Group #19128367 — subgroups

Subgroups — D₄×Dic₁₆₆ #19128367

Order 5312.

280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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×Dic₁₆₆	C₂	1	2	yes	maximal, complemented
2	#14581792	C₂	1	1	yes	maximal, characteristic, complemented
3	#14581791	C₂	1	2	yes	maximal, complemented
4	#14581771	C₂	1	2	yes	maximal, complemented
5	#14581769	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Dic₈₃	C₂	1	2	yes	maximal
8	C₄×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₁₆₆	—	2	8	no	—
10	C₂×Dic₁₆₆	C₂²	1	4	yes	—
11	C₂×Dic₁₆₆	C₂²	1	2	yes	—
12	C₂×Dic₁₆₆	C₂²	1	1	yes	characteristic
13	C₂²×Dic₈₃	C₂²	1	4	yes	—
14	C₄⋊Dic₈₃	C₂²	1	2	yes	—
15	C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₈₃	C₂²	1	2	yes	—
17	Dic₈₃⋊C₄	C₂²	1	2	yes	—
18	Dic₈₃⋊C₄	C₂²	1	4	yes	—
19	C₂²⋊Dic₈₃	C₂²	1	4	yes	—
20	C₄×Dic₈₃	C₂²	1	2	yes	—
21	C₂×D₄×C₈₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	C₂²	1	2	yes	—
23	C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
25	C₄²×C₈₃	C₂²	1	1	yes	characteristic
26	C₂³×C₈₃	C₂³	1	2	yes	—
27	D₄×C₈₃	Q₈	1	4	yes	complemented
28	C₂×C₄×C₈₃	—	2	4	no	—
29	C₂×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₈₃	C₂³	1	2	yes	—
31	C₂×C₄×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₈₃	—	2	8	no	—
34	C₂×Dic₈₃	C₂³	1	2	yes	—
35	C₂×Dic₈₃	C₂³	1	4	yes	—
36	C₂×Dic₈₃	C₂³	1	2	yes	—
37	Dic₁₆₆	—	2	4	no	—
38	Dic₁₆₆	—	2	8	no	—
39	Dic₁₆₆	D₄	1	4	yes	direct product factor
40	C₂²×C₈₃	—	2	2	no	—
41	C₂²×C₈₃	—	2	2	no	—
42	C₂²×C₈₃	C₂×Q₈	1	4	yes	complemented
43	C₂²×C₈₃	C₂⁴	1	1	yes	characteristic
44	C₄×C₈₃	—	2	2	no	—
45	C₄×C₈₃	—	2	1	no	—
46	C₄×C₈₃	C₂×D₄	1	2	yes	—
47	C₄×C₈₃	C₂×Q₈	1	2	yes	complemented
48	Dic₈₃	—	2	4	no	—
49	Dic₈₃	—	2	2	no	—
50	Dic₈₃	C₂×D₄	1	4	yes	—
51	C₂×C₈₃	—	2	4	no	—
52	C₂×C₈₃	C₂²×D₄	1	1	yes	characteristic
53	C₂×C₈₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
55	C₈₃	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
56	D₄×Q₈	—	83	1	no	maximal, Sylow
57	C₂²×Q₈	—	83	2	no	—
58	C₄⋊Q₈	—	83	2	no	—
59	C₄⋊Q₈	—	83	1	no	—
60	C₂²⋊Q₈	—	83	2	no	—
61	C₂²⋊Q₈	—	83	4	no	—
62	C₄×Q₈	—	83	1	no	—
63	C₄×D₄	—	83	2	no	—
64	C₄×D₄	D₈₃	1	1	yes	characteristic
65	C₂×Q₈	—	83	1	no	—
66	C₂×Q₈	—	83	4	no	—
67	C₂×Q₈	—	83	2	no	—
68	C₂×Q₈	—	166	8	no	—
69	C₂×D₄	C₂×D₈₃	1	1	yes	characteristic
70	C₂²×C₄	—	83	4	no	—
71	C₂²×C₄	C₂×D₈₃	1	2	yes	—
72	C₄⋊C₄	—	83	2	no	—
73	C₄⋊C₄	—	83	2	no	—
74	C₄⋊C₄	—	83	1	no	—
75	C₄⋊C₄	—	83	4	no	—
76	C₄⋊C₄	—	83	2	no	—
77	C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
78	C₂²⋊C₄	—	83	4	no	—
79	C₂²⋊C₄	C₂×D₈₃	1	2	yes	—
80	C₄²	—	83	2	no	—
81	C₄²	C₂×D₈₃	1	1	yes	characteristic
82	C₂³	C₂²×D₈₃	1	2	yes	—
83	Q₈	—	83	4	no	—
84	Q₈	—	166	8	no	—
85	Q₈	—	166	4	no	—
86	D₄	Dic₁₆₆	1	4	yes	direct product factor
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	83	2	no	—
89	C₂×C₄	—	83	2	no	—
90	C₂×C₄	—	83	4	no	—
91	C₂×C₄	—	166	8	no	—
92	C₂×C₄	C₂²×D₈₃	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₈₃	1	2	yes	—
94	C₂×C₄	C₂²×D₈₃	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₈₃	1	1	yes	characteristic
96	C₂²	—	2	2	no	—
97	C₂²	—	2	2	no	—
98	C₂²	C₂×Dic₁₆₆	1	4	yes	complemented
99	C₂²	C₂³×D₈₃	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	1	no	—
102	C₄	—	83	4	no	—
103	C₄	—	166	4	no	—
104	C₄	—	166	2	no	—
105	C₄	D₄×D₈₃	1	2	yes	—
106	C₄	C₂×Dic₁₆₆	1	2	yes	complemented
107	C₂	—	2	4	no	—
108	C₂	C₂×D₄×D₈₃	1	1	yes	characteristic
109	C₂	C₂²×Dic₁₆₆	1	1	yes	characteristic
110	C₂	#14581839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₆₆

