Group #20274234 — subgroups

Subgroups — #20274234

Order 6784.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16811028	C₂	1	1	yes	maximal, characteristic
2	#16810834	C₂	1	2	yes	maximal
3	#16810753	C₂	1	1	yes	maximal, characteristic
4	#16810370	C₂	1	2	yes	maximal
5	C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	—	2	2	no	—
7	#13133217	C₂²	1	2	yes	—
8	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	#13133151	—	2	1	no	—
10	#13133151	—	2	1	no	—
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	#13133146	—	2	1	no	—
13	#13133142	—	2	2	no	—
14	#13133127	—	2	1	no	—
15	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₀₆	—	2	2	no	—
18	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
26	#11719919	—	2	1	no	—
27	C₄×Dic₅₃	—	2	1	no	—
28	C₂×#88322	—	2	1	no	—
29	Dic₂₁₂	—	4	4	no	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	1	no	—
34	C₂×Dic₅₃	—	2	1	no	—
35	C₂×Dic₅₃	—	4	2	no	—
36	Dic₁₀₆	—	2	2	no	—
37	Dic₁₀₆	—	4	2	no	—
38	Dic₁₀₆	—	4	2	no	—
39	C₈×C₅₃	—	2	1	no	—
40	C₈×C₅₃	—	4	2	no	—
41	C₈×C₅₃	Q₁₆	1	2	yes	complemented
42	#88322	—	4	1	no	—
43	C₂²×C₅₃	#121	1	1	yes	characteristic
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₄×C₅₃	#137	1	1	yes	characteristic
48	Dic₅₃	—	4	1	no	—
49	Dic₅₃	—	4	1	no	—
50	Dic₅₃	—	8	2	no	—
51	#1694	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	#462	1	1	yes	characteristic
53	C₂×C₅₃	#469	1	1	yes	characteristic
54	C₂×C₅₃	#473	1	1	yes	characteristic
55	#500	—	53	1	no	—
56	#462	—	53	2	no	—
57	#446	—	53	1	no	—
58	#334	D₅₃	1	1	yes	characteristic
59	#332	—	53	2	no	—
60	C₅₃	#1694	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₁₆	—	106	2	no	—
62	C₄⋊Q₈	—	53	2	no	—
63	C₄×Q₈	—	53	1	no	—
64	#107	—	106	1	no	—
65	#107	—	106	1	no	—
66	C₄⋊C₈	—	53	1	no	—
67	C₄⋊C₈	—	106	1	no	—
68	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
69	#103	—	106	2	no	—
70	#97	—	106	1	no	—
71	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₂×Q₈	—	53	1	no	—
73	C₂×Q₈	—	106	2	no	—
74	Q₁₆	—	212	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	106	1	no	—
77	C₂×C₈	D₂₁₂	1	1	yes	characteristic
78	C₂×C₈	D₂₁₂	1	1	yes	characteristic
79	C₄⋊C₄	—	53	1	no	—
80	C₄⋊C₄	—	106	1	no	—
81	C₄⋊C₄	—	106	2	no	—
82	C₄⋊C₄	—	106	2	no	—
83	C₄²	—	106	1	no	—
84	C₄²	C₂²×D₅₃	1	1	yes	characteristic
85	Q₈	—	106	2	no	—
86	Q₈	—	212	2	no	—
87	Q₈	—	212	2	no	—
88	C₂×C₄	—	106	1	no	—
89	C₂×C₄	—	106	1	no	—
90	C₂×C₄	—	212	2	no	—
91	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
92	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
93	C₂×C₄	#11719949	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	212	1	no	—
97	C₈	Dic₂₁₂	1	2	yes	complemented
98	C₂²	#13133192	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	424	2	no	—
101	C₄	—	212	1	no	—
102	C₄	—	212	1	no	—
103	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
104	C₄	#13133244	1	2	yes	—
105	C₄	#13133250	1	1	yes	characteristic
106	C₂	#16810834	1	1	yes	characteristic
107	C₂	#16810877	1	1	yes	characteristic
108	C₂	#16810899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811028

C₂

yes

maximal, characteristic

#16810834

C₂

yes

maximal

#16810753

C₂

yes

maximal, characteristic

#16810370

C₂

yes

maximal

C₅₃×#334

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

—

#13133217

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

Q₁₆

yes

complemented

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

#332

—

C₅₃

#1694

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

106

—

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

106

—

#97

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

106

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719949

yes

characteristic

C₈

—

C₈

—

C₈

—

212

—

C₈

Dic₂₁₂

yes

complemented

C₂²

#13133192

yes

characteristic

C₄

—

100

C₄

—

424

—

101

C₄

—

212

—

102

C₄

—

212

—

103

C₄

C₂×Dic₂₁₂

yes

characteristic

104

C₄

#13133244

yes

—

105

C₄

#13133250

yes

characteristic

106

C₂

#16810834

yes

characteristic

107

C₂

#16810877

yes

characteristic

108

C₂

#16810899

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#334	C₂
socle	C₂²×C₅₃	#121
hypercenter	#334	D₅₃
hypocenter	C₅₃	#1694
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#13133192

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#334

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
41. C₈×C₅₃	Q₁₆	semidirect
60. C₅₃	#1694	semidirect
97. C₈	Dic₂₁₂	semidirect

H ≅ G/N

type

41. C₈×C₅₃

Q₁₆

semidirect

60. C₅₃

#1694

semidirect

97. C₈

Dic₂₁₂

semidirect

Subgroups — #20274234

Conjugacy classes of subgroups

Complements