Group #20273474 — subgroups

Subgroups — #20273474

Order 6784.

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

normal subgroups only

This group has 184 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	#16810647	C₂	1	2	yes	maximal
2	#16810376	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	2	no	—
5	C₄⋊#88322	—	2	2	no	—
6	C₄⋊#88322	C₂²	1	4	yes	—
7	C₂²×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₅₃	—	2	4	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₄²×C₅₃	Q₈	1	1	yes	characteristic
21	C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	#11719917	—	4	4	no	—
23	C₂×#88322	—	2	4	no	—
24	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	2	2	no	—
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	D₈	1	2	yes	—
31	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₈×C₅₃	—	2	2	no	—
36	C₈×C₅₃	M₁₆	1	4	yes	complemented
37	#88322	—	4	4	no	—
38	C₂²×C₅₃	—	2	1	no	—
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	#107	1	2	yes	—
41	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	2	1	no	—
44	C₄×C₅₃	#107	1	2	yes	—
45	C₄×C₅₃	#108	1	2	yes	—
46	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
47	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
48	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	#1567	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	2	no	—
51	C₂×C₅₃	#423	1	1	yes	characteristic
52	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
53	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
54	#423	—	53	2	no	—
55	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
56	#335	—	53	4	no	—
57	C₅₃	#1567	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₁₆	—	106	2	no	—
59	C₂²×C₈	Dic₅₃	1	2	yes	—
60	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
61	C₄⋊C₈	—	53	4	no	—
62	C₄⋊C₈	—	106	2	no	—
63	C₄×C₈	Dic₅₃	1	1	yes	characteristic
64	C₄×C₈	Dic₅₃	1	1	yes	characteristic
65	C₄×C₈	C₂×D₅₃	1	2	yes	—
66	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
67	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
68	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
69	M₁₆	—	212	4	no	—
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	106	4	no	—
72	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
73	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
74	C₄²	Dic₁₀₆	1	1	yes	characteristic
75	C₄²	D₂₁₂	1	1	yes	characteristic
76	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
77	C₄²	C₂²×D₅₃	1	1	yes	characteristic
78	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	D₄₂₄	1	2	yes	—
82	C₂×C₄	Dic₂₁₂	1	2	yes	—
83	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
84	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
85	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
86	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
88	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	212	4	no	—
91	C₈	#11719917	1	4	yes	complemented
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	#13133151	1	2	yes	—
95	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	#13133151	1	2	yes	—
99	C₄	#13133153	1	2	yes	—
100	C₄	C₂×#11719917	1	2	yes	—
101	C₄	C₂×D₄₂₄	1	1	yes	characteristic
102	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
103	C₂	—	2	2	no	—
104	C₂	#16810647	1	1	yes	characteristic
105	C₂	C₂×#13133151	1	1	yes	characteristic
106	C₂	C₂×#13133153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal

#16810376

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₂²×C₈×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊C₄

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

complemented

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#107

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#107

yes

—

C₄×C₅₃

#108

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

#1567

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#335

—

C₅₃

#1567

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

Dic₅₃

yes

characteristic

C₄×C₈

Dic₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄₂₄

yes

—

C₂×C₄

Dic₂₁₂

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

#11719917

yes

complemented

C₂²

—

C₂²

—

C₂²

#13133151

yes

—

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

C₄

—

C₄

—

C₄

#13133151

yes

—

C₄

#13133153

yes

—

100

C₄

C₂×#11719917

yes

—

101

C₄

C₂×D₄₂₄

yes

characteristic

102

C₄

C₂×Dic₂₁₂

yes

characteristic

103

C₂

—

104

C₂

#16810647

yes

characteristic

105

C₂

C₂×#13133151

yes

characteristic

106

C₂

C₂×#13133153

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1567
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

#1567

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16810376	C₂	semidirect
36. C₈×C₅₃	M₁₆	semidirect
57. C₅₃	#1567	semidirect
91. C₈	#11719917	semidirect

H ≅ G/N

type

2. #16810376

C₂

semidirect

36. C₈×C₅₃

M₁₆

semidirect

57. C₅₃

#1567

semidirect

91. C₈

#11719917

semidirect

Subgroups — #20273474

Conjugacy classes of subgroups

Complements