Group #20273856 — subgroups

Subgroups — #20273856

Order 6784.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811353	C₂	1	1	yes	maximal, characteristic
2	#16811002	C₂	1	1	yes	maximal, characteristic
3	#16810919	C₂	1	2	yes	maximal
4	#16810353	C₂	1	2	yes	maximal
5	C₅₃×#333	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719925	—	2	2	no	—
7	#13133217	C₂²	1	1	yes	characteristic
8	#13133216	—	2	2	no	—
9	#13133215	—	2	2	no	—
10	C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	#13133182	—	2	2	no	—
12	#13133151	—	2	2	no	—
13	#13133142	—	2	2	no	—
14	C₄×#88322	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₅₃	D₄	1	2	yes	—
18	C₂×Dic₁₀₆	—	2	1	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₁₀₆	D₄	1	2	yes	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#11719925	—	4	4	no	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	D₄	1	2	yes	—
27	#11719919	—	2	4	no	—
28	#11719919	—	2	2	no	—
29	C₄×Dic₅₃	—	2	2	no	—
30	C₄×Dic₅₃	—	2	2	no	—
31	C₂×#88322	—	2	1	no	—
32	Q₈×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
36	C₂×Dic₅₃	—	2	2	no	—
37	C₂×Dic₅₃	—	2	2	no	—
38	C₂×Dic₅₃	—	4	1	no	—
39	Dic₁₀₆	—	2	2	no	—
40	Dic₁₀₆	—	4	2	no	—
41	Dic₁₀₆	—	8	2	no	—
42	Dic₁₀₆	Q₁₆	1	4	yes	complemented
43	C₈×C₅₃	—	4	2	no	—
44	#88322	—	2	2	no	—
45	C₂²×C₅₃	#120	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	4	2	no	—
48	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
49	C₄×C₅₃	#137	1	2	yes	—
50	Dic₅₃	—	2	4	no	—
51	Dic₅₃	—	4	2	no	—
52	Dic₅₃	—	8	1	no	—
53	#1633	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	#451	1	2	yes	—
55	C₂×C₅₃	#453	1	1	yes	characteristic
56	Q₈²	—	53	1	no	—
57	#494	—	53	1	no	—
58	#477	—	53	2	no	—
59	#333	D₅₃	1	1	yes	characteristic
60	Q₈⋊C₈	—	53	2	no	—
61	C₅₃	#1633	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	106	2	no	—
63	C₄⋊Q₈	—	53	1	no	—
64	C₄⋊Q₈	—	106	2	no	—
65	C₄⋊Q₈	—	106	2	no	—
66	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
67	C₄×Q₈	—	53	2	no	—
68	C₄×Q₈	—	106	2	no	—
69	#107	—	106	2	no	—
70	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
71	#103	—	106	2	no	—
72	C₄×C₈	—	53	1	no	—
73	C₂×Q₈	—	53	2	no	—
74	C₂×Q₈	—	106	2	no	—
75	C₂×Q₈	—	106	1	no	—
76	C₂×Q₈	D₂₁₂	1	2	yes	—
77	Q₁₆	—	212	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	106	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	53	2	no	—
82	C₄⋊C₄	—	106	4	no	—
83	C₄⋊C₄	—	106	2	no	—
84	C₄⋊C₄	—	106	2	no	—
85	C₄²	—	106	2	no	—
86	C₄²	—	106	2	no	—
87	C₄²	C₂²×D₅₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	53	4	no	—
90	Q₈	—	106	2	no	—
91	Q₈	—	424	2	no	—
92	Q₈	—	212	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	106	2	no	—
95	C₂×C₄	—	106	2	no	—
96	C₂×C₄	—	212	1	no	—
97	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
98	C₂×C₄	D₄×D₅₃	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	106	2	no	—
101	C₂²	#13133187	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	106	4	no	—
105	C₄	—	424	1	no	—
106	C₄	—	212	2	no	—
107	C₄	Q₁₆×D₅₃	1	2	yes	—
108	C₄	#13133246	1	2	yes	—
109	C₂	#16810780	1	2	yes	—
110	C₂	#16810793	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811353

C₂

yes

maximal, characteristic

#16811002

C₂

yes

maximal, characteristic

#16810919

C₂

yes

maximal

#16810353

C₂

yes

maximal

C₅₃×#333

C₂

yes

maximal, characteristic

C₂×#11719925

—

#13133217

C₂²

yes

characteristic

#13133216

—

#13133215

—

C₄×Dic₁₀₆

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

Q₁₆

yes

complemented

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1633

—

maximal, Sylow

C₂×C₅₃

#451

yes

—

C₂×C₅₃

#453

yes

characteristic

Q₈²

—

#494

—

#477

—

#333

D₅₃

yes

characteristic

Q₈⋊C₈

—

C₅₃

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

106

—

#107

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

D₂₁₂

yes

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

106

—

Q₈

—

424

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

—

C₈

—

100

C₈

—

106

—

101

C₂²

#13133187

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

106

—

105

C₄

—

424

—

106

C₄

—

212

—

107

C₄

Q₁₆×D₅₃

yes

—

108

C₄

#13133246

yes

—

109

C₂

#16810780

yes

—

110

C₂

#16810793

yes

characteristic

	subgroup	G/H
center	C₂²	#13133187
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#333	C₂
socle	C₂²×C₅₃	#120
hypercenter	#333	D₅₃
hypocenter	C₅₃	#1633
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#13133187

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#333

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
42. Dic₁₀₆	Q₁₆	semidirect
61. C₅₃	#1633	semidirect

H ≅ G/N

type

42. Dic₁₀₆

Q₁₆

semidirect

61. C₅₃

#1633

semidirect

Subgroups — #20273856

Conjugacy classes of subgroups

Complements