C₂

yes

maximal, complemented

#14581792

C₂

yes

maximal, characteristic, complemented

#14581791

C₂

yes

maximal, complemented

#14581771

C₂

yes

maximal, complemented

#14581769

C₂

yes

maximal, complemented

C₄×Dic₁₆₆

C₂

yes

maximal, characteristic, complemented

D₄×Dic₈₃

C₂

yes

maximal

C₄×D₄×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂²

yes

—

C₂×Dic₁₆₆

C₂²

yes

—

C₂×Dic₁₆₆

C₂²

yes

characteristic

C₂²×Dic₈₃

C₂²

yes

—

C₄⋊Dic₈₃

C₂²

yes

—

C₄⋊Dic₈₃

C₂²

yes

characteristic

C₄⋊Dic₈₃

C₂²

yes

—

Dic₈₃⋊C₄

C₂²

yes

—

Dic₈₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₈₃

C₂²

yes

—

C₄×Dic₈₃

C₂²

yes

—

C₂×D₄×C₈₃

C₂²

yes

characteristic

C₂²×C₄×C₈₃

C₂²

yes

—

C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂²⋊C₄×C₈₃

C₂²

yes

—

C₄²×C₈₃

C₂²

yes

characteristic

C₂³×C₈₃

C₂³

yes

—

D₄×C₈₃

Q₈

yes

complemented

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₄×C₈₃

C₂³

yes

—

C₂×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₄×C₈₃

C₂³

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂³

yes

—

C₂×Dic₈₃

C₂³

yes

—

C₂×Dic₈₃

C₂³

yes

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

D₄

yes

direct product factor

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×Q₈

yes

complemented

C₂²×C₈₃

C₂⁴

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×D₄

yes

—

C₄×C₈₃

C₂×Q₈

yes

complemented

Dic₈₃

—

Dic₈₃

—

Dic₈₃

C₂×D₄

yes

—

C₂×C₈₃

—

C₂×C₈₃

C₂²×D₄

yes

characteristic

C₂×C₈₃

C₂²×Q₈

yes

characteristic

C₂×C₈₃

2₋¹⁺⁴

yes

characteristic

C₈₃

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

maximal, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×D₄

C₂×D₈₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₈₃

yes

—

C₄²

—

C₄²

C₂×D₈₃

yes

characteristic

C₂³

C₂²×D₈₃

yes

—

Q₈

—

Q₈

—

166

—

Q₈

—

166

—

D₄

Dic₁₆₆

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

166

—

C₂×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₄

C₂²×D₈₃

yes

—

C₂×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₄

C₂²×D₈₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₁₆₆

yes

complemented

C₂²

C₂³×D₈₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

166

—

104

C₄

—

166

—

105

C₄

D₄×D₈₃

yes

—

106

C₄

C₂×Dic₁₆₆

yes

complemented

107

C₂

—

108

C₂

C₂×D₄×D₈₃

yes

characteristic

109

C₂

C₂²×Dic₁₆₆

yes

characteristic

110

C₂

#14581839

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂⁴
Fitting	C₄×D₄×C₈₃	C₂
hypercenter	C₄×D₄	D₈₃
hypocenter	C₈₃	D₄×Q₈
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₃

commutator, socle

C₂²×C₈₃

C₂⁴

Fitting

C₄×D₄×C₈₃

C₂

hypercenter

C₄×D₄

D₈₃

hypocenter

C₈₃

D₄×Q₈

LCS

G ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂²×Dic₁₆₆	C₂	semidirect
2. #14581792	C₂	semidirect
3. #14581791	C₂	semidirect
4. #14581771	C₂	semidirect
5. #14581769	C₂	semidirect
6. C₄×Dic₁₆₆	C₂	semidirect
27. D₄×C₈₃	Q₈	semidirect
39. Dic₁₆₆	D₄	direct (N × H)
42. C₂²×C₈₃	C₂×Q₈	semidirect
47. C₄×C₈₃	C₂×Q₈	semidirect
55. C₈₃	D₄×Q₈	semidirect
86. D₄	Dic₁₆₆	direct (N × H)
98. C₂²	C₂×Dic₁₆₆	semidirect
106. C₄	C₂×Dic₁₆₆	semidirect

H ≅ G/N

type

1. C₂²×Dic₁₆₆

C₂

semidirect

2. #14581792

C₂

semidirect

3. #14581791

C₂

semidirect

4. #14581771

C₂

semidirect

5. #14581769

C₂

semidirect

6. C₄×Dic₁₆₆

C₂

semidirect

27. D₄×C₈₃

Q₈

semidirect

39. Dic₁₆₆

D₄

direct (N × H)

42. C₂²×C₈₃

C₂×Q₈

semidirect

47. C₄×C₈₃

C₂×Q₈

semidirect

55. C₈₃

D₄×Q₈

semidirect

86. D₄

Dic₁₆₆

direct (N × H)

98. C₂²

C₂×Dic₁₆₆

semidirect

106. C₄

C₂×Dic₁₆₆

semidirect

Subgroups — D₄×Dic₁₆₆ #19128367

Conjugacy classes of subgroups

Complements

Subgroups — D4×Dic166 #19128367

Conjugacy classes of subgroups

Complements

Subgroups — D₄×Dic₁₆₆ #19128367