Group #20277636 — subgroups

Subgroups — #20277636

Order 6784.

552 conjugacy classes of subgroups; only the 313 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 313 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	#16810522	C₂	1	6	yes	maximal
3	#16810489	C₂	1	3	yes	maximal
4	C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₅₃	—	2	8	no	—
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	6	yes	—
11	#13133124	C₂²	1	1	yes	characteristic
12	#13133124	C₂²	1	3	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	4	yes	—
18	C₂×Q₈×C₅₃	D₄	1	4	yes	—
19	C₂×Q₈×C₅₃	Q₈	1	4	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
23	C₂²×Dic₅₃	C₂³	1	6	yes	—
24	C₂²×Dic₅₃	C₂³	1	2	yes	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
26	C₄²×C₅₃	C₂×C₄	1	12	yes	—
27	C₄⋊Dic₅₃	—	2	12	no	—
28	C₄⋊Dic₅₃	—	2	12	no	—
29	C₄×Dic₅₃	—	2	12	no	—
30	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
31	Q₈×C₅₃	C₄⋊C₄	1	16	yes	complemented
32	C₂×C₄×C₅₃	—	2	6	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
38	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
39	C₂×Dic₅₃	—	2	2	no	—
40	C₂×Dic₅₃	—	2	2	no	—
41	C₂×Dic₅₃	—	2	6	no	—
42	C₂×Dic₅₃	—	2	6	no	—
43	C₂×Dic₅₃	—	2	6	no	—
44	C₂×Dic₅₃	#41	1	4	yes	—
45	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
48	C₂²×C₅₃	C₂×#41	1	2	yes	—
49	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
50	C₄×C₅₃	—	2	6	no	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
53	Dic₅₃	—	2	4	no	—
54	Dic₅₃	—	4	6	no	—
55	#2351	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
58	C₂×C₅₃	#519	1	1	yes	characteristic
59	C₂×C₅₃	#548	1	2	yes	—
60	C₂×C₅₃	#557	1	2	yes	—
61	C₂×C₄×Q₈	—	53	2	no	—
62	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
63	#389	—	53	6	no	—
64	#382	—	53	3	no	—
65	C₄×C₄⋊C₄	—	53	3	no	—
66	C₅₃	#2351	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
68	C₄×Q₈	—	106	8	no	—
69	C₄×Q₈	Dic₅₃	1	8	yes	complemented
70	C₂×C₄⋊C₄	—	53	6	no	—
71	C₂×C₄⋊C₄	—	53	6	no	—
72	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
73	C₂×C₄²	—	53	6	no	—
74	C₂×C₄²	C₂×D₅₃	1	3	yes	—
75	#95	—	53	1	no	—
76	#95	—	53	3	no	—
77	#95	—	53	6	no	—
78	C₂×Q₈	Dic₁₀₆	1	4	yes	—
79	C₂×Q₈	D₂₁₂	1	4	yes	—
80	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
81	C₂²×C₄	—	53	2	no	—
82	C₂²×C₄	—	53	6	no	—
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
85	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
86	C₄⋊C₄	—	106	12	no	—
87	C₄⋊C₄	—	106	12	no	—
88	C₄⋊C₄	C₂×Dic₅₃	1	12	yes	—
89	C₄²	—	106	12	no	—
90	C₄²	C₂×Dic₅₃	1	12	yes	—
91	C₂³	C₂³×D₅₃	1	1	yes	characteristic
92	Q₈	C₄⋊Dic₅₃	1	16	yes	complemented
93	C₂×C₄	—	2	6	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	53	4	no	—
96	C₂×C₄	—	106	2	no	—
97	C₂×C₄	—	106	2	no	—
98	C₂×C₄	—	106	6	no	—
99	C₂×C₄	—	106	6	no	—
100	C₂×C₄	—	106	6	no	—
101	C₂×C₄	C₂×Dic₁₀₆	1	6	yes	—
102	C₂×C₄	C₂×D₂₁₂	1	6	yes	—
103	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
104	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
105	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
106	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
107	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
108	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
109	C₂²	#41×D₅₃	1	2	yes	—
110	C₂²	#13133265	1	2	yes	—
111	C₄	—	2	6	no	—
112	C₄	—	2	2	no	—
113	C₄	—	106	4	no	—
114	C₄	—	212	6	no	—
115	C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
116	C₂	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
117	C₂	#41×Dic₅₃	1	1	yes	characteristic
118	C₂	#16811101	1	1	yes	characteristic
119	C₂	#16811285	1	2	yes	—
120	C₂	#16811349	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₃

C₂

yes

maximal

#16810522

C₂

yes

maximal

#16810489

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

Q₈×Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

characteristic

#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

D₄

yes

—

C₂×Q₈×C₅₃

Q₈

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

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

C₄⋊C₄

yes

complemented

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

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#41

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

—

Dic₅₃

—

#2351

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₄×#41

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#557

yes

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

#389

—

#382

—

C₄×C₄⋊C₄

—

C₅₃

#2351

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

106

—

C₄×Q₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

#95

—

#95

—

#95

—

C₂×Q₈

Dic₁₀₆

yes

—

C₂×Q₈

D₂₁₂

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

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

—

106

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

C₄⋊Dic₅₃

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

—

106

—

101

C₂×C₄

C₂×Dic₁₀₆

yes

—

102

C₂×C₄

C₂×D₂₁₂

yes

—

103

C₂×C₄

C₂²×Dic₅₃

yes

—

104

C₂×C₄

C₂²×Dic₅₃

yes

—

105

C₂×C₄

C₂²×Dic₅₃

yes

—

106

C₂²

C₂³×Dic₅₃

yes

characteristic

107

C₂²

C₂²×D₂₁₂

yes

characteristic

108

C₂²

C₂²×Dic₁₀₆

yes

characteristic

109

C₂²

#41×D₅₃

yes

—

110

C₂²

#13133265

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

106

—

114

C₄

—

212

—

115

C₄

C₂×C₄⋊Dic₅₃

yes

—

116

C₂

C₂²×C₄⋊Dic₅₃

yes

characteristic

117

C₂

#41×Dic₅₃

yes

characteristic

118

C₂

#16811101

yes

characteristic

119

C₂

#16811285

yes

—

120

C₂

#16811349

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#2351
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₅₃	C₄	semidirect
31. Q₈×C₅₃	C₄⋊C₄	semidirect
66. C₅₃	#2351	semidirect
69. C₄×Q₈	Dic₅₃	semidirect
92. Q₈	C₄⋊Dic₅₃	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₅₃

C₄

semidirect

31. Q₈×C₅₃

C₄⋊C₄

semidirect

66. C₅₃

#2351

semidirect

69. C₄×Q₈

Dic₅₃

semidirect

92. Q₈

C₄⋊Dic₅₃

semidirect

Subgroups — #20277636

Conjugacy classes of subgroups

Complements