Group #141693 — subgroups

Subgroups — #141693

Order 512.

336 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 130 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	#19604	C₂	1	1	yes	maximal, characteristic
2	#13285	C₂	1	4	yes	maximal
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#8823	C₂	1	1	yes	maximal, characteristic
5	#3170	—	2	2	no	—
6	#3160	C₂²	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	#1848	C₂²	1	2	yes	—
9	#1713	C₄	1	2	yes	—
10	#1710	C₄	1	2	yes	—
11	#1707	—	2	2	no	—
12	#1567	—	2	2	no	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1382	—	2	8	no	—
15	C₈×C₁₆	—	2	2	no	—
16	#533	—	2	2	no	—
17	C₂²×C₁₆	D₄	1	4	yes	—
18	#500	—	2	2	no	—
19	#500	C₂×C₄	1	4	yes	—
20	#499	—	2	4	no	—
21	#498	—	2	2	no	—
22	#494	—	2	2	no	—
23	#487	—	2	4	no	—
24	#467	—	4	4	no	—
25	C₄×Q₁₆	—	4	4	no	—
26	C₂×#108	—	2	4	no	—
27	#427	—	2	4	no	—
28	#423	—	2	2	no	—
29	C₂×C₄×C₈	D₄	1	2	yes	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	#335	—	4	4	no	—
34	C₈²	C₂×C₄	1	1	yes	characteristic
35	C₈²	C₂×C₄	1	1	yes	characteristic
36	C₈²	D₄	1	2	yes	—
37	C₂×Q₁₆	—	4	4	no	—
38	C₂×M₁₆	—	4	2	no	—
39	C₂²×C₈	SD₁₆	1	4	yes	—
40	C₂²×C₈	C₂×D₄	1	2	yes	—
41	C₄⋊Q₈	—	2	4	no	—
42	#125	—	4	2	no	—
43	C₂²⋊Q₈	—	4	4	no	—
44	C₄×Q₈	—	4	4	no	—
45	#117	—	4	2	no	—
46	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
47	C₂×C₁₆	—	2	4	no	—
48	C₂×C₁₆	—	2	4	no	—
49	C₂×C₁₆	Q₁₆	1	8	yes	complemented
50	#108	—	4	8	no	—
51	#107	—	2	8	no	—
52	#107	—	4	4	no	—
53	#106	—	4	4	no	—
54	C₄⋊C₈	—	4	4	no	—
55	#103	—	4	8	no	—
56	C₄×C₈	C₂²⋊C₄	1	2	yes	—
57	C₄×C₈	C₂²⋊C₄	1	2	yes	—
58	C₄×C₈	D₈	1	4	yes	—
59	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₄×C₈	C₂×D₄	1	1	yes	characteristic
61	C₄×C₈	#41	1	2	yes	—
62	C₂×Q₈	—	4	4	no	—
63	C₂²×C₄	#127	1	1	yes	characteristic
64	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
65	Q₁₆	—	8	8	no	—
66	M₁₆	—	8	4	no	—
67	C₂×C₈	—	2	8	no	—
68	C₂×C₈	—	8	2	no	—
69	C₂×C₈	#102	1	4	yes	—
70	C₂×C₈	#102	1	4	yes	—
71	C₂×C₈	C₄×D₄	1	2	yes	—
72	C₂×C₈	#121	1	2	yes	—
73	C₂×C₈	C₂×Q₁₆	1	4	yes	—
74	C₂×C₈	#135	1	4	yes	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	8	4	no	—
78	C₂²⋊C₄	—	8	2	no	—
79	C₄²	—	8	2	no	—
80	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₄²	#124	1	1	yes	characteristic
82	C₄²	C₂×D₈	1	2	yes	—
83	C₁₆	—	2	8	no	—
84	C₂³	#492	1	1	yes	characteristic
85	Q₈	—	8	8	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	8	4	no	—
88	C₂×C₄	#390	1	1	yes	characteristic
89	C₂×C₄	C₂×#102	1	2	yes	—
90	C₂×C₄	#486	1	2	yes	—
91	C₂×C₄	#493	1	1	yes	characteristic
92	C₂×C₄	#494	1	2	yes	—
93	C₂×C₄	#495	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	—	16	2	no	—
96	C₈	#359	1	8	yes	—
97	C₈	C₄×Q₁₆	1	4	yes	—
98	C₈	#467	1	4	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	#1707	1	2	yes	—
101	C₂²	#1957	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	4	no	—
104	C₄	#1709	1	1	yes	characteristic
105	C₄	#1713	1	1	yes	characteristic
106	C₄	#1969	1	2	yes	—
107	C₄	C₂×#359	1	4	yes	—
108	C₂	—	2	2	no	—
109	C₂	#9825	1	1	yes	characteristic
110	C₂	#13208	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19604

C₂

yes

maximal, characteristic

#13285

C₂

yes

maximal

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

#3170

—

#3160

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

#1848

C₂²

yes

—

#1713

C₄

yes

—

#1710

C₄

yes

—

#1707

—

#1567

—

C₂×C₈²

C₂²

yes

characteristic

#1382

—

C₈×C₁₆

—

#533

—

C₂²×C₁₆

D₄

yes

—

#500

—

#500

C₂×C₄

yes

—

—

—

—

—

—

C₄×Q₁₆

—

C₂×#108

—

#427

—

#423

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

C₄⋊C₈

—

#103

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×C₂²⋊C₄

yes

characteristic

C₄²

#124

yes

characteristic

C₄²

C₂×D₈

yes

—

C₁₆

—

C₂³

#492

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#390

yes

characteristic

C₂×C₄

C₂×#102

yes

—

C₂×C₄

#486

yes

—

C₂×C₄

#493

yes

characteristic

C₂×C₄

#494

yes

—

C₂×C₄

#495

yes

—

C₈

—

C₈

—

C₈

#359

yes

—

C₈

C₄×Q₁₆

yes

—

C₈

#467

yes

—

C₂²

—

100

C₂²

#1707

yes

—

101

C₂²

#1957

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

#1709

yes

characteristic

105

C₄

#1713

yes

characteristic

106

C₄

#1969

yes

—

107

C₄

C₂×#359

yes

—

108

C₂

—

109

C₂

#9825

yes

characteristic

110

C₂

#13208

yes

—

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1957
prime radical	C₂³	#492
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
49. C₂×C₁₆	Q₁₆	semidirect

H ≅ G/N

type

49. C₂×C₁₆

Q₁₆

semidirect

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

subgroup	index
#19604	2
#13285	2
C₂×C₈×C₁₆	2
#8823	2

subgroup

index

#19604

#13285

C₂×C₈×C₁₆

#8823

Subgroups — #141693

Conjugacy classes of subgroups

Complements

Maximal subgroups