Group #153386 — subgroups

Subgroups — #153386

Order 512.

156 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 66 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	#13638	C₂	1	1	yes	maximal, characteristic
2	#13636	C₂	1	1	yes	maximal, characteristic
3	#13172	C₂	1	1	yes	maximal, characteristic
4	#12911	C₂	1	1	yes	maximal, characteristic
5	#7477	C₂	1	1	yes	maximal, characteristic
6	#7474	C₂	1	1	yes	maximal, characteristic
7	#7469	C₂	1	1	yes	maximal, characteristic
8	C₂×M₆₄	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×#368	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	1	yes	characteristic
12	#2158	C₂²	1	1	yes	characteristic
13	C₂×#367	C₂²	1	1	yes	characteristic
14	#2155	—	2	1	no	—
15	#2108	C₂²	1	1	yes	characteristic
16	#1952	—	2	1	no	—
17	#1833	—	2	1	no	—
18	#1426	—	2	2	no	—
19	#1425	—	2	2	no	—
20	#1387	—	2	1	no	—
21	C₂×M₃₂	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂³	1	1	yes	characteristic
23	C₂×#108	—	2	1	no	—
24	C₂×#108	—	2	1	no	—
25	#428	—	4	1	no	—
26	#427	—	2	1	no	—
27	C₂×#107	—	2	1	no	—
28	#405	D₄	1	1	yes	characteristic
29	M₆₄	—	2	1	no	—
30	M₆₄	—	2	1	no	—
31	M₆₄	—	2	1	no	—
32	M₆₄	Q₈	1	2	yes	—
33	C₂×C₃₂	—	2	1	no	—
34	C₂×C₃₂	Q₈	1	2	yes	—
35	#368	—	2	1	no	—
36	#368	—	2	1	no	—
37	#368	—	2	1	no	—
38	#368	D₄	1	2	yes	—
39	#367	—	2	1	no	—
40	#367	—	2	1	no	—
41	#367	—	2	1	no	—
42	#366	—	2	1	no	—
43	#365	—	4	2	no	—
44	#346	—	2	1	no	—
45	C₄×C₁₆	—	2	1	no	—
46	#344	—	4	1	no	—
47	#343	—	4	1	no	—
48	H(3,ℤ/4)	—	4	1	no	—
49	C₂×M₁₆	—	4	1	no	—
50	C₂×M₁₆	—	4	1	no	—
51	C₂×M₁₆	D₈	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	#117	—	4	1	no	—
54	#117	D₈	1	1	yes	characteristic
55	C₂×C₄⋊C₄	—	4	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	Q₁₆	1	2	yes	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
60	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₁₆	#41	1	1	yes	characteristic
63	C₂×C₁₆	#41	1	1	yes	characteristic
64	C₂×C₁₆	#41	1	1	yes	characteristic
65	#108	—	2	2	no	—
66	#108	—	4	1	no	—
67	#108	—	4	1	no	—
68	#108	—	4	1	no	—
69	#107	—	4	1	no	—
70	#107	—	4	1	no	—
71	#107	—	4	1	no	—
72	#107	—	8	1	no	—
73	#106	—	4	1	no	—
74	#106	—	8	1	no	—
75	#97	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	C₃₂	—	2	2	no	—
78	C₃₂	—	2	2	no	—
79	C₂²×C₄	—	8	1	no	—
80	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
81	M₁₆	—	2	1	no	—
82	M₁₆	—	2	1	no	—
83	M₁₆	—	4	2	no	—
84	M₁₆	—	8	1	no	—
85	M₁₆	—	8	1	no	—
86	M₁₆	—	8	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	8	1	no	—
89	C₂×C₈	—	8	1	no	—
90	C₂×C₈	#121	1	1	yes	characteristic
91	C₂×C₈	#123	1	1	yes	characteristic
92	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₈	#137	1	1	yes	characteristic
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	8	1	no	—
98	C₄⋊C₄	—	8	1	no	—
99	C₄⋊C₄	—	8	1	no	—
100	C₂²⋊C₄	—	2	1	no	—
101	C₂²⋊C₄	—	8	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	8	1	no	—
104	C₁₆	—	2	2	no	—
105	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
106	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
107	C₁₆	#125	1	1	yes	characteristic
108	C₁₆	#125	1	1	yes	characteristic
109	C₂³	#508	1	1	yes	characteristic
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	16	1	no	—
112	C₂×C₄	—	16	1	no	—
113	C₂×C₄	—	32	1	no	—
114	C₂×C₄	#459	1	1	yes	characteristic
115	C₂×C₄	#480	1	1	yes	characteristic
116	C₂×C₄	#500	1	1	yes	characteristic
117	C₂×C₄	#508	1	1	yes	characteristic
118	C₂×C₄	#509	1	1	yes	characteristic
119	C₂×C₄	#510	1	1	yes	characteristic
120	C₈	—	4	1	no	—
121	C₈	—	8	2	no	—
122	C₈	—	16	1	no	—
123	C₈	#398	1	1	yes	characteristic
124	C₈	#462	1	1	yes	characteristic
125	C₈	#484	1	1	yes	characteristic
126	C₈	#501	1	1	yes	characteristic
127	C₂²	—	4	1	no	—
128	C₂²	#2211	1	1	yes	characteristic
129	C₂²	#2234	1	1	yes	characteristic
130	C₂²	#2253	1	1	yes	characteristic
131	C₄	—	8	1	no	—
132	C₄	—	32	1	no	—
133	C₄	—	32	1	no	—
134	C₄	#2083	1	1	yes	characteristic
135	C₄	#2210	1	1	yes	characteristic
136	C₄	#2237	1	1	yes	characteristic
137	C₄	#2255	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	—	2	1	no	—
141	C₂	#13507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13638

