Group #20277077 — subgroups

Subgroups — #20277077

Order 6784.

780 conjugacy classes of subgroups; only the 325 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 325 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×C₄×D₂₁₂	C₂	1	3	yes	maximal
2	C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal, complemented
3	#16810470	C₂	1	6	yes	maximal, complemented
4	#95×D₅₃	C₂	1	2	yes	maximal, complemented
5	C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	C₂²×C₄×D₅₃	C₂²	1	6	yes	—
8	C₄×D₂₁₂	C₄	1	24	yes	complemented
9	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
10	C₂×#11719921	C₂²	1	6	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
12	#13133124	C₂²	1	6	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
14	#95×C₅₃	C₂²	1	1	yes	characteristic
15	#95×C₅₃	C₂²	1	3	yes	—
16	C₂³×D₅₃	C₂³	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂³	1	6	yes	—
21	C₂×D₂₁₂	C₂×C₄	1	12	yes	—
22	C₂×C₄×D₅₃	—	4	12	no	—
23	C₂×C₄×D₅₃	C₂×C₄	1	24	yes	complemented
24	C₄²×C₅₃	C₂×C₄	1	12	yes	complemented
25	#11719921	C₂×C₄	1	24	yes	complemented
26	C₄⋊Dic₅₃	C₂×C₄	1	12	yes	complemented
27	C₄×Dic₅₃	—	4	6	no	—
28	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
29	C₂²×D₅₃	—	2	6	no	—
30	C₂²×D₅₃	—	2	2	no	—
31	C₂²×D₅₃	C₂²×C₄	1	12	yes	—
32	C₂×C₄×C₅₃	—	2	6	no	—
33	C₂×C₄×C₅₃	—	2	3	no	—
34	C₂×C₄×C₅₃	—	2	3	no	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
38	C₂×Dic₅₃	—	2	12	no	—
39	C₂×Dic₅₃	C₂²×C₄	1	12	yes	—
40	D₂₁₂	C₄²	1	16	yes	complemented
41	C₄×D₅₃	—	4	24	no	—
42	C₂²×C₅₃	C₂³×C₄	1	3	yes	—
43	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
44	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
45	C₂×D₅₃	—	2	12	no	—
46	C₂×D₅₃	—	2	12	no	—
47	C₂×D₅₃	C₂×C₄²	1	8	yes	—
48	C₄×C₅₃	—	2	6	no	—
49	C₄×C₅₃	—	2	6	no	—
50	C₄×C₅₃	C₂×C₄²	1	4	yes	—
51	Dic₅₃	—	2	12	no	—
52	#2275	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
54	C₂×C₅₃	#518	1	3	yes	—
55	C₂×C₅₃	#520	1	3	yes	—
56	D₅₃	—	2	8	no	—
57	C₂×C₄×D₄	—	53	3	no	—
58	C₄×C₄⋊C₄	—	53	3	no	—
59	C₄×C₂²⋊C₄	—	53	6	no	—
60	#376	D₅₃	1	1	yes	characteristic, complemented
61	C₂×#95	—	53	2	no	—
62	C₅₃	#2275	1	1	yes	characteristic, complemented, Sylow
63	C₂²×D₄	—	53	1	no	—
64	C₂³×C₄	—	53	6	no	—
65	C₄×D₄	—	53	24	no	—
66	C₂×C₄⋊C₄	—	53	3	no	—
67	C₂×C₂²⋊C₄	—	53	6	no	—
68	C₂×C₄²	—	53	6	no	—
69	C₂×C₄²	C₂×D₅₃	1	3	yes	—
70	#95	—	53	6	no	—
71	#95	C₂×D₅₃	1	1	yes	characteristic
72	#95	C₂×D₅₃	1	3	yes	—
73	C₂⁴	—	53	2	no	—
74	C₂×D₄	—	53	12	no	—
75	C₂²×C₄	—	53	6	no	—
76	C₂²×C₄	—	53	24	no	—
77	C₂²×C₄	—	212	12	no	—
78	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
79	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
80	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
81	C₄⋊C₄	—	53	12	no	—
82	C₂²⋊C₄	—	53	24	no	—
83	C₄²	—	212	6	no	—
84	C₄²	C₄×D₅₃	1	12	yes	complemented
85	C₂³	—	53	12	no	—
86	C₂³	—	106	6	no	—
87	C₂³	—	106	2	no	—
88	C₂³	C₂³×D₅₃	1	1	yes	characteristic
89	D₄	—	53	16	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	53	12	no	—
94	C₂×C₄	—	106	12	no	—
95	C₂×C₄	—	212	24	no	—
96	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
97	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
98	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
99	C₂²	—	53	8	no	—
100	C₂²	—	106	12	no	—
101	C₂²	—	106	12	no	—
102	C₂²	C₂²×C₄×D₅₃	1	3	yes	—
103	C₂²	#13133263	1	3	yes	—
104	C₂²	#13133264	1	1	yes	characteristic
105	C₄	—	2	6	no	—
106	C₄	—	2	6	no	—
107	C₄	—	106	12	no	—
108	C₄	C₄²×D₅₃	1	4	yes	—
109	C₂	—	106	8	no	—
110	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
111	C₂	#16811097	1	3	yes	—
112	C₂	#16811103	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₂₁₂

