Group #20282979 — subgroups

Subgroups — #20282979

Order 6784.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16810559	C₂	1	4	yes	maximal, complemented
2	#16810546	C₂	1	8	yes	maximal, complemented
3	#16810479	C₂	1	2	yes	maximal
4	C₅₃×#379	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	complemented
6	C₂×#11719919	C₂²	1	8	yes	complemented
7	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
8	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
10	#13133124	C₂²	1	4	yes	complemented
11	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂⁴×C₅₃	—	4	4	no	—
15	C₂⁴×C₅₃	—	4	2	no	—
16	C₂⁴×C₅₃	Q₈	1	4	yes	—
17	C₂⁴×C₅₃	C₂³	1	2	yes	—
18	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
20	C₂²×Dic₅₃	C₂³	1	8	yes	—
21	C₂²⋊C₄×C₅₃	—	4	4	no	—
22	C₂²⋊C₄×C₅₃	—	4	2	no	—
23	C₂²⋊Dic₅₃	—	4	4	no	—
24	C₂²⋊Dic₅₃	—	4	4	no	—
25	C₂²⋊Dic₅₃	—	4	4	no	—
26	C₄⋊Dic₅₃	—	4	4	no	—
27	#11719919	—	4	8	no	—
28	C₂³×C₅₃	—	2	4	no	—
29	C₂³×C₅₃	—	2	4	no	—
30	C₂³×C₅₃	—	2	4	no	—
31	C₂³×C₅₃	—	2	4	no	—
32	C₂³×C₅₃	—	2	2	no	—
33	C₂³×C₅₃	—	4	8	no	—
34	C₂³×C₅₃	—	4	4	no	—
35	C₂³×C₅₃	—	4	16	no	—
36	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
38	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	8	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×Dic₅₃	—	2	8	no	—
42	C₂×Dic₅₃	—	2	8	no	—
43	C₂×Dic₅₃	—	2	8	no	—
44	C₂²×C₅₃	—	2	4	no	—
45	C₂²×C₅₃	—	2	4	no	—
46	C₂²×C₅₃	—	2	4	no	—
47	C₂²×C₅₃	—	2	4	no	—
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	—	4	16	no	—
50	C₂²×C₅₃	—	4	8	no	—
51	C₂²×C₅₃	—	4	4	no	—
52	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
54	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
55	C₄×C₅₃	—	4	4	no	—
56	Dic₅₃	—	4	8	no	—
57	#2842	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	4	4	no	—
59	C₂×C₅₃	—	4	2	no	—
60	C₂×C₅₃	#543	1	2	yes	—
61	C₂×C₅₃	#543	1	1	yes	characteristic
62	C₂×C₅₃	#560	1	4	yes	—
63	#399	—	53	4	no	—
64	#396	—	53	8	no	—
65	#379	—	53	2	no	—
66	#379	D₅₃	1	1	yes	characteristic
67	C₅₃	#2842	1	1	yes	characteristic, complemented, Sylow
68	C₂⁵	C₂×D₅₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	53	8	no	—
70	C₂×C₄⋊C₄	—	53	4	no	—
71	C₂×C₂²⋊C₄	—	53	4	no	—
72	C₂×C₂²⋊C₄	—	53	4	no	—
73	C₂×C₂²⋊C₄	—	53	4	no	—
74	C₂×C₂²⋊C₄	C₂×D₅₃	1	4	yes	—
75	C₂×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
76	#95	—	53	4	no	—
77	C₂⁴	—	4	4	no	—
78	C₂⁴	—	4	2	no	—
79	C₂⁴	Dic₁₀₆	1	4	yes	—
80	C₂⁴	C₂²×D₅₃	1	2	yes	—
81	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
82	C₂²×C₄	—	53	8	no	—
83	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
84	C₄⋊C₄	—	212	8	no	—
85	C₄⋊C₄	—	212	4	no	—
86	C₂²⋊C₄	—	4	4	no	—
87	C₂²⋊C₄	—	4	2	no	—
88	C₂²⋊C₄	—	212	4	no	—
89	C₂²⋊C₄	—	212	4	no	—
90	C₂²⋊C₄	—	212	4	no	—
91	C₂³	—	2	4	no	—
92	C₂³	—	2	2	no	—
93	C₂³	—	2	4	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	2	4	no	—
96	C₂³	—	4	8	no	—
97	C₂³	—	4	4	no	—
98	C₂³	—	4	16	no	—
99	C₂³	C₂×Dic₁₀₆	1	4	yes	—
100	C₂³	C₂×Dic₁₀₆	1	2	yes	—
101	C₂³	C₂³×D₅₃	1	1	yes	characteristic
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	106	8	no	—
105	C₂×C₄	—	106	8	no	—
106	C₂×C₄	—	106	8	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	2	4	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	2	4	no	—
112	C₂²	—	4	16	no	—
113	C₂²	—	4	8	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
116	C₂²	#13133262	1	4	yes	—
117	C₂²	#13133262	1	2	yes	—
118	C₄	—	4	4	no	—
119	C₄	—	212	8	no	—
120	C₂	—	4	4	no	—
121	C₂	—	4	2	no	—
122	C₂	#16811249	1	2	yes	—
123	C₂	#16811249	1	1	yes	characteristic
124	C₂	#16811362	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810559

C₂

yes

maximal, complemented

#16810546

C₂

yes

maximal, complemented

#16810479

C₂

yes

maximal

C₅₃×#379

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

complemented

C₂×#11719919

C₂²

yes

complemented

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

complemented

C₂⁵×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

Q₈

yes

—

C₂⁴×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₄×C₅₃

—

Dic₅₃

—

#2842

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#543

yes

—

C₂×C₅₃

#543

yes

characteristic

C₂×C₅₃

#560

yes

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2842

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

Dic₁₀₆

yes

—

C₂⁴

C₂²×D₅₃

yes

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₀₆

yes

—

100

C₂³

C₂×Dic₁₀₆

yes

—

101

C₂³

C₂³×D₅₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²×Dic₁₀₆

yes

characteristic

116

C₂²

#13133262

yes

—

117

C₂²

#13133262

yes

—

118

C₄

—

119

C₄

—

212

—

120

C₂

—

121

C₂

—

122

C₂

#16811249

yes

—

123

C₂

#16811249

yes

characteristic

124

C₂

#16811362

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#379	C₂
hypercenter	#379	D₅₃
hypocenter	C₅₃	#2842
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #379

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#379

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #379

N	H ≅ G/N	type
1. #16810559	C₂	semidirect
2. #16810546	C₂	semidirect
5. C₂×C₄⋊Dic₅₃	C₂²	semidirect
6. C₂×#11719919	C₂²	semidirect
10. #13133124	C₂²	semidirect
67. C₅₃	#2842	semidirect

H ≅ G/N

type

1. #16810559

C₂

semidirect

2. #16810546

C₂

semidirect

5. C₂×C₄⋊Dic₅₃

C₂²

semidirect

6. C₂×#11719919

C₂²

semidirect

10. #13133124

C₂²

semidirect

67. C₅₃

#2842

semidirect

Subgroups — #20282979

Conjugacy classes of subgroups

Complements