Group #20274442 — subgroups

Subgroups — #20274442

Order 6784.

372 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂²×#11719917	C₂	1	2	yes	maximal
2	#16810388	C₂	1	4	yes	maximal, complemented
3	C₂²×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	2	no	—
5	C₂×#11719917	—	2	4	no	—
6	C₂×#11719917	C₄	1	8	yes	—
7	C₂×#11719917	C₂²	1	4	yes	—
8	C₂²×#88322	—	2	2	no	—
9	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₅₃	—	2	4	no	—
11	C₂×M₁₆×C₅₃	C₄	1	4	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₅₃	D₄	1	2	yes	—
17	C₂²×C₄×C₅₃	D₄	1	4	yes	—
18	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	M₁₆×C₅₃	—	2	8	no	—
21	C₂×C₈×C₅₃	—	2	4	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
23	#11719917	—	2	8	no	—
24	#11719917	—	2	4	no	—
25	#11719917	C₂×C₄	1	8	yes	—
26	C₂×#88322	—	2	8	no	—
27	C₂×#88322	—	2	4	no	—
28	C₂³×C₅₃	—	4	2	no	—
29	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
30	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₄²	1	4	yes	—
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
36	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₈×C₅₃	—	2	4	no	—
43	#88322	—	2	8	no	—
44	C₂²×C₅₃	—	2	4	no	—
45	C₂²×C₅₃	—	2	2	no	—
46	C₂²×C₅₃	—	4	4	no	—
47	C₂²×C₅₃	#95	1	4	yes	—
48	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₅₃	#95	1	4	yes	—
51	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
53	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	#1738	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	4	no	—
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
58	C₂×C₅₃	#430	1	2	yes	—
59	C₂²×M₁₆	—	53	2	no	—
60	C₂²×M₁₆	D₅₃	1	1	yes	characteristic
61	#341	—	53	4	no	—
62	C₅₃	#1738	1	1	yes	characteristic, complemented, Sylow
63	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
64	C₂×M₁₆	—	2	4	no	—
65	C₂×M₁₆	—	53	8	no	—
66	C₂×M₁₆	—	53	4	no	—
67	C₂×M₁₆	—	106	2	no	—
68	C₂×M₁₆	—	106	4	no	—
69	C₂×M₁₆	Dic₅₃	1	4	yes	—
70	C₂²×C₈	—	106	2	no	—
71	C₂²×C₈	C₂×D₅₃	1	2	yes	—
72	C₂⁴	Dic₁₀₆	1	1	yes	characteristic
73	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
74	C₂²×C₄	C₄×D₅₃	1	4	yes	—
75	C₂²×C₄	D₂₁₂	1	2	yes	—
76	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
77	C₂²×C₄	C₂²⋊D₅₃	1	4	yes	—
78	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	53	8	no	—
81	M₁₆	—	106	4	no	—
82	M₁₆	—	106	8	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	106	4	no	—
85	C₂×C₈	—	106	8	no	—
86	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
87	C₂³	—	4	2	no	—
88	C₂³	#11719919	1	4	yes	—
89	C₂³	C₄⋊Dic₅₃	1	2	yes	—
90	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
91	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
92	C₂×C₄	#11719919	1	4	yes	—
93	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
94	C₂×C₄	#11719921	1	4	yes	—
95	C₂×C₄	#11719921	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₅₃	1	4	yes	—
97	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
98	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	—	106	8	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	4	4	no	—
106	C₂²	#13133124	1	4	yes	—
107	C₂²	C₂×#11719919	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
109	C₄	#13133124	1	4	yes	—
110	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
111	C₄	C₂×#11719921	1	2	yes	—
112	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
113	C₂	—	2	4	no	—
114	C₂	—	2	2	no	—
115	C₂	C₂×#13133124	1	1	yes	characteristic
116	C₂	#16810672	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719917

C₂

yes

maximal

#16810388

C₂

yes

maximal, complemented

C₂²×M₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×#11719917

—

C₂×#11719917

C₄

yes

—

C₂×#11719917

C₂²

yes

—

C₂²×#88322

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

#11719917

—

#11719917

—

#11719917

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

#95

yes

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

#1738

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#95

yes

characteristic

C₂×C₅₃

#430

yes

—

C₂²×M₁₆

—

C₂²×M₁₆

D₅₃

yes

characteristic

#341

—

C₅₃

#1738

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂×M₁₆

Dic₅₃

yes

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂⁴

Dic₁₀₆

yes

characteristic

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂³

—

C₂³

#11719919

yes

—

C₂³

C₄⋊Dic₅₃

yes

—

C₂³

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₄×Dic₅₃

yes

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

#11719921

yes

—

C₂×C₄

#11719921

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

101

C₈

—

102

C₈

—

106

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#13133124

yes

—

107

C₂²

C₂×#11719919

yes

—

108

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

109

C₄

#13133124

yes

—

110

C₄

C₂×C₄×Dic₅₃

yes

characteristic

111

C₄

C₂×#11719921

yes

—

112

C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×#13133124

yes

characteristic

116

C₂

#16810672

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂²×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂²×M₁₆	D₅₃
hypocenter	C₅₃	#1738
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂²×M₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

hypercenter

C₂²×M₁₆

D₅₃

hypocenter

C₅₃

#1738

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
2. #16810388	C₂	semidirect
62. C₅₃	#1738	semidirect

H ≅ G/N

type

2. #16810388

C₂

semidirect

62. C₅₃

#1738

semidirect

Subgroups — #20274442

Conjugacy classes of subgroups

Complements