Group #19133419 — subgroups

Subgroups — #19133419

Order 5504.

192 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 87 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	D₄×C₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
2	#14582650	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#14582620	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₈	—	2	2	no	—
7	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
9	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₄₃	C₄	1	1	yes	characteristic
12	C₄×D₄×C₄₃	C₂²	1	2	yes	—
13	#107×C₄₃	C₄	1	1	yes	characteristic
14	#102×C₄₃	C₄	1	2	yes	—
15	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₄₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₄₃	—	2	2	no	—
18	D₈×C₄₃	C₈	1	4	yes	complemented
19	C₂×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
22	C₂²⋊C₄×C₄₃	—	2	2	no	—
23	C₄²×C₄₃	C₂³	1	1	yes	characteristic
24	C₂×C₄₃⋊C₈	—	2	2	no	—
25	C₂×C₄₃⋊C₈	—	2	1	no	—
26	C₂×C₄₃⋊C₈	—	4	4	no	—
27	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
28	C₂³×C₄₃	—	2	2	no	—
29	D₄×C₄₃	—	2	2	no	—
30	D₄×C₄₃	C₂×C₈	1	4	yes	complemented
31	C₂×C₄×C₄₃	—	2	2	no	—
32	C₂×C₄×C₄₃	—	4	2	no	—
33	C₂×C₄×C₄₃	—	4	2	no	—
34	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
37	C₈×C₄₃	—	2	1	no	—
38	C₈×C₄₃	C₂×C₈	1	2	yes	complemented
39	C₄₃⋊C₈	—	2	2	no	—
40	C₄₃⋊C₈	—	4	1	no	—
41	C₄₃⋊C₈	—	4	2	no	—
42	C₂²×C₄₃	—	2	4	no	—
43	C₂²×C₄₃	—	4	2	no	—
44	C₂²×C₄₃	—	4	2	no	—
45	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
46	C₄×C₄₃	—	2	1	no	—
47	C₄×C₄₃	—	4	2	no	—
48	C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₄₃	#131	1	1	yes	characteristic
50	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
51	#1580	—	43	1	no	maximal, Sylow
52	C₂×C₄₃	—	4	4	no	—
53	C₂×C₄₃	C₈×D₄	1	1	yes	characteristic
54	C₂×C₄₃	#442	1	1	yes	characteristic
55	C₂×C₄₃	#444	1	1	yes	characteristic
56	C₄×D₈	D₄₃	1	1	yes	characteristic
57	C₈×D₄	—	43	2	no	—
58	#334	—	43	1	no	—
59	D₄⋊C₈	—	43	2	no	—
60	#322	—	43	1	no	—
61	C₄₃	#1580	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	Dic₄₃	1	1	yes	characteristic
63	C₂²×C₈	—	86	2	no	—
64	C₄×D₄	C₂×D₄₃	1	2	yes	—
65	#107	Dic₄₃	1	1	yes	characteristic
66	C₄⋊C₈	—	43	2	no	—
67	#102	Dic₄₃	1	2	yes	—
68	C₂²⋊C₈	—	86	2	no	—
69	C₄×C₈	—	43	1	no	—
70	C₄×C₈	—	43	1	no	—
71	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₄₃	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	C₄₃⋊C₈	1	4	yes	complemented
75	C₂×C₈	—	43	2	no	—
76	C₂×C₈	—	86	2	no	—
77	C₂×C₈	—	86	1	no	—
78	C₂×C₈	—	172	4	no	—
79	C₂×C₈	C₂×Dic₄₃	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₄₃	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₄₃	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×C₄₃⋊C₈	1	4	yes	complemented
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
91	C₂×C₄	#10621900	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	86	2	no	—
95	C₈	—	172	1	no	—
96	C₈	—	172	2	no	—
97	C₈	C₂×C₄₃⋊C₈	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₄₃	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
105	C₄	#13063332	1	1	yes	characteristic
106	C₄	#13063349	1	2	yes	—
107	C₂	—	4	4	no	—
108	C₂	D₄×C₄₃⋊C₈	1	1	yes	characteristic
109	C₂	#14583011	1	1	yes	characteristic
110	C₂	#14583021	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×C₄₃⋊C₈

