Group #513155 — subgroups

Subgroups — #513155

Order 512.

313 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 114 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	#33734	C₂	1	2	yes	maximal
2	#13704	C₂	1	4	yes	maximal, complemented
3	#13698	C₂	1	4	yes	maximal
4	#13688	C₂	1	4	yes	maximal, complemented
5	#13627	C₂	1	1	yes	maximal, characteristic
6	#3170	—	2	2	no	—
7	C₂×Q₆₄	C₂²	1	4	yes	—
8	C₂×M₆₄	C₂²	1	2	yes	—
9	#2252	C₂²	1	2	yes	—
10	#2247	C₂²	1	2	yes	—
11	#2241	—	2	2	no	—
12	#2216	—	2	2	no	—
13	#2216	C₂²	1	4	yes	—
14	C₄×Q₃₂	—	2	2	no	—
15	C₄×Q₃₂	C₂²	1	4	yes	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1424	C₂²	1	4	yes	—
18	C₄⋊C₃₂	C₂²	1	4	yes	—
19	#1416	C₂²	1	8	yes	—
20	#533	—	4	2	no	—
21	C₂×Q₃₂	—	2	2	no	—
22	C₂×Q₃₂	C₂³	1	4	yes	—
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂³	1	1	yes	characteristic
25	#500	—	2	2	no	—
26	#494	—	2	2	no	—
27	#487	—	4	2	no	—
28	#467	—	2	4	no	—
29	#467	—	4	2	no	—
30	C₄×Q₁₆	—	2	4	no	—
31	C₄×Q₁₆	—	4	2	no	—
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	Q₆₄	—	4	4	no	—
34	M₆₄	—	4	2	no	—
35	C₂×C₃₂	C₂³	1	4	yes	—
36	#366	—	2	2	no	—
37	#366	C₂³	1	4	yes	—
38	#358	—	2	4	no	—
39	#358	—	2	4	no	—
40	C₄×C₁₆	D₄	1	2	yes	—
41	C₄×C₁₆	C₂³	1	2	yes	—
42	C₂×Q₁₆	—	2	4	no	—
43	C₂×Q₁₆	—	4	2	no	—
44	C₂²×C₈	D₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Q₈	—	4	2	no	—
47	C₄⋊Q₈	—	4	2	no	—
48	#125	—	8	2	no	—
49	C₂²⋊Q₈	—	4	4	no	—
50	C₂²⋊Q₈	—	8	2	no	—
51	C₄×Q₈	—	4	4	no	—
52	C₄×Q₈	—	8	2	no	—
53	#117	—	8	2	no	—
54	C₂×C₄²	D₈	1	1	yes	characteristic
55	Q₃₂	—	2	4	no	—
56	Q₃₂	—	2	4	no	—
57	Q₃₂	—	4	4	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	4	1	no	—
61	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	C₂×D₄	1	2	yes	—
63	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
64	#107	—	2	4	no	—
65	#107	—	4	2	no	—
66	#103	—	4	4	no	—
67	#103	—	4	4	no	—
68	C₄×C₈	D₈	1	2	yes	—
69	C₄×C₈	C₂×D₄	1	2	yes	—
70	C₃₂	—	2	4	no	—
71	C₂×Q₈	—	4	4	no	—
72	C₂×Q₈	—	8	2	no	—
73	C₂²×C₄	D₁₆	1	2	yes	—
74	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
75	Q₁₆	—	4	4	no	—
76	Q₁₆	—	4	4	no	—
77	Q₁₆	—	8	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	C₂×D₈	1	1	yes	characteristic
82	C₂×C₈	C₂×D₈	1	2	yes	—
83	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
84	C₄⋊C₄	—	4	4	no	—
85	C₄⋊C₄	—	8	2	no	—
86	C₄⋊C₄	—	8	2	no	—
87	C₄⋊C₄	—	8	4	no	—
88	C₄⋊C₄	—	8	4	no	—
89	C₄⋊C₄	—	8	2	no	—
90	C₂²⋊C₄	—	8	4	no	—
91	C₄²	—	8	4	no	—
92	C₄²	D₁₆	1	1	yes	characteristic
93	C₄²	D₁₆	1	1	yes	characteristic
94	C₄²	C₂×D₈	1	2	yes	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	2	2	no	—
97	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
98	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
99	C₂³	C₂×D₁₆	1	1	yes	characteristic
100	Q₈	—	8	4	no	—
101	Q₈	—	8	4	no	—
102	Q₈	—	16	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	8	4	no	—
107	C₂×C₄	—	8	4	no	—
108	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
109	C₂×C₄	C₂×D₁₆	1	2	yes	—
110	C₂×C₄	C₂×D₁₆	1	2	yes	—
111	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	#537	1	1	yes	characteristic
115	C₈	#575	1	1	yes	characteristic
116	C₂²	—	2	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	C₂²×D₁₆	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	2	no	—
121	C₄	—	16	4	no	—
122	C₄	—	16	4	no	—
123	C₄	#2264	1	4	yes	—
124	C₄	#3245	1	1	yes	characteristic
125	C₄	#3414	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	#33811	1	1	yes	characteristic
128	C₂	C₂×#2264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33734

C₂

yes

maximal

#13704

C₂

yes

maximal, complemented

#13698

C₂

yes

maximal

#13688

C₂

yes

maximal, complemented

#13627

C₂

yes

maximal, characteristic

#3170

—

C₂×Q₆₄

C₂²

yes

—

C₂×M₆₄

C₂²

yes

—

#2252

C₂²

yes

—

#2247

C₂²

yes

—

#2241

—

#2216

—

#2216

C₂²

yes

—

C₄×Q₃₂

—

C₄×Q₃₂

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1424

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

—

#1416

C₂²

yes

—

#533

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂³

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

Q₆₄

—

M₆₄

—

C₂×C₃₂

C₂³

yes

—

#366

—

#366

C₂³

yes

—

#358

—

#358

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

D₁₆

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂²×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₁₆

yes

characteristic

C₄²

D₁₆

yes

characteristic

C₄²

C₂×D₈

yes

—

C₁₆

—

C₁₆

—

C₁₆

H(5,𝔽₂)

yes

characteristic

C₁₆

2₋¹⁺⁴

yes

characteristic

C₂³

C₂×D₁₆

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×D₁₆

yes

characteristic

109

C₂×C₄

C₂×D₁₆

yes

—

110

C₂×C₄

C₂×D₁₆

yes

—

111

C₂×C₄

C₂²×D₈

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

#537

yes

characteristic

115

C₈

#575

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

C₂²×D₁₆

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#2264

yes

—

124

C₄

#3245

yes

characteristic

125

C₄

#3414

yes

characteristic

126

C₂

—

127

C₂

#33811

yes

characteristic

128

C₂

C₂×#2264

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂⁴

prime radical

C₂³

C₂×D₁₆

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13704	C₂	semidirect
4. #13688	C₂	semidirect

H ≅ G/N

type

2. #13704

C₂

semidirect

4. #13688

C₂

semidirect

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

subgroup	index
#33734	2
#13704	2
#13698	2
#13688	2
#13627	2

subgroup

index

Subgroups — #513155

Conjugacy classes of subgroups

Complements

Maximal subgroups