Group #20288858 — subgroups

Subgroups — C₂×C₅₃⋊#564 #20288858

Order 6784.

896 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 469 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	C₅₃⋊#564	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133209	C₂	1	2	yes	maximal
3	C₂×#13133208	C₂	1	4	yes	maximal
4	C₂×#13133207	C₂	1	4	yes	maximal
5	C₂×#13133207	C₂	1	4	yes	maximal
6	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#13133209	C₂²	1	16	yes	—
8	#13133208	C₂²	1	32	yes	—
9	#13133207	C₂²	1	32	yes	—
10	#13133207	C₂²	1	32	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂×#11719919	C₂²	1	4	yes	—
15	C₂×#11719919	C₂²	1	4	yes	—
16	C₂×#11719919	C₂²	1	4	yes	—
17	C₂×#11719919	C₂²	1	4	yes	—
18	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
19	#125×C₅₃	C₂²	1	8	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
26	C₂²×Dic₅₃	C₂³	1	4	yes	—
27	C₂²×Dic₅₃	C₂³	1	4	yes	—
28	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
29	C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
30	C₄²×C₅₃	C₂³	1	4	yes	—
31	C₄⋊Dic₅₃	C₂³	1	8	yes	—
32	C₄⋊Dic₅₃	C₂³	1	16	yes	—
33	C₄⋊Dic₅₃	C₂³	1	8	yes	—
34	#11719919	C₂³	1	16	yes	—
35	#11719919	C₂³	1	16	yes	—
36	#11719919	C₂³	1	16	yes	—
37	#11719919	C₂³	1	16	yes	—
38	C₄×Dic₅₃	C₂³	1	16	yes	—
39	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	2	8	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
44	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
46	C₂×Dic₅₃	—	2	8	no	—
47	C₂×Dic₅₃	—	2	8	no	—
48	C₂×Dic₅₃	C₂⁴	1	8	yes	—
49	C₂×Dic₅₃	C₂⁴	1	8	yes	—
50	C₂²×C₅₃	2₋¹⁺⁴	1	4	yes	—
51	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
52	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
53	C₄×C₅₃	—	2	8	no	—
54	C₄×C₅₃	—	2	4	no	—
55	C₄×C₅₃	—	2	2	no	—
56	Dic₅₃	—	2	8	no	—
57	Dic₅₃	—	2	8	no	—
58	C₂×#564	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	#564	1	4	yes	complemented
60	C₂×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
61	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
62	#564	—	53	16	no	—
63	C₂×#125	—	53	4	no	—
64	C₂×#125	—	53	4	no	—
65	C₂×#125	—	53	4	no	—
66	C₂×#125	—	53	2	no	—
67	C₂×#125	D₅₃	1	1	yes	characteristic
68	C₅₃	C₂×#564	1	1	yes	characteristic, complemented, Sylow
69	#125	—	53	32	no	—
70	#125	—	53	32	no	—
71	#125	—	53	32	no	—
72	#125	—	53	16	no	—
73	#125	C₂×D₅₃	1	8	yes	—
74	C₂×C₄⋊C₄	—	53	4	no	—
75	C₂×C₄⋊C₄	—	53	4	no	—
76	C₂×C₄⋊C₄	—	53	4	no	—
77	C₂×C₄⋊C₄	—	53	4	no	—
78	C₂×C₄⋊C₄	—	53	4	no	—
79	C₂×C₄⋊C₄	—	53	2	no	—
80	C₂×C₄⋊C₄	—	53	2	no	—
81	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
82	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
83	C₂×C₄²	—	53	4	no	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	C₂²×C₄	—	53	4	no	—
86	C₂²×C₄	—	53	4	no	—
87	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
90	C₄⋊C₄	—	53	16	no	—
91	C₄⋊C₄	—	53	16	no	—
92	C₄⋊C₄	—	53	16	no	—
93	C₄⋊C₄	—	53	16	no	—
94	C₄⋊C₄	—	53	16	no	—
95	C₄⋊C₄	—	53	8	no	—
96	C₄⋊C₄	—	53	8	no	—
97	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
98	C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
99	C₄²	—	53	16	no	—
100	C₄²	C₂²×D₅₃	1	4	yes	—
101	C₂³	C₂³×D₅₃	1	1	yes	characteristic
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	53	8	no	—
106	C₂×C₄	—	53	8	no	—
107	C₂×C₄	—	106	8	no	—
108	C₂×C₄	—	106	8	no	—
109	C₂×C₄	C₂³×D₅₃	1	8	yes	—
110	C₂×C₄	C₂³×D₅₃	1	4	yes	—
111	C₂×C₄	C₂³×D₅₃	1	2	yes	—
112	C₂²	#13133264	1	4	yes	—
113	C₂²	#13133265	1	2	yes	—
114	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
115	C₄	—	2	8	no	—
116	C₄	—	2	4	no	—
117	C₄	—	2	2	no	—
118	C₄	—	106	8	no	—
119	C₄	—	106	8	no	—
120	C₂	C₅₃⋊#564	1	4	yes	direct product factor
121	C₂	C₂×#13133264	1	2	yes	—
122	C₂	C₂×#13133265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃⋊#564

C₂

yes

maximal, direct product factor

C₂×#13133209

C₂

yes

maximal

C₂×#13133208

C₂

yes

maximal

C₂×#13133207

C₂

yes

maximal

C₂×#13133207

C₂

yes

maximal

C₂×#125×C₅₃

C₂

yes

maximal, characteristic

#13133209

C₂²

yes

—

#13133208

C₂²

yes

—

#13133207

C₂²

yes

—

#13133207

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×2₋¹⁺⁴

yes

—

C₂×C₅₃

C₂×2₋¹⁺⁴

yes

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

C₂×#564

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄²

—

100

C₄²

C₂²×D₅₃

yes

—

101

C₂³

C₂³×D₅₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

C₂³×D₅₃

yes

—

110

C₂×C₄

C₂³×D₅₃

yes

—

111

C₂×C₄

C₂³×D₅₃

yes

—

112

C₂²

#13133264

yes

—

113

C₂²

#13133265

yes

—

114

C₂²

C₂⁴×D₅₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

106

—

119

C₄

—

106

—

120

C₂

C₅₃⋊#564

yes

direct product factor

121

C₂

C₂×#13133264

yes

—

122

C₂

C₂×#13133265

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	C₂×#564
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂×#125×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
1. C₅₃⋊#564	C₂	direct (N × H)
59. C₂×C₅₃	#564	semidirect
68. C₅₃	C₂×#564	semidirect
120. C₂	C₅₃⋊#564	direct (N × H)

H ≅ G/N

type

1. C₅₃⋊#564

C₂

direct (N × H)

59. C₂×C₅₃

#564

semidirect

68. C₅₃

C₂×#564

semidirect

120. C₂

C₅₃⋊#564

direct (N × H)

Subgroups — C₂×C₅₃⋊#564 #20288858

Conjugacy classes of subgroups

Complements

Subgroups — C2×C53⋊#564 #20288858

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₅₃⋊#564 #20288858