Group #20272625 — subgroups

Subgroups — #20272625

Order 6784.

130 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810680	C₂	1	1	yes	maximal, characteristic
2	#16810647	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#432	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₄	1	2	yes	—
5	C₄⋊#88322	—	2	1	no	—
6	C₄⋊#88322	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₂²⋊#88322	C₄	1	2	yes	—
9	#13133128	—	2	1	no	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
13	#97×C₅₃	—	2	1	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₄²×C₅₃	D₄	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₄²×C₅₃	Q₈	1	1	yes	characteristic
22	#11719917	—	4	2	no	—
23	C₂×#88322	—	2	2	no	—
24	C₂×#88322	—	2	2	no	—
25	C₂³×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₈×C₅₃	—	4	2	no	—
37	#88322	—	4	2	no	—
38	#88322	—	4	2	no	—
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	#95	1	1	yes	characteristic
43	C₄×C₅₃	—	2	1	no	—
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	#100	1	1	yes	characteristic
49	C₄×C₅₃	#101	1	1	yes	characteristic
50	#1307	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	4	1	no	—
52	C₂×C₅₃	#343	1	1	yes	characteristic
53	C₂×C₅₃	#344	1	1	yes	characteristic
54	C₂×C₅₃	#344	1	1	yes	characteristic
55	#432	—	53	1	no	—
56	#432	D₅₃	1	1	yes	characteristic
57	#423	—	53	1	no	—
58	C₅₃	#1307	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	53	2	no	—
60	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	106	1	no	—
63	C₄⋊C₈	—	106	1	no	—
64	C₄⋊C₈	—	106	1	no	—
65	C₂²⋊C₈	—	53	2	no	—
66	C₂²⋊C₈	Dic₅₃	1	2	yes	—
67	#97	—	2	1	no	—
68	#97	—	106	1	no	—
69	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
70	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
72	M₁₆	—	212	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	106	2	no	—
75	C₂×C₈	—	106	2	no	—
76	C₄²	Dic₁₀₆	1	1	yes	characteristic
77	C₄²	D₂₁₂	1	1	yes	characteristic
78	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
79	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
80	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	#11719919	1	1	yes	characteristic
86	C₂×C₄	#11719919	1	1	yes	characteristic
87	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
88	C₂×C₄	#11719921	1	1	yes	characteristic
89	C₂×C₄	#11719921	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
91	C₈	—	4	2	no	—
92	C₈	—	212	2	no	—
93	C₈	—	212	2	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	#13133124	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	#13133133	1	1	yes	characteristic
104	C₄	#13133135	1	1	yes	characteristic
105	C₂	—	4	1	no	—
106	C₂	#16810391	1	1	yes	characteristic
107	C₂	#16810393	1	1	yes	characteristic
108	C₂	#16810394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, characteristic

#16810647

C₂

yes

maximal, characteristic

C₅₃×#432

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

—

—

—

—

C₄

yes

—

#13133128

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₄

yes

—

#97×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

—

—

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

#423

—

C₅₃

#1307

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₅₃

yes

—

#97

—

#97

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

C₄×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

212

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133124

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#13133133

yes

characteristic

104

C₄

#13133135

yes

characteristic

105

C₂

—

106

C₂

#16810391

yes

characteristic

107

C₂

#16810393

yes

characteristic

108

C₂

#16810394

yes

characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂³×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#432	C₂
socle	C₂²×C₅₃	#95
hypercenter	#432	D₅₃
hypocenter	C₅₃	#1307
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

#13133124

commutator

C₂³×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#432

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
58. C₅₃	#1307	semidirect

H ≅ G/N

type

58. C₅₃

#1307

semidirect

Subgroups — #20272625

Conjugacy classes of subgroups

Complements