C₂

yes

maximal, complemented

#14582650

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, complemented

#14582620

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₄₃

C₄

yes

characteristic

C₄×D₄×C₄₃

C₂²

yes

—

#107×C₄₃

C₄

yes

characteristic

#102×C₄₃

C₄

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×D₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

—

D₈×C₄₃

C₈

yes

complemented

C₂×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

C₂×C₈

yes

complemented

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

C₂×C₈

yes

complemented

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²×C₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

#1580

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₈×D₄

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₄×D₈

D₄₃

yes

characteristic

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

—

C₄₃

#1580

yes

characteristic, complemented, Sylow

C₂×D₈

Dic₄₃

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₂×D₄₃

yes

—

#107

Dic₄₃

yes

characteristic

C₄⋊C₈

—

#102

Dic₄₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×D₄

C₂×Dic₄₃

yes

—

C₂²×C₄

—

D₈

C₄₃⋊C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

C₂×Dic₄₃

yes

characteristic

C₂×C₈

C₂×Dic₄₃

yes

characteristic

C₄⋊C₄

C₂×Dic₄₃

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₄₃

yes

characteristic

C₂³

—

D₄

—

D₄

C₂×C₄₃⋊C₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₄₃

yes

characteristic

C₂×C₄

#10621900

yes

characteristic

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

C₈

—

C₈

—

C₈

—

172

—

C₈

—

172

—

C₈

C₂×C₄₃⋊C₈

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₄₃

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂²×C₄₃⋊C₈

yes

characteristic

105

C₄

#13063332

yes

characteristic

106

C₄

#13063349

yes

—

107

C₂

—

108

C₂

D₄×C₄₃⋊C₈

yes

characteristic

109

C₂

#14583011

yes

characteristic

110

C₂

#14583021

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₈
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄×D₈×C₄₃	C₂
socle	C₂²×C₄₃	C₄×D₄
hypercenter	C₄×D₈	D₄₃
hypocenter	C₄₃	#1580
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₄₃

commutator

C₄×C₄₃

C₂²×C₈

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄×D₈×C₄₃

C₂

socle

C₂²×C₄₃

C₄×D₄

hypercenter

C₄×D₈

D₄₃

hypocenter

C₄₃

#1580

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. D₄×C₄₃⋊C₈	C₂	semidirect
2. #14582650	C₂	semidirect
3. D₄⋊(C₄₃⋊C₈)	C₂	semidirect
4. #14582620	C₂	semidirect
18. D₈×C₄₃	C₈	semidirect
30. D₄×C₄₃	C₂×C₈	semidirect
38. C₈×C₄₃	C₂×C₈	semidirect
61. C₄₃	#1580	semidirect
74. D₈	C₄₃⋊C₈	semidirect
86. D₄	C₂×C₄₃⋊C₈	semidirect
97. C₈	C₂×C₄₃⋊C₈	semidirect

H ≅ G/N

type

1. D₄×C₄₃⋊C₈

C₂

semidirect

2. #14582650

C₂

semidirect

3. D₄⋊(C₄₃⋊C₈)

C₂

semidirect

4. #14582620

C₂

semidirect

18. D₈×C₄₃

C₈

semidirect

30. D₄×C₄₃

C₂×C₈

semidirect

38. C₈×C₄₃

C₂×C₈

semidirect

61. C₄₃

#1580

semidirect

74. D₈

C₄₃⋊C₈

semidirect

86. D₄

C₂×C₄₃⋊C₈

semidirect

97. C₈

C₂×C₄₃⋊C₈

semidirect

Subgroups — #19133419

Conjugacy classes of subgroups

Complements