Group #20274666 — subgroups

Subgroups — #20274666

Order 6784.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16810607	C₂	1	3	yes	maximal, complemented
2	#16810590	C₂	1	3	yes	maximal, complemented
3	C₅₃×#585	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₂²	1	3	yes	complemented
5	C₂²×#88322	C₂²	1	1	yes	characteristic, complemented
6	#13133136	—	2	3	no	—
7	#13133134	—	2	3	no	—
8	C₂²⋊#88322	—	2	6	no	—
9	2₋¹⁺⁴×C₅₃	—	2	3	no	—
10	H(5,𝔽₂)×C₅₃	—	2	3	no	—
11	H(5,𝔽₂)×C₅₃	C₄	1	4	yes	—
12	C₂×#41×C₅₃	—	2	6	no	—
13	C₂×#41×C₅₃	C₂²	1	3	yes	—
14	#41×C₅₃	—	2	4	no	—
15	#41×C₅₃	—	2	12	no	—
16	#41×C₅₃	—	2	12	no	—
17	#41×C₅₃	—	2	6	no	—
18	#41×C₅₃	—	2	6	no	—
19	C₂×Q₈×C₅₃	—	2	6	no	—
20	C₂×Q₈×C₅₃	D₄	1	3	yes	—
21	C₂×D₄×C₅₃	—	2	6	no	—
22	C₂×D₄×C₅₃	—	2	12	no	—
23	C₂×D₄×C₅₃	C₂×C₄	1	6	yes	—
24	C₂×D₄×C₅₃	D₄	1	3	yes	—
25	C₂²×C₄×C₅₃	—	2	4	no	—
26	C₂²×C₄×C₅₃	D₄	1	6	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	#11719917	—	2	6	no	—
29	C₂×#88322	—	2	3	no	—
30	C₂×#88322	—	2	3	no	—
31	C₂³×C₅₃	—	2	4	no	—
32	C₂³×C₅₃	C₂²⋊C₄	1	6	yes	—
33	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₅₃	—	2	4	no	—
35	Q₈×C₅₃	—	2	6	no	—
36	D₄×C₅₃	—	2	12	no	—
37	D₄×C₅₃	—	2	12	no	—
38	D₄×C₅₃	—	2	6	no	—
39	C₂×C₄×C₅₃	—	2	12	no	—
40	C₂×C₄×C₅₃	—	2	6	no	—
41	C₂×C₄×C₅₃	—	2	6	no	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
45	#88322	—	4	1	no	—
46	#88322	—	4	3	no	—
47	C₂²×C₅₃	—	2	6	no	—
48	C₂²×C₅₃	—	4	1	no	—
49	C₂²×C₅₃	—	4	6	no	—
50	C₂²×C₅₃	—	8	4	no	—
51	C₂²×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
52	C₄×C₅₃	—	2	6	no	—
53	C₄×C₅₃	#120	1	3	yes	—
54	C₄×C₅₃	#120	1	1	yes	characteristic
55	#1791	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	3	no	—
57	C₂×C₅₃	—	4	6	no	—
58	C₂×C₅₃	#379	1	1	yes	characteristic
59	#585	D₅₃	1	1	yes	characteristic
60	#413	—	53	3	no	—
61	#408	—	53	3	no	—
62	C₅₃	#1791	1	1	yes	characteristic, complemented, Sylow
63	2₋¹⁺⁴	—	2	3	no	—
64	H(5,𝔽₂)	—	2	3	no	—
65	H(5,𝔽₂)	Dic₅₃	1	4	yes	—
66	C₂×#41	—	2	6	no	—
67	C₂×#41	C₂×D₅₃	1	3	yes	—
68	C₂×M₁₆	—	53	3	no	—
69	C₂²×C₈	—	53	1	no	—
70	#101	—	106	3	no	—
71	#100	—	106	3	no	—
72	C₂²⋊C₈	—	106	6	no	—
73	#41	—	2	4	no	—
74	#41	—	2	12	no	—
75	#41	—	2	6	no	—
76	#41	—	2	6	no	—
77	#41	—	2	12	no	—
78	C₂×Q₈	—	2	6	no	—
79	C₂×Q₈	C₂²⋊D₅₃	1	3	yes	—
80	C₂×D₄	—	2	6	no	—
81	C₂×D₄	—	2	12	no	—
82	C₂×D₄	C₂×Dic₅₃	1	6	yes	—
83	C₂×D₄	C₂²⋊D₅₃	1	3	yes	—
84	C₂²×C₄	—	2	4	no	—
85	C₂²×C₄	C₂²⋊D₅₃	1	6	yes	—
86	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
87	M₁₆	—	106	6	no	—
88	C₂×C₈	—	106	3	no	—
89	C₂×C₈	—	106	3	no	—
90	C₂³	—	2	4	no	—
91	C₂³	C₂²⋊Dic₅₃	1	6	yes	—
92	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	2	6	no	—
95	D₄	—	2	12	no	—
96	D₄	—	2	12	no	—
97	D₄	—	2	6	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	C₂²⋊Dic₅₃	1	6	yes	—
102	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
103	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
104	C₈	—	212	3	no	—
105	C₈	—	212	1	no	—
106	C₂²	—	2	6	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	6	no	—
109	C₂²	—	8	4	no	—
110	C₂²	C₂×C₂²⋊Dic₅₃	1	3	yes	—
111	C₄	—	2	6	no	—
112	C₄	#13133186	1	3	yes	—
113	C₄	#13133186	1	1	yes	characteristic
114	C₂	—	2	3	no	—
115	C₂	—	4	6	no	—
116	C₂	#16810479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810607

C₂

yes

maximal, complemented

#16810590

C₂

yes

maximal, complemented

C₅₃×#585

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

complemented

C₂²×#88322

C₂²

yes

characteristic, complemented

#13133136

—

#13133134

—

C₂²⋊#88322

—

2₋¹⁺⁴×C₅₃

—

H(5,𝔽₂)×C₅₃

—

H(5,𝔽₂)×C₅₃

C₄

yes

—

C₂×#41×C₅₃

—

C₂×#41×C₅₃

C₂²

yes

—

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

—

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#379

yes

characteristic

#585

D₅₃

yes

characteristic

#413

—

#408

—

C₅₃

#1791

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

Dic₅₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#101

—

106

—

#100

—

106

—

C₂²⋊C₈

—

106

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂×D₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²⋊Dic₅₃

yes

—

102

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

103

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

104

C₈

—

212

—

105

C₈

—

212

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₅₃

yes

—

111

C₄

—

112

C₄

#13133186

yes

—

113

C₄

#13133186

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#16810479

yes

characteristic

	subgroup	G/H
center	C₄	#13133186
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#585	C₂
socle	C₂×C₅₃	#379
hypercenter	#585	D₅₃
hypocenter	C₅₃	#1791
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₄

#13133186

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#585

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #16810607	C₂	semidirect
2. #16810590	C₂	semidirect
4. C₂×#11719917	C₂²	semidirect
5. C₂²×#88322	C₂²	semidirect
62. C₅₃	#1791	semidirect

H ≅ G/N

type

1. #16810607

C₂

semidirect

2. #16810590

C₂

semidirect

4. C₂×#11719917

C₂²

semidirect

5. C₂²×#88322

C₂²

semidirect

62. C₅₃

#1791

semidirect

Subgroups — #20274666

Conjugacy classes of subgroups

Complements