characteristic
122	C₂×C₄	C₂×Q₃₂	1	2	yes	—
123	C₂×C₄	#508	1	2	yes	—
124	C₈	—	2	4	no	—
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	4	2	no	—
128	C₈	—	16	2	no	—
129	C₈	C₂×#101	1	2	yes	—
130	C₈	#473	1	2	yes	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#1970	1	1	yes	characteristic
134	C₂²	C₄×Q₃₂	1	2	yes	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	16	4	no	—
138	C₄	#1966	1	1	yes	characteristic
139	C₄	#1979	1	1	yes	characteristic
140	C₄	#2216	1	2	yes	—
141	C₄	#2221	1	2	yes	—
142	C₄	#2247	1	1	yes	characteristic
143	C₄	#2249	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	#9799	1	1	yes	characteristic
146	C₂	#13236	1	1	yes	characteristic
147	C₂	#13245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

#12022

C₂

yes

maximal, characteristic

#11972

C₂

yes

maximal

C₁₆×M₁₆

C₂

yes

maximal, characteristic

#9801

C₂

yes

maximal

C₂×#500

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#366

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1997

—

#1567

C₂²

yes

—

#1549

—

C₈×M₁₆

C₂²

yes

—

—

—

C₈×C₁₆

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×#107

—

C₂×#107

C₂×C₄

yes

—

#423

—

C₄×M₁₆

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

D₄

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

—

—

C₈²

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

Q₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

#121

yes

characteristic

102

C₄²

#127

yes

characteristic

103

C₄²

C₂×D₈

yes

characteristic

104

C₄²

#135

yes

characteristic

105

C₁₆

—

106

C₁₆

—

107

C₁₆

#101

yes

—

108

C₂³

C₄×D₈

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#390

yes

characteristic

117

C₂×C₄

C₄×D₈

yes

characteristic

118

C₂×C₄

#466

yes

characteristic

119

C₂×C₄

#466

yes

characteristic

120

C₂×C₄

#493

yes

characteristic

121

C₂×C₄

#495

yes

characteristic

122

C₂×C₄

C₂×Q₃₂

yes

—

123

C₂×C₄

#508

yes

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

C₂×#101

yes

—

130

C₈

#473

yes

—

131

C₂²

—

132

C₂²

—

133

C₂²

#1970

yes

characteristic

134

C₂²

C₄×Q₃₂

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#1966

yes

characteristic

139

C₄

#1979

yes

characteristic

140

C₄

#2216

yes

—

141

C₄

#2221

yes

—

142

C₄

#2247

yes

characteristic

143

C₄

#2249

yes

characteristic

144

C₂

—

145

C₂

#9799

yes

characteristic

146

C₂

#13236

yes

characteristic

147

C₂

#13245

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1970

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₄×D₈

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#2252	2
#12022	2
#11972	2
C₁₆×M₁₆	2
#9801	2

subgroup

index

C₁₆×M₁₆

Subgroups — #141534

Conjugacy classes of subgroups

Maximal subgroups