Group #20277078 — subgroups

Subgroups — #20277078

Order 6784.

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

normal subgroups only

This group has 628 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×Dic₁₀₆	C₂	1	3	yes	maximal
2	C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
3	#16810474	C₂	1	6	yes	maximal
4	#16810468	C₂	1	2	yes	maximal
5	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₀₆	C₄	1	24	yes	complemented
8	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
9	C₂×#11719919	C₂²	1	6	yes	—
10	C₂×C₄×Dic₅₃	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	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂³	1	6	yes	—
20	C₂²×Dic₅₃	C₂³	1	2	yes	—
21	C₂×Dic₁₀₆	C₂×C₄	1	12	yes	—
22	C₄²×C₅₃	C₂×C₄	1	12	yes	—
23	C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
24	#11719919	C₂×C₄	1	24	yes	—
25	C₄×Dic₅₃	—	4	6	no	—
26	C₄×Dic₅₃	C₂×C₄	1	24	yes	—
27	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	3	no	—
29	C₂×C₄×C₅₃	—	2	6	no	—
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
34	C₂×Dic₅₃	—	2	6	no	—
35	C₂×Dic₅₃	—	2	6	no	—
36	C₂×Dic₅₃	C₂²×C₄	1	12	yes	—
37	C₂×Dic₅₃	C₂²×C₄	1	12	yes	—
38	Dic₁₀₆	C₄²	1	16	yes	complemented
39	C₂²×C₅₃	C₂³×C₄	1	3	yes	—
40	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂²×C₅₃	2₋¹⁺⁴	1	3	yes	—
42	C₄×C₅₃	—	2	6	no	—
43	C₄×C₅₃	—	2	6	no	—
44	C₄×C₅₃	C₂×C₄²	1	4	yes	—
45	Dic₅₃	—	2	12	no	—
46	Dic₅₃	C₂×C₄²	1	8	yes	—
47	#2276	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
49	C₂×C₅₃	#519	1	3	yes	—
50	C₂×C₅₃	#520	1	3	yes	—
51	C₂×C₄×Q₈	—	53	3	no	—
52	C₄×C₄⋊C₄	—	53	3	no	—
53	C₄×C₄⋊C₄	—	53	6	no	—
54	#376	—	53	2	no	—
55	#376	D₅₃	1	1	yes	characteristic
56	C₅₃	#2276	1	1	yes	characteristic, complemented, Sylow
57	C₂²×Q₈	—	53	1	no	—
58	C₄×Q₈	—	53	24	no	—
59	C₂×C₄⋊C₄	—	53	6	no	—
60	C₂×C₄⋊C₄	—	53	3	no	—
61	C₂×C₄²	—	53	6	no	—
62	C₂×C₄²	—	53	6	no	—
63	C₂×C₄²	C₂×D₅₃	1	3	yes	—
64	#95	—	53	6	no	—
65	#95	C₂×D₅₃	1	1	yes	characteristic
66	#95	C₂×D₅₃	1	3	yes	—
67	C₂×Q₈	—	53	12	no	—
68	C₂²×C₄	—	53	2	no	—
69	C₂²×C₄	—	53	6	no	—
70	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
71	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
72	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
73	C₄⋊C₄	—	53	24	no	—
74	C₄⋊C₄	—	53	12	no	—
75	C₄²	—	53	24	no	—
76	C₄²	—	212	6	no	—
77	C₄²	C₄×D₅₃	1	12	yes	—
78	C₂³	C₂³×D₅₃	1	1	yes	characteristic
79	Q₈	—	53	16	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	53	12	no	—
84	C₂×C₄	—	53	12	no	—
85	C₂×C₄	—	106	6	no	—
86	C₂×C₄	—	106	6	no	—
87	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
88	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
89	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
90	C₂²	C₂²×C₄×D₅₃	1	3	yes	—
91	C₂²	#13133263	1	1	yes	characteristic
92	C₂²	#13133264	1	3	yes	—
93	C₄	—	2	6	no	—
94	C₄	—	2	6	no	—
95	C₄	—	53	8	no	—
96	C₄	—	106	12	no	—
97	C₄	C₄²×D₅₃	1	4	yes	—
98	C₂	C₂×C₄²×D₅₃	1	1	yes	characteristic
99	C₂	#16811100	1	3	yes	—
100	C₂	#16811103	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

#16810474

C₂

yes

maximal

#16810468

C₂

yes

maximal

C₅₃×#376

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

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

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

Dic₁₀₆

C₄²

yes

complemented

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

Dic₅₃

—

Dic₅₃

C₂×C₄²

yes

—

#2276

—

maximal, Sylow

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

#519

yes

—

C₂×C₅₃

#520

yes

—

C₂×C₄×Q₈

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

#376

—

#376

D₅₃

yes

characteristic

C₅₃

#2276

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

#95

—

#95

C₂×D₅₃

yes

characteristic

#95

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

212

—

C₄²

C₄×D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂²

C₂²×C₄×D₅₃

yes

—

C₂²

#13133263

yes

characteristic

C₂²

#13133264

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

106

—

C₄

C₄²×D₅₃

yes

—

C₂

C₂×C₄²×D₅₃

yes

characteristic

C₂

#16811100

yes

—

100

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	#2276
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
7. C₄×Dic₁₀₆	C₄	semidirect
38. Dic₁₀₆	C₄²	semidirect
56. C₅₃	#2276	semidirect

H ≅ G/N

type

7. C₄×Dic₁₀₆

C₄

semidirect

38. Dic₁₀₆

C₄²

semidirect

56. C₅₃

#2276

semidirect

Subgroups — #20277078

Conjugacy classes of subgroups

Complements