Group #20273350 — subgroups

Subgroups — #20273350

Order 6784.

238 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 77 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	2	yes	maximal, complemented
2	#16810363	C₂	1	4	yes	maximal, complemented
3	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	—	2	2	no	—
5	C₂²⋊#88322	—	2	2	no	—
6	#13133128	C₂²	1	4	yes	—
7	C₂×#41×C₅₃	C₄	1	2	yes	—
8	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
9	#127×C₅₃	C₄	1	1	yes	characteristic
10	#124×C₅₃	C₂²	1	2	yes	—
11	#121×C₅₃	—	2	2	no	—
12	C₄×D₄×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#41×C₅₃	—	4	4	no	—
15	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₂²⋊C₄×C₅₃	—	2	4	no	—
24	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₅₃	D₄	1	2	yes	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×#88322	—	2	4	no	—
28	C₂³×C₅₃	—	2	2	no	—
29	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	Q₈×C₅₃	—	4	2	no	—
31	D₄×C₅₃	—	4	2	no	—
32	D₄×C₅₃	—	4	4	no	—
33	D₄×C₅₃	—	4	4	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	—	4	4	no	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	#88322	—	4	4	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	—	4	2	no	—
46	C₂²×C₅₃	—	4	4	no	—
47	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	#100	1	2	yes	—
53	#1536	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	4	1	no	—
55	C₂×C₅₃	—	8	2	no	—
56	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
57	C₂×C₅₃	#421	1	2	yes	—
58	#532	D₅₃	1	1	yes	characteristic
59	#432	—	53	2	no	—
60	#329	—	53	4	no	—
61	C₅₃	#1536	1	1	yes	characteristic, complemented, Sylow
62	C₂×#41	Dic₅₃	1	2	yes	—
63	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
64	#127	Dic₅₃	1	1	yes	characteristic
65	#124	C₂×D₅₃	1	2	yes	—
66	#121	—	2	2	no	—
67	C₄×D₄	—	2	2	no	—
68	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
69	C₄⋊C₈	—	106	2	no	—
70	C₂²⋊C₈	—	106	2	no	—
71	#97	—	53	4	no	—
72	#41	—	4	4	no	—
73	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
80	C₂×C₈	—	106	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₂²⋊C₄	—	2	4	no	—
83	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
84	C₄²	C₂²⋊D₅₃	1	2	yes	—
85	C₄²	C₂²×D₅₃	1	1	yes	characteristic
86	C₂³	—	2	2	no	—
87	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	D₄	—	4	2	no	—
90	D₄	—	4	4	no	—
91	D₄	—	4	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	4	no	—
96	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
97	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
100	C₈	—	212	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	4	2	no	—
104	C₂²	—	4	4	no	—
105	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	#13133134	1	2	yes	—
111	C₂	—	4	1	no	—
112	C₂	—	8	2	no	—
113	C₂	C₂×#13133134	1	1	yes	characteristic
114	C₂	#16810640	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, complemented

#16810363

C₂

yes

maximal, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₄⋊#88322

—

C₂²⋊#88322

—

#13133128

C₂²

yes

—

C₂×#41×C₅₃

C₄

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

#127×C₅₃

C₄

yes

characteristic

#124×C₅₃

C₂²

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

—

#1536

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#100

yes

characteristic

C₂×C₅₃

#421

yes

—

#532

D₅₃

yes

characteristic

#432

—

#329

—

C₅₃

#1536

yes

characteristic, complemented, Sylow

C₂×#41

Dic₅₃

yes

—

C₄⋊Q₈

Dic₅₃

yes

characteristic

#127

Dic₅₃

yes

characteristic

#124

C₂×D₅₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

#41

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

100

C₈

—

212

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#13133134

yes

—

111

C₂

—

112

C₂

—

113

C₂

C₂×#13133134

yes

characteristic

114

C₂

#16810640

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#1536
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16810680	C₂	semidirect
2. #16810363	C₂	semidirect
61. C₅₃	#1536	semidirect

H ≅ G/N

type

1. #16810680

C₂

semidirect

2. #16810363

C₂

semidirect

61. C₅₃

#1536

semidirect

Subgroups — #20273350

Conjugacy classes of subgroups

Complements