Group #20282758 — subgroups

Subgroups — #20282758

Order 6784.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133211	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810541	C₂	1	6	yes	maximal, complemented
4	#16810533	C₂	1	6	yes	maximal, complemented
5	C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	6	yes	—
7	C₂²×D₂₁₂	C₂²	1	4	yes	—
8	#13133214	—	2	4	no	—
9	#13133211	—	2	4	no	—
10	C₂×#11719919	C₂²	1	3	yes	—
11	C₂×#11719921	C₂²	1	3	yes	—
12	C₂×#11719921	C₂²	1	12	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#95×C₅₃	C₂²	1	6	yes	—
15	C₂³×D₅₃	C₂³	1	4	yes	—
16	C₂³×D₅₃	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×Dic₅₃	C₂³	1	3	yes	—
20	C₂×D₂₁₂	—	2	12	no	—
21	C₂×D₂₁₂	—	4	12	no	—
22	C₂×D₂₁₂	—	4	12	no	—
23	C₄²×C₅₃	D₄	1	4	yes	complemented
24	#11719921	—	2	6	no	—
25	#11719921	—	4	12	no	—
26	#11719919	—	2	6	no	—
27	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
28	C₂²×D₅₃	—	2	4	no	—
29	C₂²×D₅₃	—	2	1	no	—
30	C₂²×D₅₃	—	2	12	no	—
31	C₂²×D₅₃	—	2	6	no	—
32	C₂²×D₅₃	—	2	12	no	—
33	C₂×C₄×C₅₃	—	2	6	no	—
34	C₂×C₄×C₅₃	—	2	12	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
36	C₂×Dic₅₃	—	2	3	no	—
37	C₂×Dic₅₃	—	2	6	no	—
38	D₂₁₂	—	4	24	no	—
39	D₂₁₂	—	4	16	no	—
40	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
41	C₂²×C₅₃	H(5,𝔽₂)	1	6	yes	—
42	C₂×D₅₃	—	4	12	no	—
43	C₂×D₅₃	—	4	3	no	—
44	C₂×D₅₃	—	4	4	no	—
45	C₂×D₅₃	—	4	4	no	—
46	C₂×D₅₃	—	4	12	no	—
47	C₄×C₅₃	—	2	6	no	—
48	C₄×C₅₃	—	4	4	no	—
49	Dic₅₃	—	4	3	no	—
50	#2818	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	#535	1	3	yes	—
52	C₂×C₅₃	#560	1	4	yes	—
53	D₅₃	—	8	4	no	—
54	D₅₃	—	8	1	no	—
55	C₂×#127	—	53	1	no	—
56	C₂×#126	—	53	1	no	—
57	#394	—	53	6	no	—
58	#392	—	53	6	no	—
59	#383	D₅₃	1	1	yes	characteristic, complemented
60	C₅₃	#2818	1	1	yes	characteristic, complemented, Sylow
61	C₂²×D₄	—	53	6	no	—
62	C₂²×D₄	—	53	4	no	—
63	#127	—	106	4	no	—
64	#126	—	106	4	no	—
65	C₂×C₄⋊C₄	—	53	3	no	—
66	C₂×C₂²⋊C₄	—	53	3	no	—
67	C₂×C₂²⋊C₄	—	53	12	no	—
68	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
69	#95	C₂×D₅₃	1	6	yes	—
70	C₂⁴	—	53	1	no	—
71	C₂⁴	—	53	4	no	—
72	C₂×D₄	—	106	12	no	—
73	C₂×D₄	—	212	12	no	—
74	C₂×D₄	—	212	12	no	—
75	C₂²×C₄	—	53	3	no	—
76	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
77	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
78	C₄⋊C₄	—	106	6	no	—
79	C₂²⋊C₄	—	106	6	no	—
80	C₂²⋊C₄	—	212	12	no	—
81	C₄²	D₂₁₂	1	4	yes	complemented
82	C₂³	—	106	1	no	—
83	C₂³	—	106	4	no	—
84	C₂³	—	106	12	no	—
85	C₂³	—	106	6	no	—
86	C₂³	—	106	12	no	—
87	C₂³	C₂³×D₅₃	1	1	yes	characteristic
88	D₄	—	212	24	no	—
89	D₄	—	212	16	no	—
90	C₂×C₄	—	2	6	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	106	3	no	—
93	C₂×C₄	—	106	6	no	—
94	C₂×C₄	C₂×D₂₁₂	1	6	yes	—
95	C₂²	—	212	4	no	—
96	C₂²	—	212	4	no	—
97	C₂²	—	212	12	no	—
98	C₂²	—	212	12	no	—
99	C₂²	—	212	3	no	—
100	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
101	C₂²	#13133263	1	6	yes	—
102	C₄	—	2	6	no	—
103	C₄	—	4	4	no	—
104	C₄	—	212	3	no	—
105	C₂	—	424	1	no	—
106	C₂	—	424	4	no	—
107	C₂	#16811192	1	3	yes	—
108	C₂	#16811364	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic, complemented

C₂×#13133211

C₂

yes

maximal, characteristic, complemented

#16810541

C₂

yes

maximal, complemented

#16810533

C₂

yes

maximal, complemented

C₅₃×#383

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂²×D₂₁₂

C₂²

yes

—

#13133214

—

#13133211

—

C₂×#11719919

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₂²

yes

—

C₂³×D₅₃

C₂³

yes

—

C₂³×D₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₄²×C₅₃

D₄

yes

—

—

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2818

—

maximal, Sylow

C₂×C₅₃

#535

yes

—

C₂×C₅₃

#560

yes

—

D₅₃

—

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2818

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

#127

—

106

—

#126

—

106

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

C₂×D₅₃

yes

—

C₂⁴

—

C₂⁴

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

D₂₁₂

yes

complemented

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂³×D₅₃

yes

characteristic

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

—

212

—

100

C₂²

C₂²×D₂₁₂

yes

characteristic

101

C₂²

#13133263

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

212

—

105

C₂

—

424

—

106

C₂

—

424

—

107

C₂

#16811192

yes

—

108

C₂

#16811364

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#383	C₂
hypercenter	#383	D₅₃
hypocenter	C₅₃	#2818
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#383

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
1. C₂×#13133214	C₂	semidirect
2. C₂×#13133211	C₂	semidirect
3. #16810541	C₂	semidirect
4. #16810533	C₂	semidirect
5. C₅₃×#383	C₂	semidirect
23. C₄²×C₅₃	D₄	semidirect
59. #383	D₅₃	semidirect
60. C₅₃	#2818	semidirect
81. C₄²	D₂₁₂	semidirect

H ≅ G/N

type

1. C₂×#13133214

C₂

semidirect

2. C₂×#13133211

C₂

semidirect

3. #16810541

C₂

semidirect

4. #16810533

C₂

semidirect

5. C₅₃×#383

C₂

semidirect

23. C₄²×C₅₃

D₄

semidirect

59. #383

D₅₃

semidirect

60. C₅₃

#2818

semidirect

81. C₄²

D₂₁₂

semidirect

Subgroups — #20282758

Conjugacy classes of subgroups

Complements