Group #20272800 — subgroups

Subgroups — #20272800

Order 6784.

132 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13133157	C₂	1	1	yes	maximal, characteristic
2	#16810578	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₄	1	1	yes	characteristic
5	#13133176	C₄	1	1	yes	characteristic
6	#13133157	—	2	1	no	—
7	#13133157	C₄	1	2	yes	complemented
8	C₂×#11719908	—	2	1	no	—
9	#13133127	—	2	1	no	—
10	C₈×Dic₅₃	—	2	1	no	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	M₃₂×C₅₃	—	2	1	no	—
13	M₃₂×C₅₃	—	2	1	no	—
14	C₂×C₁₆×C₅₃	C₄	1	2	yes	complemented
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²⋊Dic₅₃	C₈	1	2	yes	—
23	C₄⋊Dic₅₃	—	2	1	no	—
24	C₄×Dic₅₃	—	2	1	no	—
25	#11719917	—	2	1	no	—
26	#11719917	C₈	1	2	yes	—
27	C₂×#88322	—	2	1	no	—
28	C₁₆×C₅₃	—	2	2	no	—
29	#11719908	—	2	2	no	—
30	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄²	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	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×Dic₅₃	—	2	2	no	—
38	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	#88322	—	2	2	no	—
43	C₂²×C₅₃	—	4	1	no	—
44	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
45	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₅₃	#95	1	1	yes	characteristic
48	C₄×C₅₃	#97	1	1	yes	characteristic
49	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
51	Dic₅₃	—	4	2	no	—
52	#1378	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	#336	1	1	yes	characteristic
57	C₂×M₃₂	—	53	1	no	—
58	C₂×M₃₂	D₅₃	1	1	yes	characteristic
59	#405	—	53	1	no	—
60	C₅₃	#1378	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	53	1	no	—
62	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
63	#117	—	53	1	no	—
64	M₃₂	—	2	1	no	—
65	M₃₂	—	2	1	no	—
66	M₃₂	—	53	2	no	—
67	M₃₂	—	106	1	no	—
68	C₂×C₁₆	—	106	1	no	—
69	C₂×C₁₆	Dic₅₃	1	2	yes	complemented
70	#97	—	106	1	no	—
71	C₄×C₈	—	106	1	no	—
72	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
73	M₁₆	—	53	2	no	—
74	M₁₆	—	106	1	no	—
75	C₂×C₈	—	106	1	no	—
76	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
77	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
78	C₂×C₈	D₂₁₂	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
82	C₄⋊C₄	—	106	1	no	—
83	C₂²⋊C₄	—	53	2	no	—
84	C₄²	—	106	1	no	—
85	C₁₆	—	2	2	no	—
86	C₁₆	—	106	2	no	—
87	C₂³	C₈×D₅₃	1	1	yes	characteristic
88	C₂×C₄	—	106	2	no	—
89	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
90	C₂×C₄	#11719912	1	1	yes	characteristic
91	C₂×C₄	#11719912	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
93	C₂×C₄	#11719919	1	1	yes	characteristic
94	C₂×C₄	#11719921	1	1	yes	characteristic
95	C₈	—	106	2	no	—
96	C₈	#11719919	1	1	yes	characteristic
97	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
98	C₈	#11719921	1	1	yes	characteristic
99	C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
100	C₂²	—	4	1	no	—
101	C₂²	C₈×Dic₅₃	1	1	yes	characteristic
102	C₂²	#13133129	1	1	yes	characteristic
103	C₂²	#13133146	1	1	yes	characteristic
104	C₄	—	212	2	no	—
105	C₄	#13133124	1	1	yes	characteristic
106	C₄	#13133127	1	1	yes	characteristic
107	C₄	#13133129	1	1	yes	characteristic
108	C₄	#13133146	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	#16810377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, characteristic

#16810578

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

characteristic

#13133176

C₄

yes

characteristic

#13133157

—

#13133157

C₄

yes

complemented

C₂×#11719908

—

#13133127

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂²⋊Dic₅₃

C₈

yes

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

C₈

yes

—

C₂×#88322

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₈

yes

characteristic

C₂²×C₅₃

C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

Dic₅₃

—

#1378

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#336

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

D₅₃

yes

characteristic

#405

—

C₅₃

#1378

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

Dic₅₃

yes

complemented

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

C₄²

—

106

—

C₁₆

—

C₁₆

—

106

—

C₂³

C₈×D₅₃

yes

characteristic

C₂×C₄

—

106

—

C₂×C₄

C₈×D₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

106

—

C₈

#11719919

yes

characteristic

C₈

C₄⋊Dic₅₃

yes

characteristic

C₈

#11719921

yes

characteristic

C₈

C₂²⋊Dic₅₃

yes

characteristic

100

C₂²

—

101

C₂²

C₈×Dic₅₃

yes

characteristic

102

C₂²

#13133129

yes

characteristic

103

C₂²

#13133146

yes

characteristic

104

C₄

—

212

—

105

C₄

#13133124

yes

characteristic

106

C₄

#13133127

yes

characteristic

107

C₄

#13133129

yes

characteristic

108

C₄

#13133146

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

#16810377

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₄×C₈
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	#336
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#1378
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

C₂²⋊Dic₅₃

commutator

C₂²×C₅₃

C₄×C₈

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂×C₅₃

#336

hypercenter

C₂×M₃₂

D₅₃

hypocenter

C₅₃

#1378

LCS

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
7. #13133157	C₄	semidirect
14. C₂×C₁₆×C₅₃	C₄	semidirect
60. C₅₃	#1378	semidirect
69. C₂×C₁₆	Dic₅₃	semidirect

H ≅ G/N

type

7. #13133157

C₄

semidirect

14. C₂×C₁₆×C₅₃

C₄

semidirect

60. C₅₃

#1378

semidirect

69. C₂×C₁₆

Dic₅₃

semidirect

Subgroups — #20272800

Conjugacy classes of subgroups

Complements