Group #20290265 — subgroups

Subgroups — #20290265

Order 6784.

180 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 82 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	#16811642	C₂	1	2	yes	maximal
2	#16811486	C₂	1	2	yes	maximal, complemented
3	#16811485	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#88324	C₂²	1	4	yes	—
6	#13133274	—	2	2	no	—
7	C₄×#88324	—	2	2	no	—
8	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
10	C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	C₄×#88322	C₄	1	1	yes	characteristic
12	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
18	C₂×#88324	—	2	4	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₅₃	D₄	1	1	yes	characteristic
23	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
25	C₂×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂×#88322	C₂×C₄	1	1	yes	characteristic
27	C₈×D₅₃	—	2	2	no	—
28	C₈×D₅₃	C₈	1	4	yes	complemented
29	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₄×D₅₃	—	2	2	no	—
37	C₄×D₅₃	—	2	2	no	—
38	C₄×D₅₃	C₂×C₈	1	2	yes	—
39	C₄×D₅₃	M₁₆	1	2	yes	—
40	#88324	—	4	4	no	—
41	C₈×C₅₃	—	2	1	no	—
42	C₈×C₅₃	C₂×C₈	1	2	yes	—
43	#88322	—	2	1	no	—
44	#88322	C₂×C₈	1	2	yes	—
45	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₅₃	—	2	1	no	—
47	C₂×D₅₃	—	2	1	no	—
48	C₂×D₅₃	C₄⋊C₈	1	2	yes	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	#108	1	2	yes	—
51	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
53	Dic₅₃	—	2	1	no	—
54	Dic₅₃	C₄⋊C₈	1	2	yes	—
55	Dic₅₃	#135	1	2	yes	—
56	#1564	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₅₃	#427	1	1	yes	characteristic
59	C₂×C₅₃	C₂×#108	1	1	yes	characteristic
60	D₅₃	—	2	2	no	—
61	#431	—	53	2	no	—
62	C₂×C₄×C₈	—	53	1	no	—
63	#335	—	53	2	no	—
64	#334	—	53	2	no	—
65	C₅₃	#1564	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₈	—	53	1	no	—
67	C₂²×C₈	—	53	1	no	—
68	C₂×C₄²	—	53	1	no	—
69	C₄⋊C₈	—	53	4	no	—
70	C₂²⋊C₈	—	106	2	no	—
71	C₄×C₈	—	53	1	no	—
72	C₄×C₈	—	53	1	no	—
73	C₄×C₈	—	53	1	no	—
74	C₄×C₈	—	106	2	no	—
75	C₄×C₈	#6157	1	1	yes	characteristic
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	—	53	1	no	—
78	C₂²×C₄	—	53	1	no	—
79	C₂×C₈	—	53	1	no	—
80	C₂×C₈	—	53	4	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	106	2	no	—
83	C₂×C₈	—	106	4	no	—
84	C₂×C₈	C₂×#6157	1	1	yes	characteristic
85	C₂×C₈	C₂×#6157	1	1	yes	characteristic
86	C₄²	—	53	1	no	—
87	C₄²	—	53	1	no	—
88	C₄²	—	53	1	no	—
89	C₄²	C₂×#6157	1	1	yes	characteristic
90	C₂³	—	53	1	no	—
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	53	2	no	—
94	C₂×C₄	—	53	2	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	106	2	no	—
97	C₂×C₄	—	106	2	no	—
98	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
99	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
100	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	53	2	no	—
103	C₈	—	106	1	no	—
104	C₈	—	212	4	no	—
105	C₈	D₅₃⋊C₈	1	2	yes	—
106	C₂²	—	53	2	no	—
107	C₂²	—	106	1	no	—
108	C₂²	—	106	1	no	—
109	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	53	2	no	—
112	C₄	—	53	2	no	—
113	C₄	—	106	1	no	—
114	C₄	#13133288	1	2	yes	—
115	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
116	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
117	C₂	—	106	2	no	—
118	C₂	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
119	C₂	#16811633	1	1	yes	characteristic
120	C₂	C₂×#13133288	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811642

C₂

yes

maximal

#16811486

C₂

yes

maximal, complemented

#16811485

C₂

yes

maximal, complemented

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₄⋊#88324

C₂²

yes

—

#13133274

—

C₄×#88324

—

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

characteristic

C₄²×D₅₃

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

characteristic

C₂×#88324

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

Q₈

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

C₈

yes

complemented

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

M₁₆

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

—

#88322

—

#88322

C₂×C₈

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

#108

yes

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

C₄⋊C₈

yes

—

Dic₅₃

#135

yes

—

#1564

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#431

—

C₂×C₄×C₈

—

#335

—

#334

—

C₅₃

#1564

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²×#6157

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

106

—

104

C₈

—

212

—

105

C₈

D₅₃⋊C₈

yes

—

106

C₂²

—

107

C₂²

—

106

—

108

C₂²

—

106

—

109

C₂²

C₂×C₄⋊#6157

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

106

—

114

C₄

#13133288

yes

—

115

C₄

C₂×D₅₃⋊C₈

yes

characteristic

116

C₄

D₅₃⋊M₁₆

yes

characteristic

117

C₂

—

106

—

118

C₂

D₅₃⋊(C₄⋊C₈)

yes

characteristic

119

C₂

#16811633

yes

characteristic

120

C₂

C₂×#13133288

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1564
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₄

C₄×C₈

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #16811486	C₂	semidirect
3. #16811485	C₂	semidirect
28. C₈×D₅₃	C₈	semidirect
65. C₅₃	#1564	semidirect

H ≅ G/N

type

2. #16811486

C₂

semidirect

3. #16811485

C₂

semidirect

28. C₈×D₅₃

C₈

semidirect

65. C₅₃

#1564

semidirect

Subgroups — #20290265

Conjugacy classes of subgroups

Complements