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, complemented

#16810470

C₂

yes

maximal, complemented

#95×D₅₃

C₂

yes

maximal, complemented

C₅₃×#376

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂²×C₄×D₅₃

C₂²

yes

—

C₄×D₂₁₂

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₂²

yes

—

C₂³×D₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

complemented

C₄²×C₅₃

C₂×C₄

yes

complemented

#11719921

C₂×C₄

yes

complemented

C₄⋊Dic₅₃

C₂×C₄

yes

complemented

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

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×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

D₂₁₂

C₄²

yes

complemented

C₄×D₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₄²

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

Dic₅₃

—

#2275

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

#518

yes

—

C₂×C₅₃

#520

yes

—

D₅₃

—

C₂×C₄×D₄

—

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

#376

D₅₃

yes

characteristic, complemented

C₂×#95

—

C₅₃

#2275

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

#95

—

#95

C₂×D₅₃

yes

characteristic

#95

C₂×D₅₃

yes

—

C₂⁴

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

212

—

C₄²

C₄×D₅₃

yes

complemented

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂³×D₅₃

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂²

—

100

C₂²

—

106

—

101

C₂²

—

106

—

102

C₂²

C₂²×C₄×D₅₃

yes

—

103

C₂²

#13133263

yes

—

104

C₂²

#13133264

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

106

—

108

C₄

C₄²×D₅₃

yes

—

109

C₂

—

106

—

110

C₂

C₂×C₄²×D₅₃

yes

characteristic

111

C₂

#16811097

yes

—

112

C₂

#16811103

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₅₃×#376	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	#376	D₅₃
hypocenter	C₅₃	#2275
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Fitting

C₅₃×#376

C₂

socle

C₂³×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
2. C₄×C₄⋊Dic₅₃	C₂	semidirect
3. #16810470	C₂	semidirect
4. #95×D₅₃	C₂	semidirect
5. C₅₃×#376	C₂	semidirect
8. C₄×D₂₁₂	C₄	semidirect
23. C₂×C₄×D₅₃	C₂×C₄	semidirect
24. C₄²×C₅₃	C₂×C₄	semidirect
25. #11719921	C₂×C₄	semidirect
26. C₄⋊Dic₅₃	C₂×C₄	semidirect
40. D₂₁₂	C₄²	semidirect
60. #376	D₅₃	semidirect
62. C₅₃	#2275	semidirect
84. C₄²	C₄×D₅₃	semidirect

H ≅ G/N

type

2. C₄×C₄⋊Dic₅₃

C₂

semidirect

3. #16810470

C₂

semidirect

4. #95×D₅₃

C₂

semidirect

5. C₅₃×#376

C₂

semidirect

8. C₄×D₂₁₂

C₄

semidirect

23. C₂×C₄×D₅₃

C₂×C₄

semidirect

24. C₄²×C₅₃

C₂×C₄

semidirect

25. #11719921

C₂×C₄

semidirect

26. C₄⋊Dic₅₃

C₂×C₄

semidirect

40. D₂₁₂

C₄²

semidirect

60. #376

D₅₃

semidirect

62. C₅₃

#2275

semidirect

84. C₄²

C₄×D₅₃

semidirect

Subgroups — #20277077

Conjugacy classes of subgroups

Complements