C₂

yes

maximal, characteristic

#13636

C₂

yes

maximal, characteristic

#13172

C₂

yes

maximal, characteristic

#12911

C₂

yes

maximal, characteristic

#7477

C₂

yes

maximal, characteristic

#7474

C₂

yes

maximal, characteristic

#7469

C₂

yes

maximal, characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

C₂×#368

C₂²

yes

characteristic

#2158

C₂²

yes

characteristic

C₂×#367

C₂²

yes

characteristic

#2155

—

#2108

C₂²

yes

—

—

—

—

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

D₄

yes

characteristic

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

Q₈

yes

—

C₂×C₃₂

—

C₂×C₃₂

Q₈

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

—

C₄×C₁₆

—

#344

—

#343

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

#117

D₈

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

105

C₁₆

C₂²⋊Q₈

yes

characteristic

106

C₁₆

C₂²⋊Q₈

yes

characteristic

107

C₁₆

#125

yes

characteristic

108

C₁₆

#125

yes

characteristic

109

C₂³

#508

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#459

yes

characteristic

115

C₂×C₄

#480

yes

characteristic

116

C₂×C₄

#500

yes

characteristic

117

C₂×C₄

#508

yes

characteristic

118

C₂×C₄

#509

yes

characteristic

119

C₂×C₄

#510

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

#398

yes

characteristic

124

C₈

#462

yes

characteristic

125

C₈

#484

yes

characteristic

126

C₈

#501

yes

characteristic

127

C₂²

—

128

C₂²

#2211

yes

characteristic

129

C₂²

#2234

yes

characteristic

130

C₂²

#2253

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#2083

yes

characteristic

135

C₄

#2210

yes

characteristic

136

C₄

#2237

yes

characteristic

137

C₄

#2255

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

#13507

yes

characteristic

	subgroup	G/H
center, socle	C₂	#13507
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

#13507

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
#13638	2
#13636	2
#13172	2
#12911	2
#7477	2
#7474	2
#7469	2

subgroup

index

Subgroups — #153386

Conjugacy classes of subgroups

Maximal subgroups