Group #20277080 — subgroups

Subgroups — #20277080

Order 6784.

628 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 389 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	C₂×Q₈×Dic₅₃	C₂	1	2	yes	maximal
2	C₄⋊C₄×Dic₅₃	C₂	1	6	yes	maximal
3	C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
4	#16810469	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₅₃	C₄	1	16	yes	complemented
7	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
8	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
10	#13133124	C₂²	1	1	yes	characteristic
11	#13133124	C₂²	1	3	yes	—
12	#13133124	C₂²	1	6	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₅₃	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
17	C₂×Q₈×C₅₃	C₂×C₄	1	8	yes	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
22	C₂²×Dic₅₃	C₂³	1	2	yes	—
23	C₂²×Dic₅₃	C₂³	1	6	yes	—
24	C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
25	C₄²×C₅₃	C₂×C₄	1	12	yes	—
26	C₄⋊Dic₅₃	C₂×C₄	1	24	yes	—
27	C₄×Dic₅₃	—	4	6	no	—
28	C₄×Dic₅₃	C₂×C₄	1	24	yes	—
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	Q₈×C₅₃	C₄²	1	16	yes	complemented
31	C₂×C₄×C₅₃	—	2	3	no	—
32	C₂×C₄×C₅₃	—	2	3	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	12	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
38	C₂×Dic₅₃	—	2	6	no	—
39	C₂×Dic₅₃	—	2	2	no	—
40	C₂×Dic₅₃	—	2	2	no	—
41	C₂×Dic₅₃	—	2	6	no	—
42	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
43	C₂×Dic₅₃	C₂²×C₄	1	12	yes	—
44	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
45	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
48	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	2	6	no	—
51	C₄×C₅₃	C₂×C₄²	1	12	yes	—
52	Dic₅₃	—	2	4	no	—
53	Dic₅₃	—	2	12	no	—
54	#2276	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
56	C₂×C₅₃	#519	1	2	yes	—
57	C₂×C₅₃	#519	1	1	yes	characteristic
58	C₂×C₅₃	#520	1	2	yes	—
59	C₂×C₅₃	#520	1	1	yes	characteristic
60	C₂×C₄×Q₈	—	53	2	no	—
61	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
62	C₄×C₄⋊C₄	—	53	6	no	—
63	C₄×C₄⋊C₄	—	53	3	no	—
64	#376	—	53	3	no	—
65	C₅₃	#2276	1	1	yes	characteristic, complemented, Sylow
66	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
67	C₄×Q₈	—	53	16	no	—
68	C₄×Q₈	Dic₅₃	1	8	yes	complemented
69	C₂×C₄⋊C₄	—	53	6	no	—
70	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
71	C₂×C₄²	—	53	6	no	—
72	C₂×C₄²	—	53	6	no	—
73	C₂×C₄²	C₂×D₅₃	1	3	yes	—
74	#95	—	53	1	no	—
75	#95	—	53	6	no	—
76	#95	—	53	3	no	—
77	C₂×Q₈	C₄×D₅₃	1	8	yes	—
78	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
79	C₂²×C₄	—	53	2	no	—
80	C₂²×C₄	—	53	6	no	—
81	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
83	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
84	C₄⋊C₄	—	53	24	no	—
85	C₄⋊C₄	C₂×Dic₅₃	1	12	yes	—
86	C₄²	—	53	24	no	—
87	C₄²	—	212	6	no	—
88	C₄²	C₂×Dic₅₃	1	12	yes	—
89	C₂³	C₂³×D₅₃	1	1	yes	characteristic
90	Q₈	C₄×Dic₅₃	1	16	yes	complemented
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	53	4	no	—
95	C₂×C₄	—	53	12	no	—
96	C₂×C₄	—	106	6	no	—
97	C₂×C₄	—	106	2	no	—
98	C₂×C₄	—	106	2	no	—
99	C₂×C₄	—	106	6	no	—
100	C₂×C₄	C₂×C₄×D₅₃	1	12	yes	—
101	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
102	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
103	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
104	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
105	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
106	C₂²	#13133263	1	1	yes	characteristic
107	C₂²	#13133264	1	1	yes	characteristic
108	C₂²	#13133265	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	2	6	no	—
111	C₄	—	106	4	no	—
112	C₄	—	106	12	no	—
113	C₄	C₂×C₄×Dic₅₃	1	12	yes	—
114	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
115	C₂	#16811099	1	2	yes	—
116	C₂	#16811101	1	1	yes	characteristic
117	C₂	#16811105	1	2	yes	—
118	C₂	#16811106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₃

C₂

yes

maximal

C₄⋊C₄×Dic₅₃

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

#16810469

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

Q₈×Dic₅₃

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

characteristic

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

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

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

C₄²

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

Dic₅₃

—

Dic₅₃

—

#2276

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

#519

yes

—

C₂×C₅₃

#519

yes

characteristic

C₂×C₅₃

#520

yes

—

C₂×C₅₃

#520

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

#376

—

C₅₃

#2276

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

#95

—

#95

—

#95

—

C₂×Q₈

C₄×D₅₃

yes

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

—

C₄²

—

212

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

C₄×Dic₅₃

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

C₂×C₄×D₅₃

yes

—

101

C₂×C₄

C₂²×Dic₅₃

yes

—

102

C₂×C₄

C₂²×Dic₅₃

yes

—

103

C₂×C₄

C₂²×Dic₅₃

yes

—

104

C₂²

C₂²×C₄×D₅₃

yes

—

105

C₂²

C₂³×Dic₅₃

yes

characteristic

106

C₂²

#13133263

yes

characteristic

107

C₂²

#13133264

yes

characteristic

108

C₂²

#13133265

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

106

—

112

C₄

—

106

—

113

C₄

C₂×C₄×Dic₅₃

yes

—

114

C₂

C₂²×C₄×Dic₅₃

yes

characteristic

115

C₂

#16811099

yes

—

116

C₂

#16811101

yes

characteristic

117

C₂

#16811105

yes

—

118

C₂

#16811106

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#2276
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
6. Q₈×Dic₅₃	C₄	semidirect
14. C₄×Q₈×C₅₃	C₄	semidirect
30. Q₈×C₅₃	C₄²	semidirect
65. C₅₃	#2276	semidirect
68. C₄×Q₈	Dic₅₃	semidirect
90. Q₈	C₄×Dic₅₃	semidirect

H ≅ G/N

type

6. Q₈×Dic₅₃

C₄

semidirect

14. C₄×Q₈×C₅₃

C₄

semidirect

30. Q₈×C₅₃

C₄²

semidirect

65. C₅₃

#2276

semidirect

68. C₄×Q₈

Dic₅₃

semidirect

90. Q₈

C₄×Dic₅₃

semidirect

Subgroups — #20277080

Conjugacy classes of subgroups

Complements