Group #20272545 — subgroups

Subgroups — #20272545

Order 6784.

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

normal subgroups only

This group has 152 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₄	1	2	yes	—
5	C₄⋊#88322	—	2	1	no	—
6	C₂²⋊#88322	—	2	1	no	—
7	#13133128	—	2	1	no	—
8	#13133128	—	2	1	no	—
9	C₄×#88322	C₄	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂²⋊C₈×C₅₃	—	2	1	no	—
13	C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂×#88322	—	2	2	no	—
23	C₂×#88322	—	2	2	no	—
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	C₈	1	4	yes	complemented
26	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
37	C₈×C₅₃	—	4	2	no	—
38	#88322	—	2	4	no	—
39	#88322	—	4	2	no	—
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	#95	1	1	yes	characteristic
43	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
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	C₄×C₈	1	1	yes	characteristic
49	C₄×C₅₃	#97	1	1	yes	characteristic
50	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
51	#1275	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	2	2	no	—
53	C₂×C₅₃	#336	1	1	yes	characteristic
54	C₂×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
55	C₂×C₅₃	#338	1	1	yes	characteristic
56	#431	—	53	1	no	—
57	#431	D₅₃	1	1	yes	characteristic
58	C₂×#97	—	53	1	no	—
59	C₅₃	#1275	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	53	2	no	—
61	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	106	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	106	1	no	—
66	#97	—	106	1	no	—
67	#97	—	106	1	no	—
68	C₄×C₈	—	53	2	no	—
69	C₄×C₈	Dic₅₃	1	2	yes	—
70	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	53	4	no	—
75	C₂×C₈	—	106	2	no	—
76	C₂×C₈	—	106	2	no	—
77	C₂×C₈	—	106	2	no	—
78	C₄²	C₄×D₅₃	1	1	yes	characteristic
79	C₄²	D₂₁₂	1	1	yes	characteristic
80	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
81	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
82	C₂³	#11719919	1	1	yes	characteristic
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₈×D₅₃	1	2	yes	—
86	C₂×C₄	#11719912	1	2	yes	—
87	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
88	C₂×C₄	#11719919	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
90	C₂×C₄	#11719921	1	1	yes	characteristic
91	C₂×C₄	#11719921	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	106	4	no	—
95	C₈	—	212	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#13133124	1	1	yes	characteristic
99	C₂²	#13133146	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₅₃	1	1	yes	characteristic
105	C₄	#13133127	1	1	yes	characteristic
106	C₄	#13133129	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#16810377	1	1	yes	characteristic
109	C₂	#16810379	1	1	yes	characteristic
110	C₂	#16810381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

C₂×#13133128

C₂

yes

maximal, characteristic

C₅₃×#431

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

—

—

C₈

yes

complemented

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

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

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₈

yes

characteristic

C₄×C₅₃

#97

yes

characteristic

C₄×C₅₃

C₂²⋊C₈

yes

—

#1275

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×#97

—

C₅₃

#1275

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

Dic₅₃

yes

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

D₂₁₂

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

#11719919

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₅₃

yes

—

C₂×C₄

#11719912

yes

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

#11719919

yes

characteristic

C₂×C₄

C₄⋊Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

#13133124

yes

characteristic

C₂²

#13133146

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₅₃

yes

characteristic

105

C₄

#13133127

yes

characteristic

106

C₄

#13133129

yes

—

107

C₂

—

108

C₂

#16810377

yes

characteristic

109

C₂

#16810379

yes

characteristic

110

C₂

#16810381

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	#95
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1275
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₅₃

commutator

C₄×C₅₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#431

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
25. C₂×#88322	C₈	semidirect
59. C₅₃	#1275	semidirect

H ≅ G/N

type

25. C₂×#88322

C₈

semidirect

59. C₅₃

#1275

semidirect

Subgroups — #20272545

Conjugacy classes of subgroups

Complements