Group #20274616 — subgroups

Subgroups — #20274616

Order 6784.

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

normal subgroups only

This group has 228 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×#13133152	C₂	1	2	yes	maximal
2	M₁₆×Dic₅₃	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810388	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133152	C₄	1	8	yes	complemented
10	#13133127	—	2	2	no	—
11	C₈×Dic₅₃	—	2	2	no	—
12	C₈×Dic₅₃	C₄	1	4	yes	—
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	Q₈	1	2	yes	—
17	M₁₆×C₅₃	—	2	2	no	—
18	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₄×Dic₅₃	D₄	1	1	yes	characteristic
24	C₄×Dic₅₃	D₄	1	1	yes	characteristic
25	#11719917	—	2	2	no	—
26	#11719917	C₂×C₄	1	4	yes	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	C₂×C₄	1	2	yes	—
29	C₂×#88322	C₂×C₄	1	4	yes	—
30	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	#41	1	2	yes	—
35	C₂×Dic₅₃	—	2	4	no	—
36	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
38	C₈×C₅₃	—	2	2	no	—
39	C₈×C₅₃	—	2	4	no	—
40	#88322	—	2	4	no	—
41	#88322	C₄²	1	4	yes	—
42	C₂²×C₅₃	—	2	2	no	—
43	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
45	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₅₃	#117	1	1	yes	characteristic
47	C₄×C₅₃	C₄×D₄	1	2	yes	—
48	Dic₅₃	—	2	2	no	—
49	Dic₅₃	#108	1	4	yes	complemented
50	C₄×#108	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	2	2	no	—
52	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅₃	C₂×#108	1	2	yes	—
54	C₂×#108	—	53	2	no	—
55	C₄×M₁₆	—	53	2	no	—
56	C₂×C₄×C₈	—	53	1	no	—
57	#341	—	53	1	no	—
58	#341	D₅₃	1	1	yes	characteristic
59	C₅₃	C₄×#108	1	1	yes	characteristic, complemented, Sylow
60	C₂×M₁₆	—	53	2	no	—
61	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
62	C₂²×C₈	—	53	1	no	—
63	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
64	C₂×C₄²	—	53	1	no	—
65	#108	—	53	8	no	—
66	#97	—	106	2	no	—
67	C₄×C₈	—	53	4	no	—
68	C₄×C₈	—	106	2	no	—
69	C₂²×C₄	—	53	2	no	—
70	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	53	4	no	—
73	M₁₆	—	106	2	no	—
74	M₁₆	C₄×D₅₃	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	53	2	no	—
78	C₂×C₈	—	53	4	no	—
79	C₂×C₈	—	106	2	no	—
80	C₂×C₈	C₄×D₅₃	1	2	yes	—
81	C₄²	—	53	2	no	—
82	C₄²	—	53	1	no	—
83	C₄²	—	53	1	no	—
84	C₂³	Q₈×D₅₃	1	1	yes	characteristic
85	C₂×C₄	—	53	2	no	—
86	C₂×C₄	—	53	2	no	—
87	C₂×C₄	—	106	4	no	—
88	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
90	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
91	C₂×C₄	#11719947	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	53	4	no	—
95	C₈	—	106	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
98	C₂²	#13133182	1	2	yes	—
99	C₄	—	53	4	no	—
100	C₄	—	106	2	no	—
101	C₄	C₄²×D₅₃	1	1	yes	characteristic
102	C₄	#13133173	1	1	yes	characteristic
103	C₄	#13133177	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#16810474	1	1	yes	characteristic
106	C₂	#108×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133152

C₂

yes

maximal

M₁₆×Dic₅₃

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810388

C₂

yes

maximal, characteristic

C₅₃×#341

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133152

C₄

yes

complemented

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₄

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

#11719917

—

#11719917

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₄²

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

—

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂×#108

yes

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

#341

D₅₃

yes

characteristic

C₅₃

C₄×#108

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

C₄×D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

106

—

C₂²

—

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

C₂²

#13133182

yes

—

C₄

—

100

C₄

—

106

—

101

C₄

C₄²×D₅₃

yes

characteristic

102

C₄

#13133173

yes

characteristic

103

C₄

#13133177

yes

—

104

C₂

—

105

C₂

#16810474

yes

characteristic

106

C₂

#108×D₅₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#341	D₅₃
hypocenter	C₅₃	C₄×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#341

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #13133152	C₄	semidirect
49. Dic₅₃	#108	semidirect
59. C₅₃	C₄×#108	semidirect

H ≅ G/N

type

9. #13133152

C₄

semidirect

49. Dic₅₃

#108

semidirect

59. C₅₃

C₄×#108

semidirect

Subgroups — #20274616

Conjugacy classes of subgroups

Complements