Group #153314 — subgroups

Subgroups — #153314

Order 512.

270 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₂²×D₃₂	C₂	1	1	yes	maximal, characteristic
2	#13628	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1419	C₂	1	2	yes	maximal, complemented
4	#13245	C₂	1	2	yes	maximal, complemented
5	#7472	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₆	C₂²	1	2	yes	—
7	C₂×D₃₂	—	2	4	no	—
8	C₂×D₃₂	—	2	2	no	—
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	2	yes	—
12	C₂×#361	—	2	2	no	—
13	#2108	C₂²	1	1	yes	characteristic
14	#1978	—	2	2	no	—
15	C₄⋊C₃₂	—	2	2	no	—
16	#1419	—	2	4	no	—
17	C₂²×D₈	—	2	2	no	—
18	C₂×D₁₆	—	2	2	no	—
19	C₂×D₁₆	—	2	2	no	—
20	C₂×D₁₆	—	2	2	no	—
21	C₂×D₁₆	—	2	2	no	—
22	C₂×D₁₆	—	2	2	no	—
23	C₂×D₁₆	D₄	1	4	yes	—
24	C₂×M₃₂	D₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂³	1	1	yes	characteristic
26	C₂×#108	—	2	2	no	—
27	#418	—	4	2	no	—
28	C₂×#100	—	4	2	no	—
29	#405	D₄	1	1	yes	characteristic
30	D₃₂	—	2	8	no	—
31	M₆₄	—	2	2	no	—
32	C₂×C₃₂	—	2	2	no	—
33	C₂×C₃₂	—	2	1	no	—
34	C₂×C₃₂	D₄	1	2	yes	—
35	#368	—	2	2	no	—
36	#368	—	2	2	no	—
37	#361	—	4	4	no	—
38	#346	—	2	1	no	—
39	C₄×C₁₆	—	2	1	no	—
40	C₂²×D₄	—	4	2	no	—
41	C₂×D₈	—	2	4	no	—
42	C₂×D₈	—	4	2	no	—
43	C₂×D₈	—	4	2	no	—
44	C₂×D₈	—	4	2	no	—
45	C₂×D₈	—	4	2	no	—
46	C₂×D₈	—	4	2	no	—
47	C₂×M₁₆	—	4	2	no	—
48	C₂×M₁₆	D₈	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	#117	D₈	1	1	yes	characteristic
51	D₁₆	—	2	4	no	—
52	D₁₆	—	2	4	no	—
53	D₁₆	—	4	2	no	—
54	D₁₆	—	4	2	no	—
55	M₃₂	—	2	2	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	C₂×D₄	1	2	yes	—
58	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	#41	1	1	yes	characteristic
60	C₂×C₁₆	#41	1	2	yes	—
61	#108	—	4	2	no	—
62	#108	—	4	2	no	—
63	#102	—	8	4	no	—
64	#100	—	8	4	no	—
65	#97	—	2	1	no	—
66	C₄×C₈	—	2	1	no	—
67	C₃₂	—	2	2	no	—
68	C₃₂	—	4	1	no	—
69	C₂⁴	—	8	2	no	—
70	C₂×D₄	—	4	4	no	—
71	C₂×D₄	—	8	2	no	—
72	C₂×D₄	—	8	2	no	—
73	C₂×D₄	—	8	2	no	—
74	C₂×D₄	—	8	2	no	—
75	C₂×D₄	—	8	2	no	—
76	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
77	D₈	—	4	4	no	—
78	D₈	—	4	4	no	—
79	D₈	—	8	2	no	—
80	D₈	—	8	2	no	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	8	2	no	—
83	M₁₆	—	8	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	#121	1	2	yes	—
87	C₂×C₈	#121	1	1	yes	characteristic
88	C₂×C₈	#135	1	1	yes	characteristic
89	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
90	C₄⋊C₄	D₁₆	1	2	yes	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	—	2	1	no	—
93	C₁₆	—	4	1	no	—
94	C₁₆	#120	1	1	yes	characteristic
95	C₁₆	#123	1	2	yes	—
96	C₁₆	#124	1	1	yes	characteristic
97	C₂³	—	8	4	no	—
98	C₂³	—	16	2	no	—
99	C₂³	—	16	2	no	—
100	C₂³	—	32	4	no	—
101	C₂³	#508	1	1	yes	characteristic
102	D₄	—	8	4	no	—
103	D₄	—	8	4	no	—
104	D₄	—	16	2	no	—
105	D₄	—	16	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	#459	1	2	yes	—
108	C₂×C₄	#466	1	1	yes	characteristic
109	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
110	C₂×C₄	#509	1	2	yes	—
111	C₈	—	4	1	no	—
112	C₈	—	16	2	no	—
113	C₈	#394	1	1	yes	characteristic
114	C₈	#463	1	2	yes	—
115	C₈	#469	1	1	yes	characteristic
116	C₂²	—	2	2	no	—
117	C₂²	—	16	4	no	—
118	C₂²	—	32	4	no	—
119	C₂²	—	32	4	no	—
120	C₂²	—	32	4	no	—
121	C₂²	#2210	1	2	yes	—
122	C₂²	#2215	1	1	yes	characteristic
123	C₄	—	4	2	no	—
124	C₄	#2054	1	1	yes	characteristic
125	C₄	#2206	1	2	yes	—
126	C₄	#2218	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	32	4	no	—
129	C₂	#13448	1	1	yes	characteristic
130	C₂	#13700	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₃₂

C₂

yes

maximal, characteristic

#13628

C₂

yes

maximal, characteristic, complemented

C₂×#1419

C₂

yes

maximal, complemented

#13245

C₂

yes

maximal, complemented

#7472

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₆

C₂²

yes

—

C₂×D₃₂

—

C₂×D₃₂

—

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#361

—

#2108

C₂²

yes

characteristic

#1978

—

C₄⋊C₃₂

—

#1419

—

C₂²×D₈

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

D₄

yes

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

D₄

yes

characteristic

D₃₂

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

—

C₄×C₁₆

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

D₈

yes

characteristic

D₁₆

—

D₁₆

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

D₁₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

#120

yes

characteristic

C₁₆

#123

yes

—

C₁₆

#124

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

#508

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

#459

yes

—

108

C₂×C₄

#466

yes

characteristic

109

C₂×C₄

C₂×D₁₆

yes

characteristic

110

C₂×C₄

#509

yes

—

111

C₈

—

112

C₈

—

113

C₈

#394

yes

characteristic

114

C₈

#463

yes

—

115

C₈

#469

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#2210

yes

—

122

C₂²

#2215

yes

characteristic

123

C₄

—

124

C₄

#2054

yes

characteristic

125

C₄

#2206

yes

—

126

C₄

#2218

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

#13448

yes

characteristic

130

C₂

#13700

yes

—

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₃₂	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2215

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂²×D₃₂

C₂

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13628	C₂	semidirect
3. C₂×#1419	C₂	semidirect
4. #13245	C₂	semidirect
5. #7472	C₂	semidirect

H ≅ G/N

type

2. #13628

C₂

semidirect

3. C₂×#1419

C₂

semidirect

4. #13245

C₂

semidirect

5. #7472

C₂

semidirect

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

subgroup	index
C₂²×D₃₂	2
#13628	2
C₂×#1419	2
#13245	2
#7472	2

subgroup

index

C₂²×D₃₂

Subgroups — #153314

Conjugacy classes of subgroups

Complements

Maximal subgroups