Group #8163 — subgroups

Subgroups — C₈×#101 #8163

Order 256.

227 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 115 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#101	C₂	1	1	yes	maximal, characteristic
3	#1461	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆	C₂	1	2	yes	maximal
5	#1283	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₈×Q₈	—	2	2	no	—
8	C₈×Q₈	—	2	2	no	—
9	#431	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#101	C₄	1	2	yes	—
13	C₄×M₁₆	C₂²	1	2	yes	—
14	C₂×C₄×C₈	C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#343	C₄	1	2	yes	—
17	#324	C₄	1	4	yes	—
18	#322	—	2	2	no	—
19	C₈²	—	2	2	no	—
20	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈	—	2	2	no	—
26	C₄×Q₈	—	2	2	no	—
27	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²	C₂×C₄	1	2	yes	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	C₄⋊C₈	—	2	4	no	—
32	C₄⋊C₈	—	2	2	no	—
33	C₄⋊C₈	—	2	2	no	—
34	#101	C₈	1	8	yes	direct product factor
35	C₂²⋊C₈	C₂×C₄	1	4	yes	—
36	#97	—	2	2	no	—
37	C₄×C₈	—	2	2	no	—
38	C₄×C₈	—	2	2	no	—
39	C₄×C₈	—	2	2	no	—
40	C₄×C₈	—	2	2	no	—
41	C₄×C₈	D₄	1	2	yes	—
42	C₄×C₈	D₄	1	2	yes	—
43	C₂×Q₈	—	2	2	no	—
44	C₂×Q₈	—	2	2	no	—
45	C₂×Q₈	C₂×C₈	1	4	yes	—
46	C₂²×C₄	C₄²	1	2	yes	—
47	C₂²×C₄	C₄²	1	2	yes	—
48	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
51	M₁₆	—	2	4	no	—
52	M₁₆	C₂×C₈	1	8	yes	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	—	4	2	no	—
59	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₄⋊C₄	—	2	4	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄²	—	2	2	no	—
65	C₄²	—	2	2	no	—
66	C₄²	C₂×D₄	1	2	yes	—
67	C₄²	#41	1	2	yes	—
68	C₂³	C₂×C₄²	1	1	yes	characteristic
69	Q₈	—	2	8	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	C₄×C₈	1	4	yes	—
78	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄	#117	1	1	yes	characteristic
80	C₂×C₄	C₄×D₄	1	2	yes	—
81	C₂×C₄	C₄×D₄	1	2	yes	—
82	C₂×C₄	C₂²×C₈	1	2	yes	—
83	C₂×C₄	#131	1	2	yes	—
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	2	8	no	—
87	C₈	—	4	4	no	—
88	C₈	—	4	2	no	—
89	C₈	#101	1	2	yes	direct product factor
90	C₂²	—	2	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²	#405	1	1	yes	characteristic
95	C₄	—	2	4	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	2	no	—
98	C₄	—	4	2	no	—
99	C₄	C₂×#101	1	1	yes	characteristic
100	C₄	#413	1	1	yes	characteristic
101	C₄	C₈×D₄	1	4	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	1	no	—
104	C₂	C₈×C₂²⋊C₄	1	1	yes	characteristic
105	C₂	C₄×#101	1	1	yes	characteristic
106	C₂	#1757	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

C₄×#101

C₂

yes

maximal, characteristic

#1461

C₂

yes

maximal, characteristic

C₈×M₁₆

C₂

yes

maximal

#1283

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#431

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#101

C₄

yes

—

C₄×M₁₆

C₂²

yes

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#343

C₄

yes

—

#324

C₄

yes

—

#322

—

C₈²

—

C₂²×Q₈

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

C₈

yes

direct product factor

C₂²⋊C₈

C₂×C₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₈

yes

—

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₄²

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

C₂×C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₄²

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#101

yes

direct product factor

C₂²

—

C₂²

—

C₂²

C₄×C₂²⋊C₄

yes

characteristic

C₂²

C₂×C₄×C₈

yes

characteristic

C₂²

#405

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×#101

yes

characteristic

100

C₄

#413

yes

characteristic

101

C₄

C₈×D₄

yes

—

102

C₂

—

103

C₂

—

104

C₂

C₈×C₂²⋊C₄

yes

characteristic

105

C₂

C₄×#101

yes

characteristic

106

C₂

#1757

yes

characteristic

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

subgroup

G/H

center

C₂×C₈

C₂²⋊C₄

commutator

C₂²

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

C₄×C₂²⋊C₄

prime radical

C₂³

C₂×C₄²

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
34. #101	C₈	direct (N × H)
89. C₈	#101	direct (N × H)

H ≅ G/N

type

34. #101

C₈

direct (N × H)

89. C₈

#101

direct (N × H)

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

subgroup	index
C₂×C₈×Q₈	2
C₄×#101	2
#1461	2
C₈×M₁₆	2
#1283	2

subgroup

index

C₂×C₈×Q₈

C₄×#101

#1461

C₈×M₁₆

#1283

Subgroups — C₈×#101 #8163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#101 #8163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#101 #8163