Group #20276478 — subgroups

Subgroups — C₂×#16810444 #20276478

Order 6784.

168 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 109 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×#13133157	C₂	1	2	yes	maximal
2	#16810444	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133157	—	2	2	no	—
5	#13133157	C₂²	1	4	yes	—
6	C₂×#11719908	—	2	2	no	—
7	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
10	C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₅₃	—	2	1	no	—
13	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₈	1	2	yes	—
15	C₂×C₈×C₅₃	C₈	1	2	yes	—
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄²×C₅₃	—	2	1	no	—
24	C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	#11719908	—	2	4	no	—
26	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	4	no	—
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₈×C₅₃	C₂×C₈	1	4	yes	—
38	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
42	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
43	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₅₃	—	2	4	no	—
45	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×#364	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	2	1	no	—
52	C₂×C₅₃	#364	1	2	yes	complemented
53	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂×M₃₂	—	53	2	no	—
55	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
56	#364	—	53	4	no	—
57	C₅₃	C₂×#364	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
60	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
61	M₃₂	—	53	4	no	—
62	M₃₂	—	106	2	no	—
63	C₂×C₁₆	—	106	2	no	—
64	C₄×C₈	Dic₅₃	1	2	yes	—
65	C₄×C₈	C₂×D₅₃	1	2	yes	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
68	C₂×C₈	#88322	1	2	yes	—
69	C₂×C₈	#88322	1	2	yes	—
70	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
71	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
72	C₂×C₈	D₂₁₂	1	1	yes	characteristic
73	C₂×C₈	D₂₁₂	1	1	yes	characteristic
74	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
75	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
76	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
77	C₄²	—	2	1	no	—
78	C₄²	C₂×Dic₅₃	1	2	yes	—
79	C₁₆	—	106	4	no	—
80	C₂³	#11719917	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	C₂×#88322	1	1	yes	characteristic
84	C₂×C₄	C₂×#88322	1	1	yes	characteristic
85	C₂×C₄	#11719917	1	1	yes	characteristic
86	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
87	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
88	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
89	C₈	C₂×#88322	1	4	yes	—
90	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
91	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
92	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
93	C₈	C₂×D₂₁₂	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊#88322	1	1	yes	characteristic
96	C₂²	C₄⋊#88322	1	1	yes	characteristic
97	C₂²	C₂×#11719917	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	C₄⋊#88322	1	1	yes	characteristic
100	C₄	C₄⋊#88322	1	1	yes	characteristic
101	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
102	C₄	C₂²×#88322	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#16810444	1	2	yes	direct product factor
106	C₂	C₂×C₄⋊#88322	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal

#16810444

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

#13133157

—

#13133157

C₂²

yes

—

C₂×#11719908

—

C₂²×C₈×C₅₃

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₈

yes

—

C₂×C₈×C₅₃

C₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

#11719908

—

C₂³×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×C₈

yes

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×#364

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#364

yes

complemented

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#364

—

C₅₃

C₂×#364

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

Dic₅₃

yes

characteristic

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

C₄×C₈

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

#88322

yes

—

C₂×C₈

#88322

yes

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₄²

C₂×Dic₅₃

yes

—

C₁₆

—

106

—

C₂³

#11719917

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

C₂×#88322

yes

—

C₈

C₄⋊Dic₅₃

yes

characteristic

C₈

C₄⋊Dic₅₃

yes

characteristic

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

C₂×D₂₁₂

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

C₄⋊#88322

yes

characteristic

100

C₄

C₄⋊#88322

yes

characteristic

101

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

102

C₄

C₂²×#88322

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#16810444

yes

direct product factor

106

C₂

C₂×C₄⋊#88322

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₄⋊C₈
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#364
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

D₂₁₂

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂²×C₅₃

C₄⋊C₈

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

C₂×#364

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16810444	C₂	direct (N × H)
52. C₂×C₅₃	#364	semidirect
57. C₅₃	C₂×#364	semidirect
105. C₂	#16810444	direct (N × H)

H ≅ G/N

type

2. #16810444

C₂

direct (N × H)

52. C₂×C₅₃

#364

semidirect

57. C₅₃

C₂×#364

semidirect

105. C₂

#16810444

direct (N × H)

Subgroups — C₂×#16810444 #20276478

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810444 #20276478

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810444 #20276478