Group #20276467 — subgroups

Subgroups — C₂×#16810441 #20276467

Order 6784.

184 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 117 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×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810441	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719908	—	2	2	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	C₈×Dic₅₃	C₄	1	2	yes	—
10	C₈×Dic₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	2	yes	—
19	C₂×C₈×C₅₃	Q₈	1	2	yes	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×Dic₅₃	C₈	1	2	yes	—
22	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₂×#88322	—	2	2	no	—
24	C₂×#88322	C₂×C₄	1	2	yes	—
25	C₁₆×C₅₃	—	2	2	no	—
26	#11719908	—	2	2	no	—
27	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅₃	—	2	2	no	—
33	C₂×Dic₅₃	C₁₆	1	4	yes	complemented
34	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
35	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
36	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	#88322	—	2	2	no	—
40	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₂²×C₅₃	C₂×C₁₆	1	2	yes	—
42	C₂²×C₅₃	M₃₂	1	2	yes	—
43	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
45	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
47	Dic₅₃	—	2	2	no	—
48	Dic₅₃	C₂×C₁₆	1	4	yes	complemented
49	C₂×C₄⋊C₁₆	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	C₄⋊C₁₆	1	4	yes	complemented
51	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
53	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
54	C₂²×C₁₆	—	53	1	no	—
55	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
56	C₂×C₄×C₈	—	53	1	no	—
57	C₄⋊C₁₆	—	53	4	no	—
58	C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	53	1	no	—
60	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂×C₄²	—	53	1	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	53	2	no	—
64	C₂×C₁₆	—	106	2	no	—
65	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
66	C₄×C₈	—	53	2	no	—
67	C₄×C₈	—	53	2	no	—
68	C₂²×C₄	—	53	1	no	—
69	C₂²×C₄	—	53	1	no	—
70	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
71	C₂×C₈	—	53	2	no	—
72	C₂×C₈	—	106	2	no	—
73	C₂×C₈	Dic₁₀₆	1	2	yes	—
74	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
75	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
76	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
77	C₄²	—	53	2	no	—
78	C₄²	—	53	2	no	—
79	C₁₆	—	2	2	no	—
80	C₁₆	—	106	2	no	—
81	C₂³	C₈×D₅₃	1	1	yes	characteristic
82	C₂×C₄	—	53	4	no	—
83	C₂×C₄	—	53	2	no	—
84	C₂×C₄	—	53	2	no	—
85	C₂×C₄	—	106	2	no	—
86	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
87	C₂×C₄	#11719912	1	2	yes	—
88	C₂×C₄	#11719919	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
90	C₈	—	106	2	no	—
91	C₈	#11719919	1	2	yes	—
92	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
93	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
94	C₂²	#13133146	1	2	yes	—
95	C₂²	C₁₆×D₅₃	1	2	yes	—
96	C₂²	#13133156	1	2	yes	—
97	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
98	C₄	—	53	4	no	—
99	C₄	—	106	2	no	—
100	C₄	#13133146	1	2	yes	—
101	C₄	C₂×#11719919	1	1	yes	characteristic
102	C₄	C₂×#11719912	1	1	yes	characteristic
103	C₂	#16810441	1	4	yes	direct product factor
104	C₂	C₂×#13133146	1	1	yes	characteristic
105	C₂	C₂×C₁₆×D₅₃	1	1	yes	characteristic
106	C₂	C₂×#13133156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719908

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810441

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂×#11719908

—

C₂×#11719908

C₂²

yes

—

C₈×Dic₅₃

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₈

yes

characteristic

C₂²×Dic₅₃

C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₈

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₁₆

yes

complemented

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

—

C₂²×C₅₃

C₄⋊C₈

yes

—

C₂²×C₅₃

C₂×C₁₆

yes

—

C₂²×C₅₃

M₃₂

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂×C₁₆

yes

complemented

C₂×C₄⋊C₁₆

—

maximal, Sylow

C₂×C₅₃

C₄⋊C₁₆

yes

complemented

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₅₃

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₅₃

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

C₂³

C₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

#11719912

yes

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

106

—

C₈

#11719919

yes

—

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₂²

#13133146

yes

—

C₂²

C₁₆×D₅₃

yes

—

C₂²

#13133156

yes

—

C₂²

C₂×C₈×D₅₃

yes

characteristic

C₄

—

C₄

—

106

—

100

C₄

#13133146

yes

—

101

C₄

C₂×#11719919

yes

characteristic

102

C₄

C₂×#11719912

yes

characteristic

103

C₂

#16810441

yes

direct product factor

104

C₂

C₂×#13133146

yes

characteristic

105

C₂

C₂×C₁₆×D₅₃

yes

characteristic

106

C₂

C₂×#13133156

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₈
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²×C₈

C₂×D₅₃

commutator

C₂×C₅₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂²×C₁₆×C₅₃

C₂

socle

C₂³×C₅₃

C₂×C₈

hypercenter

C₂²×C₁₆

D₅₃

hypocenter

C₅₃

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #16810441	C₂	direct (N × H)
33. C₂×Dic₅₃	C₁₆	semidirect
48. Dic₅₃	C₂×C₁₆	semidirect
50. C₂×C₅₃	C₄⋊C₁₆	semidirect
58. C₅₃	C₂×C₄⋊C₁₆	semidirect
103. C₂	#16810441	direct (N × H)

H ≅ G/N

type

3. #16810441

C₂

direct (N × H)

33. C₂×Dic₅₃

C₁₆

semidirect

48. Dic₅₃

C₂×C₁₆

semidirect

50. C₂×C₅₃

C₄⋊C₁₆

semidirect

58. C₅₃

C₂×C₄⋊C₁₆

semidirect

103. C₂

#16810441

direct (N × H)

Subgroups — C₂×#16810441 #20276467

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810441 #20276467

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810441 #20276467